History. Optics, from the Greek word ὀπτική, which signifies to see, is the name given to that branch of natural philosophy which treats of the nature and properties of light; of the changes which it suffers either in its qualities or in its course when transmitted through bodies, when reflected from their surfaces, or when passing near them; of the structure of the eye, and the laws of vision; and of the construction of those instruments in which light is the chief agent.
HISTORY.
The early history of optics, like that of all the sciences cultivated in ancient times, is involved in much obscurity. After the art of glass-making was discovered, lenses and spheres of glass seem to have been used as burning-glasses. In Aristophanes's comedy of The Clouds a burning sphere is distinctly described. Pliny speaks of globes of glass which produced combustion when held to the sun. Lactantius informs us that a globe of glass full of water could, when exposed to the sun, kindle a fire even in the coldest weather. And it appears that globes of glass were used by the Vestal Virgins to kindle the sacred fire, and by surgeons to burn the flesh of sick persons that required to be cauterised.
Among the earliest speculators on vision were Pythagoras and Plato; the former held that bodies became visible by means of particles projected from their surfaces and entering the eye, while the latter, in order to give the eye some share in the matter, supposed that something emitted from the eye met with something emitted from the object, and was again returned into the organ of vision. The followers of Plato, however, though they had deteriorated rather than improved the conclusion of Pythagoras, were acquainted with two important facts in the science. They taught that light moved in straight lines, and that when it was reflected regularly from the surfaces of polished bodies the angle of incidence was equal to the angle of reflexion.
The earliest writer on optics was Euclid, the celebrated geometer, whose treatise on the subject is still extant. It consists of two books on optics and catoptries, and proceeds on the Platonic theory, that the visual rays pass from the eye to the object, forming a cone whose apex is in the eye and whose base is the object. He shews that the angles of incidence and reflexion are equal, and that the incident and reflected rays lie in a plane at right angles to the reflecting surface; and he discusses the apparent magnitude and form of objects, and the apparent place of the images formed by reflection from plane, convex, and concave mirrors. The book on optics contains sixty-one, and that on catoptries thirty-one theorems.
As a naturalist Aristotle made some valuable optical observations. He described, with tolerable correctness, the phenomena of rainbows, halos, and parhelia. He considered the rainbow as produced by the reflexion of the sun's rays from the drops of rain which gave an imperfect image of the sun; and he ascribes the light which appears in the sun's absence to the reflective power of the atmosphere.
The speculations of Seneca and Cleomedes derive any interest they may possess from their absurdity. Seneca noticed the magnifying power of a bottle of glass in enlarging small letters, and he observed that an angular piece of A.D. glass produced all the colours of the rainbow. Cleomedes, in his cyclical theory of motion, has given an elaborate explanation of the manner in which rays proceeding from the eye render the objects which they meet visible, but it is too stupid to demand the slightest attention.
The science of optics may be justly considered as owing its origin to the celebrated Claudius Ptolemy, the astronomer of Alexandria, who flourished at the end of the first century. His work entitled Ptolemei Opticorum Sermones quinque ex Arabico-Latine versi, was known in the time of Roger Bacon to have treated on astronomical refractions, but it had escaped the notice of philosophers, and its valuable contents were unknown until 1816, when Delambre published an analysis of it from the manuscript in the Royal Library at Paris. Montucla had, long before the discovery of the French manuscript, mentioned that a manuscript copy of Ptolemy's Optics was in the catalogue of the Bodleian Library of Oxford. This interesting manuscript, which Professor Rigaud was so kind as to examine at our request, belongs to the Savilian Library, and had been the property of Sir Henry Savile himself. As in the Parisian manuscript the first book is wanting, but it has no blank spaces like the Parisian one, and it is accompanied with a preface by the translator, containing an abstract of the work, and stating that the fifth book is imperfect. The translator mentions that the second book had been previously translated from Arabic into Latin by Amritus Eugenius, a Sicilian, from the latest of two copies of which, the new translation was made. The following abstract of this interesting work is taken from Delambre's Analysis, and from the translator's abstract as communicated to us by Professor Rigaud.
"The Optics of Ptolemy consists of five books. The first book is wanting, but from the recapitulation of it at the beginning of the second, it appears to have contained a dissertation on the relations between light and vision, founded on the idea that the visual rays issue from the eye. In the second book he shews that we see better with two eyes than with one, and that the object is not seen in the same place with one eye as with two. Vision, he says, is single, if the two axes of the pyramids of the visual rays are directed in the same manner on the object, but becomes double if the axes are not directed in a similar manner, and if the distance is a little less than the distance between the eyes. He next proceeds to find, geometrically, the circumstances which produce single or double images. He ascribes imperfection of sight in old men to a want of the visual virtue, which, like the other faculties, decays with the approach of age; and he states that those who have convergent eyes see at a less distance than those who have not such eyes. Rapidity of motion, he asserts, confounds the colours on a wheel. If the colour is in the direction of a radius the wheel will appear entirely of this colour, and if different colours are at different distances from the centre, these will..." appear on the wheel as so many concentric circles differently coloured. When after looking long at a coloured object, we direct the eye to another, we attribute to it the colour of the first.
"In the third book, which treats of reflexion from plane and concave mirrors, he shews, that in a plane mirror the object is seen in the perpendicular, drawn from the object to the plane of the mirror and continued behind it. He mentions that objects appear smaller towards the zenith and larger towards the horizon, because in the former case we see them in a position to which we are less accustomed.
"In concave mirrors the objects appear concave, and in convex ones they appear convex, and the image is seen at the point of intersection of the reflected ray, and the line drawn from the object to the centre of the sphere.
"The fourth book treats of concave and compound mirrors, and of the effects of two or more mirrors. In these mirrors an object may be reflected and rendered visible by all the parts of the mirror, or by three, or two, or even one point. The image may be either on the surface of the mirror, or before the surface, or behind the eye, or behind the mirror. When the image is behind the mirror, the distance of the object from the mirror is less than that of the image. When the image is between the eye and the mirror the distance of the object from the eye will be sometimes greater than the distance of the image from the mirror, and sometimes it will be equal to it, and sometimes less. When the object is between the mirror and the eye it will be seen in a part different from that where it really is; and if we give it a motion in one direction it will appear to move in the opposite direction.
"The fifth book is the most curious and valuable of the whole work. Ptolemy begins by explaining the experiment with the piece of money, which, when concealed behind the side of a vessel, becomes visible by filling it with water. The refraction of the visual ray in penetrating the water makes us see the piece of money out of its place, and in the prolongation of the primitive direction of the ray emitted from the eye. In order to measure this refraction at different angles Ptolemy employs a circle divided into 360°, the inferior half of which is plunged in the water, so that the refracting surface covers one of the diameters of the circle. The centre of the circle is marked by a small coloured body, and a second similar body is fitted to one of the quadrants out of the water, and at a given distance from the vertical diameter; a third coloured body slides on the lower part, which is immersed in the water. This last body is then pushed with a rod till the eye placed on the body in the air sees all the three in a straight line. The two distances of the second and third body from the vertical diameter are thus measured on the graduated circle.
"In this manner Ptolemy obtained the results in the following table, which contains the angles of refraction from air to water from 10° up to 80° of incidence.
| Angles of incidence | Angles of refraction | Ratio of the sines of incidence and refraction | |---------------------|---------------------|-----------------------------------------------| | 0° | 0° | 0-7486 | | 10° | 7° | 0-70179 | | 20° | 13° | 0-68255 | | 30° | 20° | 0-70041 | | 40° | 25° | 0-65748 | | 50° | 30° | 0-65270 | | 60° | 34° | 0-65403 | | 70° | 38° | 0-66247 | | 80° | 42° | 0-67946 |
The mean of these ratios is 0-67366, whereas the true ratio is 0-64316; but at the angles of incidence of 40°, 50°, and 60°, the ratio is very near the true one.
When the semi-cylinder of glass was placed on the surface of water, Ptolemy observed that the refractions from water into glass were less than any he had observed, because the difference of density between water and glass was less than between water and air. The following were the results which he obtained:
| Angles of incidence | Angles of refraction | Ratio of the sines of incidence and refraction | |---------------------|---------------------|-----------------------------------------------| | 0° | 0° | 0-95014 | | 10° | 9° | 0-95274 | | 20° | 18° | 0-90778 | | 30° | 27° | 0-89233 | | 40° | 35° | 0-88192 | | 50° | 42° | 0-87804 | | 60° | 49° | 0-88422 | | 70° | 56° | 0-89657 | | 80° | 62° | 0-89657 |
The mean of these ratios is 0-90, the true ratio being 0-8760, the index of refraction for water being 1.336, and that of glass 1.525; but at the angles of incidence of 50°, 60°, and 70°, the ratio is very near the true one.
Ptolemy now discusses the important subject of astronomical refraction, which he ascribes to the difference of density between ether and air. If the visual ray, he remarks, is stopped by an impenetrable body, it could not show us a body which is hid behind the first; and if the second becomes visible, it can only be on account of the flexion of the visual ray. This flexion takes place at its passage into a medium of different density; and the possibility of this flexion, he asserts, may be proved by the following phenomena. By observations on the stars, it was found that the parallels drawn through the apparent place of those which rise or set, are those nearer the north pole than the paral-
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1 This is 45° 30' in the Oxford manuscript. 2 In the Oxford MS. this is 18° 30'. Professor Rigaud supposes the real number to have been 19° 30'. lens which pass through their apparent place when they are in the meridian; and the nearer the stars are to the horizon, the greater is the approach of their parallels to the pole. By observing a circumpolar star, Ptolemy found that it was nearer the pole in its lower passage across the meridian; but when it was near the zenith, its parallel became greater in appearance, whereas in the first case it became smaller. Hence it follows that refraction raises the stars towards the zenith. In order to explain the manner in which refractions operate, Ptolemy makes use of the same figure upon which Cassini has since founded his theory. He employs almost the same reasoning in order to determine the quantity of the refraction. He remarks, that the more a star is elevated, the less will be the difference between its true and its apparent place, and that this difference will be nothing in the zenith, because a perpendicular ray experiences no flexion. He demonstrates by a figure, that in every case the refraction carries the star towards the zenith; and he states that the height of the atmosphere is unknown, but that it must begin below the sphere of the moon. From this general account of the fifth book of the Optics of Ptolemy, it will be seen that he gives a theory of astronomical refractions much more complete than that of any astronomer before the time of Cassini.
These important results, which, without any other assistance, would have enabled the optician to trace the progress of the rays of light through lenses of all forms, were not applied as they might have been, to extend the boundaries of the science. Banished from Europe, optics, along with the other sciences, found shelter in Arabia; and after a period of a thousand years, it was destined to receive fresh accessions in that favoured country.
Alhazen, who flourished about the end of the eleventh century, was the individual who gave this fresh impulse to optical science. He establishes the opinion of Pythagoras, that vision is performed by rays which proceed from the object to the eye; and he states that vision is not completed till the ideas of external objects are conveyed by the optic nerves to the brain; and after a description of the eye and its parts, he assigns to each of them the function which it performs in vision. He maintains that we see objects singly with two eyes, because we must perceive only one image when it is formed on corresponding parts of the retina. The instrument employed by Alhazen for measuring the angle of refraction, is more complex than that used by Ptolemy, and his knowledge of the refraction of the atmosphere and of fluids, is obviously inferior to that of the Alexandrian philosopher. Alhazen ascribes to refraction the twinkling of the stars, and the contraction of the diameters and distances of the heavenly bodies; and it follows from his method of reasoning, that refraction elevated the stars towards the pole and not towards the zenith, as had been sagaciously ascertained by Ptolemy. Alhazen has described seven species of mirrors, and he was the first person who determined the focus of rays after reflexion, when the place of the object is known. He has treated largely of optical illusions, whether produced in direct or in refracted and reflected vision; and he ascribes the size of the horizontal moon to the apparent form of the concavity of the sky, which is imagined to be more remote in the horizon than anywhere else. Alhazen likewise observed that objects were magnified when held close to the plane wall of the larger segments of a glass sphere; and he has given rules, which are far from being correct, for determining the apparent size of objects when seen through such spheres.
The next cultivator of optics was Vitello, whose work was first published at Nuremberg in 1535. He made a series of experiments on the angles of refraction of water and glass, which apparently exceeded those of Ptolemy in correctness, the mean ratio of the sines being nearer the truth, and the ratio for each angle of incidence coinciding more accurately with the mean ratio. The following are the results he obtained with water:
| Angles of incidence | Angles of refraction | Ratio of the sines | |--------------------|---------------------|-------------------| | 0° | 0° | 0.77658 | | 10° | 7° 45' | 0.78135 | | 20° | 15° 30' | 0.76537 | | 30° | 22° 30' | 0.75423 | | 40° | 29° 0' | 0.74875 | | 50° | 35° 0' | 0.74992 | | 60° | 40° 50' | 0.75904 | | 70° | 45° 30' | 0.77787 | | 80° | 50° | |
The mean of these ratios is 0.76414, whereas that obtained by Ptolemy was 0.76736, and the true ratio (the index of refraction being 1.3358) is 0.7486. The results for 30° and 60° are exactly the same as Ptolemy's.
The following were the measures obtained by Vitello for glass:
| Angles of incidence | Angles of refraction | Ratio of the sines | |--------------------|---------------------|-------------------| | 0° | 0° | 0.70179 | | 10° | 7° | 0.68255 | | 20° | 13° 30' | 0.66761 | | 30° | 19° 30' | 0.65748 | | 40° | 25° 0' | 0.65270 | | 50° | 30° | 0.65403 | | 60° | 34° 30' | 0.66247 | | 70° | 38° 30' | 0.67945 | | 80° | 42° | |
The mean of these ratios is 0.66976, whereas that obtained by Ptolemy is 0.68736, and the true ratio 0.64516.
In comparing this last table with the similar one given by Ptolemy, we cannot fail to be struck with their entire similarity, with the single exception of the angle of refraction at 30° of incidence, which Vitello makes 19° 30', and Ptolemy, in the Paris copy, 20° 30'. Now, in the Oxford manuscript, the numbers are 18° 30'; and Professor Rigaud conjectures that the real number has been 19° 30', the same as Vitello's. Hence we cannot on any just grounds regard the measures of refraction given by the Polish philosopher as anything else than those of Ptolemy, from whom he must have borrowed them.
By comparing the two tables for water, we are inclined to make the same unfavourable supposition. The refraction for 20°, 30°, and 50° of incidence are exactly the same in both; and Vitello's measure for 70°, viz. 45° 30', is the same as Ptolemy's in the Oxford manuscript.
But this opinion is converted into certainty when we examine Vitello's table of the refractions from water into glass, in which all the measures are identically the same with those of Ptolemy.
In the course of his experiments, Vitello was led to observe that whenever light was reflected or refracted by transparent bodies, a certain portion of it was lost, but he does not estimate the quantity, contenting himself with the observation that bodies always appear less luminous when seen by refracted and reflected light. In treating of the cause of the rainbow, he shews that refraction is as neces-
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1 Montucla has very incorrectly charged Alhazen with borrowing the greater part of his optics from Ptolemy. Delambre has refused this opinion, and rendered it probable that the Arabian philosopher never saw the work of Ptolemy. What assistance he obtained from his predecessors who flourished after Ptolemy cannot now be ascertained. See Connaissance des Temps for 1816.
2 This work has been very erroneously regarded as little more than a translation of Alhazen's treatise. On the authority of various passages in the writings of Invention friar Bacon, Mr. Molyneux is of opinion that he was ac- of Specta- quainted with the use of spectacles, and when Bacon says that "this instrument (a plano-convex glass, or large seg- ment of a sphere), is useful to old men, and to those that have weak eyes; for they may see the smallest letters suf- ficiently magnified," we are at least entitled to conclude that the particular way of assisting decayed sight which he de- scribes was known to him, though he may not have used his segment of a glass sphere in looking at objects separated by an interval from its plane side. But whether spectacles were in use or not in Bacon's time, it is quite certain that they were known and used about the time of his death, which happened in 1292. Alexander de Spina, a native of Pisa, who died in that city in 1313, having seen a pair of spectacles made by some other person, who was unwilling to communicate the secret of their construction, got a pair made for himself, and found them so useful, that he cheer- fully made the invention public. M. Spoon, to whom we are indebted for this fact, fixes the date of the invention be- tween 1280 and 1311. Signor Redi, from whom Spoon quotes the preceding fact, states that he possesses a manu- script written in 1299, Di Governo della Famiglia de Scan- dro di Pissozzo, in which the author says, "I find myself so pressed by age, that I can neither read nor write without those glasses they call spectacles, lately invented, to the great advantage of poor old men, when their sight grows weak." It is stated also in the Italian Dictionary Della Cruce, under the head of Occhiali or Spectacles, that friar Jordan de Rivalto, who died at Pisa in 1311, tells his audi- ence, in one of his sermons, which were published in 1305, "that it is not twenty years since the art of making spec- tacles was found out, and is indeed one of the best and most necessary inventions in the world." Bernard Gordon, too, a celebrated physician of Montpellier, in his Lilium Medi- cinae, published about 1305, recommends an eye salve as capable of making the patient read the smallest letters without spectacles; and Muschenbroek informs us that it is inscribed on the tomb of Salvius Armatus, a Florentine nobleman, who died in 1317, that he was the inventor of spectacles.
Before we quit the period of Friar Bacon we must notice Leonard a claim to the invention of the telescope which has been Diego- made in favour of Leonard Digges, an Englishman, because Died 1574. this claim, whatever be its amount, supports undeniably the prior claim of Bacon. The claim of Digges is founded on passages in his Pantometria and Stratiotikos. The first of these works appeared at London in 1571, and a second ed- ition of it, edited by his son Thomas Digges, Esq., was pub- lished in 1591. The Stratiotikos was published in 1579 and also in 1590. In the preface to the second edition of the Pantometria, Thomas Digges remarks: "My father, by his continual painful practices, assisted by demonstrations mathematical, was able, and sundrie times hath, by propor- tionall glasses, dely situate in convenient angles, not onely discouerred things farre off, read letters, numbred pieces of money, with the verye coyne and superscription thereof; cast by some of his freinds of purpose upon downes in the open fields, but also seven miles off declared what hath beene doone in priuate places."
In the twenty-first chapter of the first book, Leonard Digges himself says, "But marvellous are the conclusions that may be performed by glasses (mirrors) concave and convex, of circular and parabolic forms, using for multiplications of beams, sometimes the aid of glasses transposed, which, by practice, should unite or dissipate the images or figures presented by the reflection of others. By these kind of glasses, or rather frames of them placed in due angles, yee may not only set out the proportion of an whole region, yea, represent before your eye the lively image of every house, village, &c., and that in as little or great space or plan as ye will presente; but also augment or dilate any parcel thereof, so that, whereas, at the first appearance a whole town shall present itself so small and compact together that ye shall not discover any difference of streets, yee may, by application of glasses in due proportion, cause any peculiar house or room thereof dilate and shew itself in as ample form as the whole towns at first appeared, so that yee shall discern any trifle, or read any letter lying there open, especially if the sun beams may come into it, as plainly as if you were corporeally present, although it be distant from you as far as eye can discier. But of these conclusions I mind not here more to introduce, having at large, in a volume by itself, opened the miraculous effects of perspective glasses.
Now it is a curious fact that Thomas Digges expressly says that his father's knowledge of optics "partly grew by the aid he had by one old written book of Bakon's Experiments, that by strange adventure, or rather destinie, came to his hands."
In support of the opinion that the telescope was known in England more than forty years before 1609 or 1610, when it was supposed to have been invented in Holland, we may quote a passage or two from the celebrated John Dee's mathematical preface to Euclid, written at Mattoke on the 9th of February 1570, the year in which it was published.
"Is it not," says he, "greatly against the sovereignty of man's nature to be so overshot and abused with things before his eyes? things far off to seem near, and near to seem far off; small things to seem great and great to seem small; one man to seem an army, or a man to be constantly afraid of his own shadow; yea, so much to fear, that if you, being alone, near a certain glass, and prefer with dagger or sword to foyne at the glasse, you shall suddenly be moved to give back by reason of an image appearing in the air between you and the glass, with like hand, sword, or dagger, and with like quickness foyning at your eye, likewise as you do at the glasse." He then mentions that such a glass was in the possession of a gentleman famous and honourable for his good service done to his country. Dee likewise speaks of having seen once or twice in company with Orontius at St. Denis in 1551; "The lively image of another man in the air aloft, walking to and fro or standing still," but the most remarkable passage is that in which he speaks of the means of ascertaining the numbers of an enemy's army; "The herald, pursuivant, sergeant royall, captain, or whosoever is careful to come near the truth herein; besides the judgment of his expert eye, his skill of ordering tactically, the help of his geometrical instrument; ring or staffe astronomical, commodiously framed for carriage and use. He may wonderfully help himself by perspective glasses, in which I trust our posterity will prove more skilful and expert, and to greater purposes than in these days can be credited to be possible."
When polite learning began to revive in Europe some of the more abstract sciences began to be cultivated with success. Maurolycus, a teacher of mathematics at Messina, A.D. was particularly distinguished by his optical researches, of which he published an account in his Theoremata de Luminis et Umbra et Diaphanorum Partes, seu libri tres. In the first of these works, which was completed in 1525, but not published till 1575, Maurolycus treats of the measure of light, or the illumination of bodies, and he particu-
1 By a glass, he means "any thing from which a beam reboundeth." 2 The yellow which is in the sun's direct rays is observed when the sun's rays are reflected from the sky or from clouds. which held its sittings in his own house, and which numbered among its members all the virtuosi in Naples. Each member was bound to contribute to the common stock something not commonly known, and in this way he obtained the materials for his Magia Naturalis which appeared in the year 1560, when he was only about fifteen years of age. This work was speedily translated into French, Hebrew, Spanish, and Arabic, and went through numerous editions in different parts of Europe. The Papal court viewed with jealousy the proceedings of a society which devoted so much energy to the spread of knowledge, and, though Baptista Porta was a Roman Catholic, the meetings of the academy were prohibited by the ecclesiastical power. Although Baptista Porta was well acquainted with the writings of his predecessors, yet the principal invention recorded in his Natural Magic is that of the camera obscura, which he seems to have brought to great perfection. He remarks, in the 17th chapter of this work, that if a small aperture is made in the shutter of a dark room, distinct images of all external objects will be depicted on the opposite wall in their true colours; and he further adds that if a convex lens be fixed in the opening, so that the images are received on a surface at the distance of its focal length, the pictures will be rendered so much more distinct that the features of a person standing on the outside of the window may be readily recognised in his inverted image. Various attempts seem to have been made to obtain an erect image of objects in the camera obscura, but the contrivances for this purpose had the effect, as Baptista Porta assures us, of making the pictures so obscure that there was no pleasure in viewing them. He observes, however, that the image may be best rendered erect by receiving it "upon a concave mirror properly adapted to the convex lens and held at a great distance from the hole." The effect of this expedient, he says, cannot be sufficiently admired. Baptista Porta applied his instrument to the representation of eclipses of the sun, and of hunting scenes, battles, and other events produced by moveable pictures and drawings. In this way he magnified small objects and drawings, and produced the effects of the magic lantern by the light of the sun in place of that of a lamp. He considered the eye as a camera obscura, the pupil as the hole in the window contracting and dilating with different lights, and the crystalline lens as the principal organ of vision, though he seems to have regarded it not as his convex lens but as the tablet on which the images of external objects were formed, the cornea being, no doubt, in his estimation, the part of the eye which formed the picture. Baptista Porta was doubtless acquainted with what may be called the simplest form of the refracting telescope, namely, that in which a convex lens is the object-glass, and the eye placed six inches behind its focus, the eye-glass. He found that when his eye was thus placed behind a convex lens, he could read a letter which he could not read with his naked eye. His lens must have exceeded in focal length, the distance at which his eye saw distinctly, for the magnifying power of such a telescope is equal to the focal length of the object-glass divided by that distance.
In another place Porta, after mentioning the effects produced by a concave and a convex lens separately, remarks, "that if you knew how to combine one of each sort rightly, you would see both far and near objects larger and more clearly." "If Porta," says Mr. Drinkwater Bethune, in his admirable life of Galileo, "had stopped here, he might more securely have enjoyed the reputation of the invention, but he then professes to describe the construction of his instrument, which has no relation whatever to his previous remarks." "I shall now endeavour to show in what manner we may contrive to recognise our friends at the distance of several miles, and how those of weak sight may read the most minute letters from a distance. It is an invention of great utility, and grounded on optical principles, nor is it at all difficult of execution; but it must be so divulged as not to be understood by the vulgar, and yet be clear to the sharp-sighted." The description which follows, seems far enough removed from the apprehended danger of being too clear; and indeed every writer who has hitherto quoted it, has merely given the passage in its original Latin, apparently despairing of an intelligible translation. With some alterations in the punctuation, which appear necessary to bring it into any grammatical construction, it may be supposed to bear something like the following meaning:—"Let a view be contrived in the centre of a mirror, where it is most effective. All the solar rays are exceedingly dispersed, and do not in the least come together (in the true centre); but there is a concourse of all the rays in the central part of the said mirror, half way towards the other centre, where the cross diameters meet. This view is contrived in the following manner: A concave cylindrical mirror placed directly in front, but with its axis inclined, must be adapted to the focus; and let obtuse angled, or right angled triangles be cut out with two cross lines on each side drawn from the centre, and a glass (specillum) will be completely fit for the purposes we mentioned." If it were not for the word specillum, which, in the passage immediately preceding this, Porta contrasts with speculum, and which he afterwards explains to mean a glass lens, it would be very clear that the foregoing passage, supposing it to have any meaning, must be referred to a reflecting telescope; and it is a little singular, that whilst this obscure passage has attracted universal attention, no one, so far as we are aware, has taken any notice of the following unequivocal description of the principal part of Newton's construction of the same instrument. It is in the fifth chapter of the seventeenth book, where Porta explains by what device exceedingly minute letters may be read without difficulty, "Place a concave mirror so that the back of it may lie against your breast; opposite to it and within the burning point, place the writing; put a plane mirror behind it, that may be under your eyes. Then the images of the letters which are in the concave mirror, and which the concave has magnified, will be reflected in the plane mirror, so that you may read without difficulty."
On these grounds Porta claimed for himself the invention of the telescope; and his death which took place in 1615, at the age of eighty, is said to have been hastened by the exhaustion of writing a work on that instrument.
At a more advanced age, Baptista Porta composed another work entitled, De Refractione Optices parte, libri novem, which appeared in 1593, but contains nothing that is deserving of particular notice.
The subject of the rainbow, which had hitherto been a question vexata among philosophers, now began to excite notice, as much from the absurdity of the theories which were advanced to explain it, as from the native interest of the subject. Clichotovus, whom Dr. Priestley supposes to Clichot be the person who distinguished himself by his opposition, died to Luther, had maintained that the second rainbow was a reflected image of the first, not only from the faintness of its light, but from the inversion of its colours; an opinion which our celebrated countryman, Dr. Gilbert has justly ridiculed, alleging, that the whole form of the bow ought to be inverted as well as its colours, and that it ought to have its convex side downwards.
A more correct theory of the rainbow was about this time proposed by J. Fleschier of Breslau, in his treatise entitled, De Iridibus doctrina Aristotelis et Vitellionis, which A.D.157 was published in 1571. He supposes the rays to suffer two refractions, one on entering, and the other on emerging. from the drop, but after one ray had thus been separated into a coloured beam by these refractions, he supposed that this beam was reflected to the eye from another drop.
These views, imperfect as they are, paved the way for the true theory of the rainbow. Antonio de Dominis, archbishop of Spalatro, first broached this theory in his treatise De Radiis Visus et Lucis, which was published in 1611. He justly asserts that two refractions in a drop of water, and one intermediate reflection, were sufficient to bring back to the eye of the spectator the rays of light by which the bow was formed. An experiment with a globe of glass enclosing water, either suggested to him or confirmed this opinion. In following out this experiment, however, our author committed several mistakes. He explained the exterior bow by the same number of refractions and one reflexion, but he supposed that the rays which formed it were returned to the eye by a part of the drop lower than that which transmitted the red of the interior bow. In addition to this mistake, he supposed that the rays which went to form one of the bows came from the upper part, and that which went to form the other bow from the under part of the sun's disc. Notwithstanding these mistakes, De Dominis is entitled to be regarded as the true discoverer of the cause of the primary rainbow.
The treatise containing these discoveries was not published till after the use of the telescope by Galileo, but Bartolo, who published it, informs us in the preface, to use the words of the author of the Life of Galileo, "that the manuscripts was communicated to him from a collection of papers written twenty years before, on his enquiring the archbishop's opinions with respect to the newly discovered instrument, and that he got leave to publish it, 'with the addition of one or two chapters.' The treatise contains a complete description of a telescope, which, however, is proposed merely to be an improvement on spectacles; and if the author's intention had been to interpolate an after-written account in order to secure to himself the undeserved honour of the invention, it seems improbable that he would have suffered an acknowledgment of additions, previous to publication, to be inserted in the preface. Besides, the whole tone of the work is that of a candid and truth-seeking philosopher, very far indeed, removed from being, as Montucla calls him, conspicuous for ignorance even among the ignorant men of his age. He gives a drawing of a convex and concave lens, and traces the passage of the rays through them; to which he subjoins, that he has not satisfied himself with any determination of the precise distance to which the glasses should be separated according to their convexity and concavity, but recommends the proper distance to be found by actual experiment, and tells us that the effect of the instrument will be to prevent the confusion arising from the interference of the direct and refracted rays, and to magnify the object by increasing the visible angle under which it is viewed."
From the great liberality of his sentiments, and his conversion to the protestant faith, this eminent ecclesiastic was obliged to leave Italy, and to take refuge in London, in 1616, where he lived some years. Having been induced to return to Italy, his imprudence exposed him to new persecutions, and having been imprisoned by Urban VIII., he died of poison in the prison of the Inquisition. Sentence was passed upon him after his death, and his body, with all his books and papers were publicly burnt in the Campo de Fero, in the year 1624.
We now approach the time when the telescope was unquestionably invented. We have no doubt that this invaluable instrument was invented by Roger Bacon or Baptista Porta, in the form of an experiment, though it perhaps had not in their hands assumed the maturity of an instrument made for sale, and applied to useful purposes both terrestrial and celestial. If a telescope is an instrument by means of which things at a distance can be seen better than by the naked eye, then Baptista Porta's concave lens with his eye looking at the image which it formed, and reading a letter too remote to be otherwise legible, was a real telescope; but if we give the name to a tube having a convex object glass at one end, and a convex or a concave lens at the other, placed at the distance of the sun or the difference of their focal lengths, then we have no distinct evidence that such an instrument was used before the beginning of the seventeenth century.
In his Treatise on Dioptrics, Descartes has ascribed the invention of the telescope to James Metius, a citizen of Alkmaer in Holland; but Huygens in mentioning this claim says, that "to his certain knowledge telescopes were made before this, at Middleburg in Zeeland, about the year 1609, either by John Lippersheim, whom Sirtorus mentions, or by Zacharias (Jansen), whom Borellus makes the first inventor of them in his book de Vero Telescopii inventore. The telescope which they made did not exceed a foot and a half long. But much earlier than both, Joannes Baptista Porta, a Neapolitan, had delivered the rudiments of this art in his book on dioptries and natural magic, published fifteen years before telescopes appeared in our Belgium. In these books he speaks of his specilla as showing things placed at a distance as if they were nigh, and also of the construction of concave and convex lenses. But that he made no great progress in this art is hence evident, that in all that time it did not become famous; and that he did not discover any of those things in the heavens that were observed afterwards. And this shews that his invention was not owing to his skill, but to some accidental experiments. For though he had some degree of knowledge in mathematicks, yet he did not understand those fundamental principles and abstruse theorems in dioptries, that are necessary to invent a telescope by reason; and by much less did those illiterate mechanicks that I mentioned before. But no wonder that good luck and casual experiments should produce this effect; after spectacles and concave glasses, for defective eyes of both sorts, had been commonly used for above three hundred years, it is rather surprising that so obvious a thing should be so long unknown."
In this passage Huygens leaves the claims of his two Dutch friends on the same level; but though Borellus adopts the conclusion that Jansen was the inventor, yet it has been ingeniously suggested, that Jansen's claim as the inventor of the microscope, has been mixed up with the invention of the telescope, on the evidence adduced by Borellus. On this hypothesis Lippersheim is supposed to have invented the telescope by accident in 1609, and that Jansen possessing an instrument so like it, had been able, after hearing of Lippersheim's contrivance, to make a similar instrument without having seen the telescope of his rival.
Much light has recently been thrown on the history of the telescope by Professor Moll, who has discussed the claims of the various competitors with much sagacity and fairness. It appears from the official acts and journals of the States General of Holland, still existing among the archives at the Hague, that on the 2d of October 1608 that body took into consideration a petition from Hans (John) Lippershey, a native of Wesell, and spectacle maker at Middleburg, praying, that an instrument which he had invented for seeing at a distance, might be rewarded, either by granting an exclusive privilege of making it for thirty years, or an annual pension to enable him to make these instruments for Holland alone. It was resolved, that a commit- tee should communicate with the petitioner, and enquire if he could not so improve the instrument so as to enable one to look through it with both eyes. Lippershey offered to make three telescopes of rock crystal for one thousand florins each, (about £83 each) but the committee was instructed to get him to moderate his charge, and promise never to transmit his invention to any body. On the 6th of October a bargain was made that Lippershey should construct one instrument of rock crystal for the States, at the price of 900 florins, (£75) 300 florins to be paid down, and 600 when the telescope was completed and approved of. On the 16th December the committee report, "that they examined the instrument invented by Lippershey to see at a distance with two eyes, and that they approved of it." But, in reference to the exclusive privilege, they "resolved that, whereas it appears that many other persons have a knowledge of this new invention to see at a distance, it is expedient to refuse the prayer of the petitioner for an exclusive privilege, but that he will be commanded to make, within a certain time, two other instruments of his invention for seeing with two eyes, at the same price." These two new instruments were delivered before the 13th February 1609.
While these transactions were going on, Jacob Adriaansz, sometimes called Metius of Alkmaer petitioned the States-General on the 17th of October 1608, for an exclusive privilege for a similar instrument. He was the third son of Adrianus Anthonisz, or Metius, who discovered the approximate ratio of the diameter of a circle to its circumference. His petition still exists among the manuscripts of Huygens, in the library at Leyden. He alleges, that he began his researches as far back as 1606; that the invention was accidental, and when he was making other experiments; and that, in 1608, when he sent in his petition, his instrument was made of bad materials. He at the same time readily admits, that a spectacle maker of Middleburg had offered before him a similar instrument to the States, which had been tried by Prince Maurice, and other persons.
With regard to the claims of Zacharias Jansen, or rather Tansz or Zansz, they cannot be supported by any evidence, and there is reason to believe, as we shall afterwards see, that his invention of the microscope was mistaken for the invention of the telescope. The following is Professor Moll's summary of the facts which he has established by authentic documents.
"That on the 21st of October 1608, John, or Hans Lippershey, a native of Wezel, a spectacle maker of Middleburg, in Zeeland, was actually in possession of the invention of telescopes.
"That, on the 17th of October, of the same year 1608, Jacob Adriaansz, sometimes called Metius of Alkmaer in Holland, also was in possession of the art of making telescopes, and that he actually made those instruments; but that either from disgust or some other reason, he afterwards concealed his invention, and thus actually gave up every claim attached to the honour of it.
"That there is little reason to believe that either Hans, or his son Zacharias Zansz, were also inventors of the telescope; but there is every probability that this Hans, or John, or his son Zacharias Zansz, invented a compound microscope about 1590.
"That this Lippershey used rock or mountain crystal in the construction of telescopes, and that he is the inventor of the Binoculars."
When Galileo was at Venice in 1609, he heard rumours Galileo's that an instrument which represented distant objects as if constructed were near, had been invented by a Dutch spectacle maker. This rumour was confirmed by a letter which he received from James Badorere at Paris, and Galileo, who asserts that he had never seen one of the instruments, set himself to discover the principle of their construction, and to make one for his own use. It has become a question, though one of no interest, and affecting the veracity more than the reputation of Galileo, whether the Italian philosopher had actually seen one of the new instruments. We cannot hesitate for a moment in believing Galileo's assertion; and even if we confide in the statement made by Fucarius, that he had himself seen one of the Dutch telescopes, which at that time had been brought to Venice, it by no means follows that Galileo saw it. It is quite certain, indeed, that previous to the 31st August 1609, one of the new perspective glasses had been sent from Flanders to the Cardinal Borghese, and Lorenzo Pignoria, on the authority of whose letter of the above date this fact rests, adds, "we have seen some here, and truly they succeed well."
The following is Galileo's account of the matter, from a letter which he wrote in March 1610: "It is about ten months ago that it came to our ears, that a glass had been worked by a Belgian, by the help of which, visible objects, though at a great distance from the eye of the observer, may be seen distinctly. (In the Italian of the Saggiatore it is added, ne pia aggiunto, no more was added, or this was all.) And some experiments were related of the admirable effects of this instrument, which some believed, and others not. A few days afterwards the same was confirmed by letters of a noble Frenchman, Jacob de Badorere, from Paris; all which occasioned me to apply myself wholly to enquire into the cause of this, and to think on the means by which the invention of a similar instrument might be brought about; in which I succeeded in a short time, assisted by the doctrine of refraction; and I first procured a leaden tube, (an organ pipe) at the end of which I adapted spectacle glasses, both plane on one side, the one convex on the other side, the second concave. Bringing the eye near the concave glass, I saw the objects large, and near enough; they appeared three times nearer, and nine times larger, than if seen with the naked eye.
"Afterwards I made another instrument, which made objects appear sixty times larger.
"Finally, sparing neither industry nor expense, I succeeded so far as to make an instrument of such excellence, as to make the objects seen through it appear a thousand times larger, and more than thirty times nearer, than if seen with the natural power of the eye."
Galileo's first telescope must have been made in May or June 1610. Viviani says, that it was in April or May 1609, that the rumour of the invention of the telescope reached Venice, when Galileo was there, and that, with this information only, Galileo returned to Padua and succeeded in finding out the principle in the following night.
The new instrument long went by the names of Galileo's tube, the perspective, and the double eye-glass, the History. more appropriate names of telescope and microscope having been afterwards given to these instruments by Demisiano.
Telescopes Telescopes were early and eagerly imported into England, and known by the name of trunks and cylinders; A.D. 1619, and so soon as July 1609, we find that our countryman Harriot was directing them to the lunar disc, and had begun two full drawings of that luminary, which he afterwards completed. Harriot's earliest observations on Jupiter's satellites were made on the 1st October 1610, nine months after their discovery by Galileo. The earliest telescope in England must therefore have been obtained from Holland; and in a letter from Sir William Lower to Harriot, dated the longest day of 1610, from Traventi in Caernarvonshire, he says, "we are here so on fire with these things, that I must render my request and your promise, to send more of all sorts of these cylinders. My man shall deliver you monie for anie charge requisite, and contente your man for his pains and skill. Send me so many as you think needful unto these observations: in requital I will send you store of observations. Send me also one of Galileus bookes, if anie yet be come over, if you can get them." In a letter dated July 6, 1610, Sir Christopher Heyden writes to his friend Camden, "I have read Galileus, and to be short, do concur with him in opinion, for his reasons are demonstrative; and of my own experience with one of our ordinary trunks, I have told eleven stars in the Pleiades, whereas no age ever remembers above seven, and one of these, as Virgil testifieth, not always to be seen?" From this and other facts, Professor Rigaud infers "that it is perfectly clear that Harriot and his friend had been in the habit of using telescopes before the discoveries of Galileo were known to them; and it appears likewise that in 1610 they were manufactured in England."
The magnifying power of some of the telescopes used by Harriot were \( \frac{3}{2}, \frac{5}{4}, \frac{7}{8}, \frac{9}{10}, \frac{11}{12}, \frac{13}{14}, \frac{15}{16}, \frac{17}{18}, \frac{19}{20} \). In a letter from Sir William Lower to Harriot, dated Traventi, 6th July 1610, he says, "I have received the perspective cylinder that you promised me, and am sorry that my man gave you not more warning, that I might have had also the two or three more that you mentioned to chuse for me. Henceforward he shall have orders to attend you better, and to defray the charge of this an others, for he confesseth to me that he forgot to pay the worke man.
According as you wished, I have observed the moone in all his changes. In the new I discover manifestie the earthshine a little before the dichotomie; that spot which represents unto me the man in the moone (but without a head) is first to be seene. A little after, neare the brimme of the gibbous parts, towards the upper corner, appeare luminous parts like starres, much brighter than the rest; and the whole brimme along lookes like unto the description of coasts in the Dutch bookes of voyages. In the full she appears like a tarte that my cooke made me the last weeke. Here a vaine of bright stuffe, and there of darke, and so confusedlie al over. I must confess I can see none of this without my cylinder; yet an ingenious younge man that accompanies me here often, and loves you and these studies much, sees manie of these things, even without the helpe of the instrument, but with it sees them most plainlie, I mean the young Mr. Protheroe."
It is highly probable that the first Dutch telescopes had their eye-glass concave, like Galileo's, though this supposition is opposed by the traditional story of a large and inverted image of a weathercock having been seen through the earliest of them, in which case the eye-glass must have been convex. Even so late as the period when Descartes published his Dioptrics, which was in 1637, no other telescope but a Galilean one had been described, excepting in Kepler's Dioptrica, which appeared at Frankfort in 1611. In his 86th proposition he explains the theory of the telescope, and has shown how an instrument which produces the same effects might be made, by substituting for the usual concave eye-glass one or more convex eye-glasses. Kepler, however, does not seem to have constructed such a telescope, and father Scheiner seems to have been the first person who embodied the plan in an actual instrument, which has ever since been known by the name of the astronomical telescope, in consequence of the inversion of the images not being disagreeable in astronomical observations.
The real inventor of the compound microscope is as little known as the inventor of the telescope. It would be vain to inquire into the history of the single microscope, for the magnifying power of globes was known to the ancients; and no individual ambition or national partiality has endeavoured to assign the honour of inventing it to any person whatever. We agree with Professor Moll, that Zacharias Zansz, or Jansen, has the best claims to be considered as the inventor and constructor of the compound microscope. He seems to have made one so early as 1590, and to have presented one to the Archduke Albert of Austria, who gave it to Cornelius Drebell, who lived, as mathematician to the king, at the court of our James the First. William Borell, the envoy to England from the States of Holland, saw in England, in 1619, and in the hands of Cornelius Drebell, the very microscope which Tanzz had given to the archduke. This account of its history was given by Drebell himself. The microscope in question was eighteen inches long, consisting of a tube of gilt copper two inches in diameter, supported by two sculptured dolphins, resting on a base of ebony, upon which the objects were placed. M. Fontana, a Neapolitan, first described the compound microscope, consisting of two convex lenses, in his work entitled Nova Terrestre et Celestium Observationes, which appeared in 1646, but claims to have made the discovery so early as 1618, though he does not adduce any evidence whatever of this fact. Huygens, on the contrary, says, "It does not appear that these microscopes were made in the year 1618, because Sirturus, who published a book that year about the origin and construction of telescopes, would hardly have been silent upon so remarkable an invention, if it had been thus known. Fontana, indeed, lays claim to it from the year 1618, in his book of Observations, published in 1646; but the testimony of Lyrsalis, there printed, goes no higher than the year 1625. But that my countryman Drebelius made these compound microscopes at London in the year 1621, I have often been informed by several eye-witnesses, and that he was then reckoned the first inventor of them."
This testimony of Huygens in favour of Drebell, is in direct contradiction to the statement said by Borell to have been made to the Dutch envoy in 1619.
In consequence of this conflicting evidence, Galileo may be regarded as having the best claim to the invention of the compound microscope. Viviani distinctly informs us in his Life of Galileo, that he was led to the invention of the microscope by that of the telescope, and that in the year 1612, he actually sent a microscope in a present to Sigismund King of Poland. Having been dissatisfied with the performance of this instrument, he seems to have devoted himself twelve years afterwards to its improvement; and in a letter to P. Frederigo Cesii, he says that he had delayed to send him the microscope, the use of which he describes, as he had only then brought it to perfection, owing to the difficulty he experienced in making the glasses. In his Magic of Nature, Schottus mentions a singular accident which took place with one of the newly invented microscopes.
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1 Rigaud's Supplement to Bradley's Miscellaneous Works, p. 20, 21. 2 Camden, Epistles, p. 129, quoted by Professor Rigaud. 3 Before February. 4 Rosa Ursina, 1650. A Bavarian philosopher, when travelling in the Tyrol, was taken ill on the road and died. The village authorities found a little glass instrument in his pocket, which happened to contain a flea fixed in the focus of the microscope. Upon looking into the eye-glass, they were struck with terror at the sight of the gigantic animal, and the remains of the poor philosopher, who was then proved to be a sorcerer, were pronounced unworthy of Christian burial. Some bold sceptic, however, explored the mystery, and produced the giant which had alarmed them.
The name of Kepler, though associated principally with astronomical discovery, will ever be venerated by the cultivators of optical science. His researches, which relate principally to vision and refraction, are contained in his *Paralipomena ad Vitellionem*, published at Frankfort in 1604, and in his *Dioptrica*, already referred to. His discoveries respecting vision, though founded to a certain degree on the views of Maurolycus and Baptista, are nevertheless to a great extent original. He was the first person who actually shewed that distinct and inverted images of external objects are formed upon the retina, as in the camera obscura, by the foci of pencils emanating from every point of the object. He explained all the phenomena of distinct and indistinct vision, and shewed how that indistinctness could be removed by the use of convex and concave glasses. Although D'Alambert has asserted that all optical writers before him had assumed it as an axiom that every visual point is seen in the direction of its visual ray, yet, as Dr. Wells has observed, this assertion is not well founded, for Kepler had long ago maintained that objects are perceived not along the visual rays, but along lines which pass from their pictures on the retina through the centre of the eye; an opinion in which he has been followed by Dechales and Dr. Porterfield, to the last of whom Dr. Reid has by mistake ascribed the discovery of this law. Hence Kepler was led at once to the true theory of erect objects being seen from inverted images, which he considered as the business of the mind, which, when it judges of an impression made on the lower part of an inverted image on the retina, considers it as made by rays proceeding from the higher parts of an erect object, a necessary consequence of his opinion that objects are perceived in lines passing through the centre of the retina. Kepler has wisely declared his ignorance of the manner in which the mind perceives images on the retina, and he blames Vitellio for attempting the solution of a question which does not belong to optics. In order to explain the adaptation of the eye to different distances, Kepler supposed that the ciliary processes draw the sides of the eye towards the crystalline lens, by which change the globe of the eye is elongated, and the retina placed at a greater distance from the crystalline, so as to accommodate the eye to the distinct vision of near objects.
The refraction of light in its passage through different media, is treated at great length, but very unsatisfactorily, by Kepler. Although he failed in his attempts to discover the law of refraction, yet he arrived at certain rules of refraction for glass, which enabled him to discover many of the leading principles of convex and concave lenses. He found, for example, that below $30^\circ$ of incidence, the angle of refraction was nearly two-thirds of the angle of incidence; that at $90^\circ$ of incidence the angle of refraction was $42^\circ$; and that if the refracted ray fell at a greater obliquity than $42^\circ$, upon the interior surface of glass, it would be totally reflected back again into the glass at an angle equal to that of incidence. He then shews, by applying these principles, that plano-convex lenses of glass have their foci at a distance from the lens equal to the diameter of the sphere of which their convex surface is a portion, and that equi-convex lenses have their focal length equal to the radius of the sphere of which their convexities are a portion. When the lens has its surface unequally convex, he makes the focal length equal to a mean of the radii of the two spheres. The same properties being proved in reference to concave lenses, Kepler proceeds to find the focus of refracted rays, when they radiate from points at different distances from the lens. He proved also that rays issuing from the focus of a lens will emerge on the other side of it parallel; that if they issue from a point between the focus of the lens, they will diverge after refraction, while those which issue from a point beyond the focus will converge; and, finally, that when the distance of the radiant point is equal to twice the focal length of the lens, the distance of the image will be equal to the distance of the object.
In treating of the refraction of the atmosphere, Kepler remarked that the quantity of refraction would alter if the atmosphere varied in weight, and that it would be different at different temperatures. In one of his letters to Bregger, dated in 1605, on the colours of the rainbow, he makes the following observation:—“The sun's rays are not coloured except with a definite quantity of refraction. Whether you are in the optical chamber, or standing opposite glass globes, or walking in the morning dew, every where it is obvious that a certain and definite angle is observed, under which, when seen in dew, in glass, in water, the sun's splendour appears coloured, and under no other angle. There is no colouring by mere reflexion, without the refraction of a clearer medium.”
Although Tycho and Kepler made many ineffectual attempts to discover the law of refraction, yet the honour of discovering that great discovery was reserved for Wilibrord Snellius, the law of Professor of Mathematics at Leyden, who died at the age of thirty-five, leaving behind him a manuscript work on the subject. The doctrine of refraction having become more important after the invention of the telescope, Snellius devoted himself to its investigation, and “after many troublesome experiments and attempts,” succeeded in his research.
Supposing $AB$ to be the refracting surface of water, an object under the water at $D$ appeared as if it were raised and seen in the line $RC$. He then produced $RC$ till it intersected at $E$ a line $DK$ drawn parallel to the perpendicular $MN$, and he asserted that at every angle at which the object $D$ was viewed, it would appear at $E$, and that $CD$ was to $CE$ in a given ratio, such as $4$ to $3$, when the refracting body was water. Now this is a true geometrical expression of the law of refraction, though the same truth may be better enunciated in other two ways. If we continue the lines $CE$, $CD$, till they meet $Ad$, a line perpendicular to $AD$, in the points $f$ and $d$, then on account of the parallels $Ad$, $KD$; $CD$ is to $CE$ as $Cd$ is to $Cf$; but $ACd$ is the complement of the angle of refraction, and $ACf$ the complement of the angle of incidence, and $Ad$, $Af$ are their secants. Hence it follows from Snellius' result, that the cosecants of the angles of incidence and refraction are in a constant ratio, which is a correct mathematical expression of the law of refraction. Again, in the triangle $CDE$, the sides $CD$, $CE$ are to one another as the Sines of the opposite angles, that is, as the Sines of the angles $DEC$ or $KEC$, or $ECN$ or $RCM$, and of $CDE$ or $DCN$; that is the sines of the angles of incidence and refraction are
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1 Life of Galileo, Libr. Usef. Knowledge, p. 27. 2 Life of Kepler, Libr. Usef. Knowledge, p. 17. 3 Opuscules Mathematiques, tom. i. p. 265. in a constant ratio, which is the usual and most distinct expression of the law of refraction. In giving this law of Snellius', Huygens has in our opinion forgotten his usual courtesy, when he states that Willebrord Snellius did not "thoroughly comprehend his own invention," and "never imagines that the ratio was the ratio of the Sines." Now we cannot conceive it possible that a man like Snellius, who was a good geometer, was ignorant of the two simple trigonometrical expressions of his geometrical law; and we do not doubt that he preferred his own for two distinct reasons. In the first place, it connects itself with the leading physical phenomena of the apparent rise of the refracted object from D to E, and by substituting CF for CD it furnishes us with a much more simple and accurate method of obtaining by projection the refracted ray from the incident one. If RF for example, is the incident ray, we have only to divide CP into two parts, CE, EF, so that CF is to CE in the constant ratio belonging to the refracting body.
But whether we are right in this conjecture or not, it is an unquestionable truth that Snellius discovered the true law of refraction, though he did not express it in trigonometrical language.
In the year 1637, about eleven years after the death of Descartes, Descartes published his Dioptries, in which, without ever mentioning the name, or alluding to the labours of Snellius, he announces the true law of refraction expressed in terms of the Sines, as the result of his own enquiries. As Snellius's work existed only in manuscript, it was quite possible that Descartes knew nothing of its contents, but Vossius, in his work De Natura Lucis, states, that the heirs of Hortensius had communicated freely to Descartes the manuscripts of that professor, among which was that of Snellius's work, and Huygens confirms this allegation when he states in his Dioptries that he had himself seen the whole manuscript volume of Snellius, and had heard that Descartes had also seen it, and that it was perhaps from hence that he deduced (elucuerit) that measure which consists in the Sines. We should not have entered so minutely into this subject, had not M. Biot thrown into entire oblivion the labours of Snellius, and ascribed to Descartes the undoubted discovery of what he calls "this great property of light." The same eminent philosopher likewise ascribes to Descartes the discovery that the incident and refracted rays are always in the same plane, a truth which was well known to Ptolemy, and which is clearly included in Snellius's expression of the law of refraction. In opposition to the opinion of M. Biot, we must place those of Huygens, Montucla, Bossut, Priestley, David Gregory, Smith, Hutton, Robison, Young, and Playfair; and we shall dismiss the subject after giving the admirable reasons which induced Professor Playfair to decide against Descartes. "There is no doubt, therefore," says he, "that the discovery was first made by Snellius; but whether Descartes derived it from him, or was himself the second discoverer, remains undecided. The question is one of those, where a man's conduct in a particular situation can only be rightly interpreted from his general character and behaviour.
"If Descartes had been uniformly fair and candid in his intercourse with others, one would have rejected with disdain a suspicion of the kind just mentioned. But the truth is, that he appears throughout a jealous and imperious man, always inclined to depress and conceal the merit of others. In speaking of the invention of the telescope, he has told minutely all that is due to accident, but has passed carefully over all that proceeded from design, and has incurred the reproach of relating the origin of that instrument without mentioning the name of Galileo. In the same manner, he omits to speak of the discoveries of Kepler, so nearly connected with his own; and, in treating of the rainbow, Blaise has made no mention of Antonio de Dominis. It is impossible that this should not produce a favourable impression; and hence it is that the warmest admirers of Descartes do not pretend that his conduct towards Snellius can be completely justified.
"Descartes would have conceived his philosophy to be disgraced if it had borrowed any general principle from experience, and he therefore derived, or affected to derive the law of refraction from reasoning or from theory."
Even if Descartes had liberally taken from his optical wreath the law of the Sines, and generously placed the hallowed branch on the brow of the accomplished Snellius, he would have added to his reputation as a man, without shortening his immortality as a philosopher. His Dioptries consists of ten chapters. The first treats of light, the second of refraction, the third of the eye, the fourth on lenses in general, the fifth on the images formed on the bottom of the eye, the sixth on vision, the seventh on the mode of perfecting vision, the eighth on the figures which transparent bodies require to turn the rays by refraction, suited to all modes of vision, the ninth on microscopes, and the tenth on the mode of polishing glasses. The inability of spherical surfaces to converge rays to one point or focus, had been long known to opticians, and Kepler, though he conjectured that surfaces generated by the revolutions of the conic sections, might have such a property, left the subject just as he found it. Descartes, however, has discussed it in a most ingenious manner, in the eighth chapter of his Dioptries. He has shewn how parallel, and converging, and diverging rays may be brought to accurate foci by means of ellipsoidal and hyperboloidal surfaces, so that if such surfaces could be executed by opticians, all optical instruments would receive the highest degree of perfection which they could attain from the removal of spherical aberration. In order to carry this system into effect, he contrived machines for grinding elliptical and hyperbolical lenses, and in the tenth chapter of his Dioptries he has given perspective drawings and descriptions of them. In the years 1627 and 1628, when he was residing at Paris, M. Mydorgius, with whom he lived on the most intimate habits, urged him to undertake the grinding of hyperbolical and elliptical lenses, and he soon became a great master of the art of glass grinding. He found it necessary, however, to associate with himself in this undertaking an eminent artist, M. Ferrier, who, as an optical instrument maker, was well acquainted both with the theory and the practice of his art. After many failures, a tolerably good hyperbolic convex lens was completed; but the concaves were found to be more difficult; and in consequence of M. Ferrier refusing to accompany Descartes to France, and having occasioned him much needless expense in the erection of his laboratory, a quarrel took place, and the great practical object which they had in view was for a while abandoned. Descartes, however, was sanguine in his expectations, and not aware that there was another aberration more difficult to overcome than that of spherical figure, he expected to be able to make the greatest discoveries in the heavens by means of his new lenses. With the assistance of M. Huygens, the father of the celebrated philosopher, he induced some Dutch artists to renew the attempts of Ferrier; but these and his subsequent endeavours to construct such lenses have failed, though we cannot allow ourselves to think that the attempt is a hopeless one.
Descartes made some interesting observations upon vision, particularly on the method by which we judge of the distances and magnitude of objects; but his principal discovery in physical optics relates to the theory of the rain- He discovered the true cause of the exterior rainbow, and in his *Traité des Météores*, proves that it was produced by two refractions, and two intermediate reflections within the drop, thus explaining most satisfactorily the faintness of its illumination, and the inversion of its colours. He has clearly shewn also, why the interior bow is 42° in diameter, while the exterior one is 52°; though he did not understand the true origin of the colours. We regret to add, that Descartes gives his explanations of both the interior and exterior bows without ever mentioning the name of Antonio de Dominis, who was the real discoverer of the cause of the rainbow; and our regret is increased, when we are compelled to add, that M. Biot has, contrary to the opinion of all philosophers, given his aid to Descartes in depriving the Italian philosopher of the only discovery which has immortalised his name.
The science of optics is under considerable obligations in so far as he proved by direct experiment that the pictures of external objects were distinctly delineated on the retina. By paring away the coats from the back of the eyes of sheep and oxen, and also the human eye, he made the inverted pictures distinctly visible, and exhibited the experiment publicly at Rome in 1625. In his work entitled *Oculus*, published in 1652, he speaks of the great resemblance of the eye to the camera obscura, and gives various contrivances for erecting the images. He adopts the theory of Kepler respecting the visible direction of objects, and he observed the interesting fact that the pupil of the eye is dilated in viewing distant, and contracted in viewing near objects. In measuring the refractive powers of the humours of the eye, he makes that of the aqueous humour differ little from that of water, and that of the crystalline humour differ little from that of glass, ascribing to the vitreous humour an intermediate refractive power. By tracing the progress of the visual rays through all the humours of the eye, he demonstrates that the retina, and not the crystalline lens, is the seat of vision; and he describes some interesting experiments respecting vision through one or more small apertures. We owe also to Scheiner the interesting experiment of exhibiting on the wall of a darkened room the disc of the sun with all its spots by means of a telescope. When Kircher afterwards describes this appearance he represents the spectator as "fixed with the utmost astonishment."
A new and very interesting branch of optics had begun to excite the attention of philosophers, namely that of the double refraction of light. Erasmus Bartholinus, a physician at Copenhagen, and the author of several excellent works on geometry, received from some Danish merchants that frequented Iceland, "a crystal stone like a rhombic prism, which, when broken into small pieces, kept the same figure." With this substance, which was called *Iceland spar*, from its locality, Bartholinus made a number of experiments both chemical and optical, and he has published an account of the optical results which he obtained in a small volume which appeared at Copenhagen in 1669, under the title of *Erasmii Bartholini Experimenta Crystallari Islandici, Dissociationis quibus mira et insolita Refractio detectur*, and is dedicated to Frederick III, king of Denmark. In seventeen experiments and twelve propositions this able and sagacious philosopher has presented us with an excellent summary of the more prominent phenomena of double refraction. He has shewn that Iceland spar has the property of double refraction, that is, of giving two images of all objects seen through it, whether its faces are parallel or inclined, like those of a prism; that the incident light is equally divided between these two pencils; that one of these refractions is performed according to the law of Snellius, the ratio of the Sines being as 1 to 1.667, but that the other is performed according to an extraordinary law which had not previously been observed by philosophers. He observed also a position in which the object appears six-fold, but he did not discover that this took place only in some specimens which were composite or irregular crystals.
These discoveries of Bartholinus having been communicated to the Royal Society of London, and printed in No. of Huygens' Transactions, they attracted the notice of others. Born Christian Huygens, a celebrated Dutch philosopher of the 1629, died 1695, genius and the highest attainments. Having given a new theory of refraction, he wanted to repeat Bartholinus's experiments principally with the view of ascertaining if they opposed any difficulties to that theory. His work on this subject, entitled *De l'extraordinaire Réfraction du Cristal d'Iceland*, which forms the 5th chapter of his *Traité de la Lumière*, was written in 1678, and was read to Cassini, Roemer, and De la Hire, and to several other members of the Royal Academy of Sciences, which he had been invited to join by the liberality of the French king; but it was not published till 1690, when he was resident at the Hague. After giving Bartholinus the credit of having discovered some of the principal phenomena of double refraction, he describes the general properties of Iceland spar in forming two images of objects, and he shews that all the phenomena are related to the axis, or that diagonal of the rhomb, in the direction of which the crystal has no double refraction. He proves that the double refraction, or separation of the two images, gradually increases as the inclination of the refracted ray to the axis increases, and becomes a maximum in a plane at right angles to the axis. In the few preceding chapters of his *Traité de la Lumière* he had explained all the phenomena of reflexion and refraction upon a new theory, in which he supposes light to be produced in the same manner as sound, by means of undulations propagated in an elastic etherial medium, an hypothesis revived by Euler and extended by Dr. Young, and now almost universally embraced under the name of the undulatory theory. In applying the same theory to explain the phenomena of double refraction, he supposes the ray produced by the ordinary refraction of the medium to be produced by spherical undulations propagated through the crystal, while the ray formed by the extraordinary refraction is produced by spheroidal undulations, the ratio of the two refractions, determining the form of the generating ellipse. Huygens then proceeds to show that this theory affords, by calculation, results agreeing very exactly with those which he had obtained by direct experiment. This discovery is perhaps the most splendid which has occurred in the history of optical science.
When Huygens had finished his researches on double refraction, he discovered what he calls a "wonderful phenomenon," and, though he acknowledges that we cannot find the cause of it, yet he thinks it proper to indicate the light phenomenon that others may inquire into it. This discovery is that of the *polarisation of the light*, which forms the two pencils of Iceland spar, and he confesses that he must add to his theory other suppositions in order to explain it, though he thinks that a theory confirmed by so many proofs will still preserve its plausibility (*avaisemblance*). Huygens had naturally supposed that the light which composed the two pencils was like all other light, but upon transmitting the two rays formed by one rhomb of calcareous spar through another rhomb, he was astonished to perceive that when the two rhombs were similarly placed as if they had formed one larger one, neither of the rays suffered double refraction in
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1 In his Optics, book ix. p. 147, Sir Isaac gives almost the whole merit of the explanation of the rainbow to De Dominis. He says, "at the same explication Descartes hath pursued in his Meteors, and mendeth that of the exterior bow." 2 "Un phénomène merveilleux, que j'ai découvert après avoir écrit tout ce que dessus." (Traité de la Lumière, p. 88.) passing through the second rhomb, the ordinary ray from the first being only ordinarily refracted by the second rhomb, and the extraordinary ray only extraordinarily refracted. The same thing took place when one of the rhombs, the second, for example, was turned round 90°, with this difference, that the ordinary ray of the first rhomb suffered only extraordinary refraction, and the extraordinary ray only ordinary refraction from the second rhomb. But in all other positions of the second rhomb, excepting these two rectangular ones, the ordinary and extraordinary rays of the first rhomb were each divided into two by the second rhomb, so that there were now four rays sometimes of equal but generally of unequal brightness, and such that the light of all the four never exceeded that of the single ray incident on the first rhomb.
Huygens discovered also the double refraction of quartz, or rock crystal, but he committed a great mistake in supposing that its double refraction was regulated by an entirely different law, the light being in this case propagated through it in two spherical waves, one of which was a little slower than the other. This result he mentions in his preface as having been obtained after he had read his work to his colleagues in the Academy of Sciences. It is, however, founded on an incorrect observation, as the extraordinary refraction of rock crystal is produced by spheroidal undulations like that of Iceland spar, with this difference only, as afterwards discovered by M. Biot, that the spheroid is a prolate one.
Even if Huygens had not immortalised his name by these great discoveries, his treatises on dioptrics and on halos, and his construction of refracting telescopes of immense size, would have given him the highest reputation. His treatise on dioptrics, which was not published till 1703, among his posthumous works, and which he had begun to prepare at an early period of his life, was particularly admired by Sir Isaac Newton. It contains a copious explanation of the properties of lenses of all forms; and their spherical aberration is treated with much perspicuity, having previously, in the 6th chapter of his Traité de la Lumière, published an interesting discussion respecting the figures of transparent bodies for refracting and reflecting light to a single focus. The subject of vision, and the method of assisting long and short-sighted persons by lenses, is ably discussed, and nearly the latter half of the work is devoted to the theory of telescopes, telescopic eye-pieces and microscopes.
Many of these theoretical views Huygens submitted to the test of experiment. Having acquired great expertise in the art of grinding lenses, he executed refracting telescopes 12 and 24 feet in focal length, and afterwards one of 120 and another of 123 feet, with which he discovered Saturn's ring, and the fourth of his satellites. These two last object-glasses he presented to the Royal Society; but as it was impracticable to use tubes of such enormous length, Huygens contrived a method of mounting them without tubes at the top of a long pole. The practical knowledge which he had thus acquired, was published along with his Dioptrics in a work entitled, Commentarii de formandis polien-dique vitris ad Telescopia, a considerable part of which was published by Dr. Smith in his optics. Among his posthumous works appeared his Dissertatio de Cornuis et Parhelis, a work of great merit, in which he ascribes these phenomena generally to crystals of ice in the upper atmosphere, and a translation of the whole of which Dr. Smith has published in the first volume of his optics.
Among the eminent men who gave an impulse to optical discovery, we must assign a considerable place to our countryman James Gregory. This eminent mathematician in confirming the experiments of Vitello and Kircher on the angles of refraction, discovered the true law which had previously been found by Snellius. He made the refractive power of water $1:3347$, which coincides exactly with that of the middle ray between the lines D and E of Fraunhofer. Having discovered before the publication of his work that Descartes' dioptrics contained the law of refraction, he mentions the circumstance, and ascribes his being unacquainted with that work to the "want of new mathematical books," in the library of the college of Aberdeen. Although Battista Porta appears from the extract which we have already given from his Natural Magic, to have made the nearest possible approach to the invention of the Newtonian reflecting telescope, or rather microscope, yet his experiment excited no notice, and no instrument could be said to have been invented. James Gregory, however, has described what is now known by the name of the Gregorian Reflecting Telescope, at the end of his Optica Pratana, published in 1663.
It consisted of a parabolic concave mirror perforated at the centre, and having in front of it a small concave elliptic speculum, at a distance a little greater than the sum of their focal lengths. The parallel rays emitted by a remote object formed an image of that object in front of the great mirror, and in its focus; and in the conjugate focus of the small speculum, behind the great speculum, there was formed another image of the object, which was magnified by an eye glass. In 1664, Messrs Rives and Co., English opticians, attempted to construct a six foot Gregorian telescope, under the superintendence of its inventor, but, after a rough trial of it, Mr. Gregory, not aware of the nice adjustments which it required, conceived that the figure of the speculum was defective, and, being on the eve of going abroad, he never even made a tube for the mirrors. Stimulated by the failure of his friend, Newton, "altered," as he says, "the design of the instrument," and "placed the eye glass at the end of the tube rather than at the middle;" and, therefore, he was obliged to reflect the rays to a side by an oval plane speculum. Sir Isaac actually constructed one of these instruments with his own hands, and described it in a letter to a friend, dated the 23rd February 1668-9. The aperture of the speculum was one inch, its focal length six inches, the eye glass, which was a plano-convex lens, about $\frac{3}{5}$ths of an inch in focal length, and the magnifying power 39 times. He considered it as equal to a three or four feet refractor, and it shewed distinctly the four satellites of Jupiter, and the phases of Venus. Encouraged by his success, he completed another telescope in 1671, which was better than the first, and which is preserved in the library of the Royal Society. The next Newtonian reflecting telescope of any importance, was executed by Mr. John Hadley in 1719 or 1720, with a speculum six inches in diameter, and five feet in focal length; but for a long time the Gregorian form was the most popular in England. About 1672 M. Cassegrain substituted a convex speculum for the small concave one of Gregory, which had the advantage of shortening the tube of the telescope without diminishing the power of the instrument.
Other claimants have arisen for the honour of inventing the reflecting telescope. Father Mersenne in a letter to Descartes in 1637, suggested the idea of using concave mirrors in reflecting telescopes; but Descartes endeavoured to convince him that his views were not likely to succeed. At a later period Fontenelle, in the History of the Academy of Sciences, for 1700, has very recklessly ascribed the invention of this instrument to Father Zucchi, an Italian Jesuit, who published, at Lyons in 1652, a volume, entitled, Opus Philosophae. In this work he says, that he thought of substituting concave specula for object glasses, and having
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1 Traité &c. § 20, 21. "Cette double refraction semblait demander une double émission d'ondes de lumière, toutes deux sphériques (car les deux refractions sont régulières) et les unes seulement un peu plus lentes que les autres."
2 Brewster's Life of Sir Isaac Newton, p. 26, 27 He succeeded in splitting mica into plates of extreme tenuity, as to give the most brilliant colours, one giving a yellow, another a blue, and the two together a deep purple. In his Micrographia, printed about seven years before any of Newton's experiments were made on the same subject, Dr. Hooke has published the following remarkable explanation of these phenomena, which coincides in a singular manner, with that which is now universally received: "It is most evident, says he, that the reflection from the under or further side of the body, is the principal cause of the production of these colours." Let the ray fall obliquely on the thin plate, part thereof is reflected back by the first superficies—part refracted to the second surface, whence it is reflected and refracted again. So that, after two refractions and one reflection, there is propagated a kind of fainter ray, and by reason of the time spent in passing and repassing, this fainter pulse comes behind the former reflected pulse; so that hereby, (the surfaces being so near together that the eye cannot discriminate them from one) this confused or duplicated pulse, whose strongest part precedes, and whose weakest follows, does produce on the retina the sensation of a yellow. If these surfaces are further removed asunder, the weaker pulse may become coincident with the reflection of the second, or next following pulse, from the first surface, and lagg behind that also, and be coincident with the third, fourth, fifth, sixth, seventh, or eighth; so that, if there be a thin transparent body, that from the greatest thinness requisite to produce colours, does by degrees grow to the greatest thickness,—the colours shall be so often repeated, as the weaker pulse does lose paces with its primary or first pulse, and is coincident with a subsequent pulse. And this, as it is coincident, or follows from the first hypothesis I took of colours, so upon experiment have I found it in multitudes of instances that seem to prove it."
Galileo, and the philosophers of the Accademia del Cimento, had proposed to measure the velocity of light by means of a base on the surface of the globe; but such an attempt was utterly hopeless, and it was only in a wider range that this problem could be solved. Baffled in finding an explanation of some irregularity in the emersion of the first satellite of Jupiter, Cassini and Roemer had concluded that it depended on the distance of Jupiter from the earth, and Born 1644, that in order to explain it was necessary to suppose, that the Died 1710 light of the satellite required ten or eleven minutes to move across the earth's orbit. This happy idea seems to have first occurred to Cassini, but he speedily abandoned it, while Roemer persistuously cherished the hypothesis, and at last immortalised himself by demonstrating in the most rigorous manner that light moves through the diameter of the earth's orbit, a distance of 190 millions of miles, in eleven minutes.
Passing over the valuable researches of Tschirnhausen, a Saxon nobleman, on caustic curves, which had been previously discovered, and the discoveries of Mariotte and de la Hire, respecting the seat of vision, which have not terminated in any satisfactory conclusions, we are brought to one of the most brilliant periods of optical discovery.
In the year 1665, Sir Isaac Newton, when only 23 years Discoveries of age, bought three prisms, but he does not seem to have of Newton made any particular experiments with them. In 1666, Born 1642, however, he bought another, with which he proposed to repeat Grimaldi's experiment, on the elongation of the sun's image produced by the prism. In the course of this and the two or three subsequent years, he made and perfected his great discovery of the different refrangibility of light, which he communicated to the Royal Society on the 6th of February 1672, having, on the 18th of January, announced it as "the oldest if not the most considerable detection..." which hath hitherto been made in the operations of nature." Having found that refraction could not be produced without colour, he was led to direct his attention to the perfection of the reflecting telescope, and produced the instruments which we have already mentioned. Another result of this discovery was the completion of the theory of the rainbow, the origin of the colours of which had hitherto perplexed philosophers.
The next optical discovery made by Sir Isaac Newton, related to the colours of thin plates, or of thin transparent bodies, such as the soap bubble. We have already seen that Dr. Hooke had made some progress, both in observing the phenomena and in investigating the cause of such colours; but it is to Newton that we owe an elaborate analysis of the subject. In a letter from Sir Isaac to Dr. Hooke, dated 5th February 1676, he acknowledges that the latter had previously observed "the dilatation of the coloured rays by the obliquity of the eye, and the opposition of a black spot at the contact of two convex glasses, and at the top of a water bubble," (soap bubble). In the course of his experiments on thin plates, Newton was led to the discovery of the colours of thick plates, and he devised a theory for explaining both classes of phenomena, known by the name of the theory of fits of easy reflection and transmission. This theory, remarkable for its ingenuity, is now no longer an expression of the phenomena, and has given way to the theory of undulations, which Hooke had the sagacity to anticipate as affording the true cause of the colours of thin plates.
Early in 1676, Newton communicated to the Royal Society his Theory of the Colours of Natural Bodies, in which he ascribes all the varieties of colour exhibited in nature to the circumstance "that the transparent parts of bodies, according to their several sizes, reflect rays of one colour and transmit those of another, on the same grounds that thin plates or bubbles do reflect or transmit those rays." This theory, perhaps the finest of all Newton's optical speculations, has met with much opposition. It was controverted by Mr. Delaval, Sir John Herschel, and others, in its leading propositions; but some recent discoveries afford us reason for believing that the principal objections to it are groundless, and that with some modifications it may yet be placed among the finest generalizations in science.
Sir Isaac Newton's experiments on the inflexion of light were never finished by their author. His observations were limited, and his theory incorrect; and indeed it was only from the hands of those who adopted the undulating system that a true explanation of the phenomena could be expected.
The experiments of our author on the refractive powers of bodies, from which he anticipated that the diamond "was probably an unctuous substance coagulated," have on this account been regarded with high favour, while his few observations on the double refraction and polarisation of light have almost disappeared from the history of optics.
The next great step in the history of optical discovery is the invention of the achromatic telescope, or telescopes which are free from colour. When Sir Isaac Newton found that he could not produce refraction without colour, he abandoned the improvement of the refracting telescope as hopeless, and devoted himself to the construction of reflectors.
The opinion at which he had arrived respecting the impracticability of refracting light without colouring it, was, however, an erroneous one, which he had deduced from an incorrect observation of the relative length of the prismatic spectra formed by different bodies, when the mean refraction was the same. In less than two years after Newton's death, namely, in 1729, Mr. Chester More Hall, of More Hall in Essex, was led by the study of the human eye, which he erroneously conceived to be achromatic, to consider the possibility of constructing a telescope by an analogous combination of media. After many experiments, he found two kinds of glass capable of producing, by their combination, refraction without colour. About 1733 he completed several such object-glasses, which, with a focal length of twenty inches, bore an aperture of more than two-and-a-half inches, one of which was long afterwards in the possession of the Rev. Mr. Smith of Charlotte Street, Rathbone Place, and was found to be achromatic. Another of Mr. Hall's telescopes was in the possession of Mr. Ayscough, optician in Ludgate Hill, in 1754. Mr. Hall, however, kept his invention a secret; none of his instruments were either sold or exhibited for sale, and those into whose hands they fell do not seem to have discovered either their principle or their value.
Without calling in question the merits of Mr. Hall, we must do justice to those of Mr. Dollond, an undoubted inventor of the achromatic telescope, who, unacquainted with the instruments of Mr. Hall, proceeded step by step, in a scientific progression, till he invented and constructed the achromatic telescope in 1757. To this eminent individual, and the other members of his family, we owe the construction of many of the finest instruments by which the science of astronomy has been so much promoted. Mr. Peter Dollond, the son of John Dollond, first suggested and used the triple object-glass, in which a better correction of the spherical aberration was effected, by placing the concave flat glass between two convex lenses of crown glass.
The mathematical world owes many obligations to Euler, Clairaut, D'Alembert, and Boscovich, for their able investigations of the theory of achromatism, but their investigations did not prove of any practical value; and it has been justly stated by Sir John Herschel, "that from all the abstruse researches of Clairaut, Euler, and D'Alembert, and other celebrated geometers, nothing hitherto has resulted beyond a mass of complicated formulas, which, though confessedly exact in theory, have never yet been made the basis of construction for a single good instrument, and remains therefore totally inapplicable, or at least unapplied in practice."
No attempt had hitherto been made to measure the intensity of different lights emanating either directly from luminous bodies, or when transmitted through or reflected from different bodies. This subject, to which the name of Photometry has been given, was begun by Huygens and P. F. Marie, who describes an instrument called a lucimeter; but it is to M. Bouguer and M. Lambert that we owe the most scientific and complete investigation of this class of facts.
Bouguer's earliest experiments were published in 1729, in his Optical Essay on the Gradation of Light, which was republished in 1760, much augmented and improved, under the title of Traité d'Optique sur la Gradation de la Lumière.
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1 We must refer our readers for a full and elaborate account of Newton's optical discoveries to a new work, entitled Memoirs of the Life, Writings, and Discoveries of Sir Isaac Newton, in which Sir David Brewster is now occupied, having had access to the interesting collection of the papers of Sir Isaac, in the possession of the Earl of Portsmouth.
2 Mr. Hall might have got this hint from David Gregory's Catoptrics, published at Edinburgh in 1713. "But if," says he, "on account of physical difficulties in grinding and polishing proper specula, we should still use lenses, it would perhaps be useful to employ media of different density to compose the object-glass, as we see done by nature in the structure of the eye, where the crystalline humour, (of almost the same refractive power as glass), is joined by nature, who does nothing in vain, with the aqueous and vitreous humours, (not unlike water in their refractive power), to paint the image as distinctly as possible in the bottom of the eye." Gregory's Catoptrics, prop. xxiv. Sciolium. Dr. Brown's translation of the preceding passage is very incorrect.
3 Phil. Trans. 1821, p. 222. Bouguer was followed in this inquiry by M Lambert, an able German mathematician, who published an account of his researches at Augsburg in 1760, in a duodecimo volume of 547 pages, entitled *Photometria seu de Mensura et Gra- dibus Luminis, colorum et umbrae*. It is divided into seven parts: 1. On the modifications and degrees of direct light, and of its brightness and illuminating power; 2. Experiments and calculations on the modifications of light depending on transparent bodies, but chiefly glass; 3. Experiments and calculations respecting the modifications of light depending on the opacity of bodies; 4. Calculations and experiments on the sense of light, and its apparent brightness; 5. On the dispersion of light passing through diaphanous media, chiefly the earth's atmosphere; 6. Calculations respecting the illumination of the planetary system; and, 7. On the modifications and degrees of heterogeneous and relative light, or the light of colours and shadow.
Passing over the minor labours of Porterfield, Turner, Mazzea, Dautour, Buffon, Scheffer, Darwin, Melville, Mitchell, and others we come to the period of Sir William Herschel. Since the discovery of the belts and nearest satellites of Saturn, no discovery of any importance had been made respecting the natural history of the heavens. At the age of 36, when Sir William was residing at Bath, he devoted much of his time to the construction of telescopes; and the following account of his progress is too interesting to be given in any other language than his own.—"When I resided," says he, "at Bath, I had long been acquainted with the theory of optics and mechanism, and wanted only that experience which is so necessary in the practical part of these sciences. This I acquired by degrees at that place, where, in my leisure hours, by way of amusement, I made for myself several two-feet, five feet, seven feet, ten feet, and twenty feet Newtonian telescopes, besides others of the Gregorian form, of eight inches, twelve inches, two feet, three feet, five feet, and ten feet, focal length. My way of doing these instruments at that time, when the direct method of giving the figure of any one of the conic sections to specula was still unknown to me, was to have many mirrors of each sort cast, and to finish them all as well as I could, then to select by trial the best of them, which I preserved; the rest were put by to be re-polished. In this manner, I made no less than two hundred seven-feet, one hundred and fifty ten-feet, and about eighty twenty-feet, not to mention those of the Gregorian form, or of the construction of Dr. Smith's reflecting microscope, of which I also made a great number. My mechanical amusements went hand in hand with the optical ones. The number of stands I invented for these telescopes it would not be easy to assign. I contrived and delineated them of different forms, and executed the most promising of the designs. To these labours we owe my seven-feet Newtonian telescope stand, which was brought to its present convenient construction about 1778."
By means of these instruments, with which he surveyed the heavens with unwearied diligence, he discovered the Georgian Sidus, with six satellites, two new satellites circulating round Saturn, the quintuple belt and double ring of the same planet, and various other astronomical phenomena of the highest interest. In 1783 he finished a twenty-feet reflector, with an aperture of 18.7 inches, and formed the design of constructing a still larger instrument. On the recommendation of Sir Joseph Banks, His Majesty George III. agreed to defray the expense of a large telescope, and under his munificent patronage, which has never since been imitated by his successors, Sir William began in 1785, and completed on the 27th August 1789, a reflecting telescope forty feet in focal length, having its great speculum four feet in breadth three-and-a-half inches thick, and weighing, when newly cast, 2118 pounds. On the 28th of August, the day after this gigantic instrument was erected, Sir William discovered a new satellite of Saturn, and in the same year another satellite, both of which were nearer the body of the planet than the other five discovered by Huygens and Cassini. In this manner the telescope, which was a toy in the hands of Galileo, became with Sir William Herschel a vast machine, carrying the observer himself, and directed and moved by appropriate mechanism.
An improvement in the achromatic telescope, of great value, though not yet brought into practical use, was made A.D. 1787, by Dr. Robert Blair. Although in the achromatic telescope Died 1829, composed of crown and flint glass, the colour was completely corrected as it was possible to do with such lenses, yet it had long been observed that there were residual colours, which formed what are called a secondary spectrum, and which arise from the coloured spaces in the spectrum, produced by crown glass not having the same size as those in a spectrum of equal length produced by flint glass. Various attempts had been made in vain to obtain other substances, in which this irrationality, as it was called, of the coloured spaces did not exist; and Dr. Blair was hence led to attempt the removal of the secondary spectrum by other means. The plan which he adopted was the following. He made each lens of his compound object-glass achromatic, but in such a way that the secondary spectrum produced by the one should be corrected by the secondary spectrum produced by the other. Such an object-glass required two fluid media and three lenses of glass, and Dr. Blair succeeded in constructing them so as to be perfectly free from all secondary colour. In the course of his experiments, however, he was fortunate enough to discover that the muriatic acid mixed in proper proportions with metallic antimony, or butter of antimony, as it was called, gave a spectrum, in which the colours had exactly the same proportion as crown glass; and hence, by enclosing this fluid between two lenses of crown glass, the one next the object being plano-convex and the other a meniscus, he obtained an object-glass in which the rays of different colours were bent from their rectilineal course with the same equality and regularity as in reflexions. To such an object-glass he proposed to give the name of *aplanatic*, to indicate the entire removal of all aberration. Dr. Robison informs us that one of these telescopes, which did not exceed fifteen inches in length, equalled in all respects, if it did not surpass, the best of Dollond's achromatic telescopes forty-two inches long. After the death of Dr. Blair, his son, Mr. Archibald Blair, attempted in vain to produce instruments of the same perfection. Had this young man lived, he might have executed something better, but he was cut off at an early age, and has left to the Royal Society of Edinburgh an account of his father's methods, which we hope may prove useful to science.
Hitherto the undulatory theory of light as proposed by Dr. Thomas Young and supported by Hooke and Euler, had met with few adherents; and the reputation of Newton had given to Born 1773, the theory of emission an adventitious authority to which it died 1829, was not entitled. Dr. Young, however, boldly threw down the gauntlet and maintained the theory of Huygens with the greatest ingenuity and talent. In his paper of 1800, entitled *Outlines of Experiments and Observations on Sound and Light*, he shews that light has a strong analogy with sound, and that it is produced by the undulation of a highly elastic etherial medium which pervades all nature. In another paper, which he published in 1801, *On the Theory of Light and Colours*, he applies the theory of undulations to the explanation of natural phenomena; and lays down the following hypotheses: 1. That a luminiferous ether pervades the universe, rare and elastic in a high degree. 2. That undulations are excited in this ether whenever a body becomes luminous. 3. That the sensation of different colours depends on the different frequency of vibrations excited by light in the retina; and, 4th, that all material bodies are to be considered, with respect to the phenomena of light, as consisting of particles so remote from each other as to allow the etherial medium to pervade them with perfect freedom, and either to retain it in a state of greater density and of equal elasticity, or to constitute together with the medium an aggregate which may be considered as denser but not more elastic. He then proceeds to demonstrate nine propositions, some of the leading truths in the theory, applying them in corollaries to the colour of striated surfaces, the colours of thin plates, the colours of thick plates, and the colours produced by inflexion. In 1802, Dr. Young published *An account of some causes of the Production of Colours not hitherto observed*. The cases described in this paper are the colours of delicate fibres and of mixed plates. The first he explains by the interference of two portions of light, one reflected from the fibre and the other bending round its opposite side, and at last coinciding nearly in direction with the former portion. The colours of mixed plates are those produced when moisture, butter, or tallow, are placed between two plates of glass, so that portions of air are intermixed with these substances. A candle seen through such a medium is surrounded with a sort of halo, and Dr. Young considers the colours as produced by the light which passes through one of the media, moving with greater velocity so as to anticipate the light which comes more slowly through the other.
In 1803, Dr. Young published what may be considered as his principal paper, entitled *Experiments and Calculations relating to Physical Optics*, in which he has given an experimental demonstration of the general law of interference. By intercepting the rays which passed on one side of a body which formed fringes by reflexion, the fringes disappeared whether the interception was made on one side or the other of the body. This admirable experiment established the truth of his law of interference, and paved the way for those splendid generalizations respecting the undulatory theory which have so widely enlarged the boundaries of optics.
In April 1814, in a review of Malus, Biot, and Brewster's *Experiments on Light*, which he contributed to the *Quarterly Review*, he first published his explanation of the colours of crystallised plates produced by polarised light by the law of interference, an explanation which is now universally admitted. In the article on *Chromatics*, which Dr. Young contributed to this work, the reader will find a full account of the discoveries to which the law of interference has been so successfully applied. One of the most important applications of the undulatory theory was published in that article for the first time. Dr. Young has there given an expression of the velocity of reflected light at a perpendicular incidence from bodies of various refractive powers, which is a simple function of the index of refraction.
In giving an account of Sir William Herschel's discoveries, we have not mentioned his discovery of invisible heating rays beyond the red extremity of the spectrum, because we have ourselves succeeded in discovering the luminous rays at that part of the spectrum. In repeating Sir W. Herschel's experiments, M. Ritter of Jena placed muriate of silver in different parts of the spectrum, and found that it soon became black beyond the violet extremity, less black in the violet rays, becoming still less black in the blue and green, and so on till the blackness vanished. When he used muriate of silver, slightly blackened or disoxygated, its white or original colour was partly retained by the red, and still more by the supposed invisible rays beyond it. In these experiments of Ritter's, as well as in those of Sir W. Herschel, the solar spectrum, when seen by the eye, as thrown upon paper, is extremely short. A great part of the violet extremity as well as the red extremity is invisible, so that when the thermometer and the muriate of silver seemed to be wholly out of the spectrum, they were completely within the violet and the red spaces, as we have placed beyond a doubt by comparing the length of a spectrum on paper with that which can be rendered visible by directly looking through a telescope at a highly magnified one.
Without knowing of the experiments of Ritter, Dr. Wollaston discovered the chemical effects which exist at the violet end of the spectrum; but the merit of this experiment decidedly belongs to Scheele, who discovered that muriate of silver was more blackened in the violet rays than in any other part of the spectrum. The principal discovery in optics, which we owe to Dr. Wollaston is his method of observing the spectrum, and his discovery of fixed lines in it. The following is his own description of it: "I cannot conclude these observations on dispersion without remarking that the colours into which a beam of white light is separable by refraction, appear to me to be neither seven, as they usually are seen in the rainbow, nor reducible by any means (that I can find) to three, as some persons have conceived; but that, by employing a very narrow pencil of light, four primary divisions of the prismatic spectrum may be seen with a degree of distinctness that, I believe, has not been described nor observed before.
"If a beam of day-light be admitted into a dark room by a crevice of an inch broad, and received by the eye at a distance of ten or twelve feet, through a prism of flint glass free from veins, held near the eyes, the beam is seen to be separated into the four following colours only, red, yellowish green, blue, and violet; in the proportion represented in the figure.
"The line A that bounds the red side of the spectrum is somewhat confused, which seems, in part owing to the want of power in the eye to converge red light. The line B, between red and green, in a certain position of the prism, is perfectly distinct; so also are D and E, the two limits of violet. But C, the limit of green and blue, is not so clearly marked as the red; and there are also, on each side of this limit, other distinct dark lines, f and g, either of which in an imperfect experiment, might be mistaken for the boundary of these colours.
"The position of the prism in which the colours are most clearly divided is when the incident light makes about equal angles with two of its sides. I thus found that the spaces AB, BC, CD, DE, occupied by them, were nearly as the numbers 16, 23, 36, 25.
"Since the proportions of these colours to each other have been supposed by Dr. Blair to vary according to the medium by which they are produced. I have compared with this appearance the coloured images caused by prismatic vessels, containing substances supposed by him to differ most in this respect, such as strong but colourless nitric acid, rectified oil of turpentine, very pale oil of sassafras, and Canada balsam also nearly colourless. With each of these I have found the same arrangement of the four colours, and, in similar positions of the prisms, as nearly as I could judge, the same proportions of them."
"But, when the inclination of any prism is altered so as to increase the dispersion of the colours, the proportions of them to each other are thus also changed, so that the spaces AC and CE, instead of being as before 39 and 61, may be found altered as far as 42 and 58." These interesting observations are appended to his *Method of examining Refractive and Dispersive Powers by Prismatic Reflection*, which was published in the *Phil. Trans.* for 1802.
In the year 1800, Dr. Wollaston published in the *Philosophical Transactions* some interesting experiments and observations *On Double Images caused by Atmospheric Refraction*, and in the same work for 1802, he communicated a series of measures *On the Oblique Reflection of Iceland Crystal* in different planes, which he found, as the measures taken by Huygens had done before,
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The observations in this paragraph are quite incorrect. Dr. Blair's results to which they stand opposed have been placed beyond the reach of doubt. These researches of Dr. Wollaston, but particularly the discoveries of Dr. Young, had about this time drawn the attention of the philosophers of France to the subject of double refraction. Laplace had considered the deviation of the extraordinary ray, as due to the action of the attractive and repulsive forces by which Newton and his successors had endeavoured to explain the ordinary refraction and reflection of light; and the Institute of France were thus led in 1808 to propose the double refraction of light as the subject of a prize to be adjudged in 1810. Among the few memoirs which were sent in competition for this prize, that of E. L. Malus, colonel of the imperial corps of engineers was the successful one. After his return from the fatal expedition to Egypt he had composed his "Traité d'Optique," a work of great merit; but as soon as the subject of double refraction was announced, he devoted himself to the inquiry with equal ardour and success. Residing in the Rue des Enfants, in Paris, he happened to view, through a doubly refracting prism, the windows of the palace of the Luxembourg, which were then reflecting to his eye the rays of the setting sun, and, on happening to turn round the prism, he was surprised to perceive that one of the two images of each window vanished in every quadrant of the rotation of the prism. In perceiving this remarkable experiment he was conducted to the splendid discovery which forms an epoch in the history of optics, that when a pencil of light is reflected by a surface of glass at an angle of $54^\circ 35'$, or of water at an angle of $52^\circ 45'$, the reflected light possesses all the characters of one of the pencils formed by double refraction. Hence the pencil was said to be polarised by reflexion. When a pencil thus reflected was made to fall on another surface of the same kind at the same angle, but so that the plane of the second reflexion was at right angles to the plane of the first, then not a single ray of the light suffered reflexion, the whole pencil suffering refraction only. When the light fell upon a plate of glass the light reflected from the second surface acquired the same property.
On the 11th of March 1811, Malus announced to the Institute, that, when a pencil of light was thus polarised by reflexion, the light which was at the same time transmitted through the surface consisted of a portion of light polarised in an opposite direction, and proportional to that which was reflected, and of another portion not modified, which preserves the properties of direct light. This last portion becomes less and less by transmitting the ray through a number of plates in succession till the transmitted pencil is wholly polarised in one direction.
Malus likewise made several experiments on the polarisation of light by metals, and he was led to the conclusion, that the difference between transparent and metallic bodies was, that the former refract all the light polarised in one direction, and reflect all that is polarised in the other, while metallic bodies reflect what they polarise in both directions.
In a series of experiments on crystals and organised substances, communicated to the Institute on the 19th August 1811, Malus found that they all depolarised a pencil of polarised light; that is, a pencil of polarised light which refused to be reflected by another surface properly placed, recovered its power of being reflected after being transmitted through crystals and organised substances. All crystals which did not crystallise in the form of the cube or the regular octahedron were found to possess the property of depolarisation; and the organised substances which he found to possess the same property, were the transparent and fibrous portions of leaves and flowers, the pellicles which cover the hazel, silken and woollen fibres, white hairs, scales, horn, ivory, feathers, the skins of quadrupeds and fishes, shells, and the whiskers of a whale. Malus intended to prosecute this subject to a greater extent, but his brilliant career of discovery terminated by his death on the 7th February 1812.
The loss of Malus, great as it was felt to be, was immediately supplied by his distinguished colleague in the Institute of M. Arago, who has added to this and other depart-ments of science so many brilliant discoveries. On the 17th of August 1811, before the death of Malus, M. Arago communicated to the Institute a memoir "On a particular modification which the luminous rays experience in their passage through certain transparent bodies." Upon exposing thin plates of sulphate of lime, mica, and rock-crystal, to polarised light, and subsequently analyzing the light which they transmitted by a prism of calcareous spar, M. Arago observed the most splendid complementary colours changing with every variation in the inclination of the plate. When the light was incident perpendicularly, and the plate turned round in its own plane, the colours were in all positions the same, though they varied in intensity. He found two positions at right angles to each other, in which the crystal gave no colour, and these positions were those in which the principal section of the crystal was perpendicular to, or coincident with the plane of primitive polarisation. Setting out from these positions, the intensity of the ray gradually increased, and became a maximum at an angle of $45^\circ$ to that plane. When the crystallised plate was fixed, and the analysing plate turned round so as to vary the inclination of the plane of reflexion from it, to that of the fixed plate which polarised the light primitively, the change in the colours was most beautiful. M. Arago observed that the colour reflected in any one position of the analysing plate was complementary to the colour reflected in the perpendicular position. He also found that the power of depolarising the different colours diminished with the thickness of the plate, and he reduced mica to such a degree of thinness that it depolarised no colours at all. In studying the same phenomena in sulphate of lime and rock crystal, M. Arago was led to the conclusion that the colours depended on some other cause than that of the thinness of the plate. M. Arago likewise discovered the depolarising property in a piece of flint glass, about three quarters of an inch in thickness.
We owe also to M. Arago the discovery of circular polarisation in quartz, which he made in 1811. By transmitting polarised light along the axis of the prism he observed the tints to be different in their nature from the ordinary tints of the mineral, although they increased and diminished with the thickness of the plate. When analysed with a prism of Iceland spar, he observed that the true images had complementary colours as in the ordinary tints, and that the colours changed, descending in Newton's scale as the prism was turned round, so that if the colour of the extraordinary image was red, it became in succession orange yellow, green, and violet; and hence he drew the important inference that the differently coloured rays had been polarised in different planes in passing along the axis of the crystal. M. Arago's duties in the Observatory prevented him from pursuing these valuable discoveries with that continuity of labour which they demanded.
A very important discovery was made by M. Arago respecting the colours of thin plates. When the rings of thin plates were examined through a rhomb of Iceland spar, M. Arago discovered that when the principal section of the rhomb was parallel and perpendicular to the plane of incidence, the intensity of the light in one of the images varied with the incidences, and that this image vanished altogether when the pencil of light was inclined $35^\circ$ to the surface, or when it was incident at the maximum polarising angle. This result was the same, whether he examined the reflect-
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1 Mémoires de l'Institut 1811, part i. p. 93-134. 2 Mémoires d'Arcueil, tom. iii. Sur les Couleurs des Lames Minces. ed or the transmitted rings. Hence he inferred that the light of both the systems of rings was polarised in the plane of incidence, at the polarising angle for glass. M. Arago has likewise shewn that the colours of the reflected and transmitted rays are complementary, and that their intensities are exactly equal, completely neutralising each other or forming white light when they are superposed. The most interesting experiment, however, made by M. Arago, is that in which he examined the rings when a convex lens was pressed upon a metallic reflector. When viewed with a prism of Iceland spar, as before, one of the two images vanished at the maximum polarising angle of the glass, but the phenomena were different above and below this angle. At less angles, the dimensions and the colours of the rings were the same in both images, which differed only in the quantity of their light; but at greater angles, the rings in the two images had their colours complementary, the one beginning from a white centre, and the other from a black one.
We have already seen that Dr. Young first applied the principle of interference to explain the colours of crystallised plates; but he did not explain why these colours are not produced excepting with polarised light. MM. Arago and Fresnel entered upon this inquiry, and obtained a satisfactory solution of the difficulty. They found that two rays of light polarised in the same plane, produced fringes by their interference as in common light; that no interference at all takes place when these planes are at right angles to each other; and that at an intermediate inclination, the interference is diminished, and the fringes decrease in intensity. In following out these interesting results, they found that two oppositely polarised pencils will not interfere even when their planes are made to coincide, unless they belong to a pencil which had been wholly polarised in one plane.
We owe also to M. Arago the beautiful discovery that the quantity of polarised light in the reflected and transmitted pencils of common plates are exactly equal.
We are indebted likewise to M. Arago for some important results respecting the interference of light. We have already seen that the interior fringes formed by diffraction disappear when the light which passes by one side of the inflecting body is stopped. M. Arago observed that these fringes were displaced by making the same light pass through a thin plate of some transparent substance, and that the bands were always shifted to the side on which the plate was placed. The amount of this displacement determines the velocity of light in the interposed medium, and consequently gives us a measure of the refractive power of that body with the highest degree of accuracy.
In conjunction with M. Biot, M. Arago published a valuable series of experiments on the density and refractive power of nine gaseous bodies measured in relation to atmospheric air taken as unity. Hydrogen stood at the head of the table, with a refractive index equal to 6.61436, (or 7.0335 if we use the density given by Berzelius) whilst oxygen stood at the foot of the table with a refractive index of 0.8616.
Among the most successful cultivators of physical optics, M. Biot holds a distinguished place. His attention was first directed to the colours of crystalline plates discovered by M. Arago, and by nice instruments and indefatigable labour he determined the general laws of the phenomenon in reference to the thickness of the plates, and the composition of the tints as observed in sulphate of lime and rock crystal, calcareous spar, and aragonite. He observed that at a perpendicular incidence the two colours correspond to those seen by reflexion and transmission in thin plates of air, and he concluded that the thicknesses at which these colours were developed, were proportional to the thickness of the plate of air which gave the same tint in Newton's scale. These thicknesses were found to vary with the nature of the crystal, and were always much greater than the thicknesses of thin plates which gave the same tints. He had at first supposed that at oblique incidences the changes of colour followed the same law as in thin plates, but he afterwards found that the tint depended on the thickness of the crystal traversed by the refracted ray, and as the square of the sine of the angle which the direction of the ray formed with the optic axis. In these experiments M. Biot considered aragonite, sulphate of lime, topaz, and mica, all having one axis of double refraction like calcareous spar.
In order to explain these various phenomena, M. Biot communicated to the Institute in 1812 his ingenious theory of Moveable Polarisation. In this theory the particles of a polarised ray are supposed to preserve their primitive polarisation till they reach a certain depth in the crystal, when a succession of isochronous oscillations round their centre of gravity take place, the axes of polarisation being carried alternately to each side of the axis of the crystal. The depth through which the particle is carried during each of these oscillations, is assumed to be twice the depth through which it has passed before the oscillations began. When the ray emerges from the crystalline plate, the oscillations are supposed to stop, and the ray assumes a fixed polarisation, (in which the axes of the particles are arranged in two rectangular directions) as if the last oscillation had been completed when it quitted the plate.
The remarkable colours discovered by M. Arago along the axis of quartz, were carefully studied by M. Biot, and he and M. Seebeck, nearly about the same time discovered the existence of the very same colours in several essential oils and solutions, such as oil of turpentine, oil of laurel, oil of lemons, syrup of sugar, the two first turning the planes of polarisation from right to left, and the two last from left to right. In a memoir laid before the Institute in 1818, he has shewn that the angular rotation of the plane of polarisation is directly proportional to the thickness of the plate and inversely to the square of the length of the fits as given by Newton. He then concludes that this property of turning the particles of light round their centres of gravity resides in the ultimate particles of solid or fluid bodies, that it is necessary to their very existence, and that it is entirely independent of their mutual distances and mode of aggregation.
M. Biot afterwards resumed this subject, and extended his researches to a great variety of substances; and he has still more recently employed circular polarisation in detecting the constituents of particular vegetable substances, where chemical analysis had partly or wholly failed. He has shewn that the soluble portion of plants, or the farinaceous matter of grain and roots, to which he has given the name of dextrine, and which M. Rasplai had considered to be of the nature of gum, turns the planes of polarisation more powerfully to the right (hence the name dextrine) than the syrups of cane sugar; and that all the gums, and the syrups of the sugar of grapes, turn the planes of polarisation to the left. M. Biot has also applied the same method of research in ascertaining the changes which take place in the sap of trees, and in analysing the processes of vegetation which are concerned in the growth of wheat and rye. His researches are of great practical value in an agricultural point of view, and ought to impress on those whom it most concerns, the important truth, that the most recondite discoveries in science will sooner or later find an useful application.
One of the most important discoveries made by M. Biot was the true nature of the double refraction and polarisation in quartz, which Huygens had been unable to develop. Biot found that it differed from that of calcareous spar, in having the phenomena regulated by a prolate in place of an oblate spheroid, the least refracted image being the ordinary ray in quartz, and the extraordinary one in Iceland spar.
Whilst these valuable researches were carrying on in France, Sir David Brewster was occupied with the same subject in Scotland. In his Treatise on New Philosophical Instruments, published in the beginning of 1813, he has shewn that chromate of lead and realgar exceed the diamond in refractive power; that diamond, phosphorus, and sulphur have their high refractive powers in the order of their inflammabilities; that fluor spar and cryolite have their refractive powers below all solid substances, (excepting talusheer,) and lower dispersive powers than all other bodies; and that all doubly refracting crystals have a double dispersive power. He showed that oil of cassis had the least, and sulphuric acid the greatest action upon green light; that a tertiary spectrum is formed, when prisms of the same substance but different angles are made to correct the dispersion, by the inclination of one of them; and that achromatic combinations may be effected by prisms and lenses of the same kind of glass.
In the year 1812, he began to study the subject of the polarisation of light, in consequence of having become acquainted with Malus's celebrated discovery of the polarisation of light by reflexion. He discovered the remarkable property of the agate, by which it gives only a single distinct image polarised in one plane; the property of depolarisation possessed by almost all minerals, and by many animal and vegetable substances; the polarised colours produced by thin plates of mica and topaz; the partial polarisation of light by polished metals; and the complete polarisation of the exterior and interior rainbows.
In the course of these inquiries our author discovered the two beautiful systems of elliptical coloured rings, which we see by transmitting polarised light along the two optical axes of topaz; and in consequence of his using a conical place of a parallel beam of light in these experiments, he was led to observe the same system of rings, under different modifications, in various other bodies. While examining the depolarising effect of a plate of mica at an oblique incidence, he was led to the discovery of the polarisation of light by oblique transmission through bundles of crystallised or uncrystallised plates; and though Malus had anticipated him in this discovery, yet he had determined the law of the phenomena, which had escaped the notice of that skilful observer.
Hitherto no idea had been formed of the mechanical condition of bodies in which the polarising and doubly refracting structure were exhibited; but in the years 1814 and 1815, a new light was thrown upon the subject, by three discoveries made by Sir David Brewster, namely, that the polarising structure could be produced in glass by heat, and also by rapid cooling; that Prince Rupert's glass drops, formed by rapid cooling, possessed that structure; and that by means of simple pressure, that species of crystallisation could be communicated to soft and indurated jellies, which forms two apparently polarised images, and exhibits the complementary colours by polarised light.
We have already seen that Malus considered the property of polarisation by reflexion as independent of the other modes of action which bodies exercise upon light. In order to investigate this subject, Sir David Brewster made an extensive series of experiments to determine the angles of maximum polarisation by reflexion, from the surfaces of bodies, and from the separating surfaces of different media.
After encountering many difficulties, he was led to the discovery of the very simple law that the index of refraction is the tangent of the angle of polarisation, which is rigorously true for all separating surfaces, and for rays of all refrangibilities; and hence we obtain an immediate explanation of the perplexing fact, that at the maximum polarising angle the polarisation of the ray is never complete. When this law is expressed geometrically, it informs us that when a ray of light is polarised by reflexion, the reflected ray forms a right angle with the refracted ray; that the sum of the angles of reflexion and refraction is a right angle, or counting from the surface, that the angles of reflexion and refraction are equal.
In the same paper, our author has shewn that light may be completely polarised by two, three, or more reflexions, at angles all above or all below, or partly above and partly below the angle of complete polarisation, a greater number of reflexions being required, as the incident ray approaches either to the refracting surface, or to a line perpendicular to it. In the same paper, it is shewn that every ray of light polarised by reflexion has been acted upon by the refracting force; and that total reflexion exercises an analogous action upon light with metallic surfaces.
The influence of heat in producing a transient polarising structure in glass, led our author to an elaborate examination of the subject in 1815. When the edge of a thick plate of glass is laid on a bar of hot iron, the heat gradually propagates itself along the plate, and its path is marked by the most beautiful fringes of polarised light; but no sooner has the heat entered the plate of glass, than similar fringes appear on its upper edge, where there is no heat at all. After a certain period, the whole surface of the glass is covered with coloured fringes, which are arranged in two similar polarising structures at the edges, separated by two dark lines or axes from an opposite structure in the middle. When the glass is removed from the iron, the fringes gradually disappear, and are extinguished when the heat is uniformly diffused over the glass. If the plate of glass is made very hot in boiling oil, and is then allowed to cool with its edges against a plate of cold iron, it will exhibit in a fainter degree the fringes above described; but they are now all reversed, the middle structure having the same character as the external structures had formerly, and vice versa. If, when a plate of glass is covered over with the polarised tints, it is suddenly cut in two by a diamond in the direction of its length, the whole structure is instantly changed, and each piece has the same properties and structure as the whole, exactly like a portion detached from the end of a magnet. The same properties he found in muriate of soda, fluor spar, obsidian, semi-opal, horn, tortoiseshell, and various animal and vegetable bodies. The tints thus developed by heat exhibit, by their being made to cross one another and by other modifications, a series of the most brilliant phenomena within the whole range of optics.
In continuing these experiments, our author found that when the plate of glass, after being brought to a red heat, was allowed to cool quickly, it exhibited permanently the same coloured fringes, a discovery which had likewise been made by Dr. Seebeck of Nuremberg.
This paper is followed by another, published in the same
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1 In all tables of refractive powers of solids and fluids, diamond stood at the head, and water and ice at the bottom. Our author, however, placed various substances above diamond, and talusheer far below ice. 2 Phil. Trans. 1813, p. 27. 3 These discoveries were communicated to the Royal Society of Edinburgh, and owing to the state of communication between France and England, neither the author nor any of the members of the Society were acquainted with the previous discovery of M. Arago of the colours of crystalline plates, or with those of Malus on depolarisation and metallic polarisation. 4 Phil. Trans. 1814, p. 1. 5 Phil. Trans. 1816, p. 46, and Edinburgh Transactions, vol. viii. p. 383, where the phenomena are represented by formulas. In 1816 he communicated to the Royal Society his experiments on mother-of-pearl, explaining the origin of its fine superficial colours, and shewing that they could be communicated to wax, isinglass, the fusible metals, and even to lead, by hard pressure. In 1815 he published a paper on the multiplication of images, and the colours which accompany them, in some specimens of calcareous spar; a subject which had exercised the sagacity of Huygens, Benjamin Martin, Brougham, Robison, and Malus. The last of these philosophers ascribed the multiplication of the images to the interception of the pencils by fissures within the crystal, and the colours to the thin plate of air which it enclosed; but Sir David Brewster discovered the true cause of the phenomena, and proved that there were no fissures nor plates of air; that the multiplication of the images arise from one or more veins of calcareous spar, which divided the rhomb into prisms, so as to form composite crystals having the axes of crystallization and of double refraction of the two contiguous crystals turned round $180^\circ$, so that each of the two pencils formed by double refraction are subdivided in passing from the one crystal to the other. He has shewn also that the colours are the polarised tints produced by thin plates crystallized, these plates giving the regular system of coloured rings seen along the axis of calcareous spar, the light being polarised by the first prism of spar, and analysed by the last. Hence it follows that the polarised tints of crystals were seen and studied by Huygens and his successors, without having any idea of what they were.
The crystalline lenses of animals had hitherto been supposed to increase regularly in density, from the circumference to the centre, for the purpose of correcting the spherical aberration. Sir David Brewster, however, shewed, in 1815, that there were often three structures in such lenses, a structure increasing in density from the centre, being accompanied with another diminishing in density, and that these structures displayed themselves in circular rings of polarised tints, traversed by a rectangular black cross, the tints themselves sometimes rising to a bright yellow of the first order. About the same time he discovered the remarkable property of the diamond, of exhibiting irregular patches of doubly refracting structures, as if it had been in the state of gum, subject to irregular pressure, or induration; and having afterwards discovered gaseous cavities in the same gum, in which the expansive pressure of the included gas had communicated to the surrounding parts a regular doubly refracting structure, he corroborated his supposition that it had a vegetable origin, which he has more recently confirmed by discovering that many diamonds consist of strata of different refractive powers, a property not possessed by any mineral body.
In the beginning of February 1815, when examining the action of metals upon polarised light, our author discovered that complementary colours were produced by one or more reflexions from plates of gold, silver, and other metals; that analogous colours were produced by total reflection; that common light was wholly polarised in the plane of incidence by a number of metallic reflexions, this number being greater in silver and gold than in the other metals. Though examined with much care both by our author and M. Biot, the nature and law of these phenomena were still veiled in obscurity.
Hitherto all crystals were believed to have one axis of double refraction, like Iceland spar or quartz; but in the year 1817, Sir David Brewster discovered that the greater number of crystals, and among these arragonite, sulphate of iron, sulphate of barytes, sulphate of strontian, topaz, felspar, and nitre, had two axes of double refraction, which he called resultant axes, and which were more or less inclined to each other, as the intensity of the real axes more or less approached to equality. By measuring the deviation of the two pencils in different planes, he found that the double refraction was the same in every part of the same ring, disappearing along the resultant axis, and increasing with the value of the tints; and by projecting the coloured rings, and measuring the angular distances from the axis at which the same tints were produced, he was led to the true physical law of the tints, and of the deviation of the extraordinary ray. This general law, when applied to the polarised tints, is thus expressed: The tint produced at any point of the sphere, by the joint action of two axes, is equal to the diagonal of a parallelogram whose sides represent the tints produced by each axis separately, and whose angle is double of the angle formed by two planes passing through that point of the sphere and the respective axis. When the law is applied to the phenomena of double refraction, it may be thus expressed: The increment of the square of the velocity of the extraordinary ray produced by the action of two axes of double refraction, is equal to the diagonal of a parallelogram whose sides are the increments of the square of the velocity produced by each axis separately, and calculated by the law of Huygens, and whose angle is double of the angle formed by two planes passing through the ray and the respective axis. When the two rectangular axes are of equal intensity and of the same character, the preceding law gives the very same results as the law of Huygens does for one axis placed at right angles with the other two. When the crystal has three equal rectangular axes, their forces are in equilibrium in every part of the sphere, and there is neither double refraction nor polarisation. When the first of these laws is applied to the mysterious actions of sulphate of lime, with the origin and classification of which M. Biot had been so much perplexed, and which he has represented by complicated and empirical formulas, the whole mystery disappears, and all the diversified and capricious variations of tint which he had ascribed to secondary forces, become the legitimate and calculable results of two axes of double refraction.
During these laborious researches, our author was led to the law by which the primitive forms of minerals are connected with the number of their axes of double refraction, and to point out the connexion between the optical structure and chemical composition of crystals.
The absorption of common light, in virtue of which crystals exhibit different colours, or shades of colour, in different directions, had been long observed by Wollaston, Cordier, Bournon, De Drée, and others; but Sir David Brewster discovered that in a great number of coloured crystals, both with one and two axes of double refraction, polarised light was absorbed, according to regular laws depending on the inclination of the ray to the axis or axes of the crystal, and that in such crystals the two pencils are always differently coloured, the difference of colour disappearing in the direction of the axis, and rising to a maximum at right angles to it. These phenomena are finely seen in super-acetate of... copper, dichroite, Brazilian topaz, and angite. These properties are shown by our author to be singularly modified by heat, and even communicated to crystals which do not naturally possess it. Absorbing crystals have been called dichroitic, and the property itself dichroism.
The subject of circular polarisation was likewise examined by our author in quartz and amethyst. He found that heat entirely removed from quartz the power of producing circular polarisation, when the substance was reduced to fusion; he discovered it near the resultant axis of chrysoberyl, and in certain specimens of unannealed glass. In examining the proportions of the amethyst, he found that this interesting mineral combines the opposite structures of the two kinds of quartz, being composed of alternate strata of right and left-handed quartz, these two opposite actions destroying each other at their junction, where the colouring matter of the amethyst is principally apparent.
In all the experiments on the polarisation of light by reflection from crystallised surfaces, their action was supposed to be the same as that of common solids and fluids, and Malus had distinctly stated it as the result of experiment, that Iceland spar had the same polarising angle on all its surfaces and in every azimuth, its action being "independent of the position of the principal sections," "that its reflecting power extends beyond the limit of the polarising forces of the crystal, and that as light is only polarised by penetrating the surface, the force which produces extraordinary refraction begins to act only at this limit." Doubting the accuracy of these results, Sir David Brewster instituted a series of experiments on the action of crystallised surfaces upon light, by which he has established the remarkable fact that the angle of complete polarisation varies from 57° 14' to 59° 32', on the surface of the rhomb of calcareous spar, being a minimum in the plane of the principal section, and a maximum in a plane perpendicular to it. But it is not merely the polarising angle that is changed. When the ordinary reflecting force is weakened by causing the reflexion to be made from the refracting surface of oil of cassia and Iceland spar, he found that the light was no longer polarised in the plane of reflexion, and that the deviation from this plane depended on the inclination of the ray to the axis of the crystal, the deviation becoming less and less as the refractive power of the fluid was diminished. In the same paper he has shown that by altering the mechanical condition of the surfaces of crystals, and making the ray enter this surface from fluids of different refractive powers, the ordinary or the extraordinary image may be weakened or extinguished at pleasure.
In the year 1816, Sir David Brewster discovered a remarkable system of coloured rings in apophyllite, a singularly constituted crystal, one part of which has one axis, whilst another part has two axes of double refraction. Different parts of this crystal possess different degrees of double refraction, with the same thickness, and at the same inclination to the axis; and the beautiful and symmetrical figure which a perfect crystal exhibits by polarised light, delineated in the most splendid colours, is perhaps the finest sight which the mineral kingdom can present to us. By the aid of polarised light, our author discovered also a singular structure in certain crystals of chabasite, in which the double refraction gradually diminishes in successive strata, then vanishes, and reappears with an opposite sign, the one double refraction being positive and the other negative; but the most remarkable of all these structures was that of analcime, in which he discovered a new species of double refraction.
History. In which the phenomena are related to planes in place of axes, in which the double refraction disappears.
In the year 1829, our author communicated to the Royal Society of London six papers. The first of these was on the reflection and decomposition of light at the separating surfaces of media of the same and of different refractive powers. The second, entitled, on periodical colours produced by the grooved surfaces of metallic and transparent bodies, contains an account of a new series of periodical colours exhibited by grooved surfaces, which succeed each other in a plane at right angles to that in which the usual spectra are seen, and producing a singular modification of these spectra; phenomena not yet brought within the pale of the undulatory theory. In the third paper, on the double refraction produced by pressure in the molecules of bodies, he has shown that the axis of pressure is a regular axis of double refraction, and that the doubly refracting properties, which are not inherent in the molecules themselves, are produced by the pressure caused by the forces of aggregation, which generally differ in intensity in the direction of the three rectangular axes. The same author has more recently succeeded in producing regular double refraction by the expansive presence of heat. The other three papers treat of the laws of the polarisation of light by reflexion and refraction, and on the action of the second surfaces of transparent plates upon light. In these papers he has given formulae for computing the quantities of light polarised at all incidences, both by reflexion and transmission; and has explained all the phenomena of partial polarisation, and those produced by any number of successive reflexions and transmissions. In the last of these three papers our author shows that the quantity of polarised light in the light reflected and transmitted by a transparent plate is not equal at all angles of incidence, and that the proposition is true when applied to surfaces alone, the apparent equality in the first case being produced by unexcluded light polarised perpendicular to the plane of reflexion.
We have already seen that Malus, Biot, and Sir David Brewster had been baffled in their attempts to unravel the complex phenomena of metallic polarisation. The last of these authors had at various times resumed the investigation; but it was not till February 1830, that he communicated the result to the Royal Society, in a paper entitled, on the phenomena and laws of elliptic polarisation, as exhibited in the action of metals upon light. All the phenomena of metallic polarisation are shown to be those of elliptical polarisation, connecting the phenomena of circularly polarised light with those of plane polarised light, the action of silver approaching nearest to that of totally reflecting surfaces by which circular polarisation is produced, and that of galena to transparent bodies or those not metallic, by which plane polarisation is produced. The colours accompanying these phenomena have no relation to those of crystallised plates, and in the case of silver and gold are extremely beautiful and splendid.
Hitherto the analysis of solar light by Sir Isaac Newton had been regarded as complete, and incapable of any further development. From this analysis he himself deduced the conclusion, that to the same degree of refrangibility ever belonged the same colour, and to the same colour ever belonged the same degree of refrangibility. So early as 1822, in a paper, on the monochromatic lamp, &c., Sir David Brewster showed that some of the colours of the spectrum... were compound, capable of being analysed by absorbing media, and that different colours had the same refrangibility. This result stood in direct opposition to the Newtonian doctrine, and our author, in order to support it, undertook an elaborate series of experiments the results of which were communicated to the Royal Society of Edinburgh in 1831, in a paper entitled, "A new analysis of solar light, indicating three primary colours forming coincident spectra of equal lengths." In these three overlapping spectra the intensity of each colour is a maximum at that point where the same colour is most intense in the compound spectrum. Hence it follows that all the colours in the solar spectrum are compound, consisting of red, yellow, and blue light in different proportions, so that if at any point we separate as many rays of each colour as is necessary to produce white light, by absorbing the excess at that point, we should exhibit the strange phenomenon of white light incapable of being decomposed by the prism. This has actually been done by Sir David Brewster, by means of absorbing media. But in a very recent communication made to the Royal Society, (in June 1837), he has explained a new method of effecting this by perfectly transparent media, which he calls analysis by dissection, and of which we shall give some account in another part of this article. This paper was followed, in 1833, by another, "On the colours of natural bodies," in which our author shews that they have not the same composition as those of thin plates, and demonstrates the truth of this opinion by a special analysis of the green colour of plants, the most prevalent tint in nature, and the one which Newton had pronounced to be of the third order. In the same year our author published his Observations on the Lines of the Solar Spectrum, and on those produced by the earth's atmosphere, and by the action of nitrous acid gas, but we must reserve our notice of this paper till we come to describe the discoveries of Fraunhofer.
The late Dr. Thomas John Seebeck of Nuremberg, was an active and successful cultivator of the science of Optics. His first experiments on this subject were published in Schweigger's Journal for April 1813 and December 1814. In 1811 M. Arago observed the polarising structure in thick pieces of flint, and in 1812 Sir David Brewster had noticed the same property in some pieces of plate glass. In Dr. Seebeck's paper of 1813, he observed the regular figure produced by polarised light, when the glass had the regular form of cubes and cylinders. In cubes of an inch in diameter, he found them to be indistinct, and not produced by fluor spar or rock salt. In his second paper of December 1814, he shews that a plate of glass made red hot, and set upon its edges to cool, exhibits at the part which cools first a series of coloured fringes, which spread over the whole plate, the structure which produces them remaining permanently fixed in the glass. These experiments are posterior to those made in Scotland on the effects of heat upon glass, and on the polarising structure of glass cooled in water.
Early in 1816, Dr. Seebeck discovered the property of certain essential oils in producing the polarised tints, the property of single refraction possessed by tourmaline, and the system of coloured rings produced by Iceland spar; but these discoveries he was anticipated, as we have seen, by others, though he is entitled to all the merit of a second discoverer.
In 1809 Dr. Seebeck communicated to the Academy of Sciences at Berlin, an interesting memoir on the unequal production of heat in the prismatic spectrum, in which he showed that the place of maximum heat varied with the substance of which the prism was made, being in the yellow rays in the spectra formed by water, (and according to Wunsch, in alcohol and oil of turpentine); in the orange in concentrated sulphuric acid, and solution of sal ammoniac and corrosive sublimate; in the middle of the red in crown and plate glass, and beyond the red in flint glass. Dr. Turner ascribed these results to the different powers of these media to refract the rays of solar heat; but Sir David Brewster explained them by supposing that colourless transparent bodies exercise the same variety of absorptive action upon heat that coloured bodies do upon light, the body in the last case becoming coloured in consequence of that action. Hence the maximum ordinate of heat will shift its position with the nature of the body, and we shall no doubt find media several maxima and minima, and points of no heat at all, according as we increase the size of the prism or the thickness which the heat traverses. The best way to carry such researches is to use a prism of glass whose curve of heat is well ascertained, and then to determine the changes which take place in the curve by interposing thick plates of transparent solids and fluids.
This eminent philosopher would have done still more for the science of optics, had he not been attracted to the study of thermo-electricity, in the creation and extension of which he has immortalised his name.
We are indebted to Dr. A. Seebeck for a series of instructive and accurate experiments on the polarising angle of different substances, which confirm the accuracy of the law of the tangents, and another on the polarising angle of calcareous spar in different azimuths.
We come now to that auspicious period in the history of optics, when this science was destined to receive the grandest accessions from the genius of M. A. Fresnel, engineer of roads and bridges. What Newton did for astronomy, Fresnel did for physical optics; and all Europe will, we are persuaded, confirm the decision which places him pre-eminently above all other cultivators of this branch of science. The discoveries of Fresnel, however, are so connected with theoretical considerations, that it is impossible, in a historical sketch, to give any thing like an idea of their magnitude and importance. The phenomena of rotatory polarisation in quartz, which had so much perplexed philosophers, have been completely explained by Fresnel. He found that they arise from the interference of two circularly polarised pencils propagated with different velocities along the axis of quartz, the one revolving from right to left, and the other from left to right, and that a plane polarised ray is equivalent to two circularly polarised rays of half the intensity. These facts he verified experimentally, by an achromatic combination of right and left-handed prisms of quartz, so disposed as to double the refraction of the images.
M. Fresnel had also found that light was circularly polarised by two total reflexions from glass at an angle of about $54^\circ 37'$, and by placing between two rhombs of glass, each of which polarised the light circularly and had their planes of reflexion at right angles to each other, a crystalised plate, he observed the light transmitted through this system exhibited phenomena analogous to those seen along the axis of rock crystal. The rhomb of glass so cut, that when the incident rays enter and leave it perpendicularly, they have suffered two reflexions at an angle of $54^\circ 37'$, is well known by the name of Fresnel's rhomb.
Fresnel's theory of double refraction and polarisation, one of the finest efforts of genius, conducted its author to many important results which had escaped the notice of the most diligent observers. Hitherto it had been taken for granted by all, and appeared to be proved by Biot's experiments on topaz, that in biaxial crystals one of the rays
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1 Edin. Trans. vol. xii. p. 129, or Edin. Journ. of Science, new series, vol. v. p. 197. 2 Id. Id. 3 Id. Id. 4 Treatise on New Phil. Inst. p. 333. 5 Berlin Memoirs, 1818–1819, p. 305, or Edin. Journ. of Science, vol. i. p. 358, No. 2, Oct. 1824. 6 Chemistry, 3d edit. p. 84. 7 Second Report of the British Association, 1832, p. 294. 8 Edin. Journ. of Science, N. S. vol. v. p. 99. followed the ordinary law of the sines; but it followed from Fresnel's theory that it did not, and by a series of the nicest and most difficult experiments he determined, that neither of the two rays have a constant velocity, both being performed according to a new law. What had been called the extraordinary ray, he found by his theory to be regulated by the law discovered by Sir David Brewster, and simplified in its mathematical expression by M. Biot, and he showed that all the phenomena of double refraction could be accurately calculated. The axes of elasticity in Fresnel's theory are the same as the axes of double refraction in Sir David Brewster's, and the laws of composition and resolution of such axes in uniaxial, biaxial, and tesserul crystals, which have no double refraction previously given by the latter, are all necessary results of the same theory.
But the most remarkable part of Fresnel's theory is his explanation of the polarisation of light. The hypothesis of transversal vibrations first presented itself to Dr. Young while considering the law of extraordinary refraction in biaxial crystals, as communicated to him by Sir David Brewster. M. Fresnel, however, showed that it was a necessary consequence of the laws of interference, and that the vibrations of a polarised ray are on the surface of the wave, and perpendicular to the plane of polarisation. In unpolarised light they also only on the surface of the wave, and this species of light is conceived to consist of a rapid succession of systems of waves polarised in every possible plane, passing through the normal to the front of the wave. Hence light is polarised by resolving the vibrations into two sets in two rectangular directions.
We have already slightly noticed the fine discoveries of MM. Arago and Fresnel on the interference of polarised light, and we can now only refer with admiration to the beautiful series of experiments by which the phenomena of moveable polarisation were properly explained, and brought under the dominion of the undulatory theory. When a polarised ray proceeding from a luminous point was transmitted through two rhomboids of Iceland spar of equal thickness, whose principal sections were inclined 45° to the plane of primitive polarisation, the emergent light will diverge as if from two near points, and the two portions will be oppositely polarised. MM. Arago and Fresnel found that the light formed by the union of these pencils was plane circularly, or elliptically polarised, according to the difference of the paths traversed when they met. Following out this principle, MM. Arago and Fresnel were led to an experimentum crucis, to determine the accuracy of the theory of moveable polarisation. A homogeneous ray of polarised light was transmitted through a plate of sulphate of lime, having its principal section inclined 45° to the plane of primitive polarisation, and of such a thickness that it should be circularly polarised according to the undulatory theory, and plane polarised according to the other; and the result was decisive against the theory of moveable polarisation.
We owe also to M. Fresnel the true theory of the inflexion or diffraction of light. The Institute of France made this the subject of their physical prize for 1818, and the memoir of our author was the successful one. He had at first adopted and extended the theory of Dr. Young, that the fringes arise from the interference of the direct and inflected light; but he was afterwards obliged to admit, that rays passing at a sensible distance from the reflecting body, deviate from their primitive direction, and interfere with the direct light. This interesting effect he ascribes to a number of elementary waves sent from each portion of the surface of the principal wave when it reaches the reflecting body, and he determines the resultant of all the elementary waves sent by these portions to a given point. Upon applying this theory to various cases of inflexion, he found it to agree so well with observation, that, with the exception of the case of diffraction by narrow apertures, the theory did not err more than the 2500th part of an inch.
Among the many important discoveries of Fresnel we must enumerate the theory of the reflexion of light. Dr. Young had shewn on the undulatory theory, that at a perpendicular incidence the intensity of the reflected light was a very simple function of the index of refraction. M. Poisson had arrived by another process at the same result, without knowing, we believe, what had been done by Dr. Young and he afterwards extended his inquiries to different incidences. The conclusions, however, at which this distinguished mathematician arrived, were inconsistent with observation; and Fresnel had the good fortune to give a complete solution of the problem, by combining the doctrine of transversal vibrations with the theory of waves. He assumes, that the elasticity of the ether in the two media are equal, but their density different, though he also solved the problem on the more general assumption, that the elasticity was different in the two media. He thus obtained formulae for all incidences and all refractive powers, and the law of the tangents, as well as that of the equality of pencils, polarised by reflexion and transmission, became the consequences of these formulae. At a perpendicular incidence the formula coincides with that of Young and Poisson, and at 90° the whole light is reflected, a result which has been verified by observation.
Contemporary with the discoveries of Fresnel were those of Fraunhofer of the late M. Fraunhofer of Munich, who made several important observations on the solar spectrum, on the diffraction of light, on refractive and dispersive powers, and on the refrangibility of the light of the fixed stars. By using fine prisms entirely free of veins, he discovered that the solar spectrum was crossed by about 590 black lines, and he executed a beautiful drawing of the spectrum, in which the most important of these are projected. Fraunhofer was not aware that Dr. Wollaston had previously discovered seven of these lines; but this slight anticipation does not in the least degree diminish the singularity of this splendid discovery. He discovered similar lines in electric light, and in the spectra of the Moon, Venus, Mars, Castor, Pollux, Sirius, Capella, Betelgeus, and Procyon; but none whatever in artificial white flames. These lines he found to have a fixed position in relation to the coloured spaces, and, by measuring accurately the distance of prominent lines in the different coloured spaces, he obtained measures of the refractive and dispersive powers of bodies with a degree of accuracy hitherto unknown. Fraunhofer considered these lines as having their origin in the nature of the sun's light; but Sir David Brewster, who by particular methods has discovered more than twice the number of lines reckoned by Fraunhofer, has established the curious fact, that many of them are produced also by the action of the earth's atmosphere. In his researches on this subject Sir David Brewster discovered the remarkable property possessed by nitrous gas of producing analogous lines in great numbers, increasing in width with the thickness of the gas, or with an augmentation of its temperature. "The power of heat alone," says this author, "to render a gas, which is almost colourless, as red as blood, without decomposing it, is in itself a most singular result; and my surprise was greatly..." increased, when I afterwards succeeded in rendering the same pale nitrous acid gas so absolutely black by heat, that not a ray of the brightest summer sun was capable of penetrating it. Professors Miller and Daniel afterwards discovered numerous fixed lines disposed at equal distances, in the vapour of bromine and iodine, and Sir David Brewster has very recently discovered hundreds of lines under very singular circumstances, in the spectrum of an artificial substance, resembling mother-of-pearl; but what is most interesting, these lines are moveable, shifting their place in the spectrum by varying the incidence, and are produced by the periodical action of thin plates enclosed in the substance. He has also discovered, that broad dark bands like those produced by absorbing media, but entirely different from the nearly equidistant bands formed by single thin plates, are produced by a number of thin plates in a state of combination.
Considering the lines of the spectrum as produced by interference, Fraunhofer was induced to make a complete series of experiments on the inflexion of light, particularly on the splendid colours produced by gratings of wires, and grooved surfaces, which were published in the year 1822, in the Memoirs of the Royal Bavarian Academy of Sciences. He afterwards repeated these experiments with a finer apparatus, and communicated an account of them to the Academy of Sciences at Munich, on the 14th June 1823. The science of optics owes also to Fraunhofer the art of making the finest glass for achromatic telescopes and prisms, and such was the perfection at which he arrived, that, in a letter to the author of this article, he expresses his willingness to undertake an achromatic object glass eighteen inches in diameter. Our author wrote also a treatise on halos, parhelia, &c., in which he ascribes the small solar and lunar halos to the inflexion of light, by particles of vapour in the atmosphere, and the great halos of 45° to the refraction of hexagonal prisms of ice.
Among the most distinguished contributors to optical discovery, Sir John Herschel occupies a high place. The deviations of the polarised tints from the colours of thin plates, or those of Newton's scale, had been discovered by Sir David Brewster in acetate of lead, tartrate of potash and soda, apophyllite, topaz, and various other minerals. He had divided these crystals into two classes, viz. those that had the red ends of the rings inwards, and the blue ends outwards; and those that had the blue ends of the rings inwards, and the red ends outwards. In his paper of 1818, he states, that "in almost all crystals with two axes, the tints in the neighbourhood of the resultant axes, when the plate has a considerable thickness, lose their resemblance to those of Newton's scale, as will be more minutely described in another paper." Conceiving that these deviated tints arose from the superposition of systems of rings of different colours, Sir John Herschel examined the coloured rings by homogeneous light, and established the important fact that the inclination of the resultant axes varied in the different colours of the spectrum, the poles or centres of the rings approaching to each other in red, and receding in violet light, in some crystals; while in others they receded from each other in red, and approached in violet light. In tartrate of potash and soda, for example, the inclination of the axes was 75° 42' in red, and only 55° 14' in violet light. These various axes all lie in the same plane, excepting in borax. In the paper containing this discovery, and in other
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1 Edin. Trans. vol. xii. 2 Phil. Trans. 1837. 3 Id. Id. 4 Vol. viii. 1822. 5 Edin. Journal of Science, No. xiii. p. 101, 251. 6 See Phil. Trans. 1814, p. 204, 205, and 1820, p. 95. 7 Edin. Jour. of Science, No. xii. p. 348. 8 Phil. Trans. 1820, p. 43. 9 Cambridge Trans. vols. i. and ii.
two, communicated to the Cambridge Philosophical Society, he has described various interesting phenomena which he discovered in different specimens of apophyllite and in hyposulphate of lime, and which led him to some interesting conclusions respecting the law of proportional action of these crystals on the different colours of the spectrum.
We have already seen that the force which produces circular polarisation had been deemed a property of the ultimate particles of bodies, and totally unconnected with their mode of aggregation. In 1820 Sir John Herschel made the beautiful discovery, that the direction of the circular polarisation in quartz was invariably the same with that of the plagiedral planes round the summit, the direction of the polarisation being retrograde or direct, according as these planes leant forward or backward round this summit.
We owe also to Sir John Herschel an interesting inquiry into the aberrations of compound lenses and object-glasses, a series of curious experiments on the phenomena produced by diaphragms or apertures of various shapes, variously applied to mirrors and object-glasses, and a great number of original views and valuable experiments, which are contained in his Treatise on Light, one of the most valuable and original works on science which has appeared during the last century.
M. Fresnel was, we believe, the first person who observed the change produced by heat on the tints of sulphate of lime. It is to M. Mitscherlich, however, that we owe the most complete investigation of this subject. He found that heat expands crystals differently in different directions. Iceland spar is expanded by it in the direction of its axis, while it is in a slight degree contracted in directions perpendicular to the axis. The rhomb thus approaches to the cube, and the double refraction is diminished. M. Mitscherlich also found that the inclination of the optical or resultant axes, which is about 60°, diminishes with heat till they actually form one axis, when by a farther increase of heat they again separate, and open out, as it were, in a plane at right angles to that of the laminae. We have repeated this experiment, and enjoyed the remarkable sight of observing the one system of rings marching towards the other in the plane of the laminae, and changing their form and size as they advanced. An analogous, and even a more remarkable property, was discovered by Sir David Brewster in glauberite. At the freezing point glauberite has two optical axes for all the colours of the spectrum, the inclination of the axes being greatest in red, and least in violet light. When heat is applied, the two axes approach, and those of different colours unite successively, the crystal possessing the remarkable property of being a uniaxial one for red, and a biaxal one for violet light. By increasing the temperature, the optical axes open out in the same order, but in a plane at right angles to that in which they formerly lay, and long before the temperature has reached that of boiling water the planes of the axes in all the prismatic colours are perpendicular to their first position. Such a crystal would form a delicate chromatic thermometer. M. Marx has discovered an analogous property in topaz, in which the two axes separate with heat, the variation being greater in the coloured than in the colourless varieties. Sir David Brewster has discovered that regular double refraction is produced in some soft substances by the application of heat.
Some very excellent and interesting results have been obtained by M. Rudberg, on the effect of heat upon doubly birefringent crystals. He found that the extraordinary ray in calcareous spar (the line F was used) had its deviation increased 2° 34′, as the refractive index increased 0.00043, by a rise of temperature equal to 64°; the refracting angle of the prism being 59° 53′ 9″, whereas the refractive power for the ordinary ray either does not change at all, or decreases with the temperature by a quantity extremely small. In rock crystal he found the deviation to be 42″, or 0.00027, both on the ordinary and extraordinary ray, the angle of the prism being 45° 20′ 5″. In aragonite he found that the double refraction decreased a little with the temperature.
We owe also to M. Rudberg a series of valuable experiments on the refractive actions of the differently coloured rays in crystals with one and two axes of double refraction. His measures were taken in reference to the fixed lines in the spectrum, and the minerals he employed were rock crystal, calcareous spar, aragonite, and colourless topaz. He confirmed the existence of two dispersive powers in doubly refracting crystals, announced long before by Sir David Brewster; and the variation of the inclination of the optic axes with the different colours of the spectrum, which had also been previously discovered by Sir John Herschel. M. Rudberg was no doubt unacquainted with the previous labours of these authors, otherwise he would not have passed them over without notice.
Like every other branch of physical science, optics owes much to the profound researches of M. Poisson, which are in general of too recondite a nature to find a place in a popular treatise. The theory of the colours of thin plates was left incomplete by Dr. Young. The two interfering portions from the upper and under surface of the plate were obviously unequal, and therefore could not destroy one another wholly by interference, as they are found to do. M. Poisson remedied this defect by shewing that there must be an infinite number of partial reflexions within the plate, at each of which a very small portion of light was reflected, so that the sum of all these portions of light makes up for the defect of one of the pencils, and makes the interfering pencils equal. Hence M. Poisson has shewn that at a perpendicular incidence, and at points where the effective thickness of the plate is an exact multiple of the length of half an undulation, the intensity of the reflected and transmitted light will be the same as if the plate were suppressed altogether, and the bounding media in absolute contact, so that when these media have the same refractive power, no light will be reflected and the whole transmitted. By the aid of the property discovered by M. Arago, that the light is reflected in the same proportion at the first and second surfaces of a plate, M. Fresnel extended M. Poisson's conclusions to all incidences.
In treating of the subject of diffraction, M. Poisson was led to the curious result that the centre of the shadow of a small opaque circular disc, exposed to light diverging from a single point, is as much illuminated by the diffracted light as it would be by the direct light, if the opaque disc were removed. By cementing a small metallic disc upon a plate of pure and homogeneous glass, M. Arago verified this remarkable deduction of theory.
M. Poisson's researches on the propagation of motion in elastic fluids, and their application to light, are too profound to admit of a brief and intelligible analysis.
In two memoirs read to the Academy of Sciences in 1828, M. Ampère has made a valuable addition to the theory of Fresnel. By an indirect and not very rigorous process, M. Fresnel had been led to the equation of the wave surface; but M. Ampère obtained a direct demonstration of it, deducing the equation in the manner which Fresnel had merely indicated, and he derived from this equation the elegant geometrical construction obtained indirectly by Fresnel.
The undulatory theory of light has been greatly advanced by the researches of M. Cauchy, a French mathematician of distinguished eminence. In determining the law of propagation of a plane wave, he shews that a disturbance originally limited to a given plane will give rise to three pairs of plane waves with uniform velocities, and parallel to the original plane, the two waves of each pair moving in opposite directions, but with equal velocities. He shews that the separate pairs will move with velocities represented by the reciprocals of the axes of an ellipsoid, the form of which is regulated by the position of the plane wave, and the nature of the system, the absolute displacement of the molecules being parallel to the direction of these axes. Hence a system of plane waves superposed at the point of original disturbance, will be divided into three corresponding systems, and these will generate by their superposition a curved surface of three sheets, each sheet being touched by all the plane waves of the system. If these principles are established, it will follow as a necessary consequence that a single ray of light will be divided into three polarised rays, one of which will in all cases have little intensity. M. Cauchy, as Professor Lloyd remarks, has not pointed out the method of discovering this ray, or stated the precise physical condition on which its existence depends; but it "would seem to arise from the circumstance that the vibration normal to the wave is not absolutely insensible, so that the actual vibrations are not accurately in the plane of the wave." The Triple results of M. Cauchy's general theory," continues Professor Lloyd, "embrace and confirm those of Fresnel: and the mathematical laws of the propagation of light are shewn to be particular cases of the more general laws of the propagation of vibratory motion in any elastic medium composed of attracting and repelling molecules. Considered, however, simply with reference to the theory of light, the solution given by M. Cauchy cannot, I conceive, be considered as a complete physical solution. In other words, the phenomena of light are not connected directly with any given physical hypothesis; but are shewn to be comprehended in the results of the general theory, in virtue of certain assumed relations among the constants which that theory involves. If, indeed, we were able to assign the precise physical meaning of these equations of condition, we should have nothing more to desire in the general theory of light; for these equations must necessarily express the characteristic properties of the vibrating medium. In this point of view, their discussion becomes a subject of the highest interest; and it is probable that the important conclusions of which we have yet to speak, may in this manner be confirmed and extended."
Before quitting this subject, however, we ought to mention that there is an essential difference between the theories of Fresnel and Cauchy. In the former a ray is said to be polarised in or parallel to any plane, when the vibrations of the molecules of ether are perpendicular to that line or plane; whereas, in Cauchy's theory, a ray is said to be polarised in or parallel to any plane, when the vibrations of the ether are performed in a parallel to that plane.
The inability of the undulatory theory to explain the dispersion of light, was long one of the few exceptions to its universal application. Dr. Young supposed that the material particles of bodies are incapable of permanent vibrations; that these vibrations will retard those of the ether; and that this retardation will be proportional to their frequency. The Rev. Mr. Challis, adopting Dr. Young's idea, has endeavoured to explain the manner in which the undulations of the ether within bodies are modified by their material atoms. He supposes that a sensible reflexion takes place at every interruption of continuity in the medium; and he infers that the mean effect produced by a retarding cause proportional to the reflective power of the atoms, will be to make the condensation corresponding to a given velocity greater in a certain proportion than in free space, and to diminish the velocity of propagation in the same proportion. Mr. Airy has more recently endeavoured to remove this difficulty, by supposing that in refracting media there may be something depending on time which alters their elasticity, in the same manner as in air the elasticity is greater with a quick than with a slow vibration of particles.
An anonymous writer in a very recent number of The Philosophical Magazine, has proposed another hypothesis for obtaining a difference of elasticity. He supposes that the ether accumulates itself round the particles of transparent media, and forms spheres of a density increasing towards their centres; and he infers that a succession of vibrations communicated through a medium thus constituted will give rise to new vibrations propagated with various velocities corresponding to those of the different rays in the spectrum.
The complete removal of this difficulty from the undulatory theory has been effected by the skill of M. Cauchy. Regarding the sphere of action of the ethereal molecules as indefinitely small, in comparison with the length of an undulation, it had been inferred that the velocity of the undulations must be constant in the same medium; but this restriction being removed as a groundless one, M. Cauchy has considered the problem in a more general manner, and has arrived at the result, that there exists a general relation between the length of the undulations and the velocity with which they are propagated, or the index of refraction; and consequently that rays of different colours will have different degrees of refrangibility. This relation is expressed by an equation involving two arbitrary constants, depending on the nature of the medium, and determinable by two values of the index of refraction for two waves of a known length. The refractive index for waves of other lengths may then be computed. Professor Powell has done this for several media, whose refractive indices for the fixed lines in the spectrum have been determined by Fraunhofer, Rudberg, and himself; but though there is a general coincidence with the theory, the differences are in some cases rather inauspicious.
M. Cauchy has more recently deduced from his general theory the remarkable fact, that in a particular case of reflexion the reflected pencil exceeds the incident one in intensity, a result which has been confirmed by direct experiment.
In examining the two rays produced by the double refraction of quartz, Mr. Airy was led to a discovery which we consider as one of the most important in its results, and one of the most beautiful in its phenomena, that has yet been made in this branch of optics. The circular polarisation of the two rays along the axis of quartz had been studied by different philosophers, and had been explained by Fresnel with singular ingenuity, on the principles of the undulatory theory. No attempt, however, had been made to account for the existence of this property only in the rays which pass near the axis of the crystal, or to define the limit where the circular polarisation ended, and the plane polarisation commenced. Fresnel, and all who have written on the subject, seem to have shrunk from this difficulty; but Mr. Airy thought that the two kinds of polarisation must have some connecting link, and by the aid of theory and experiment he succeeded in discovering it. In place of the two rays in quartz consisting of plane polarised light, as was universally believed, Mr. Airy has shewn that they both consist of elliptically polarised light, the greater axis of the ellipse for the one ray being in the principal plane of the crystal, and the greater axis of the other perpendicular to that plane. One of the rays he found to be right-handed elliptically polarised, and the other left-handed elliptically polarised. The proportion of the axes of the ordinary ray is more nearly one of equality than the proportion of the axes of the extraordinary ray, each proportion being one of equality when the direction of the ray coincides with the axis, and becoming more unequal with the inclination, according to a law not yet discovered. The results calculated from the theory are in perfect accordance with those which Mr. Airy has obtained from very nice and difficult experiments; so that we may regard this beautiful and singular property of the two rays of quartz as perfectly established.
Without knowing of the beautiful experiments of M. Arago, already referred to, Mr. Airy was led to make the same experiment on the coloured rings formed between a lens and a metallic reflector, and to draw the same conclusion from it in favour of the undulatory theory. From a consideration of the formulae of Fresnel, Mr. Airy was led to expect that if the rings were formed between two substances of different refractive powers, such as plate glass and diamond, the light being polarised perpendicular to the plane of incidence, they should have a black centre at incidences less than the polarising angle of the glass, and greater than the polarising angle of the diamond; while they should have a white centre at all intermediate angles. These anticipations Mr. Airy confirmed by experiment; and in the course of his observations he observed certain peculiarities in the phenomena, from which he has drawn the following conclusions, viz.
1. When the angle of incidence is less than the maximum polarising angle of the diamond, the nature of its reflexion is similar to that of metallic reflexion; the phase of vibration in the plane of reflexion being more retarded than that of vibrations perpendicular to the plane of reflexion, but perhaps by a smaller quantity than in reflexion from metals.
2. In the neighbourhood of the polarising angle, the nature of the reflexion is different from any that has hitherto been described. The vibrations in the plane of reflexion do not vanish, but on increasing the angle of incidence by three or four degrees, the phase of vibration is gradually retarded by about $180^\circ$. In the reflexion of light whose vibrations are perpendicular to the plane of reflexion, there is no striking difference between the effects of diamond and those of glass.
3. For angles of incidence greater than the polarising angle, there is no sensible difference between the effects of diamond and those of glass.
It would be desirable to ascertain if the diamond used by Professor Airy consisted of strata of different reflective and refractive powers, a structure lately discovered in some specimens by Sir David Brewster; or if it was a heterogeneous crystal, containing particles of different degrees of density and double refraction, a structure very common in this gem.
Hitherto the mathematical theory of light owed almost all its development to the distinguished members of the Institute of France—to Malus, Arago, Fresnel, Poisson, Ampère, and Cauchy; but it was now destined to receive a powerful impetus from those eminent members of Trinity College, Dublin, who have nobly sustained the honour of their country by their genius and discoveries. In his Essay on the Theory of Systems of Rays, Sir William Hamilton has given an elegant analytical form to that part of the Hamiltonian theory of Fresnel which relates to the determination of the velocity and polarisation of a plane wave; and he has demonstrated... duced the velocity and direction of the ray from that of the wave, and consequently the form of the wave surface. In these researches Sir W. Hamilton was conducted to the discovery of some new geometrical properties of the wave surface. He found that this surface has four conoidal cusps at the extremities of the resultant or optical axes, at each of which the wave is touched by an infinite number of tangent planes, forming a tangent cone of the second degree, while at the extremities of the lines of single wave velocity, there are four circles of plane contact, in every part of each of which the wave surface is touched by a single plane. These cusps and circles, the existence of which does not seem to have been suspected by Fresnel, have led Sir W. Hamilton to some remarkable theoretical conclusions respecting the laws of refraction in biaxial crystals. To this new property he has given the name of conical refraction, because a single ray is refracted into an infinite number, forming a kind of cone. This conical refraction is of two kinds, external and internal. In external conical refraction, one internal cusp ray corresponds to an external cone of rays; and in internal conical refraction, an external ray incident at an angle corresponding to the line of single wave velocity within, is connected with an internal cone of rays.
Sir W. Hamilton requested Professor Lloyd, of Trinity College, Dublin, to inquire experimentally into the existence of these two kinds of conical refraction. For this purpose he selected arragonite, a crystal of great biaxial energy, and having its optic axes inclined about 20°. It was cut with parallel faces perpendicular to the line bisecting the two optic axes. Upon looking at the light of a distant lamp through the crystal, and in the direction of one of the optical axes, Professor Lloyd saw a point more luminous than the space immediately about it, and surrounded by something resembling a stellar radiation. Hence the direction of the optical axes may be determined by this modification of common light. When the adjustment was perfected, and the light transmitted in the exact direction of the cusp ray, there appeared at first a luminous circle, with a small dark space in the centre, and in this dark central space were two bright points, separated by a narrow and well-defined dark line. These appearances rapidly changed in shifting the minute aperture next the eye. On examining the emergent cone with a plate of tourmaline, Professor Lloyd was surprised to observe that only one radius of the circular section vanished in a given position of the tourmaline, and that the vanished ray ranged through 360°, while the tourmaline was turned through 180°. Hence it follows that all the rays of the cone are polarised in different planes. On a more attentive examination of this phenomenon, Professor Lloyd discovered the remarkable law, "that the angles between the planes of polarisation of any two rays of the cone is half the angles between the plane containing the rays themselves and the axis." This law he found to be in perfect accordance with the theory.
The verification of the second kind of conical refraction Professor Lloyd found to be more difficult. The angle of the cone of rays which theory indicated, should be seen within the crystal when a single external ray corresponding with a ray refracted along an optical axis, was 18°55' in arragonite. The external ray was divided into two, but when the critical incidence was gained, after much care in the adjustment, Professor Lloyd "at last saw the two rays spread into a continuous circle, whose diameter was apparently equal to their former interval.
"This phenomenon was exceedingly striking. It looked like a small ring of gold viewed upon a dark ground; and the sudden and almost magical change of the appearance from two luminous points to a perfect luminous ring, contributed not a little to enhance the interest.
"The emergent light, in this experiment, being too faint to be reflected from a screen, I repeated the experiment with the sun's light, and received the emergent cylinder upon a small piece of silver paper. I could detect no sensible difference in the magnitude of the circular sections at different distances from the crystal.
"When the adjustment was perfect, the light of the entire annulus was white, and of equal intensity throughout. But when there was a very slight deviation from the exact position, two opposite quadrants of the circle appeared more faint than the other two, and the two pairs were of complementary colours. The light of the circle was polarised, according to the law which I had before observed in the other case of conical refraction. In this instance, however, the law was anticipated from theory by Professor Hamilton."
In addition to these interesting results, Professor Lloyd has published an account of a new case of interference, in which the experimental exhibition of the fact is much more manageable than in the experiment of two slightly inclined mirrors given by Fresnel. Professor Lloyd causes the light reflected at an angle of 90° from the surface of a single piece of plate glass or a metallic reflector, to interfere with the direct light that passes parallel to the reflecting surface and near it. A screen placed on the other side of the mirror receives the direct and the reflected pencils, which, meeting under a small angle, after having traversed paths differing by a small amount, interfere. Professor Lloyd also received the two pencils upon an eye-piece placed at a short distance from the reflector, and saw a very beautiful system of bands, in every respect similar to one half of the system formed by the two mirrors in Fresnel's experiment.
Professor Lloyd has more recently, in 1836 and 1837, communicated to the Royal Irish Academy the results of his researches, On the propagation of light in uncrystallised media. His object was to simplify and develop that part of M. Cauchy's theory, which relates to the propagation of light in an etheral medium of uniform density, and to extend the same theory to the case of the ether enclosed in uncrystallised substances, taking into account the action of the internal molecules. In the first part of his memoir, Professor Lloyd has given good reason for concluding that the theory in its present form is insufficient to explain the phenomena of light in bodies, and that it becomes necessary to take into account the action of the material molecules. In doing this he limits himself to the comparatively simple case, in which the molecules of the ether and the body are uniformly diffused. In the expression for the velocity of propagation, each term consists of two parts, one of which is due to the action of the ether, and the other to that of the body. "It is not improbable," says Professor Lloyd, "that there may be bodies for which the first or principal term is nearly nothing, the two parts of which it is composed being of opposite signs, and nearly equal. In this case the principal part of the expression for the velocity will be that derived from the second term; and, if that term be taken as an approximate value, it will follow, that the refractive index of the substance must be in the subduplicate ratio of the length of the wave nearly. Now, it is remarkable that this law of dispersion, so unlike any
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1 The wave surface is a geometrical surface employed to determine the direction and the velocity of reflected and refracted rays. It is spherical in a singly refracting medium; a double surface or one of two sheets in a doubly refracting medium; and a surface of three sheets, on the supposition that there is a triple refraction. It has always a centre round which it is symmetrical; and the radii drawn from this centre in different directions represent the velocities of rays to which they are parallel.—Macculloch's Irish Trans., vol. xvii.
2 Phil. Trans. 1839, p. 325, or Edin. Journal of Science, N.S., No. viii. p. 259. thing observed in transparent media, agrees pretty closely with the results obtained by Sir David Brewster in some of the metals. In all these bodies the refractive index (inferred from the angle of maximum polarization) increases with the length of the wave. Its value for the red, mean and blue rays, in silver, are 3.865, 3.271, 2.824, the ratios of the second and third to the first, being .85 and .73. According to the law above given, these ratios should be .88 and .79.
We are indebted also to Professor Lloyd for an admirable history Of the progress and present state of physical optics, published in the Fourth Report of the meeting of the British Association, held in Edinburgh; a history not less characterised by its candour and truth, and absence of all national partiality, than by the profound and accurate knowledge of the subject which it everywhere displays. The only cultivator of physical optics to whom Professor Lloyd has done injustice is himself; and we are glad of the opportunity which we here enjoy, of giving a brief and imperfect account of his original and valuable researches.
It is with no less pleasure that we proceed to give an account of the optical discoveries of another Irish philosopher, who, at an early period of life, has placed himself in a distinguished position both as a mathematician and a natural philosopher. We have already seen that M. Ampère gave a direct demonstration of Fresnel's construction for finding the surface of the wave. His solution, however, was extremely difficult and complicated. Mr. James MacCullagh was led in 1829 to believe, from the simplicity and elegance of the results, that there must be some simpler method of arriving at them, and, upon considering the subject with attention, he was led to a concise demonstration of the same theorem, and of some of the other leading points of Fresnel's theory. He has demonstrated a geometrical construction for finding the magnitude and direction of the elastic force arising from a displacement in any direction, and by his construction, with the aid of a few lemmas, he is immediately led to all the conclusions established by M. Fresnel. The magnitude and direction of this force are represented by means of an ellipsoid, having for its semiaxes the three principal indices of the medium, these axes coinciding in direction with, and being inversely proportional to, the axes of Fresnel's generating ellipsoid.
The properties of the wave surface and its use in determining the directions and velocities of reflected and refracted rays, seem to have been discovered independently by Sir W. Hamilton, M. Cauchy, and Mr. MacCullagh, and in a paper entitled, Geometrical propositions applied to the wave theory of light, he has applied the properties of that surface to the geometrical development of the theory of double refraction.
Hitherto the remarkable laws of the double refraction of quartz, developed by the successive labours of Arago, Biot, Fresnel, and Airy, were merely a set of independent facts unconnected by any theory; but Mr. MacCullagh in a paper On the laws of the double refraction of quartz, sent to the Royal Irish Academy, in February 1836, has shown how they may be explained hypothetically, by introducing differential coefficients of the third order into the equations of vibratory motion.
The theory of the action of the metals upon light having been long among the desiderata of physical optics, Mr. MacCullagh thought that it would be important to represent the phenomena of elliptic polarisation, discovered by Sir David Brewster, by means of empirical formulae, in a manner analogous to that employed by Fresnel in the case of total reflexion. Mr. MacCullagh has applied his formulae to steel, and in computing from it the intensity of light reflected when common light is used, he obtained the remarkable result, that the intensity decreases very slowly up to a large angle of incidence (less that 75°), and then increases up to 90°, where there is total reflection. This result entirely accords with the remarkable fact discovered by Mr. Potter, that the intensity decreases with the angle of incidence as far as 70°. Mr. MacCullagh conceives that experiments alone can decide whether the subsequent increase indicates a real phenomenon, or arises from an error in the empirical formulæ.
Mr. MacCullagh deduces also from his formulæ the phenomenon observed by Mr. Airy in the diamond; and he has applied it successfully to the phenomena discovered by M. Arago, respecting the rings formed between a transparent and a metallic surface. In this experiment Mr. MacCullagh and Prof. Lloyd have both discovered a curious appearance unnoticed by any other author. Through the last twenty or thirty degrees of incidence, the first dark ring surrounding the central spot, which is comparatively bright, remains constantly of the same magnitude, though the other rings dilate greatly by an increase of incidence.
Hitherto the undulatory theory had been unable to give any explanation of the variation of the polarising angle, when the light was reflected in different azimuths from calcareous spar, and other doubly refracting surfaces. Mr. MacCullagh, however, was induced to exercise his mathematical skill on this interesting subject; and so early as 1834, he communicated to Professor Lloyd an expression for the angle of polarisation at the surface of crystallised media, when the plane of reflexion coincides with the principal section of Fresnel's ellipsoid; and he found that the law, which he extended by analogy to all cases, provided with much exactness the observations of Sir David Brewster. In a subsequent paper, on the laws of reflexion from crystallised surfaces, he has explained the principles upon which his formula is founded. He was obliged to adopt the view of Cauchy, that the vibrations of polarised light are parallel to its plane of polarisation, and being embarrassed by his third ray, he altered Cauchy's six equations of pressure, so as to make them afford only two rays, and give a law of refraction exactly the same as Fresnel's.
It appears, from a subsequent paper of Mr. MacCullagh's, that M. Seebeck had solved the same problem long before, namely, in the case where the plane of incidence coincides with the principal section of the crystal, and had confirmed its accuracy by experiment. M. Seebeck had also pointed out a defect in Mr. MacCullagh's formulæ, No. 2 and 3, which induced the latter to resume the subject; and in a new paper read to the Irish Academy on the 9th January 1837, a solution of the following problem is given for the first time:—"Supposing a ray of light, polarised in a given plane, to fall on a doubly refracting crystal, it is required to find the plane of polarisation of the reflected ray, and the proportion between the amplitudes of vibration in the incident, the reflected, and the two refracted rays." The hypotheses employed by our author are these, viz.
1. The density of the ether is the same in all media. 2. The vibrations are parallel to the plane of polarisation. 3. The vis viva is preserved. 4. The vibrations are preserved; that is, the resultant of the incident and reflected vibrations are the same as the resultant of the refracted vibrations. "This theory," says the author, "represents very accurately the experiments of Sir David Brewster and M. Seebeck, on the light reflected in air from a surface of Iceland spar."
We owe also to Mr. MacCullagh some interesting views respecting the nature of the light transmitted by the diamond and by gold leaf. He conceives that there is a change of phase produced by refraction, as well as by reflexion, from these bodies, the change being different according as the light is polarised in the plane of incidence, or perpendicular to it. If the incident ray, therefore, is polarised in any intermediate plane, the refracted ray should be elliptically polarised, which was found to be the case in gold leaf. He conceived that the same remark explains the appearance of double refraction in specimens of the diamond which give only a single image, and that other precious stones are likely to have similar properties. Our author has obtained a general formula for the difference of phase between the two component portions of the refracted light, one polarised in the plane of incidence, and the other perpendicular to it. He finds from this formula, that the difference of phase, which is nothing at a perpendicular incidence, increases until it becomes equal to the characteristic, at an incidence of 90°; and when the light emerges into air, the difference of phase is doubled. In submitting this formula to the test of experiment, which Mr. Macculagh has not yet done, it will be requisite to ascertain that the diamond is not composed of strata of different refractive powers.
M. Lamé, an eminent professor in the Polytechnic School, has recently endeavoured to determine the laws according to which the molecules of ether and of bodies act upon each other. He arrives at the conclusion that "the action of ponderable matter upon the ether varies in the inverse ratio of the square of the distance, and that the elasticity of the ether itself is proportional to its density." He concludes that the retardation of the vibratory motion, in penetrating into a dense body, will be greater the less the length of an undulation; so that the refraction will be greater for the shorter undulations, a result which he regards as the true explanation of dispersion.
"M. Lamé," says Professor Lloyd, "has endeavoured to connect the phenomena of double refraction with an assumed constitution of the etheral fluid. He takes the case in which the ether is supposed to be distributed round the molecules of the body in confocal ellipsoidal shells; and he concludes that a vibratory movement, propagated from a vacuum into a body so constituted, will be separated, at its entrance, into two component movements, which will advance with different velocities.
"The two component vibrations, he finds, will be at right angles, and parallel to the lines of greatest and least curvature of the elementary ellipsoids. Thus, the bifurcation of a ray of light on entering a crystallised medium, and the opposite polarisation of the two pencils, are found to be consistent with a molecular constitution such as that described.
"These results are of the highest interest, and will, no doubt, receive an early examination from those engaged in the same department of analysis. Their author seems to be persuaded that his methods will lead him to the mathematical laws of other phenomena, which he conceives to depend, in like manner, on the motions of the etheral fluid."
In a memoir more recently communicated to the Academy of Sciences, M. Lamé has sought to determine the mode of vibration of the etheral particles which surround the material molecules in concentric spherical shells of decreasing density. He considers transparent homogeneous bodies as consisting of a multitude of such molecules, distributed uniformly in space, and at distances incomparably greater than their diameter; and he conceives that the waves propagated from the molecules adjoining to the surface of emergence, will produce by interference phenomena resembling the fixed lines in the spectrum. This opinion, however, is in our opinion, incompatible with the actual phenomena of fixed lines.
The narrowness of our limits will not permit us to record in this historical sketch many other discoveries which have been recently given to the world, though many of them will be detailed under the heads to which they respectively belong. These discoveries have been made principally by Mr. H. F. Talbot, Mr. Potter, Mr. Dove, and other philosophers.
Having thus given a condensed sketch of the history of optical discovery, from the earliest to the present times, we shall now proceed to the proper subject of this article. As the nature of this work requires that the subject be treated in a very popular manner, we shall pass briefly over those branches of Optics which are generally treated mathematically, and which occupy a prominent part in all ordinary treatises, and occupy our limited space with the more interesting departments of Chromatics, Physical Optics, the Double Refraction and Polarisation of Light, the Explanation of Natural Phenomena, the Laws of Vision, and the Construction of Optical Instruments.
INTRODUCTION.
The ancients confounded the phenomena of vision with those of light, by supposing that when we see external objects something passes from the eye to the object. The phenomena of light, however, are totally independent of those of vision, and have a real existence in nature, whether we suppose them to be objects of vision or not.
1. Light is the element by means of which we see external bodies. These bodies may be divided, in reference to light, into two classes, self-luminous and non-luminous, or dark bodies. The first class include the sun, the stars, flames of all kinds, and bodies which become luminous by friction, heat, and electrical and magnetical action. Such bodies become visible by the light which they themselves emit, and we then obtain a knowledge of their apparent form. The sun, for example, is seen to be round, and the flame of a candle to be of a conical shape. The second class of bodies, however, are never visible but when placed in the light of self-luminous bodies. It includes the moon and all the primary and secondary planets, of which we see only those portions upon which the sun's light directly falls, and all the other objects upon our own globe. When we bring a lighted candle into a room its light falls upon all the objects in the apartment, and they become visible. These bodies reflect or throw back the light of the candle, and they scatter it in all directions, because they are, generally speaking, visible, wherever we place our eye. But objects also become visible by the light thrown off by non-luminous. When the moon has the form of a sharp crescent, we see the obscure part of its circular disc by the light thrown upon it from the earth, which is at that time almost fully illuminated by the sun. In like manner in the room lighted with a candle, objects are seen in corners and places upon which the light of the candle does not fall. These objects, however, are illuminated by the light of the candle thrown back by the white ceiling and walls of the apartment; and hence the reason why the ceilings and apartments should always be white, and why the walls should be white if we wish to obtain the greatest quantity of light from a given flame.
2. The light thrown off from all bodies whether self-luminous or non-luminous, is of the same colour as themselves. A red hot body, or a stick of red sealing wax, will make a sheet of white paper appear red if held near them.
3. But though coloured bodies throw off light of the same colour with themselves, bodies do not appear of the same colour as that of the light which falls upon them. All bodies which are white in white light, appear of the same colour as that of the light which falls upon them; but other bodies, such as red wax, appear red even in white light, a property which they derive from a peculiar structure acting upon the different colours of which white light is composed. Bodies of this kind when illuminated with lights of different colours, always appear brightest in light of the same colour which they exhibit in white light. Thus a stick of yellow wax is more luminous than a stick of red wax, but the yellow wax will be less luminous than the red wax if we illuminate them both with red light.
4. Bodies in their relation to light, are divided into two classes, opaque and transparent. An opaque body is one that stops the light that falls upon it, such as a piece of coal, or a plate of silver; and a transparent body is one which transmits the light through it, such as glass, water, and air. The most opaque body, however, may be made transparent by making it sufficiently thin, and the most transparent one may become opaque by making it sufficiently thick.
5. The opacity of bodies, or their power of intercepting light, gives rise to what is called the shadows of bodies. As the shadows of bodies are of the same size as the bodies, we thence deduce the fundamental optical fact, that light moves in straight lines. The same fact may be proved in a thousand ways, but most simply by placing three small holes in a straight line. In this case the light will pass through them, but if any one of them deviates from the straight line, the light will be stopped. The same thing is finely seen without any experiment, by admitting light into a dark room through an aperture of an inch wide. Its path, marked out by the floating dust which it illuminates, will be seen to be a straight line.
6. Light issues or radiates in every direction and from every point in the surface of luminous and visible bodies. This fact is proved by the circumstance, that we see such bodies wherever we place our eye. However much we may magnify the bright part of the sun's disc through a telescope, or a sheet of white paper through a microscope, we shall never see any points destitute of light.
7. Light consists of separate and independent parts, which, when reduced to the smallest magnitude, are called rays of light. A beam of light transmitted into a dark room may be actually divided into smaller portions in a variety of ways. The smallest portion that we can allow to pass may be called a ray of light, and possesses the same properties as the larger beam.
8. Light moves at the rate of 192,000 miles in a second. This extraordinary property of light has been deduced by direct calculation from the immersions and emersions, in eclipses of Jupiter's satellites, which become visible to us nearly a quarter of an hour earlier when the earth is nearest Jupiter, than when it is farthest from that planet. The exact velocity of light obtained in this manner, is 192,500 miles in a second; whereas, Dr. Brinkley and M. Struve have found it to be 191,515 miles in a second, from the phenomena of aberration. This last determination is undoubtedly the most correct. The mean, however, of 192,000 is most easily remembered.
The velocity with which light travels is so inconceivable, that we require to make it intelligible by some illustrations. It moves from the sun to the earth in 7½ minutes, whereas a cannon ball fired from the earth, would require seventeen years to reach the sun.
Light moves through a space equal to the circumference of the earth, or about 25,000 miles in about the 8th part of a second. The swiftest bird would require three weeks to perform this journey.
Light would demonstrably require five years to move from the nearest fixed star to the earth, and probably many thousand years from the most remote star seen by the telescope. Hence if a remote visible star had been created at the time of the creation of man, it may not yet have become visible to our system.
9. When light falls upon any body, whether rough or smooth, coloured or uncoloured, a part of the incident light enters the body, and is either lost within it, or transmitted through it; and part of it is reflected from its surface, either in the same or in a different direction from that in which it came. The light which enters the body and is lost, the light which is transmitted through the body, and the light which is reflected from it, suffer certain changes in its direction, and in its physical properties. It belongs to the geometrical or mathematical part of optics, to assign the laws which regulate the change of direction which light experiences when it is transmitted through, or reflected from bodies, whose density is uniform, and whose surfaces have a geometrical form; and to physical optics, to explain the changes in the physical properties which light acquires in passing through bodies, in passing near them, or in being reflected from their surfaces.
The laws or rules which regulate the reflexion of light, constitute that branch of optics which is called catoptries, and the laws which regulate the changes of deviation which light experiences when transmitted through bodies, is called dioptries.
PART I. CATOPTRIES, OR THE REFLEXION OF LIGHT.
The word catoptries, derived from the Greek words κατά, from, and ὄπτον, to see, signifies that department of optics which treats of the reflexion of light from the polished and regularly formed surfaces of bodies, such as water, glass, and the metals.
The name of speculum or mirror, has commonly been given to bodies that have regularly formed and highly polished surfaces. The word speculum is generally applied to polished metals, and mirrors to reflectors made of glass and covered with an amalgam of tin and mercury, to increase their power of reflecting light.
There are four kinds of specula used in optics, namely, plane, convex, concave, and cylindrical, and when light falls upon any of these specula, which we shall always consider to be formed of polished metal, it is reflected according to the same law.
General Law of Reflexion.
Let AD be a ray of light which falls upon a plane speculum MN, and strikes it at the point D, this ray will be driven back in the direction DB, so inclined to the original ray AD that if we raise from the point D a line DE perpendicular to MN, the angle BDE will be equal to the angle ADE. The ray AD is called the incident ray, DB the reflected ray, ADE the angle of incidence, and BDE the angle of reflexion. The two rays AD, DB, and the perpendicular DE, all lie in the same plane AEBD. This plane is sometimes called the plane of incidence, and sometimes the plane of reflexion, and it is always at right angles to the reflecting surface MN.
When the reflecting surface is concave as MN in fig. 2, and is part of a sphere, whose centre is C, a ray of light AD falling upon any point D, will be reflected in a direction DB, so as to form the same angle BDC with a line CD drawn from the cen-
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1 See Aberration, vol. i. p. 31. Reflexion of diverging rays. When diverging rays, Catoptrics AD, A'D', A''D''; fall upon a speculum MN, they will be reflected in directions DB, D'B', D''B'' found by making the angles BDE, B'D'E', B''D''E'' respectively equal to ADE, A'D'E', A''D''E'', and the reflected rays being continued back till they meet, they will be found to meet at a point A' so that the line AA' is at right angles to MN, and AN equal to A'N. Hence the rays will have the same divergency after reflexion, as before it, and as if they came from A'; the reflected beam being inverted, as in the preceding case.
Reflexion of converging rays. When converging rays AD, A'D', A''D'', fall upon a speculum MN, they will converge after reflexion to a point B', so situated, that if BB' is at right angles to MN, B'M will be equal to BM. The reflected rays DB, D'B', D''B'', will be found by making the angles EDB', E'D'B', E''D''B'', respectively, equal to the angles of incidence, ADE, A'D'E', A''D''E''.
On the Reflexion of Rays from Concave Mirrors.
Reflexion of parallel rays. Let MN be a concave mirror whose centre of concavity is C, and let parallel rays AD, AD', A''D'', fall upon the mirror, the central ray AD passing through the centre C. From C draw the lines CD', CD''. Then since CD' is perpendicular to the mirror at D', the ray AD' will be reflected in the same direction DF', so that the angle of reflexion CD'F' is equal to the angle of incidence AD'F'. In like manner, the ray AD will be reflected to E, and the central ray AD will be reflected back to F also; all intermediate rays being likewise reflected to F.
If the curvature of MN is not deep, and if the points D', D'' are taken near D, it will be found by making the angles of reflexion equal to the angles of incidence, that the rays all meet accurately at F, which is called the focus of the mirror for parallel rays, or its principal focus. This focus is in all mirrors exactly half-way between the centre C, and the surface of the mirror.
The point F derives its name of focus from its being the burning point of a mirror, or the point where the parallel rays, issuing from the sun are most condensed, and therefore occasion the most powerful heat.
Reflexion of diverging rays. Let AD, AD', AD'', be diverging rays issuing from A and falling upon the mirror MN, whose centre is C, and principal focus O. Then if we make the angles of reflexion equal to the angles of incidence, as in the last case, we shall find that the rays will be reflected to a point F, between the centre C of the mirror, and its principal focus O. If the radiant point A is removed from the mirror, and the rays fall on the same and consequently the reflected rays which meet at F, will also be removed farther from them. Hence F will approach to O, and when A is infinitely distant, and the rays parallel, F will coincide with O. But if A approaches to the mirror, the incident rays will approach to the perpendiculars, and as the reflected rays will do the same, their point of concourse F will approach to C. When A reaches C, the focus F will also reach C, and the reflected ray will coincide with the incident ray.
If A still advances towards the mirror, the incident rays will get within the perpendicular, and therefore the reflected rays will be without it, and their point of concourse F will advance from C outwards, in proportion as the radiant point advances from C inwards. When A reaches the principal focus O, the reflected rays will be parallel, as seen in fig. 7; and when A comes still nearer the mirror, the reflected rays will diverge as if they proceeded from some point behind the mirror, this point being called the virtual or imaginary focus of such rays.
In all the preceding cases, the points A, from which the rays issue, and the point F where they are collected by reflection, are called conjugate foci, because if we make A the radiant point, F will be the focus, if we make F the radiant point, A will be the focus.
The conjugate foci of a concave mirror may be easily found by projection. The following rule will give the focus more accurately when the rays AD', AD'' are not far from the central one AD. Multiply the distance of the radiant point from the mirror, or AD, by the radius CD, and divide this product by the difference between double the distance AD, and the radius CD, and the quotient will be the conjugate focal distance required, or FD. If twice AD is less than CD, the conjugate focal point will not be before the mirror, but behind it, the focus being in that case a virtual one.
Reflexion of converging rays. Let AD, AD' AD'' be rays converging to a point a behind the mirror MN, whose centre is C. Having drawn CD' and CD'', make the angles of reflection CD'/f, CD''/f respectively equal to the angles of incidence AD'C, AD''C; and D'/f, D''f will be the reflected rays having their focus at f, between the mirror and its principal focus F. If the point of convergence a of the rays, or the conjugate focus, approaches to the mirror, the other conjugate focus f will also approach to it, and if it recedes from the mirror, the focus f will also recede, reaching F when a is infinitely distant, in which case AD', AD'' are parallel, as in Fig. 7. The following is the rule for finding the conjugate foci when one of them is given:
Multiply the distance of the point of convergence from the mirror, or aD by the radius of the mirror or CD, and divide this product by the sum of double the distance aD and the radius CD, and the quotient will be the conjugate focal distance required, namely, fD, the focus f being in front of the mirror.
On the Reflexion of Rays from Convex Mirrors.
Reflexion of parallel rays. Let parallel rays AD, AD', AD'' be incident upon the convex mirror MN, whose centre is C. Draw the perpendicular CE, CE passing through D' and D'', and making the angles of reflection EDB, ED'B equal to the angles of incidence, AD'E, AD''E, the reflected rays will be D'B, D''B, whose virtual focus F is behind the mirror, and so situated that FD is equal to FC.
Reflexion of diverging rays. If we suppose the rays AA'A to diverge from any point in the line or axis AD, they will recede from CE, CE, consequently the reflected rays D'B, D''B will recede also; that is, will become more divergent, as if they came from a focus between F and D, the virtual focus approaching to D as the radiant point A approaches to D.
Reflexion of converging rays. In like manner, if the rays AAA widen at A, that is, converge to some point behind the mirror MN, they will approach to CE and CE, so that the reflected rays D'B, D''B will also approach to CE, and consequently diverge less, or have their virtual focus between F and C. When the converging rays coincide with CE, CE, they will be reflected back in the direction in which they came, having C for their virtual focus. When the converging rays pass CE, the reflected rays will also pass to the opposite side, and converge less after reflexion, having their virtual focus beyond C. When they converge to F, the reflected rays will be parallel as in fig. 10, where we may suppose BD' BD'' the incident, and D'A, D''A the reflected rays. When the rays converge to a point between F and D, the reflected rays will converge to a point in the axis, and as the point of convergency of the incident rays approaches to it on the one side, the point of convergency of the reflected rays will approach to it on the other.
It would have been easy by the simplest elements of geometry, to have demonstrated the preceding truths; but the demonstration would have been rigorous only when the rays fell upon the mirror at points infinitely near D in the axis AD. By finding from projection the foci of rays of all kinds, and falling upon the mirror at all degrees of obliquity, the reader will acquire more substantial knowledge of the subject, than he can do either from geometrical or algebraic demonstrations. The same observation is applicable to the results obtained in the following section.
Sect. II.—On the Formation of Images by Plane, Concave, and Convex Mirrors.
Formation of Images by Apertures.—In optics an image is a luminous resemblance or picture of any object whatever, formed either on a white ground, such as a sheet of paper, or suspended in the air.
In order to understand how images are formed, let us suppose that a soldier is standing on the outside of an open window, with a red coat and blue trousers, strongly illuminated by the sun. The white wall opposite the window is illuminated by all the light which enters the window, the blue light of the sky, the green foliage, the red coat, and the blue trousers, so that it has no distinct colour, but a mixture of all these. If we close the shutters, so as to allow no Let AB, fig. 12, be a concave mirror, C its centre, and Catoptries MN an object placed before it. Of all the rays which flow from every part of this object in every direction, we shall consider only those which issue from its extremities MN. The rays from M radiate in every direction, but those which fall upon the mirror, namely, the pencil or cone MAB, are the only ones which require our notice. This pencil of diverging rays will have its focus at a point m farther from the mirror than its principal focus, and in like manner the pencil NAB will have its focus at some point n, pencils intermediate between M and N having their foci at points intermediate between m and n. These points may be found by projection, as already described, or by the rule given for diverging rays.
The image mn is obviously an inverted picture of the object MN, and its size is to that of the object as the distance of the image from the mirror is to the distance of the object from it, that is, as mA is to NA, as may be found from projection, or from an experimental measurement of the distance, when a mirror is actually used.
From the doctrine of reflected diverging rays it follows, and may be proved by projection, that as the object MN approaches to the mirror, the image mn will recede from the mirror, till the object and image meet one another at the centre C, where they will have the same size. If MN still moves towards the mirror within C, the image mn will move outwards beyond C, and the image will now be larger than the object. If the object comes to the place mn, and is of the same size as mn, the image of it will be formed at MN, and will have the same size as MN. If the object goes still nearer the mirror the image will go still farther off than MN, increasing in size. When the object reaches the principal focus half way between C and D, the image will be infinitely distant; and when the object goes still nearer the mirror, as in fig. 13, where it is placed at MN, between the principal focus.
Formation of Images by Concave Mirrors.—The effect of a concave mirror in forming an image is the same as that of an aperture; but it produces a finer effect, and acts upon a different principle.
F and the mirror AB, the rays will diverge in front of the mirror, and form an inverted virtual image, mn, behind the mirror. As the image MN approaches the mirror, the virtual image mn also approaches to it.
If we take a concave mirror of some size, and place before it any highly luminous or strongly illuminated object, such as a plaster of Paris cast, we may obtain an interesting experimental proof of the preceding results. When the image is formed in front of the mirror, it will appear suspended in the air, and the effect of this will be greatly heightened if it is received on a cloud of thin blue smoke raised from a chafing dish below the place of the image. By considering that as the object moves from MN to C, fig. 12, the image mn advances to C, we obtain an explanation of the celebrated experiment with the dagger mentioned by John Dee, in our history of the science, in which a person with a drawn dagger striking at the mirror is met by another person, viz., his own image, returning the stroke. If the object MN is the sun, a small image of his disc will be formed at mn, in which are collected all the rays of light which fall upon the surface of the mirror. It will therefore have such a degree of heat as to melt even the hardest gems and metals. Such a mirror is called a burning mirror, from its effects.
**Formation of Images by Convex Mirrors.**—As convex mirrors often form a part of household furniture, we are more familiar with their properties. They always form erect images of objects, which appear at a distance behind them.
If AB is a convex mirror whose centre is C, and principal focus F, and MN an object placed before it, it is obvious, from our description of fig. 10, that the diverging pencils MAB, NBA, will diverge more after reflexion, as if they came from virtual foci mn behind the mirror, so that our eye receiving such diverging rays will see an erect image mn of the object MN placed behind the mirror, and between its principal focus F and D. If MN approaches to AB, mn will approach it also, and if MN recedes from the mirror, mn will also recede from it, their relative sizes varying as their distances. When the object touches the mirror, the image also touches it; and they are then exactly of the same size.
**Formation of Images by Plane Mirrors.**—Every person is familiar with the effects of a plane mirror, or looking-glass. The image of any object placed before it is seen behind it of the same size, in the same position, and at the same distance from the mirror. In order to understand this, let AB be the mirror, MN the object, and E the eye of the observer, situated in any given position. Rays from M and N fall upon every part of the mirror, but MC, MD are the only ones from M which can reach the eye E, as all the rest are reflected either above or below the eye E. In like manner, the rays NF, NG are the only ones from N which can enter the eye. The extremity M of the object will therefore be seen in the direction Em, and the concourse or virtual focus of the reflected rays will, as shown in fig. 6, be at a point m, so situated, that if MAm is at right angles to the mirror, Am will be equal to AM. For the same reason, the point N will be seen at n, as far behind the mirror as N is before it; and it is obvious, from the parallelism of Mn and Nn, and the equality of the distance of M, m, and N, n from AB, that mn is equal to MN.
If two plane mirrors are inclined to each other, as AC, B, fig. 16, and an object MN placed between them, an eye situated so as to receive the reflected rays, will see a series of images of MN all arranged symmetrically. Behind AC, for example, an image mn will be formed, and behind BC another image M'N'. But as the rays which form these images again fall upon the mirrors, we shall see images of mn and M'N' formed by AC and BC, thus m'n' will be the image of mn formed by BC, and M''N'' an image of M'N', formed by AC. In like manner, m''n'' will be the image of m'n' formed by AC, and the image of M''N'' formed by BC will also lie upon m''n'', so that we shall have two images at m''n'' overlapping each other, and forming one exactly, if the angle ACB is exactly one-sixth part of 360°, or 60°; but if it is not, the compound image m''n'' will be seen double and imperfect. The five images above described, reckoning the double one at m''n'' as only one, will, together with the object MN, to which all the images are equal and similar, constitute a perfect equilateral triangle, so that if MN is a coloured and an irregular object, the symmetrical figure composed by it, and all its images, will be highly beautiful and agreeable to the eye. If MN, in place of being perpendicular to the mirror BC, had been inclined to it, no pair of images would have formed a straight line, as in the figure, and the combination would have been more beautiful. This is the principle of the Kaleidoscope, in as far as the multiplication and arrangement of the images is concerned; but this instrument has already been so admirably described by an eminent writer, in our article KALEIDOSCOPE, that we must refer the reader to it for further information.
**Formation of Images by Cylindrical Mirrors.**—It is not easy, in a diagram, to represent the progress of rays in the formation of an image by a cylindrical mirror. As a cylinder is in one direction a plane mirror, in another a convex mirror, and in all others an elliptical one, the eccentricity of the ellipse passing through all degrees, from a circle to a straight line, different parts of a regular figure presented to such a mirror will appear of different sizes, and at different distances behind it. Part of the figure will have the same form and position as in a plane mirror, part as in a convex mirror, and the other parts of the image will have intermediate sizes and positions. Hence the image will be completely distorted. If the mirror is placed horizontally, the human face will appear of the right size from ear to ear, but contracted, as in a convex speculum, from brow to chin. Hence, if a distorted picture is properly drawn, and properly presented to the mirror, that is, if the cylinder is placed vertically before the picture, the image of the distorted picture will be rectified; the length between the ears will be contracted into the same proportional size as the shortness between the brow and the chin, and their shortness will remain unaffected. Such a distorted picture will afterwards represented in the part of this article on Optical Instruments.
**PART II. DIOPTRICS, OR THE REFRACTION OF LIGHT.**
**Dioptrics,** from διόπτρα, through, and ὁράω, to see, is that branch of optics which treats of the passage of light through transparent bodies, and, consequently, of the changes which it experiences in entering and quitting such bodies.
**Sect. I.—On the Refraction of Light.**
If we hold a drop of pure water or an irregular piece of clear glass in the sun’s rays, each will have a sort of shadow like opaque bodies. Hence it follows, that light has not passed freely through them, and must, therefore, have suffered some change in its direction, either while entering these bodies, or passing through them, or emerging from them. The change which it has suffered is called refraction, and the nature of this change will be discovered by observing the For this purpose let AB, fig. 17, be the surface of water in a vessel, and RC a ray or pencil of light proceeding from a candle or from the sun, through a small hole, and falling upon the water at C. Part of this light will be reflected in the direction CR, so that the angle rCP, is equal to RCP, PQ being a line perpendicular to the water at C; but the greater part of the light will enter the water at C, and in place of going straight on to e, it will be bent or refracted at C, or the ray R will be broken back at C, and proceed in a straight line to E. Drawing a circle PAQB round C as a centre, and from the point E, where the refracted ray cuts it, drawing EK parallel to PQ, it was found by Snellius that CD was to CE or Ce as 3 to 4, and we have shewn in the history of optics, that if RF and EF are drawn perpendicular to PQ, CD is to Ce as EF is to RF. But RF is the sine of the angle of incidence RCP, and EF is the sine of the angle ECQ, which is called the angle of refraction. Now, Snellius discovered by numerous experiments, that whatever was the magnitude of the angle of incidence RCP, the magnitude of the angle of refraction was such that CD was to Ce as 3 to 4, or in a constant ratio. Hence it follows, that the sines of the angles of incidence and refraction RF and EF, are in the case of water in the constant ratio of 4 to 3.
Snellius in his manuscripts did not mention the constant ratio of the sines, but merely the constant ratio of CD and Ce, which is the same. The ratio of the sines was first mentioned by Descartes, but there can be no doubt that Snellius knew it perfectly, and that he preferred the use of the ratio of CD to Ce, for the following reason: When a luminous body is placed at E below water, and its light passes through a small aperture at C, it is found to be refracted or bent into the direction CR, so as to be seen by an eye at R, in the direction Re. It will, therefore, be seen, as it were, elevated from E to D. Hence Snellius preferred giving the law of this elevation, to any other law not connected with the phenomena.
As the incident ray RC approaches to the perpendicular PQ, the refracted ray CE approaches also to the perpendicular, and CD becomes less and less, and when RC coincides with PC, or when the ray is incident perpendicularly, the refracted ray CE will coincide with CQ, or the incident ray will suffer no refraction at C. When the angle of incidence RCP increases, and RC approaches to the surface of the water CB, the angle of refraction ECQ will also increase, the line CD will increase, and the refracted ray approach also to the surface CA, and when RC coincides with BC, Ce will coincide with CA, and no light whatever will enter the water but it will all be reflected. When Ce coincides with CA, CD will be 3, and D will coincide with K.
Such are the phenomena and law of refraction when light passes from a rare medium such as air, into a dense medium such as water, the ray being always refracted from the perpendicular, according to the fixed law already described. Let us now suppose, that the ray of light passes from a dense medium, such as water placed above AB, into a rare medium, such as air placed below AB, and let PQ be a perpendicular to the surface of the water at C. It is found by experiment that the ray neither goes straight on to e, nor is refracted towards the perpendicular as before, but is refracted from the perpendicular into the direction CE, so that if the line KED is drawn through E, parallel to PQ, and cutting the original direction of the ray Re prolonged, in the point D, CD will be to CE or Ce as 4 to 3, and in a constant ratio, or RF the sine of the angle of incidence will be to EF the sine of the angle of refraction in the constant ratio of 4 to 3. When the ray RC coincides with PC, so that the angle of incidence is nothing, the angle of refraction will also be nothing, and the refracted ray CE will coincide with CQ, the incident ray having gone straight on without experiencing any refraction; but when the angle of incidence increases, and RC approaches towards BC, the refracted ray CE will approach to CA, which it will reach long before R reaches B. When CE reaches CA, the ray RC will no longer emerge from the water into the air, but will suffer what is called total reflexion at C, and at every angle of incidence beyond that at which this total reflexion commences, the light will continue to be totally reflected till RC coincides with RB.
If we repeat all the above experiments with plate or crown glass, in place of water, we shall find the very same phenomena reproduced, with this difference only, that the constant ratios of CD to Ce, or of the sines RF to EF, in place of being as 3 to 4 in one case, (see fig. 17,) and 4 to 3 in the other (see fig. 18,) will be as 2 to 3, and as 3 to 2, or in the case of water the ratio will be as 1 to 1-333, and in glass as 1 to 1-500. The number 1-333 is called the index of refraction for water, and 1-500 the index of refraction for glass. In like manner it is found that the index of refraction for tabasheer is 1-111, being less than that for water; for flint glass 1-600, for diamond 2-500, and for chromate of lead about 3-00. Hence it follows, that bodies refract light in different degrees, measured by their indices of refraction. In order to have an ocular representation of the different degrees of refraction, we have drawn in fig. 19, the different refracted rays, corresponding to a given incident ray RC, supposing the surface AB to be first air, (the medium above it being a vacuum), then tabasheer, then flint glass, then diamond, and, lastly, chromate of lead.
When the index of refraction of any body is known, we can easily ascertain the progress of a ray of light which falls upon such a body, and its direction after quitting the body. The following example of this we shall give for plate glass.
Let AB be the surface of a piece of plate glass whose index or ratio of refraction is as 2 to 3, or as 1 to 1-50, and let a ray of light RC fall upon it at C. Prolong RC to e, and upon a scale of equal parts take in the compasses CD, equal to 10 of these parts, and Ce equal to 15, or CD equal to 2, and Ce to 3 parts. Upon C as a centre with the radius Ce describe the semicircle AeB, and through D draw KDE, perpendicular to AB, and meeting the semicircle in E, join CE, and CE will be the refracted ray. When the ray passes from a denser to a rarer medium, as in fig. 18, Ce is made equal to 10, and CD to 15, and DK being drawn perpendicular to AB, and a line Dioptries drawn from C to the point E, where DK cuts the circle, CE will be the refracted ray. This method, which is obviously much more simple and elegant than when we use the sines of the angles, is, so far as we know, new, and we shall use it in future for determining the place of the refracted ray. It is obvious, that when D and K coincide with A, fig. 18, DK becomes a tangent to the circle at A, and the light suffers total reflexion.
If the preceding experiments are repeated with various solids and fluids, it will be found, that the same law of refraction takes place with all of them, the index of refraction varying more or less in each, the refractive power being least in the gases, and less in fluids, generally speaking, than in solids, as will be seen in the following table of refractive powers, collected from various authors, and determined by methods possessing various degrees of accuracy.
### Table of the Refractive Powers of Gases, Fluids, Solids, and Metallic Bodies.
| Index of Refraction | Index of Refraction | |---------------------|---------------------| | A vacuum | 1·0000 | | Hydrogen | 1·000198 | | Oxygen | 1·000272 | | Atmospheric air | 1·000294 | | Azote | 1·000500 | | Nitrous gas | 1·000362 | | Carbonic oxide | 1·000340 | | Ammonia | 1·000385 | | Carburetted hydrogen| 1·000443 | | Carbonic acid | 1·000449 | | Nitric acid | 1·000451 | | Nitrous oxide | 1·000503 | | Sulphuretted hydrogen| 1·000644 | | Sulphurous acid | 1·000665 | | Olefiant gas | 1·000678 | | Chlorine | 1·000779 | | Protosulphuretted hydrogen| 1·000789 | | Cyanogen | 1·000834 | | Muriatic ether | 1·001095 | | Phosgene | 1·001150 | | Vapour of sulphuret of carbon | 1·001500 | | Vapour of sulphuric ether, (boiling point at 35° cent.) | 1·001530 |
All the preceding observations were made by M. Dubong, excepting that on atmospheric air, which we owe to M. Biot.
### FLUIDS AND SOFT SOLIDS.
| Index of Refraction | Index of Refraction | |---------------------|---------------------| | Ether expanded by heat to thrice its volume | 1·0570 Br. | | Volatile new fluid discovered by Sir D. Brewster in cavities in topaz | 1·1311 Br. | | Volatile new fluid discovered by Sir D. Brewster in amethyst, at 84° Fahr., 1·2106 Br. | | Saturated solution of salt | 1·375 Br. | | Muriatic acid | 1·392 He. | | Alcohol | 1·37 W. | | Oil of wine | 1·379 Br. | | Sweet spirit of nitre | 1·384 He. | | Cornea of a lamb | 1·386 | | Malic acid | 1·395 Br. | | Pus | 1·404 Br. Y. | | Nitrous oxide, liquefied by pressure | 1·4098 Bl. | | Nitric acid | 1·406 Br. | | Crystalline lens of man, outer coat | 1·4767 Br. | | middle coat | 1·4786 Br. | | centre | 1·4790 Br. | | lamb's eye, outer coat | 1·486 Br. Y. | | middle coat | 1·489 Br. Y. | | centre | 1·490 Br. Y. | | pigeon | 1·496 Br. Y. | | haddock's eyes, outer coat | 1·410 Br. Y. | | centre | 1·439 Br. Y. | | Juice of orange peel | 1·403 Br. Y. |
1 Taken when the temperature was 32° Fahr., and the barometer at 29·222. 2 The fixed line E is given in this Table for several substances, as it is in the green space, and nearly the mean ray. | Index of Refraction | Oil of beech | 1-471 Br. | |---------------------|--------------|-----------| | Oil of juniper | 1-478 Br. | | | Butter, cold | 1-474 Br. Y. | | | Palm oil | 1-480 W. | | | Oil of rape seed | 1-475 Br. Y. | | | Naphtha | 1-475 Br. Y. | | | Essence of lemon | 1-476 W. | | | Oil of dill seed | 1-477 Br. | | | Oil of thyme | 1-487 Br. Y. | | | Oil of cajeput | 1-486 Br. Y. | | | Niples soap | 1-483 Br. | | | Oil of mace, melted | 1-479 Br. Y. | | | Oil of spearmint | 1-481 Br. Y. | | | Oil of lemons | 1-481 Br. | | | Oil of pennyroyal | 1-482 Br. | | | Linseed oil, sp. gr. 0-932 | 1-482 N. | | | Oil of savine | 1-485 W. | | | Oil of juniper | 1-482 Br. | | | Train oil | 1-491 Br. | | | Oil of wormwood | 1-485 Br. Y. | | | Castor oil | 1-489 Br. | | | Florence oil | 1-490 Br. | | | Oil of fenugreek | 1-487 Br. | | | Oil of hyssop | 1-487 Br. | | | Windsor soap | 1-487 Br. Y. | | | Nut oil, perhaps impure | 1-491 He. | | | Tallow, cold | 1-492 Br. Y. | | | Oil of caraway seeds| 1-483 Br. Y. | | | Oil of marjoram | 1-491 Br. | | | Oil of nutmeg | 1-491 Br. Y. | | | Oil of angelica | 1-491 Br. Y. | | | Beef wax | 1-492 Br. Y. | | | Cold | 1-492 Br. Y. | | | White wax, cold | 1-492 Br. Y. | | | Balsam of sulphur | 1-494 Br. | | | Honey | 1-495 Br. Y. | | | Grass oil | 1-496 Br. | | | Treacle | 1-500 W. | | | Oil of beech nut | 1-500 Br. | | | Oil of rhodium | 1-505 Br. | | | Spermaceti, cold | 1-503 Br. Y. | | | Oil of pimento | 1-510 Br. Y. | | | Oil of amber | 1-507 Br. | | | 3rd lime | 1-507 Br. Y. | | | Oil of sweet fennel seeds | 1-506 Br. | | | Balsam of copaiba | 1-515 Br. Y. | | | Canada balsam | 1-532 Br. Y. | |
| Index of Refraction | Oil of cinnamon | 1-508 Br. | |---------------------|-----------------|----------| | Oil of mace | 1-519 Br. Y. | | | Oil of sassafras | 1-532 Br. | | | Oil of cloves | 1-535 Br. Y. | | | Oil of cashew nut | 1-536 Br. Y. | | | Oil of anise seed | 1-536 Br. Y. | | | Petroleum | 1-544 Br. | | | Oil of tobacco | 1-544 Br. Y. | | | Balsam of styrax | 1-544 Br. | | | Oil of cinnamon | 1-589 Br. Y. | | | Balsam of Peru, mean | 1-600 Br. | | | Essential oil of bitter almonds | 1-603 Br. | | | Oil of cassia | 1-611 Br. Y. | | | Sulphuret of carbon | 1-678 Br. | | | Muriate of antimony, variable, about | 1-8 W. | | | Tabasbeer from Vellore | 1-1111 Br. | | | Nagpore | 1-1454 Br. | | | ditto | 1-1503 Br. | | | whitest kind | 1-1825 Br. | | | Ice | 1-3085 W. | | | Cryolite | 1-344 Br. | | | Carbonate of potash, lowest refr. | 1-379 Br. | | | Gluten of wheat, dried | 1-426 Br. Y. | | | Fluor spar | 1-433 W. | | | Alum | 1-436 Br. | | | sp. gr. 1-714 | 1-457 W. | | | Sulphate of magnesia | 1-463 Br. | | | Borax, sp.gr. 1-714 | 1-475 N. C. | | | Fluellite | 1-47 W. | | | Gum Arabic, sp. gr. 1-375 | 1-476 N. | | | Geminite | 1-474 Br. | | | Opal, partly hydrophanous | 1-479 Br. | | | Arseniate of soda, least | 1-481 Ha. | | | Sulphate of ammonia and magnesia | 1-483 | | | Camphor | 1-487 Br. | | | sp. gr. 1-0996 | 1-500 C. N. | | | Obsidian | 1-488 Br. | | | Iceland spar, ext. | 1-488 W. | | | ditto | 1-483 M. | | | fixed line E. do. | 1-4887 Ra. | | | ord. | 1-6357 W. | | | ord. | 1-6543 M. | | | fixed line E, ord. | 1-6636 Ra. | | | ord. | 1-667 N. | | | ord. | 1-665 Br. | | | Sulphate of iron, greatest | 1-491 Br. | | | Sulphate of potash | 1-475 W. | | | Rochelle salt, mean, green | 1-4985 He. | | | red | 1-4929 He. | | | tartrate of potash and soda | 1-515 Br. | |
| Index of Refraction | Yolk of an egg, dry | 1-500 Br. Y. | |---------------------|---------------------|--------------| | Triple oxalate of chromium and potash, least | 1-506 Br. | | | greatest | 1-605 Br. | | | Glass plate, English | 1-500 Br. | | | French | 1-50 W. | | | English, ext. | 1-5183 He. | | | red ray | 1-514 Bose. | | | Dutch | 1-517 W. | | | crown | 1-525 W. | | | crown, prism Dollond, ext. red ray | 1-526 He. | | | crown, prism Dollond, 1-5109 Br. | | | crown, Fraunhofer, No. 13, sp. gr. 2-355, fixed line E. | 1-5314 Fr. | | | do., No. 9, fixed line, sp. gr. 2-355 | 1-5350 Fr. | | | Fraunhofer, sp. gr. 2-756, fixed line E. | 1-5631 Fr. | | | bottle | 1-582 Br. | | | Starch, dry | 1-594 Br. Y. | | | Stilbite | 1-508 Br. | | | Gum scammony | 1-510 Br. Y. | | | Gum Arabic | 1-512 Br. | | | not quite dry | 1-514 W. | | | Human cuticle | 1-517 W. | | | Nitre, least index | 1-335 Br. | | | greatest | 1-514 Br. | | | sp. gr. 1-9 | 1-224 N. | | | Dantzig vitriol, sulphate iron | 1-315 N. | | | Nadelstein from Faroe | 1-3153 Br. | | | Mesotype, least | 1-516 Br. | | | greatest | 1-525 Br. | | | Sulphate of zinc, ordin. refr. | 1-517 Br. | | | Myrrh gum | 1-517 Br. Y. | | | Tartaric acid, least | 1-518 Br. | | | greatest | 1-575 Br. | | | Wavelite | 1-52 W. | | | Gum dragon, (tragacanth), 1-520 Br. | | | Glass—borax, 1; silex, 2; 1-528 W. | | | Gum or shell lac | 1-525 Br. Y. | | | Caoutchouc | 1-524 W. | | | Selenite, sulphate of lime | 1-525 W. | | | refraction | 1-536 Br. | | | A selenites, sp. gr. 2-292 | 1-488 N. | | | Citric acid | 1-527 Br. | | | Leucite | 1-527 Br. | | | Crystalline of ox and fish, dried | 1-530 W. | | | Pitch | 1-531 W. | | | Sulphate of copper, least refraction | 1-531 Br. | | | Olibanum gum | 1-544 Br. | | | Glass of phosphorus, phoric acid fused | 1-532 Br. | | | Solid phosphoric acid | 1-544 Br. | | | Glass of borax, fused borax | 1-532 Br. | | | Manna | 1-533 Br. Y. | | | Arragonite, ext. index | 1-5348 M. | | | ord. | 1-6931 M. | | | 1st axis of elasticity | 1-5926 Ru. | | | 2d | 1-6863 Ru. | | | 3d | 1-6908 Ru. | | | Arseniate of potash | 1-585 W. | |
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1 Dr. Wollaston informed us that he had mistaken Dragon's Blood for Gum Dragon. | Substance | Index of Refraction | |---------------------------|---------------------| | Dioptrics | | | Fahlanite | 1.535 Br. | | Eleni gum | 1.547 Br. Y. | | Mastic gum | 1.549 Br. Y. | | Anime gum | 1.546 Br. Y. | | Copal gum | 1.549 Br. | | Sugar, white | 1.536 W. | | Felspar | 1.536 Br. | | Mellite, least | 1.538 Br. | | Juniper gum | 1.538 Br. | | Carbonate of barytes, least| 1.540 Br. | | Boxwood | 1.542 W. | | Colophony | 1.543 W. | | Apatite | 1.543 He. | | Dichroite, Idite | 1.544 | | Rock salt, sp. gr. 2-143 | 1.545 N. | | Chlo turpentine, mean | 1.551 Br. | | Sagapenum gum | 1.545 | | Turpentine | 1.545 Br. Y. | | Burgundy pitch, mean | 1.558 | | Gum thus, mean | 1.550 Br. | | Rock crystal, ord. | 1.547 W. | | Amethyst, ord. ray | 1.562 M. | | Quartz, ord. ray | 1.564 M. | | Resin, mean | 1.554 Br. | | Glue, nearly hard | 1.553 Br. Y. | | Chalcedony | 1.553 Br. | | Comptonite | 1.553 Br. | | Opium | 1.559 Br. Y. | | Hypocalcium of lime, mean red ray | 1.561 He. | | Dragon's blood | 1.562 Br. Y. | | Horn | 1.565 Br. | | Wernerite, ext. | 1.568 | | Baryto-calcite, least | 1.565 Br. | | Glass, pink coloured | 1.570 | | Assafetida | 1.575 Br. Y. | | Flint glass, var. specimens | 1.576 Br. | | Glass, extreme red | 1.596 Br. | | Fraunhofer, No. 3, line E | 1.6145 Fr. | | No. 30, line E, 1.6374 Fr.| | | No. 23, ditto | 1.6405 Fr. | | No. 13, ditto | 1.6420 Fr. | | Prussiate of potash | 1.586 Br. | | Anhydrite, ord. | 1.5772 Bi. | | Gumammoniac | 1.585 | | Hyposulphite of lime, least | 1.583 He. | | Emerald | 1.585 Br. | | Benzoic gum, mean | 1.591 W. | | Tortoise shell | 1.591 Br. | | Guineam gum | 1.600 Br. Y. | | Beryl | 1.598 Br. | | Balsam of Tolu, mean | 1.60 W. | | Siliceo-carbonate of zinc and iron, least | 1.6005 Br. | | Hopeite, ord. | 1.601 | | Glass, ruby red | 1.601 | | Meionite | 1.606 Br. | | Iron sinter | 1.606 | | Glass, purple coloured | 1.608 | | Resin of jalap | 1.608 Br. Y. | | Hyposulphite of strontian, least | 1.608 He. | | Topaz, colourless | 1.6102 Bi. | | 1st axis of elasticity | 1.6145 Ru. | | 2nd ditto | 1.6167 Br. | | 3rd ditto | 1.6241 Ru. | | Brazil, ord. | 1.6223 Bi. | | blue Aberdeen | 1.636 | | yellow | 1.638 Br. | | red | 1.652 | | Siliceo-carbonate of zinc from Aachen, least | 1.6173 Br. | | greatest | 1.6395 Br. | | from Bohemia, least | 1.600 | | greatest | 1.848 | | Glass, bright green | 1.615 Br. | | Castor | 1.629 Br. | | Sulphate of barytes, ord. | 1.6352 M. | | ext. | 1.6468 M. | | ord. yel. | 1.6201 Bi. | | low, green rays | 1.6460 He. | | another specimen, redrays | 1.6459 He. | | yellow, green rays | 1.6491 | | a pseudo-topazus, sp. gr. 4-27 | 1.643 N. | | do. do. | 1.646 W. | | Muriate of ammonia | 1.625 Br. | | Aloes | 1.634 Br. Y. | | Glass, opal coloured | 1.635 Br. | | Euclase, ord. | 1.6429 Bi. | | Sulphate of strontian | 1.649 | | Glass, hyacinth red | 1.647 | | Mother-of-pearl | 1.653 | | Spargelstein | 1.657 | | Epidote, least | 1.661 Br. | | greatest | 1.703 | | Tourmaline | 1.668 | | Chrysolite, least | 1.660 | | Chloruret of sulphur | 1.667 | | Nitrate of bismuth, least | 1.67 He. | | Glass, orange coloured | 1.695 Br. | | Boracite | 1.701 Br. | | Glass, tinged red with gold | 1.715 Br. | | deep red | 1.729 | | Eucroite, least | 1.799 Br. | | Nitrate of silver, least | 1.729 | | greatest | 1.788 | | Hyposulphite of soda and silver, least | 1.735 He. | | greatest | 1.785 He. | | Axinite | 1.735 | | Nitrate of lead | 1.758 Br. | | Cinnamon stone | 1.759 Br. | | Chrysoberyl | 1.760 | | Spinelle | 1.756 He. | | Spinel | 1.761 Br. | | Felspar, greatest refr. | 1.764 Br. | | Sapphire, white | 1.768 W. | | blue | 1.794 Br. | | Rubellite | 1.768 He. | | Ruby | 1.779 Br. | | Zircon, orange coloured | 1.782 Br. | | Glass lead (lint) | 1.787 Ze. | | Pyrope | 1.791 Br. | | Labrador hornblende | 1.80 Br. | | Arsenic | 1.813 W. | | Carbonate of lead, least | 1.813 Br. | | greatest | 1.884 Br. | | Garnet | 1.815 Br. | | Bonite of lead, fused, extreme red ray | 1.866 He. | | Sulphate of lead | 1.925 Br. | | Withamite, least | 1.931 Br. | | greatest | 1.960 Br. | | Glass, lead 2—sand 1 | 1.987 W. | | Zircon | 1.995 W. | | least refraction | 1.961 Br. | | greatest | 2.015 Br. | | Sulphur, native | 2.008 Br.Y. | | Calomel | 2.04 W. | | Tungstate of lime, least | 2.070 Br. | | greatest | 2.129 | | Glass, lead 3—flint 1 | 2.028 Ze. | | Scalp oxide of iron | 2.1 Y. | | Glass of antimony | 2.15 Po. | | Silicate of lead, atom to atom, extreme red | 2.123 He. | | Phosphorus | 2.125 Ge.Y. | | Blendé | 2.224 Br. | | Nitrate of lead, biaxial, in six-sided prisms, ord. ray | 2.229 Ha. | | Diamond, sp. gr. 3-4 | 2.470 Br. | | brown coloured | 2.487 Br. | | Plumbago | 2.494 W. | | Chromate of lead | 2.479 | | another kind, least refraction | 2.503 Br. | | another kind, do. 2-308 Br. | | | greatest refraction | 2.571 | | another kind, do. 2-928 Br. | | | Octohedrite | 2.500 | | Realgar, artificial | 2.549 Br. | | Red silver ore | 2.564 |
In the preceding Table, the letter \( N \) affixed to any index of refraction, indicates that the observation was made by Newton; \( H \), Hooke; \( E \), Euler; \( M \), Malus; \( C \), Cavallo's table; \( R \), Rudberg; \( B \), Biot; \( P \), Potter; \( Z \), Zeiller; \( B \), Boscovich; \( F \), Fraunhofer; \( H \), Sir John Herschel; \( F \), Mr. Faraday; \( H \), Haidinger; \( W \), Wollaston; \( B \), Sir David Brewster; and \( Y \), by Dr. Young, who calculated the indices from Sir David Brewster's observations.
1 Deduced from its polarising angle, which was 63°. If light is regarded as consisting of material particles, it must move with greater velocity in bodies than in vacuo, in the proportion of the sines to which the refraction of these bodies is proportional. The power of bodies, therefore, to refract and reflect light, must be inversely proportional to their specific gravities; for if a body of small specific gravity has the same index of refraction as a body of great specific gravity, the former must have exercised a greater absolute force upon light than the latter.
On the hypothesis of emission, it has been shewn by Sir Isaac Newton, that the absolute refractive power of bodies is proportional directly to the square of the cosine of their Dioptries maximum angle of refraction, and inversely to their specific gravity; that is, calling R the absolute refractive power, m the index of refraction, and D the density of the body, we shall have \( R = \frac{m^2 - 1}{D} \), a formula by which the following table of absolute refractive powers has been computed. The numbers marked Dulong, were, we believe, computed by Sir John Herschel, from the refractive indices given by Dulong in the preceding table.
### Table of Absolute Refractive Powers
| Index of Refraction | Index of Refraction | Index of Refraction | |---------------------|---------------------|---------------------| | Tabaseer............| 0.0976 | Ammonia.............| 1.0082 | | Cryolite............| 0.2742 Brewster | Alcohol, rectified..| 1.0121 Dulong | | Fluor spar..........| 0.3426 | Carbonate of potash.| 1.0227 | | Oxygen..............| 0.3799 Dulong | Chromate of lead....| 1.0496 Brewster | | Sulphate of barytes.| 0.3829 Dulong | Olefiant gas........| 1.0654 Dulong | | Nitrous acid gas... | 0.4455 | Muriate of ammonia..| 1.0788 Brewster | | Nitrous gas.........| 0.4491 Dulong | Carburetted hydrogen.| 1.2204 Dulong | | Air.................| 0.4528 | Camphor.............| 1.2251 | | Hydrocyanic acid....| 0.4580 Biot | Oil of olives.......| 1.2607 | | Carbolic acid gas...| 0.4537 | Oil of linseed......| 1.2819 Newton | | Azote...............| 0.4734 Dulong | Spirit of turpentine.| 1.3222 | | Chlorine............| 0.4813 | Bees' wax...........| 1.3308 Malus | | Glass of antimony...| 0.4864 Newton | Amber...............| 1.3654 Newton | | Nitrous oxide.......| 0.5078 Dulong | Octohedrite.........| 1.3816 | | Phosphorus..........| 0.5188 | Bi-sulphuret of carbon.| 1.4294 Brewster | | Selenite............| 0.5386 Newton | Diamond.............| 1.4366 Newton | | Carbonic oxide......| 0.5387 Dulong | Oil of cinnamon.....| 1.4944 | | Quartz..............| 0.5415 Malus | Oil of cassia'......| 1.6184 | | Rock crystal........| 0.5450 Newton | Resin..............| 1.6666 | | Common glass........| 0.5436 Newton | Ambergris...........| 1.7000 Brewster | | Muriatic acid gas...| 0.5514 Dulong | Sulphur.............| 2.2000 | | Sulphuric acid......| 0.6124 Newton | Phosphorus..........| 2.8857 | | Calcareous spar.....| 0.6424 Malus | Hydrogen............| 3.0653 Dulong |
The results given in the preceding tables are susceptible of increased accuracy, not only by taking accurate measures of the indices of refraction of the bodies, in relation to the fixed line E of the spectrum, but also by obtaining more accurate measures of their specific gravities.
### Sect. II.—On the Refraction of Rays by bodies with plane and spherical surfaces.
Having shewn how to find the refracted ray, when the incident ray is given, and the constant ratio of refraction which belongs to any transparent body, we may trace the progress of rays through bodies of any form whatever, provided we have the lines given which are perpendicular to the surface of the body at the points where the rays fall upon it. In all spherical surfaces this perpendicular is a line drawn through the point of incidence and the centre of the spherical surface; and in all other cases it is a line perpendicular to a line touching the surface at the point of incidence.
The names of prisms and lenses have been given to those transparent bodies which are most useful in optical experiments, and in the construction of optical instruments. Sections of these different refracting bodies are shewn in the annexed diagram.
1. A prism, represented in the figure at A, is a solid piece of glass, having three plane surfaces, AR, AS, RS, which are called its refracting faces, the light passing through any two of them.
2. A plane lens, B, is a lens, the centre of whose surfaces are infinitely distant. Its sides are therefore parallel like a piece of plane glass.
3. A spherical lens, C, is a lens whose surfaces have the same centre, and is consequently a sphere or a part of one.
4. A double convex lens, D, has two convex spherical surfaces, whose centres are on opposite sides of the lens. It is said to be equally convex when the radii of its two surfaces are equal; and unequally convex when the radii are unequal.
5. A plano-convex lens, E, is a lens which has one of its surfaces flat or plane, and the other convex.
6. A double concave lens, shown at F, is a solid bounded by two concave spherical surfaces. It is equally concave when its surfaces have the same radius, and unequally concave when they have different radii.
7. A plano-concave lens, G, has one of its surfaces concave and the other plane.
8. A meniscus lens, H, has one of its surfaces concave and the other convex, the two surfaces meeting if continued. The convexity predominates, and it acts as a convex lens.
9. A concavo-convex lens, I, differs from the meniscus only in the circumstance that the two surfaces do not meet if continued. Hence the concavity predominates, and the lens acts as a concave one.
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1 Assuming the specific gravity to be 1.044, the same as oil of cinnamon. 10. A cylindrical lens is shewn at M; it is merely a cylinder of glass, or any other transparent body.
11. A plano-cylindrical lens, shewn at N, has one of its surfaces plane and the other cylindrical.
12. A transverse cylindrical lens, shewn at O, resembles two plano-cylindrical lenses, plane transversely, or with their lengths at right angles to each other, and joined together by their plane surfaces at a b c d.
Refraction through Prisms.
As prisms are essential parts of optical instruments, and are of peculiar value in experiments on light, it is necessary to have a correct idea of the phenomena which they exhibit, in refracting rays of light.
Let ABE be a prism of two equal sides BA, BE, and made of glass whose index of refraction is 1·500, or whose ratio of refraction is as 1·500 to 1, or as 3 to 2, and let RC be a ray incident on its first surface at C. It is required to determine the path of this ray after it has suffered refraction at both its surfaces AB, BE.
From any scale set off CG equal to 10 divisions, and CR equal to 15, and through G draw GD perpendicular to AC. From the point C, and on the line CD, set off CD equal to CR, and through D and C draw DCe, Ce will be the refracted ray. From a scale on which Ce is 10, set off CeG equal to 15 parts, and drawing gd perpendicular to BE, make cd equal to eC, and draw through d and e the line der, er will be the path of the ray refracted by the second surface BE of the prism. When the radius Ce will not reach the perpendicular gd, the ray Ce will not be refracted at all, but will suffer total reflexion. When total reflexion commences, the point d will fall in the line EB, and the perpendicular gd will touch the circle described with the radius eC round e, at the point d. At all greater angles of incidence the ray Ce will be totally reflected.
The sine of the angle of incidence at e, when the ray Ce is not able to emerge from the prism, but suffers total reflexion, will be found in the case of plate glass (whose index of refraction is 1·500), to be equal to \(\frac{1}{1·500}\) or \(\frac{2}{3}\), or 0·666; the angle corresponding to which is 41° 48'.
The total reflexion which thus takes place within transparent bodies, is a very remarkable and highly interesting phenomenon. The light is far more brilliant than what is obtained from the brightest silver, which gives more reflected light than any other metal; and it possesses curious physical properties, which will be explained in a subsequent part of the article. The phenomenon of total reflexion may be finely seen by filling a tumbler-glass with water, and placing it above the head so as to see the image of a candle reflected from the lower side of its surface when at rest. The brilliancy of the image surpasses that of every other species of reflection. Diamonds, precious stones, and the glass ornaments of chandeliers, &c. &c., are often cut so as to send to the eye light that has suffered total reflexion. The brilliant white lustre of dew-drops arises from totally reflected light.
To a person under perfectly still water, the vision of objects either out of the water or on the bottom, must be very singular. The whole visible heavens, in place of being a hemisphere, will appear like a cone, with an angle of 97°. "All objects," says Sir John Herschel, "down to the horizon, will be visible in this space, and those near the horizon much distorted and contracted in dimensions, especially in height. Beyond the limits of this circle will be seen the bottom of the water and all subaqueous objects, reflected, and as vividly depicted, as by direct vision. In addition to these peculiarities, the circular space above mentioned will appear surrounded with a perpetual rainbow of faint but delicate colours." In order to understand this, let MN be the surface of the water, and E an eye at the bottom. Let DE be the direction in which a horizontal ray ND, would be refracted at D, and CE the direction in which MC would be refracted at C. Then it is clear that all objects on the horizon will be seen in the directions ED, EC, and as the same is true in every azimuth, ACEDB will be a section of the cone, which will comprehend within it all objects in the visible horizon. The sun and moon will appear to rise at A and set at B. They will have the appearance of ovals, with their smaller diameters vertical. They will quit the horizon, and descend to it again very slowly, as the angle of refraction varies very slowly from 90° of incidence downwards. If a man fishing near N stands up to his knees in water, his knees will just be seen above the water, in the direction EB, and his body standing within the cone BEA, while his legs will be seen bright, and inverted in the direction of about EN, by the total reflexion of the lower surface MN, of the water. If we draw Ce and Dd, making the angles eCe, dDE, equal to CED, then all objects in the water, to the right hand of d, and to the left hand of e, will be seen by total reflexion from the inner surface MN of the water, in the space surrounding the cone AEB. An object at e will be seen by reflexion from the point C, in the direction EC, and an object at d by reflexion from D; but none of the objects between e and d will be seen by reflexion to the eye at E. Hence we see the reason why the fisherman's legs, like other objects under water, will be seen by total reflexion in a direction near to EN. The circular rainbow, or rather fringe of colours, which separates the objects out of the water from those which are beneath it, and seen by total reflexion, is that band of colour which always bounds light that is totally reflected.
It frequently happens, both in optical experiments and in optical instruments, that light is refracted at the surfaces of two media placed in contact, such as water and glass, and in compound lenses of flint and crown glass, either touching one another or united by a cement. In all such cases, it is necessary to determine the refraction which light experiences at their refracting surface. It is found by experiment, and may be proved theoretically, that the index of refraction for the separating surfaces of media is equal to the quotient of the most refractive, divided by the least refractive medium. Thus, the index of refraction for the separating surface of water and plate glass will be \(\frac{1·500}{1·336}\), or 1·122, which is nearly the same as that of tabasheer. In ord.r, therefore, to find the refracted ray in this case, let MN be... A ray RC will be refracted in the direction Ce', and may be found by the method formerly given. In order to find the change produced in the direction of the ray at c', take a point g' in the line e'C, so that if c'C is 1:22, g'c' shall be 1:000; then drawing g'd' perpendicular to the refracting surface, make c'd' equal to c'C; and having drawn through the points d', c' the line d'c'd'c, c'e' will be the refracted ray. This ray being incident on the second surface of the glass plate at e, will be refracted in a direction er, which may be found by the method formerly described. It will be found both by projection and by experiment, namely, by looking through the compound plate MNOP, and observing any distant object, that the finally refracted ray cr is parallel to the incident ray RC.
If the angle RCA, fig. 23, the complement of the angle of incidence, is increased, the point e, where the refracted ray emerges from the side BE of the prism, approaches to E, and the angle reE diminishes, till at a particular inclination of the incident ray, the angle RCA will become equal to the angle reE. When this happens, the refracted ray Ce, will be equally inclined to the refracting faces of the prism BA, BE, and will be parallel to the base AB. This will be obvious by considering Ce as an internal ray incident on both sides of the prism, and at equal angles to each, in which case it will suffer equal degrees of refraction, and therefore be equally inclined to the refracting faces.
If the eye is placed at r to receive the refracted ray er, it will see the luminous body, such as a candle, from which the ray RC proceeds, in the direction re, and the angle which this ray re forms with RC, will be the deviation of the ray produced by the refraction of the prism. Let us now suppose the candle to be fixed, and the prism turned round, so that the angle RCA may be increased, it will be found experimentally, and may be easily proved by projection, that the deviation of the ray re is least when the angle RCA is equal to reE, or when Ce is parallel to AE, and increases when Ce deviates on either side from this mean position. Now this position may be easily ascertained by placing the eye behind the face BE, and turning the prism till the refracted image of the candle, or other object, becomes stationary. When this takes place, Ce is parallel to AE or CeA is an equilateral triangle; and it may easily be shown, by similarity of triangles, or by projection, that the angle of refraction at the first surface is equal to half the angle of the prism, or \( \frac{1}{2} \) ABE. Hence we obtain the following simple rule for finding the index of refraction, after having measured, with a goniometer or otherwise, the angle of incidence, or the complement of the angle RCA. Divide the sine of the angle of incidence by the sine of half the angle of the prism, and the quotient will be the index of refraction.
For the purpose of measuring indices of refraction, we do not require regular prisms of considerable size. Two small pieces well ground and tolerably polished, are sufficient for this purpose. They need never be larger than the pupil of the eye, and will answer well enough if they are of the size of a pin's head. If we wish to measure the index of refraction of fluids, we have only to place a drop of the fluid at the angular point A, of two pieces of parallel glass fixed at any angle by a piece of wood or ax BE. Enough of the fluid for the purpose will be retained, by capillary attraction at the point A, and after measuring the angle BAE of the prism, and the angle of incidence at which the image of the candle becomes stationary, the index of refraction will be found as before.
**Refraction through Plane Glasses.**
Every person is acquainted with the fact that light which passes through plane glasses, or glasses which have their two surfaces flat and parallel, like MN, in the annexed figure, does not suffer any very perceptible change, either in its general direction, or in the parallelism, convergency, or divergency of its rays. If AB, A'B', for example, be two parallel rays incident on the plate of glass MN, they will suffer equal refractions at B, B', because they are incident at the same angles, and the refracted rays BC, B'C', will therefore be parallel. These parallel rays again falling upon the second surface at C, C' will suffer equal refractions there, and will emerge parallel in the lines CD, C'D'. Hence we conclude that parallel rays after transmission, at any obliquity, through a plane glass, will emerge parallel. But as the rays DC, D'C', will, to an eye at D and D', be seen in the directions DCa, D'C'a, their absolute directions in space are altered, and the difference between the real and the visible direction will increase with the obliquity of the rays AB, A'B', and with the thickness of the plate of glass. If we suppose MN to be part of a looking-glass, silvered on its lower side CC', then the refracted rays BC, B'C', will, after reflection at C, C', in the directions Ce, C'e', be refracted at c, c', into the parallel directions cd, c'd'. But the rays AB, A'B', will be reflected, though in a much fainter degree, in the directions Ab, A'b', so that an eye placed so as to receive these rays, will see the bright image reflected from the silvered surface, in the direction de, and the faint image reflected from the first surface, in the direction dB, at a distance from each other depending on the obliquity of the reflexion, and on the thickness of the plate. A candle, for example, will be seen double at a short distance from the mirror; but a larger object, in order to be seen double, must be viewed at a greater distance. At great obliquities, and when the objects are very luminous, such as gas-burners, &c., other images will be seen by reflexions at c, c', and subsequent reflexions from the other side of the plate. If the two faces of the plate are not exactly parallel, the bright and faint images above described, will change their distance, sometimes overlapping each other, and sometimes separating, according to the part of the plate on which they fall, though the angle of incidence may remain the same.
When diverging and converging rays pass through a plane glass, their degree of divergency and convergency is altered by it,
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We had once a looking-glass of this kind sent to us as a curiosity by a gentleman, who valued it on account of its remarkable properties. It differed from all the rest in his possession, only in its being the worst. Dioptics, and also the position of the points of divergency and convergency. Let ABB' be a pencil of rays diverging from A, and incident upon the plane glass MN. The emergent rays CD, C'D', will, after their second refraction at C, C', proceed as if they had come from the point b, and their angle of divergency will now be bBB'. Hence a plane glass brings the divergent point of diverging rays nearer to it, and increases the angle of divergency. For the same reason, if Dbd' is a converging beam of light, its point of convergency b, will be removed to A by the plane glass, or its angle of convergency will be diminished.
When there is only one refraction, as in the case of standing water, whose surface is BB', and bottom CC', then the very reverse will be the result. A diverging beam ABB' will have its divergent point removed to a, and a converging beam would have it brought nearer the surface.
Refraction of Rays by Spheres.
When a ray of light falls upon a curved surface of any kind, the infinitely small part of the surface which it occupies may be considered as coinciding with the tangent to the surface, or with a plane surface touching the curve at the point of incidence. When the surface is spherical, this tangent plane is perpendicular to the radius, or the line drawn from the centre of the sphere to the point of incidence. Hence it is always given when the centre is given.
Let MN be the section of a sphere of glass, whose index of refraction is 1·500, as before, RSf a ray passing through its centre S, and therefore unrefracted, because it is incident perpendicularly on both surfaces, and RC, RC, other rays parallel to, and equidistant from, RSf; it is required to find the path of one of these RC, through the sphere. Join SC, which will be perpendicular to the surface at C. From a scale on which RC is three parts, set off RC equal to 1, and draw CD parallel to CS, (which is the same as drawing it perpendicular to the elementary surface, or tangent to the sphere at C). Make CD equal to CR, and through D and C draw the line DCf, meeting the posterior surface of the sphere at c, and the axis of the sphere at f. The point f would have been the focus, had there been no second surface to refract the ray Ce a second time. On a scale in which Ce is two parts, set off Cg equal to 1 part, and having joined Sc, draw gd parallel to Sc, and make cd equal to Cc. Through c draw deF, meeting the axis of the sphere in F. As the ray RC, below RSf, falls in the very same manner on the sphere MN, it will have its refracted ray in a similar direction, and the two rays will meet at F, which is called the focus of the sphere for parallel rays, or the principal focus of the sphere.
If we determine the path of the ray RC, and find the foci f and F for both surfaces; by using different indices of refraction, we shall find that in every case the distance EF of the principal focus of the sphere is exactly one-half of the distance Ef of the focus for the first surface, and that FS is to ES as the sine of incidence or the index of refraction is to the difference between twice the sine of incidence and twice the sine of refraction; that is, in glass as 1·500 is to 3·000—2·000, or as 1·500 to 1·000.
Hence we have for different refractive bodies the following results:
| Index of refraction | |---------------------| | Tabasheer,...........FS is to ES as 1·111 is to 0·222 | | Water................FS is to ES as 1·336 is to 0·672 | | Glass................FS is to ES as 1·500 is to 1·000 | | Zircon................FS is to ES as 2·000 is to 2·000 |
Hence it appears that in the case of zircon, and all other bodies whose index of refraction is 2·00, the focus F falls exactly on the posterior surface of the sphere at E, and it therefore follows that in diamond, phosphorus, &c., and all bodies whose refractive power exceeds 2·00, the principal focus falls within the sphere, the focus advancing from E towards S, as the index of refraction increases, and reaching the centre of the sphere S, when the index of refraction becomes infinite.
It may be interesting to trace the distances E of the principal focus F from the sphere, in bodies of various refractive powers, supposing the radius of sphere to be one inch, and placed in vacuo.
| Distances E.F. | Feet. Inches | |---------------|-------------| | Hydrogen......| 3623 inches | .301 11 | | Oxygen........| 1838 ------ | .153 2 | | Atmospheric air.| 1701 ------ | .141 9 | | Phosgen.......| 432 ------- | .36 0 | | Tabasheer.....| 4 --------- | .04 0 | | Water.........| 0·98 ------ | 1 nearly | | Glass.........| 0·50 ------ | 0 1/4in. | | Zircon........| 0·00 ------ | 0 | | Diamond.......| within the sphere |
In spheres of diamond and other substances of high refractive power, a refracted ray Ce may fall so obliquely upon the inner surface of the sphere, that it would be totally reflected, and would therefore be carried round the surface of the sphere, without the possibility of making its exit. If the length of the refracted ray Ce should cut off an arc which is an aliquot part of a circle, the ray would describe a regular polygon, being always reflected from the same points; but if it was not an aliquot part of a circle, the points of reflection would vary in every revolution of the ray.
The following is the rule for finding the principal focus of a sphere, or its focus for parallel rays:—Divide the index of refraction by twice its excess above unity, and the quotient is the distance of the principal focus from the centre of the sphere.
When the rays RC, RC, in place of falling on the sphere in directions parallel to the axis, or to one another, proceed from a near object, and always from a point in the axis RSE, their focus may be found by the very same method which we have already given. When the point from which the rays diverge is very distant, the focus of such rays will be a little farther from the sphere than F, and as the radiant point approaches to the sphere, the focus F will recede from it, as will be more fully explained when we treat of the progress of rays through lenses.
Refraction of Rays by Convex Lenses.
The action of an equally convex lens in refracting the rays of light, is exactly the same as that of a sphere, with this difference only, that the two surfaces are brought nearer each other, and in consequence of this the ray refracted by the first surface falls upon the second surface, at a different point, and at a different angle, the effect of which is to produce a change in the position of the focus. Let LL be a double and equally convex lens of glass, a line AF passing through the centre C, or middle point of its greatest thickness, is called its axis. Let parallel rays AB, A'B', fall upon the first surface, at the points B, B'; these will be refracted in directions BD, B'D', which will be determined by the method shown in fig. 28. Had there been no second surface, these rays would have converged to a focus at f, but as they meet the second surface of the lens at D and D', they will there be refracted, as shown in fig. 28, for the sphere, so as take the directions DF, D'F', and have their principal focus at F.
The following is the rule for finding the principal focus of a glass lens unequally convex:—Multiply the radius of the one surface by the radius of the other, and divide twice this product by the sum of the same radii.
If the glass lens is equally convex, and has its index of refraction 1:500, the distance CF, or its principal focal distance, will be equal to the radius of any of its surfaces.
The following is the rule for finding the principal focal distance of a plano-convex lens of glass. When the convex side is exposed to parallel rays, the focal distance, reckoned from the plane side, will be equal to double the radius of the convex surface, diminished by two-thirds of the thickness of the lens. When the plane side of the lens is exposed to parallel rays, the distance of the focus from the convex side will be equal to twice the radius.
When the rays AB, AB are oblique to the axis, the middle ray AC passing through the centre C, will obviously suffer refraction at B, but as it falls upon the second surface at the same angle, it will be refracted a second time in an opposite direction, so that it will proceed in a direction df parallel to AB. The rays AB, AB will suffer refraction at the points B, B', and also at the points D and D', and it will be found by projection that they meet in a focus F in the axis df.
In the preceding case the parallel rays are supposed to issue from some very distant object; but if the object from which the rays proceed is near or not very distant from the lens, its focus will recede from the lens, in proportion as the object or point of divergence approaches to it. This fact scarcely requires to be proved, for it is manifest that as the radiant point approaches to the lens, the rays fall more and more obliquely on the first surface, and less and less obliquely on the second, so that the deviation produced by refraction is not sufficient to bring them to a focus so near the lens as the point F, in fig. 30a. This will be better understood from fig. 32, where LL is a convex lens, whose focus for parallel rays is F. Let RL, RL be rays diverging from a candle or other body, at R, then, if we trace the refracted rays by the method already given in fig. 27, we shall find that they will meet at a point f, farther from the lens than F, and that if the point R advance to R', the focus f will advance to f', and so on, the focus f receding from the lens as R approaches to it. When the distance RC is equal to twice CF, or twice the principal focal distance, the distance of the focus f' from the lens will be equal to the distance of the radiant point from it, or Cf' will be equal to CR'. When R comes nearer C, f' goes rapidly away from it, and when R comes to F', which is called the anterior focus, CF' being equal to CF, the rays will be parallel, or what is the same thing, the focus f will Dioptries have retired to an infinite distance. When R comes nearer to C than F', the rays will diverge, after passing through the lens, as if they came from some point in front of the lens, and this point, or virtual focus, as it is called, will approach to the lens as R approaches it, in moving from F' towards C. The points R and f, and R' and f', are called conjugate foci, because it may be shewn that rays diverging from f will be refracted to R, and rays diverging from f' to R'.
It is indeed a general truth in all the phenomena of refraction and reflexion, that if the refracted rays are supposed to be the incident ones, the incident rays will be the refracted ones; for the ray experiences the very same action in an inverse order, by retracing its path.
The following is the rule for finding the focus f, or the conjugate focal length of a convex lens of glass for diverging rays: Multiply twice the product of the radii of the two surfaces of the lens, by the distance of the radiant point or RC, for a dividend. Multiply the sum of the two radii by the same distance RC, and from this product subtract twice the product of the radii for a divisor. The quotient of the dividend divided by the divisor will be the focal distance Cf required.
When the lens is equally-convex, multiply the distance of the radiant point RC, by the radius of the surfaces, and divide that product by the difference between the same distance and the radius, and the quotient will be the focal distance Cf required.
If the lens is plano-convex, divide twice the product of the distance of the radiant point RC, multiplied by the radius of the convex surface, by the difference between that distance and twice the radius, and the quotient will be the distance of the focus from the centre of the lens.
When converging rays fall upon a convex lens, they are always refracted to a point between the lens and their point rays of convergence. Let RL, RL be rays converging to any point r, behind the lens LL, it is very evident that refraction must always make them cross the axis RC of the lens somewhere between r and the lens, and always between the principal focus F and the lens. The exact point may be found by the methods already given. As the point of convergence r recedes from the lens, the focus f will approach to the principal focus F, and when r is infinitely distant, the rays RL, RL become parallel, and f will coincide with F. When r approaches to C, f will also approach to it.
The focus of a double convex glass lens, when its thickness is small, for converging rays may be found by the following rule: Multiply twice the product of the radii of the two surfaces by the distance rC of the point of convergence for a dividend. Multiply the sum of the two radii by the same distance rC, and add to this product twice the product of the radii for a divisor. The quotient obtained by dividing the above dividend by this divisor, will be the focal distance fC required.
When the lens is equally-convex, multiply the distance rC by the radius of the surfaces, and divide that product by the sum of the same distance and the radius, and the quotient will give the focal distance fC required.
In plano-convex lenses we must divide twice the product of the distance rC multiplied by the radius of the convex surface, by the sum of that distance and twice the radius, and the quotient will give the focal distance required.
Refraction of Rays by Concave Glasses.
In order to show how to find the refracted ray when the When \( fC \) is equal to twice the principal focal distance \( CF \), the virtual focus of divergence \( f' \) will be at the same distance on the left hand of \( C \) as the point of convergence \( f'' \) is distant on the right hand. When \( f' \) approaches the lens on the right hand, the virtual focus \( f'' \) will recede from it on the left. When \( f' \) reaches \( F \), the virtual focus will be infinitely distant, or the refracted rays will be parallel; and when \( f' \) advances from \( F \) to the lens, the refracted rays will converge on the right hand of the lens, and the focus will advance towards the lens, as the point of convergence advances towards it.
The rule for finding the conjugate focus of a converging beam, for a doubly concave lens, is the same as that for diverging rays in a doubly convex lens. If the lens is plano-concave, the rule is the same as for diverging rays falling upon a plano-convex lens.
**Refraction of Rays through Meniscuses, and Concavo-Convex Lenses.**
It would be quite unprofitable to trace the progress of different rays through these various forms of glasses, both because they are but little used, and because the very same methods which are applicable to convex and concave surfaces, are applicable also to them. When used by themselves and for ordinary purposes, these lenses are inferior to the common convex and concave lenses, and therefore are seldom met with. We shall therefore content ourselves with giving the rules for finding their foci.
In a meniscus the focus for parallel rays is obtained by dividing twice the product of the two radii by their difference, and the quotient will give the focal distance.
In the same kind of lens the focus for diverging rays will be thus found: Multiply twice the distance of the radiant point, by the product of the radii of the two surfaces, for a dividend. Multiply the same distance by the difference between the two radii, and to their product add twice the product of the two radii for a divisor. The quotient arising from dividing the dividend by the divisor, will be the focal distance of the meniscus. This rule will answer also for converging rays.
In concavo-convex lenses the very same rules will apply, but the rays have a virtual focus in front of the lens, as in concave lenses.
In treating of the passage of oblique rays through a double convex glass, as shown in fig. 31, we have stated that there is a point \( C \), called the centre of the lens, through which the ray that passes suffers the same refraction at both surfaces, or emerges parallel to its original direction. In equally double convex lenses, this centre \( C \) is accurately in the middle part of the thickness of the lens; but in other forms of lenses it is not. Hence it is necessary to point out the method of finding this centre. In double convex or concave lenses, the centre \( C \), see fig. 31, 37, and 38, lies within the two surfaces of the lens. In plano-convex and plano-concave lenses, it is coincident with the vertex of the convex or concave surfaces, and in meniscuses and concavo-convex lenses it lies without the thickness of the lens, and nearest to the surface which has the greatest curvature. Let \( R, r \), figs. 37—40, be the centres... Every intermediate portion of the lens Dioptres will have a similar focus somewhere in the line FF', and if we suppose all the rays to proceed from a distant object, such as the sun, there will be an image of the sun, or a luminous focus in every point of the line FF', and FF' will be a brilliant line of light.
This property of a cylindrical lens to form a bright line of light has been ingeniously applied by Captain Kater in the construction of his azimuth compass, which we have described in our article MAGNETISM, vol. xiii. p.765.
In cylindrical lenses diverging and converging rays will have the same foci as in common and concave lenses of the same curvature; and therefore the rules for finding their foci are applicable also to them.
Cylindrical lenses have been recently applied by Sir David Brewster for improving the vision of objects that are rectilineal, such as the defective lines in the solar spectrum. When these lines are not visible, or are very imperfectly visible, on account of the imperfections of the telescope, the application of a cylindrical lens, either solid or fluid, renders them more visible when the axis of the cylinder or cylindrical surface is accurately perpendicular to the lines. A prism has a similar effect. Both of them act in filling up the irregularities of the edges of the line by a succession of images of other parts of the line. If we look, for example, at a screw nail, or a twisted or rough rope, through a prism or cylinder, whose length is perpendicular to the screw or rope, the edges of both will be as smooth as if they were polished cylinders.
A patent was taken out several years ago by a Parisian artist for a transverse cylindrical lens similar to that shewn in fig. 22 O; which differs from the cylindrical lens in fig. 41 in this, that the second cylindrical surface has the axis of the cylinder of which it is a part, perpendicular to the axis of the cylinder of which the first surface is a part. The effect of this combination is exactly equivalent to a plano-convex glass of the same radii of curvature, and therefore it does not possess any superior properties, as was believed by its inventor.
If we cross two cylindrical lenses, such as that in fig. 41, at right angles, we shall have all the effect of a double convex lens. Or if we cross two good test tubes filled with water or any other fluid, in the same manner, we shall also obtain a rude imitation of the effect of a spherical lens, which may answer for the common purposes of a microscope.
The application of a cylindro-spherical lens by Mr. Airy to Mr. Airy's purpose of remedying imperfect vision in his own eye de-sphero-ey-serves to be more particularly noticed. He found that his eye lindrical refracted rays to a nearer or shorter focus in a vertical than spectacles in a horizontal plane, so that his eye was completely useless. Hence he concluded that the curvature of his cornea was greater in a vertical than in a horizontal plane, and he ingeniously proposed to correct this defect by cylindrical refraction. As the eye was shortsighted, he required concave surfaces to correct the general defect of a too convex cornea. He therefore had a lens constructed, which was doubly concave, one of the surfaces being spherically concave, and the other cylindrically concave, and of such a curvature as to bring to the same point the vertical and horizontal foci of the cornea. An artist of the name of Fuller, at Ipswich, constructed for Mr. Airy lenses of the proper dimensions, which enabled him to read the smallest print at a considerable distance with his defective eye, as well as he could do with the other. He found that vision was most distinct when the cylindrical surface was turned from the eye, and he placed the lens as near the eye as possible. There is another application of cylindrical lenses which we believe has not hitherto been made. In
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This was the case also in Dr. Thomas Young's eye, but it did not injure his vision. EL Nat. Phil. vol. ii. pp. 578, 9.
Dioptries - all preparations of natural history, objects which are generally preserved in cylindrical bottles or vessels containing fluids, the objects are always seen distorted, being magnified to the greatest extent in a plane perpendicular to the length or axis of the cylinder, while in a rectangular direction, the object is not magnified at all. In order to see the objects of their true shape, and have them equally magnified in all directions, a cylindrical lens of a suitable focus should be employed, so that the axis of the cylinder may be at right angles to the cylindrical axis of the vessel.
Sect. III.—On the Formation of Images by Lenses, and on the vision of objects through them.
In the preceding section we have treated of the formation of images by rays transmitted through small apertures, and have considered the formation of images by reflecting surfaces.
In order to explain the formation of images by convex lenses, whether double, or plano-convex, or meniscuses, let LL be a convex lens, MN an object farther from it than its principal focus. Let MLL be a cone of divergent rays proceeding from M, and having their focus at m behind the lens; and NLL another similar cone from the other extremity N of the object, and having their focus at n. Every other part of the object will send out rays in all directions; but only those which fall upon the circular surface of the lens LL, will be refracted by it, and they will all have their focus between m and n. These refracted pencils, however, cannot be shown in the figure without crossing one another. As every part of the object MN, will therefore send to corresponding points of the image mn rays of their own colour, an image of MN resembling it in all respects will be formed at mn, and as the rays from the upper part M of the object go to m, and from the lower part to n, this image will be an inverted one, and if we draw lines through the centre of the lens from M to m, and N to n, it will be evident from the similarity of triangles that the size of the image will be to the size of the object, as the distance of the image from the lens is to the distance of the object.
If we place the eye behind this image mn, we do not see it suspended in the air at mn, but it appears as if it were in front of the lens. That the image, however, is formed at mn, may be proved by viewing it on smoke raised at that place, or on a piece of ground glass, or semi-transparent paper; or if we bring the eye in front, we shall see it distinctly painted on any white ground, such as a piece of white paper. We shall suppose it, however, to be seen on smoke by the eye placed behind it at A or B. It will be seen exactly at mn, as if it were a real object; and in order to see it distinctly, the eye must view it at the same distance as it views other objects, and it may be viewed as any other object is, through a pair of spectacles or a magnifying glass.
As all the rays from M, N cross each other at the points m, n of the image, the very same rays radiate from those points that radiated from M, N, and consequently the very same effect must be produced in the eye as if these rays proceeded from a real object at mn. Hence by placing another convex lens at a proper distance behind mn, a distance greater than its principal focus, we may form another image of this image, in the conjugate focus of the second lens.
If we wish to form a magnified image of an object by any lens, we have only to place the object nearer the lens, and it follows from the rules for conjugate foci that the image will increase. If MN, for example, is brought nearer LL, the image mn will recede from the lens, and increase in size. When ML is equal to twice the principal focal distance of the lens, the distance of the image mL, and the size of the image will be the same as that of the object MN. If MN comes still nearer the lens, mn will recede still farther, and continue to increase in size till it becomes infinitely large and infinitely distant. When this happens, the rays which form it will have become parallel. If during all these changes the eye is withdrawn from the lens so as to be at least six inches behind the place where the image is formed, it will observe the image distinctly before it. But when the rays become parallel, the eye may then be placed immediately behind the lens, and it will see the object distinctly in the anterior principal focus of the lens, and magnified in proportion to the shortness of the focal distance of the lens.
In the preceding paragraphs, we have described the principles of the camera obscura, the compound microscope, and the operation of the single microscope. When the image mn is distinctly formed on paper, the lens LL acts as in the camera obscura, painting all objects before it in their natural colours, in their just proportions, and with all their movements, on a white ground placed behind it. When the image mn has become greater than the object MN, by the advance of the latter to the lens, the eye views this magnified picture, and the effect is the same as in the compound microscope, whose object-glass is LL, and whose eye-glass has a focal distance equal to that of the eye. When the image is infinitely distant, and the rays enter the eye parallel, the object being then in the anterior principal focus of the lens LL, and the eye behind it, the lens is then acting as a single microscope.
When objects are within our reach, such as microscopic objects, or near objects presented before a camera obscura, it is always in our power to illuminate them with artificial light, and thus make dark objects give brighter images; but when this cannot be done, in consequence of the objects being out of our reach, we can increase the brightness of the image by increasing the area or superficies of the lens. If the area of the lens LL, for example, were doubled, it would collect twice the quantity of rays that flow from every point of an object, and concentrate them at the corresponding points of the image mn.
In order to understand the principle of the telescope and single microscope, we must be acquainted with what is called the apparent magnitudes of objects. If we hold a sixpence A at the distance of six or eight inches from the eye E, then it will exactly cover or appear equal to a shilling placed at B, a half-crown placed at C, and a crown at D. If we remove the sixpence A, the shilling will just cover the half-crown. If we remove the shilling, the half-crown will just cover the crown. Hence all these coins placed as they are in the figure, are said to have an eye placed at E the same apparent magnitude, because they are all seen under the same angle DEF, and would all cover the same portion of the sky, or of any distant object.
If the sixpence A is brought thrice as near the eye E as in fig. 44, its angle of apparent magnitude will now be GEF three times as great as DEF, fig. 43, and it will appear thrice as large as DF. The sixpence has therefore been magnified; and if we interpose a lens between it and the eye, so as to make the rays refracted by the lens parallel, it will appear distinctly magnified, and the lens which we interpose will be a single microscope.
Objects within our reach, and capable of being placed where we please, may be therefore magnified to any extent, by placing them very near the eye, and in the anterior focus of a small convex glass, which, by making the diverging rays parallel, render the object as distinctly seen under a great angle, as if it were a large object placed at a distance, and subtending the same angle at the eye.
But when objects are at a distance, and beyond our reach, such as remote terrestrial objects, and the planets and stars, we can magnify them, or represent them to our eye under a greater angle of apparent magnitude, by a different principle. If the object is the dial-plate of a clock, at the distance of 12,000 feet, we place a lens whose focal distance is six feet, in the end of a tube about six feet long, and having directed it to the dial-plate, a distinct inverted image of the dial-plate will be formed in the focus of the lens, at a distance of six feet from it, and if we view this image with our eye placed six inches behind it, we shall see the image of the dial-plate distinct and magnified. Now, as the distance of the dial-plate is 12,000 feet, and that of the image only six feet from the lens, the image will be $\frac{12,000}{6}$, or 2000 times smaller in diameter than the object, not in apparent magnitude, but by real measurement; and if we were to take the image and place it beside the dial-plate, and view them both at the distance of 12,000 feet, their apparent magnitudes would, like their real magnitudes, be in the proportion of 2000 to 1. But the image is fortunately within our reach, and we can do with it what we choose. Let us first view it with the naked eye, which, generally speaking, sees objects most distinctly at a distance of six inches, and as we see it at the distance of six inches, it will appear as much greater as it would have done at the distance of 12,000 feet, as 12,000 feet is to six inches, or as 24,000 is to 1. Hence it follows, that though the image is diminished in the focus of the lens 2000 times, yet it is magnified from its proximity to the eye 24,000 times, that is, it is magnified on the whole $\frac{24,000}{2000}$, or twelve times. Now, this magnifying effect will be found under all circumstances to be equal to the focal length of the lens employed, divided by the focal distance of the eye, or the distance at which it sees small objects most distinctly, which is six inches, that is, in the present case $\frac{6}{\text{six inches}}$, or $\frac{72}{\text{inches}}$, or twelve times. A short-sighted person, whose eyes have a focus of only three inches, would be able to see the same image of the dial-plate at the distance of three inches, and in his case the magnified effect would be $\frac{72}{3}$ inches, or 24 times, and an old person, or one whose eyes were long-sighted, so as not to be able to see objects distinctly, nearer than 12 inches, would see the dial-plate magnified only six times. But both these persons could put on highly magnifying spectacles, so as to see the image at very short distances, or what is the same thing, to look at the image of the dial-plate through a magnifying glass, which would enable them to see it at the distance of one inch. In this case the magnifying effect would be $\frac{72}{1}$, or 72 times.
But the instrument which we have now fitted up is precisely a telescope, the large lens being its object-glass, and the small one used by the observer its eye-glass, and hence the magnifying power of such an instrument is always equal to the focal length of the object-glass divided by the focal length of the eye-glass. The image formed by such a telescope is inverted, which is of no consequence when we look at the heavenly bodies, and it is therefore called an astronomical telescope; but in looking at the dial-plate, and at terrestrial objects, the inversion would be disagreeable, and it is therefore usual to make the image erect, by using either a concave eye-glass, or three or more convex eye-glasses. In the former case it is called the Galilean telescope, and in the latter a terrestrial telescope.
When the distance of the object is not very great, or when the focal length of the lens bears a considerable proportion to the distance of the object, the magnifying power of a lens, when the eye views the image formed by the lens at the distance of six inches, is the following. Subtract the focal distance of the lens in feet from the distance between the image and the object, and divide the remainder by the same focal distance. By this quotient divide twice the distance of the object in feet, and the quotient will express the magnifying power, or the number of times that the object has been increased in apparent magnitude by the lens.
The very same observations apply to images formed by concave mirrors, and hence a single concave mirror becomes the simplest form of the reflecting telescope, the eye viewing the image which it forms. In the case of such images the body or the head of the observer must be placed between the object and the image, so that in order to use a single concave mirror, we must either make the mirror so large that the observer's head will not obstruct all the light, or we must make the reflection a little obliquely, or, what is done in practice, we must by means of a small plane mirror or a prism reflect or refract the rays to one side, so as to allow the observer to look at the image formed by the concave mirror, without obstructing the rays in their passage from the object to the mirror, the quantity obstructed by the plane mirror or prism being too small to do any injury. If we view the image through a convex lens, so as to magnify it still more, the mirror and the lens will constitute a reflecting telescope.
PART III.—ON SPHERICAL ABERRATION AND CAUSTIC CURVES.
The rules which we have already given for finding the Spherical foci of lenses and mirrors, are strictly applicable only to rays aberration, that pass near the axis of the lenses and mirrors; and this may be readily proved by the method of finding the refracted and reflecting ray which we have explained and used.
SECT I.—On Spherical Aberration of Lenses.
In order to prove and illustrate the preceding truth, we shall suppose parallel rays to be incident on a mass of glass MNOP, in which there is only refraction at its first surface, and we do this both to avoid the confusion of lines, and because it is perfectly sufficient for the purpose of explanation. Let RS be the axis of the spherical surface MN, passing Spherical through S, its centre of curvature; and if we consider it a ray, also, it will go on to F without any refraction.
Let RB be a ray falling on the refracting surface at a distance from the axis RS, and parallel to it. From the point of incidence draw BS, which will be perpendicular to the surface at B, and take BG three fourths of BR, BG being to BR as 1 is to 1:500, the index of refraction. From G draw GD parallel to BS, and making BD equal to BR, through the points D and B, draw BC for the refracted ray. Do the very same thing for the ray RB' falling on the point B', and parallel to RS, and equidistant from it, and B'F will be the refracted ray.
If we now take two rays r, r' near the axis, and parallel to and equidistant from it, and apply the same method of projection to them, we shall find the refracted rays to be bF, b'F crossing the axis, and converging at a point F more remote from the refracting surface than f. If we draw through F a line AEC, perpendicular to the axis, then A and C being the points where the marginal or most remote rays which fall on the surface MN, and F being the focus of those nearest the axis, the distance fF is called the longitudinal spherical aberration, and AC the lateral spherical aberration of the lens.
These results may be obtained experimentally by covering up with a circle of black paper, all the central parts of the spherical surface, leaving a clear marginal ring corresponding with BB'. If the surface thus limited is exposed to the solar rays, we shall find a pretty distinct picture or image of the sun formed at f, which, from a cause which we shall soon explain, will be highly coloured at the edges. If we now remove the black circle from the surface MN, and cover the outside surface with black paper, excepting a small opening in the vertex of the lens, where the axis RS cuts it, and expose the lens to the sun's rays, we shall find the image of the sun distinct at F, and it will be less coloured than the image formed at f from another cause.
If we now expose successive rings of the surface MN to the sun's light, shutting up all the rest of the lens, we shall find that the ring nearest the axis will have its focus near f; between f and F; the second ring, its focus still nearer F; the third, its focus still nearer F; and so on, till the last ring will give its image of the sun close to F. Hence it follows, that there will be distinct images of the sun formed by each ring, and occupying the whole space FF'; and, therefore, if we expose the whole surface MN to the solar rays, the image of the sun must be extremely confused and indistinct; and if received upon a sheet of white paper placed at AC, it will consist of a bright disc at F, surrounded with a broad halo of light, becoming fainter and fainter towards A and C.
As this is true of every spherical surface whatever, it follows, that every image formed by a spherical surface or lens, and every object seen through it, must be indistinct, from the confusion of rays produced by spherical aberration.
As this indistinctness increases with the aperture of the lens, or the distance of the marginal rays from the axis, we may remove it to a certain degree by limiting the aperture, or using smaller lenses; but excepting in the case of the sun or any highly luminous body, this diminution of the aperture would injure vision, from the want of light, especially in microscopes and telescopes; and hence it becomes an object of the highest importance in optics, and it is one which has occupied much attention, to discover methods of diminishing or correcting the spherical aberration of lenses.
Philosophers have, therefore, been led to calculate with accuracy the amount of spherical aberration in lenses of different forms, and having different sides exposed to the incident rays. The following are the results which they have obtained, and they may be readily verified either by experiment, or by tracing the refracted rays through large diagrams of lenses of different shapes.
1. In a plano-convex lens, (such as that shewn at E, fig. 21.) whose plane side is turned to parallel rays, or to distant objects, if it is intended to form an image of them least in its focus; or with its plane side turned to the eye, if it is to be employed as a single microscope or magnifier, the spherical aberration is $4\frac{1}{2}$ times its thickness, or the greatest that can be obtained from it. This is called its worst position.
2. In a plano-convex lens, whose convex side is turned to parallel rays, the spherical aberration is only $1\frac{1}{16}$ths of its thickness, or the least that can be obtained from it. This is called its best position.
3. In double convex lenses with equal convexities, the spherical aberration is $1\frac{3}{8}$ths of its thickness, greater than that of a plano-convex lens in its best position.
4. In double convex lenses, having their radii as 2 to 5, the spherical aberration will be the same as in a plano-convex lens in its worst position, if the flattest side, or that which has its radius 5, is turned towards parallel rays; and it will be the same as that of a plano-convex lens in its best position, if the surface whose radius is 2 is turned to parallel rays.
5. The lens of least spherical aberration is a double convex one, the radii of whose surfaces are as 1 to 6, having the surface whose radius is 1 turned towards parallel rays. In this, which is its best position, the aberration is only $1\frac{1}{16}$ths of its thickness. But if the side with the radius 6 is turned towards parallel rays, the aberration will be $3\frac{1}{16}$ths of its thickness.
If we determine the virtual focus of the central and marginal rays for a concave surface, as in fig. 45, we shall find that the spherical aberration is exactly the same for concave as for convex lenses; and hence, all the preceding results are equally applicable to them.
If we suppose that the lens of least spherical aberration, as in art. 5, has an aberration expressed by unity, the comparative aberrations of other lenses will be as follows:
Double convex or concave with radii as 1 to 6, in best position .................................................. 1:000 Plano-convex or concave in best position .................................................. 1:081 Double equi-convex or equi-concave .................................................. 1:561 Plano-convex or concave in worst position .................................................. 4:206
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1 "The effect of aberration," says Sir John Herschel, "may be very prettily exhibited by covering a large convex lens with a paper screen full of small round holes, regularly disposed, and exposing it to the sun, receiving the converged rays on a white paper behind the lens, which should be first placed very near it, and then gradually withdrawn. The pencils which passed through the holes will form spots on the screen, and their disposition will become more and more unequal over the surface, as the screen is farther removed; those at the circumference becoming crowded together before the central ones. The manner in which the several spots corresponding to central rays blend together into an image at the focus, and those formed by the exterior ones are scattered round it, gives us a very good idea of the variation of density of the rays in the circle of aberration at or near the principal focus; and if the white screen be waved rapidly to and fro in the cone of rays, so as to pass over the focus at each oscillation, the whole cone will be seen as a solid figure in the air, and the place of the circle of least aberration will become evident to the eye, forming altogether a very pleasing and instructive experiment." Treatise on Light, § 317. As a general rule for all lenses already made, and whose focus it is inconvenient to alter, the most convex surface should always be placed towards parallel rays when the lenses are used singly.
The preceding results are calculated on the supposition, that the lens is made of glass, whose index of refraction is 1:500; but the numerical results vary greatly when we use transparent media of higher and lower refractive powers. When the index of refraction, for example, is 1:6861, which is nearly that of some of the metallic glasses, and of several precious stones, and of sulphuret of carbon nearly, the lens of least spherical aberration, is not one which has its radius as 1 to 6, but one which is plano-convex; and when we come to higher refractive powers, such as those of sapphire, ruby, garnet and diamond, of which lenses are now made for microscopes, and ought to be made for the eye-glasses of powerful telescopes, one of the surfaces of the lens of least spherical aberration must be concave. This will be seen from the following results which we have calculated from Sir John Herschel's Formula, viz.: \[ R'' = \frac{2\mu^2 - \mu - A}{R' + 2\mu^2 + \mu} \] where \( R'' \) and \( R' \) are the radii of the surfaces of the lens of least spherical aberration, and \( \mu \) the index of refraction.
| Index of Refraction | Ratio | |---------------------|-------| | Vacuum | 1:000...1 to 1:00 equi-convex. | | Tabasheer | 1:100...1 to 1:31 | | New fluid in amethyst | 1:111...1 to 1:35 | | Second do. in topaz | 1:200...1 to 1:76 | | Ice | 1:300...1 to 2:43 | | Water | 1:368...1 to 2:77 | | Cryolite | 1:350...1 to 2:93 | | Flour spar | 1:400...1 to 3:60 | | Plate glass | 1:500...1 to 6:00 | | Quartz, Topaz | 1:600...1 to 14:00 | | Chrysolite | 1:686...1 to infinity, plano-convex. | | Sulphuret of carbon | 1:700...1 to 93 meniscus. | | Garnet, Ruby | 1:800...1 to 12 | | Glass—lead 2½, flint | 1:900...1 to 7 | | Zircon | 2:000...1 to 5 | | Diamond, Octohedrite | 2:500...1 to 2:5 | | Chromate of lead | 3:000...1 to 2:1 | | | 3:500...1 to 1:6 | | | 4:000...1 to 1:5 equal radii. | | infinite 1 to 1 |
But it is not merely the curvature of the lens of least aberration that changes its character and its magnitude—the aberration itself suffers a very great variation. This will appear from the table already referred to in our article Microscope; but we shall repeat that part of it which shows the amount of this change of the aberration, as the index of refraction changes from 1:4 to 2:0.
| Index of refraction | Spherical aberration when least in parts of the thickness of the lens, which is supposed | |---------------------|------------------------------------------------------------------------------------------| | Flour spar | 1:4...1:096 | | Glass | 1:5...1:071 | | Quartz | 1:6...0:933 | | Epidote | 1:7...0:666 | | Sapphire | 1:8...0:357 | | Sulphate of Lead | 1:9...0:166 | | Zircon | 2:0...0:062 |
In the case of diamond the aberration must be next to nothing; but in order to obtain this great advantage, its second surface must be very concave, which diminishes greatly its magnifying power. We have no doubt that artificial glasses or other solids will yet be made by art, and that mineral bodies will be discovered which will have such a high refractive power, as to enable opticians to remove spherical almost wholly the spherical aberration of single lenses.
Hitherto we have spoken only of the aberration of parallel rays, the effect of which is invariably to shorten the focus Aberration of the marginal or exterior rays; but when the incident of rays not rays converge or diverge, the aberration diminishes, and the focus of the marginal rays continues to be nearer the surface than that of central rays, till the focus of convergence or divergence comes up to two particular points in the axis, at the first of which, as will be presently seen, the aberration disappears, and at the second of which, namely, the focus of parallel rays in the convex side, it is infinite. When the focus of convergence or divergence is situated between these points, the effect of aberration is to lengthen the focus of marginal rays, and shorten that of central rays, the focal distance of the latter being now shorter than that of the former. These results are true for all curvatures and all indices of refraction.
Sir John Herschel has given the following general rule for all double convex or concave lenses, and for all meniscuses and concavo-convex lenses in which the sum of the curvatures of their surfaces is greater than \( \sqrt{2\mu + 3\mu^2} \) times their difference, \( \mu \) being their index of refraction. The effect of aberration will be to throw the focus of marginal rays more towards the incident light than that of central ones, when the lens is of a positive character, or makes parallel rays converge; but more from the incident light if of a negative character, or if it cause parallel rays to diverge.
We have mentioned above, that there is a point in the axis, at which rays which diverge from it, and fall upon a concave surface, will have no spherical aberration. This will be understood from fig. 46, where BB' is the first concave surface of a medium, C its centre, RCD its axis, and A a point in the spherical surface, where it meets the axis beyond the centre C. If we take two points R, F, such that RC is equal to the radius AC of the surface multiplied by the index of refraction, or RA to AF as the index of refraction is to unity, then all rays diverging from R, whether marginal or central, and falling upon the concave surface BDB', will be refracted at BB' in directions Br, B'r, which will proceed from the virtual focus F without any spherical aberration. This may be readily proved by the projection of the rays. Hence if upon F as a centre with any radius FE greater than FD, we describe a circle MEN, we shall have the second surface of a concavo-convex lens, which will be entirely free of spherical aberration. This is evident, as the rays refracted by the first surface BDB' fall perpendicularly on the second surface, and suffer therefore no refraction.
As there is a concavo-convex lens without aberration, for rays diverging from one point of its axis r, so there is a meniscus without aberration, for rays converging to a particular point in its axis. Let RB, R'B' be rays converging to a point f in the axis R'C' of a convex refracting surface BDB', whose centre is C. If we take fc, so that it is to the radius CD as the index of refraction is to unity, then it may be shewn by projection or calculation that the refracted rays RB, rB, whether marginal or central, will be refracted in lines BF, bF having the same focus F without any spherical aberration. Hence if with F as a centre we describe
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1 Phil. Trans. 2 See our article Microscope for a table of the results obtained by Mr. Coddington, from indices of refraction from 1:4 up to 2:0. 3 These aberrations are computed by Mr. Coddington. 4 Sir John Herschel's Treatise on Light, § 288. 5 Id. 299. Spherical any circle, having its radius $FD'$ less than $FD$, we shall have aberration. The second or concave surface of a meniscus, which will have no aberration, because the rays $BF$, $BP$ will pass through that surface perpendicularly, and suffer no refraction.
Owing to the injurious effect of spherical aberration on the performance of telescopes and compound microscopes, philosophers have sought to correct the spherical aberration of convex lenses, by the opposite aberration of concave ones. We have already given drawings of three doublets without spherical aberration, according to the calculations of Sir John Herschel; and we shall now give an account of the method used by Dr. Blair of correcting the spherical aberration in his compound object-glasses, and as they possess some historical interest, we shall give the same diagrams which he employed. Let $AB$, fig. 48, represent a convex lens receiving a pencil of diverging rays from the object $S$, and let $D$ be the focus of marginal and $F$ that of rays incident near the axis, such as $ST$. The greatest longitudinal aberration will therefore in this case be $DF$. Let $GH$, fig. 49, be now a concave lens, upon which are incident parallel rays $SHRK$. Let $P$ be the virtual focus of marginal rays, such as $SH$, and $N$ the focus of rays near the centre, such as $RK$, so that $PN$ will be in this case the longitudinal aberration. The convex lens in fig. 48 is in the position which gives the least spherical aberration, and the concave lens in fig. 49 is in the position which gives the greatest aberration. Hence in order to make the aberrations equal, we must make the focal distance of the convex glass much shorter than that of the concave one, and if it is requisite to have the distance of the points $F$ and $N$ from the convex and concave lenses the same as it is shewn in the figures, the object must then be placed much nearer the convex lens. Hence the image of the near object $S$ is placed at the same distance from the convex lens in fig. 48, or the virtual focus of the concave lens in fig. 49, where it is shewn as refracting parallel or infinitely distant rays.
When the focal distance $KN$, therefore, for parallel rays is equal to the distance $TF$ for rays diverging from $S$, and when the aberration $DF$ and $PN$ are equal, then if the two lenses are combined, as in fig. 50, parallel rays $SH$, $RK$ falling upon them, will be refracted to the focus $S$, without any spherical aberration. For if we suppose all the rays from $S$, which the convex lens, fig. 48, converges to $D$ and $F$, to be returned back from these points to the lens, they would be refracted accurately to $S$. But the parallel rays $SH$, $RK$, after refraction by the concave lens, fig. 49, in the directions $HX$, $KV$, are exactly in the same relative situation as the rays which we have supposed to be returned directly back from $F$ and $D$ are in at their incidence on the convex lens. Hence when these lenses are combined, as in fig. 50, parallel rays falling on the concave lens, and after refraction incident upon the convex lens, will be refracted accurately to $S$ without any spherical aberration.
The difficulty of getting rid of the aberration of spherical surfaces, induced opticians at a very early period to propose the construction of lenses that were not spherical, and that had such forms as to be entirely free of spherical aberration. As the marginal parts of spherical lenses refract the rays which fall upon them too much, the spherical aberration would obviously be removed or diminished, by giving the surface any curvature, in which the marginal parts become less convex. Now, this is the character of two well known curves, viz. the ellipse and the hyperbola, which, as Descartes discovered, may be employed in the formation of lenses in the following manner:
If a lens $LL'$, fig. 51, has the form of a meniscus in which the convex surface $LAL'$ is part of a prolate spheroid formed by the revolution of an ellipse, whose greater axis $AD$ is to its eccentricity or distance between its foci $F$, $f$ as the index of refraction is to unity, and if the other surface $LaL'$ is concave, and part of a sphere whose centre is $F$, the remoter focus of the spheroid, then rays $RB$, $RA$, $RB$ parallel to the axis $RF$ of the spheroid or the ellipse will be refracted by the convex spheroidal surface alone to the remoter focus $F$, and as these rays fall perpendicularly upon the second surface $LaL'$, they will suffer no change whatever by its action, and continue their progress to the focus $F$.
The same property belongs to a concavo-convex lens $L'L$, whose anterior or concave surface is part of a prolate spheroid of the same dimensions as for the meniscus, and having its convex side of any radius. In this case, all rays $R'a$, $R'b$ parallel to the axis and incident at $b$, $b'$, will be made to diverge from the same virtual focus $F$, and will suffer no change of direction in passing out of the second or convex surface, as they all fall upon it perpendicularly.
These truths may be proved by the most elementary principles of the conic section, or by drawing a tangent to the elliptical surfaces at $B$, $b$, and determining the refracted rays by the method already described.
Upon the same principles a plano-convex lens may be constructed without spherical aberration, as shewn in fig. 52, provided its posterior surface $LaL'$ is part of a hyperboloid formed by the revolution of a hyperbola $LaL'$, whose greater axis is to the distance between the foci as the index of refraction is to unity. parallel rays RB, RA, R'B', will suffer no refraction at the plane surface BAB', but at the points of the convex surface b, b' will be refracted accurately to F, the further focus of the hyperboloid.
In like manner, a plano-concave lens having its concave-surface part of a hyperboloid will diverge all rays so as to have their virtual focus in one of the foci of the hyperboloid.
The following elegant method of determining the figure of a refracting surface which shall refract marginal and central rays to the same focus is due to Huygens. Let R be the focus of diverging rays, and F the focus to which it is required to refract them with accuracy. Take any point A as the vertex of the refracting surface required; the surface must be such that the incident and refracted rays RB, BF have such a ratio to each other, that the excess of RB above RA shall be to that of FA above FB, as the index of refraction is to unity. In order to find the curve BAB which possesses this property, take in the axis RF any point D, and let DA be divided at the point C in such a manner that AC is to AD as unity is to the index of refraction, and from R and F as centres, describe the arches of circles BDB', BCB', with the radii RD, FC, their points of intersection B, B' will be in the required curve. In like manner, any other points in the curve may be found, and in order to convert it into a lens, we have only to describe a spherical surface LAL round the focus F as a centre, so that the refracted rays RB, B'F may suffer no change by passing through it perpendicularly. The curve BAB gradually approaches to an ellipsoid as the radiant point R becomes more distant from the lens, and when it is infinitely distant, the curve is the section of an ellipsoid, whose further focus is in F.
Various attempts have been made to execute lenses of other forms than spherical, but without decided success. Descartes has described machines for this purpose in the tenth chapter of his Dioptries, but though he succeeded to a certain degree in his experiments, yet the art has never been acquired of producing figures sufficiently accurate for fine telescopes and microscopes.
Sect. II.—Spherical Aberration of Mirrors.
It has already been stated under Catoptries in this article, that in all reflexions from spherical surfaces it is only for the ray near the axis that the rules for finding their foci are correct, those which fall farther and farther from the axis having their foci nearer the reflecting surface. Hence all the images which are formed by spherical surfaces are indistinct from spherical aberration, like those formed by lenses, with this difference, that the images are not confused with the different colours which always accompany the refraction of lenses.
If MN, fig. 54, is a concave mirror, by which parallel rays RR are reflected from its margin and r, r, from near its axis, it will be found from a simple projection of the reflected rays that R, R will invariably be reflected to a focus F nearer the mirror than the focus f of the central rays. The space Ff is called the longitudinal or linear spherical aberration, and it will obviously become greater as the diameter of the mirror is increased, its focal length or its curvature remaining the same, and with its curvature when its diameter or aperture remains the same.
In all cases but one, the marginal rays have a shorter focus than the central ones, or what is the same, have their focus nearest the reflecting surface. This case takes place when the radiant point is situated between the surface and the principal focus on the concave side of the mirror, in which case the focus of marginal rays is farther from the mirror than that of central rays.
There are only two cases in which spherical reflecting surfaces have no spherical aberration, namely, when diverging rays radiate from the centre of a concave mirror or spherical surface, in which case they are reflected back without aberration to the point from which they came, without any aberration, and when they converge to the centre of a convex mirror or spherical surface, in which case they will be reflected back in lines diverging from the centre or virtual focus behind it without any aberration.
One of these cases, namely, the first, is not an ideal one, but is actually applicable to practical purposes. For example, if rays diverging from F the centre of curvature, (not the focus) of the reflecting mirror MN, fall upon the mirror, they will be reflected to F, and pass through F towards a lens LL, which will refract them into a parallel beam LLRR; if FL is the focal length of the lens, or into a converging beam, so as to illuminate strongly any near object, if FL is greater than the focal length of the lens. This contrivance has been proposed for light-house illumination, where, in addition to the beam FLL radiating directly from F the lens LL, receives also the other beam FMN, both of which it unites in one parallel beam LLRR. In the accurate illumination of objects for the microscope, this contrivance is also applicable; and hence for this purpose a spherical mirror is better than a mirror of any other form.
As we cannot in the case of reflectors diminish their spherical aberration as we did in lenses, by giving a different shape to the two surfaces, it becomes of great importance to form the reflecting surface in such a manner as to remove the spherical aberration altogether. It is evident from the inspection of fig. 54, where CB is a perpendicular to the mirror at B, RMC the angle of incidence, and CBF the angle of reflexion, that if the reflecting surface should be such that the line BF drawn to a fixed point, F should always form equal angles with a line CB perpendicular to the mirror at the point of incidence, the parallel rays would all converge to the point F. Now the parabola is a curve which possesses this property as shown in fig. 56. Let AEB be a parabola which form a reflecting surface by its revolution round its axis RFE, and let R, R, R be parallel rays incident upon the paraboloidal surface at the points A, E, B. Then if F is the focus of the parabola, and GH a line touching the curve at A, it is a property of the parabola, that the angle GAR is equal to HAF, but GAR is the complement of the angle of incidence, and therefore HAF will be the complement of the angle of reflexion, and
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1 See Descartes' Dioptries, cap. viii. 2 See Newton's Lectures Optics, part i. sec. iv. and Huygen's Traité de la Lumière, p. 111, &c. &c. 3 See Conic Sections, vol. vii. p. 292, prop. liii. cor. 3. Spherical consequently AF the reflected ray. As this is true for every aberration, ray parallel to the axis RFE, it follows that all parallel rays incident upon the surface of a paraboloidal mirror will be reflected accurately to the focus of the paraboloid.
It may be shewn, in like manner, that convex paraboloidal reflectors will reflect parallel rays, so as to make them diverge from the virtual focus of the paraboloid. If, in fig. 56, we continue the line RA to R, and also FA to rC, it follows, from the above reasoning, that RAH is equal to raC, and that the reflected ray is Ar, diverging accurately from the focus F.
When we wish to reflect diverging rays to a focus, without aberration, we must have recourse to another solid of revolution, namely, a prolate spheroidal surface formed by the revolution of an ellipse round its greater axis. In this case, rays diverging from one of its foci will be reflected accurately, without aberration, to the focus F. This will be understood from fig. 57, where R, F are the foci of an ellipsoid, AEB a section of the ellipsoidal surface, and GH a line touching the ellipse at A; then if rays diverging from one of its foci R, fall upon the reflecting surface at A, E, and B, they will be reflected accurately to the focus F, or if they radiate from F, they will be reflected to R. As it is a property of the ellipse, that the angle GAR is equal to HAF, then since GAR is the complement of the angle of incidence, HAF must be the complement of the angle of reflection, and AE the reflected ray. As the same is true of every other point of the ellipsoidal surface, it follows that all rays incident upon it from one focus, will be converged without aberration to the other focus.
In like manner, it may be shewn, by producing RA to R', and RB to R', that rays falling upon a convex ellipsoidal mirror, and converging to one focus, will be reflected as if they diverged accurately from the other focus. That is, rays R'AR, R'BR, converging to R, will be reflected in diverging directions Ar, Br, as if they diverged from the focus F, or rays rA, rB converging to F, will be reflected in directions AR', BR', as if they diverged from R as their virtual focus.
If the concave surface of a mirror is a portion of a hyperboloid, a solid generated by the revolution of a hyperbola about its axis, rays converging to one focus will be reflected to the other focus. Let AEB be a section of the hyperboloid, and RAR, RBR', rays converging to its focus; these rays will be reflected to its other focus F. Let GH be a tangent to the hyperbola at A, then by a well-known property of the hyperbola, the angle GAK is equal to HAF; but the former being the complement to the angle of incidence, and the latter the complement to the angle of reflection, AF will be the reflected ray. For the same reason, if the rays diverge from the focus F, they will, after reflexion, diverge in the directions AR, BR, as if they came from the other focus R'.
In a similar manner, it may be shewn, that in a convex hyperboloidal mirror AEB, rays diverging from one focus R', will be reflected in directions Ar, Br, as if they diverged from the other focus F.
The preceding truths are of great practical use in the construction of optical instruments. In all reflecting telescopes, where parallel rays are required to be reflected to a single focus, it is necessary that the figure of the reflecting surface should be that of a paraboloid; and as in the specula of such telescopes the portion of the paraboloid which is requisite, does not differ much from the same portion of a spherical surface that has the same focal length, artists have contrived particular methods by which the marginal parts of the spherical surface shall be worn down in the act of polishing, so as to convert the spherical into a paraboloidal surface.
In the reflectors of light-houses, where a large surface is required to be used, a copper plate thickly plated with silver is hammered by means of a gauge to as correct a paraboloidal figure as possible, and a lamp being placed in its focus, the light which it radiates is reflected in a beam of considerable brilliancy.
In the construction of reflecting microscopes, where the image of a small object placed in one spot has a magnified image of it formed in another point, an ellipsoidal speculum is used; and Mr. Cuthbert, an eminent London artist, has succeeded in giving to such small specula an accurate ellipsoidal form. Mr. Potter has also succeeded, as we have mentioned elsewhere, in giving specula a true ellipsoidal form, and has published an account of the method by which he was able to effect this important object.
**Sect. III.—On Caustic Curves formed by Spherical Reflecting and Refracting Surfaces.**
When two or more rays of light cross one another at any point, they illuminate any reflecting substance placed in that point with their united light. Hence it follows, that when spherical surfaces converge the rays which fall upon them to different foci, these different foci must form so many illuminated points, if they are received on smoke, on white paper, or on water with any reflecting particles suspended in it. The lines which pass through these luminous foci, or rather the lines formed by the union of a great number of them, are called caustics, or caustic curves. As these curves are in reality a visible representation of the phenomena of spherical aberration, they possess considerable interest, as experimental illustrations of that class of facts.
When diverging rays fall upon a spherical mirror, whose surface exceeds a hemisphere, the caustics formed by reflexion are exceedingly beautiful. Let ACB be the section of such a spherical surface, whose centre is C, and whose principal focus for parallel and central rays is at f. Let a beam of light RAC, diverging from R, be incident on the upper part AC of this mirror, the beam consisting of the individual rays R1, R2, R3, &c., up to R10; and let the reflected rays 1, 1; 2, 2; 3, 3, &c., be found by making the angle of reflexion which they form with the perpendiculars.
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1 See Conic Sections, vol. vii. p. 229, prop. v. cor. 4. 2 Id. Id. p. 241, prop. v. 3 Art. Microscope, vol. xv. p. 42. 4 Edin. Jour. of Science, N. S., No. 12, p. 228. drawn from 1, 2, 3, &c., to C, equal to the angles of incidence which they form with the same perpendiculars. We shall then have the directions, and also the foci and intersections of all the rays. The ray 10,10 does not meet the axis RC at all, but falling on the mirror at the point 3, will there suffer a second reflexion. The ray 9, 9 has its focus exactly at C, the vertex of the mirror, where it will suffer a second reflexion, and so on with all the rest, up to 14, which is the last which will suffer a second reflexion. All the reflected rays, after 9, 9, cross the axis, or have their foci at points gradually approaching to f, which is the focus of the central ray R1. As all the rays proceeding from R, but not drawn in the figure, which fall upon the other half CB of the mirror, at points corresponding to R1, R2, &c., will have their foci in the same points between C and f, there will be along that line a series of foci constituting a line of light becoming more intense towards f.
But the rays R10, R9, R8, cross each other after reflexion, and before they reach the axis, as shewn in the figure, and hence there will be a beautiful curve of light Af, called a caustic, formed by the intersection of these rays. The other half of the mirror CB will form a similar caustic, and the projecting points f are called the cusps of these caustics, and Cf their tangent.
If a small pencil of light, consisting of two contiguous rays, moves from RA towards the position RB, being incident successively at 9, 8, 7, 6, &c., the conjugate focus of this pencil, or that formed by the intersection of the two rays of which it consists, will move along the caustic curve Af, while the points where it crosses the axis RC, or its focus formed by its union with a similar pencil similarly incident on the other half of the mirror, will advance from C to f.
If we now consider ACB as a concave spherical surface, and place the radiant point R as far to the right of the vertex C as it is to the left of it in the figure, and if we project the reflected rays, we shall find that when traced backwards, they will intersect the axis and each other, in the very same manner as they do in the figure, forming an imaginary or virtual caustic, in place of a real one, the two being in every respect the same.
If, while the radiant point R remains as in the figure, we suppose the convex surface ASB to receive the incident rays, it will then be found, by projecting the reflected rays, that they will form an imaginary caustic AΦB, less than AfB, and joining it at the points A, B. This difference in size arises from the radiant point being in this case much nearer the convex surface than before.
Let us now suppose that the radiant point R recedes from the concave mirror ACB, the point f of the cusps will gradually approach to F, the tangent Cf diminishing at the same time; and when R is infinitely distant, or the rays parallel, the point f will coincide with F, the focus of parallel rays. The same will take place in the case of the convex mirror ACB; but in the case of the convex mirror ASB, the point Φ of the cusp will approach to F', and will coincide with it when R is infinitely distant.
If, in the case of the concave mirror ACB, the radiant point R now approaches to the mirror, the cusps f will approach to the centre E of the mirror, the caustic curve Af becoming flatter and flatter, and when R reaches C, there will be no caustic at all, in consequence of all the rays being reflected back to the centre, all their foci and intersections having united in that point.
In the case of the imaginary caustic AΦB, when R approaches to S, Φ will also approach to S, the caustic approximating in form to the circular arch AS; and when R reaches S, Φ will also reach S, the caustic disappearing when it has reached that limiting form.
All that we have said is obviously applicable only to one section of the spherical mirror; but as the same is true of every section whatever, the caustic will not be a curve, but a surface formed by the revolution of the curves AfB round its axis fC, all the reflected rays being tangents to this surface.
We shall now consider the change in the appearance of the caustic, when the radiant point comes within the sphere of which the reflecting surface is a part, and when the mirror becomes a concave polished sphere. The effect thus produced is shewn in fig. 60, RE being less than RS. In this case a remarkable double caustic will be formed, composed of a short one of the kind shewn in fig. 59, and another with two long branches, one of which is shewn at 1, 2, 3, 4, 5, the dotted line below the axis SE shewing the other halves of the caustic, the long branches converging behind the mirror. Had R been placed nearer S than E, the branches 1, 2, 3, &c., would have diverged behind the mirror, having their virtual foci within the mirror. When R is half-way between S and E, the curved branches become parallel lines. When R comes nearer E the branches 1, 2, 3, &c., shorten; the smaller caustics also shorten; they both approach to the centre E, the long branches moving quickest till at E, as we have already seen, all the rays from R are reflected back to the same point, and the caustics all disappear.
M. A. Delarive, in his ingenious dissertation on caustic curves, has shewn that caustics generated by parallel rays are epicycloids, formed by one circle rolling upon a fixed circle concentric with that of the mirror, and having a radius equal to half of its own. Dr. Smith has shewn that when the radiant point is at S, the caustic is an epicycloid, whose generating circle is two-thirds of the radius of the mirror, and the fixed circle one-third of that radius.
When the radiant point passes the centre E, the caustics shift their place to the opposite side of E, and present the same phenomena as before.
There is a curious property, however, involved in these phenomena, which we have represented in fig. 61, where the radiant point is supposed to be at F, a very little within the principal focus of a spherical mirror ACB. We have supposed the rays to diverge from a point a little within the principal focus, because it is only in this case that the rays F1, F1, at a little distance from the axis, may be reflected in directions 1, 1, 1, 1, exactly parallel. The rays F2, F2, falling at a greater distance from the axis, will be made to converge to a focus at f2,
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1 Dissertation sur la Partie de l'Optique qui traite des courbes dites Caustiques, p. 84. Geneve, 1823. This interesting dissertation contains an account of the labours of preceding mathematicians, including Malus and Gergonne, and merits the attention of those who wish to prosecute the subject mathematically.
Complete System of Optics, vol. i. p. 174. between the rays F1, F1, and the axis FC, these rays will diverge, because they radiate from a point a little within the principal focus, and hence we have one spherical mirror which has, under these circumstances, the paradoxical property of rendering a faint cone of diverging rays parallel, converging, and diverging, after reflexion.
All that we have said of the caustics formed by the sections of spherical surfaces, are true of cylindrical surfaces, and by means of surfaces of this kind, the phenomena of caustic curves may be beautifully exhibited. They are indeed often presented to the eye at the bottom of china vessels of a cylindrical form, when exposed to the rays of the sun or to the light of a candle. Owing to the depths of such vessels, the obliquity of the rays prevents the effect from being well seen, but we may take shallow cylindrical vessels, or make deep ones shallow by an artificial bottom of paper or pasteboard, or by filling them nearly with milk, or any other fluid with a white opacity, or with a fine white powder pressed into a smooth surface. In order to shew the caustics, when the radiant point is placed within the cylindrical surface, a piece of card should be made to float upon oil in the cylindrical vessel, and a very minute wick inserted in the card at the points of the axis where we wish the radiant point to be placed. This wick, when lighted, will be the radiant point R in fig.60, and the caustics will be beautifully formed on the surface of the white card.
The following method, however, of exhibiting caustic curves, we have found very convenient and instructive, and it has the advantage of allowing the radiant point to come within the cylinder. A piece of steel-spring, highly polished, such as a watch-spring, is bent into a concave form, like AB, fig. 62, and is placed vertically with its lower edge resting upon a piece of card or white paper. It is then exposed to the solar rays, or those of any artificial light, so that the plane of the card MN passes through the luminous body, and the caustic curves will be seen finely displayed, varying with the distance of the radiant point, and with the reflecting arch AB. By altering too the curvature of the arch and bending it into different known curves, either by applying a portion of its breadth to the required curves delineated upon a piece of wood, and either cut or burned sufficiently deep in the wood to allow the edge of the thin strip of metal to be inserted in it, a great variety of interesting phenomena may be observed. The brightest reflector is a thin strip of polished silver or plated copper. Gold and silver foil will also answer, or a strip of mica. A cylindrical section of a wide glass tube or a bottle, especially if a piece is cut out of them to allow the incident rays to pass to the reflecting surface in the plane nearly of the base of the cylinder, will produce the caustic curves in great perfection. The caustic curves produced by a highly gilt or polished metallic ring, such as the ring of a bell handle, are exceedingly beautiful, the phenomena of a convex and a concave surface being here united.
Caustics formed by Refracting Surfaces.
It is evident, from what has been said of caustics formed by reflexion or Catacaustics, that analogous curves must also be formed by spherical refracting surfaces, which has been called Diaucaustics. In order to explain these curves, we shall take the case of diverging rays falling upon a spherical surface, as shown in fig. 63, where DBDF is the spherical surface, C its centre, R the radiant point, RD, RD two extreme rays touching the sphere, and refracted in the directions Df', Df, and RB, RB other rays nearer the axis, and refracted in the directions BF, BF'. If we join CD, CD, and drawing the semicircles DEC, DEC make the lines DE, CE' in the same proportion to CD as unity is to the index of refraction, the caustic will begin at EE' and extending in the directions E, F', F will approach the axis RC till it meets it at the principal focus F.
The caustics formed by the two refractions of a sphere, or of a cylinder (in a plane perpendicular to its axis) are shown in fig. 64, where ACB is the spherical section, E its centre, R the radiant point, and RC the ray which touches the spherical surface. This ray will be refracted by the first surface in the direction 6,6, and by the second surface of the sphere at 6, the other rays R5, R4, &c. will be all refracted in the directions indicated by the numerals 5, 4, &c., and their various intersections will form the caustic 6, 4, 3, 2, 1, f, each ray crossing the next ray before it cuts the axis, f being the focus or the point where the rays nearest the axis cut it. The luminous figure bounded by the intersection of the successive rays, is composed of the two bright caustic curves. Within these caustics there is also much light arising from the intersections between the caustics and the axis; but as there are no intersections without the caustics they are bounded by absolute darkness.
When parallel rays fall upon the spherical section ACB, the caustic commences at the extremity of a diameter perpendicular to the axis of the section, because the extreme ray suffers refraction at that point, and will intersect the nearest a little within it, and they extend, as in fig. 64, to the principal focus of the sphere for rays near the axis. The real caustic will be the surface formed by the revolution of the curves round the axis Ef, the section of this curve will be a luminous point at f, but at the posterior surface it will be a luminous circle vividly depicted on the sphere. M. Delarive has pointed out a method of determining the index of refraction of solid spheres, or of hollow spheres containing different fluids, by measuring the diameter of this luminous circle, which is smaller in fluids of high than in those of low refractive power. The phenomena of caustics formed by refraction, may be distinctly exhibited by exposing to the rays of the sun, or a strong artificial light, a globe of glass filled with any fluid, or a solid transparent sphere, or the widest part of a round glass decanter. With all these bodies the whole of the luminous figure will be clearly seen. If we use a cylinder full of wa- PART II. ON THE REFRACTION OF COMPOUND LIGHT, OR THE DOCTRINE OF COLOURS AND THE PRISMATIC SPECTRUM.
In the preceding pages we have considered white light, whether emanating directly from the sun or from artificial flames, or consisting of the same rays reflected and modified by other bodies, as a simple element all the particles of which had the same index of refraction, or suffered the same change of direction when refracted by any transparent body. This, however, is not the nature of light. White light as emitted by the sun or other luminous bodies, is a very compound element, all the parts of which possess very different properties, and these properties are of a very remarkable and interesting kind. The power which causes the reflexion of light from polished metallic bodies is not capable of decomposing it, unless when it enters the substance of the metal; but the power which produces refraction is peculiarly influential in separating compound white light into its elements. The same decomposition may be effected by the interference of rays of light by absorption, and by another principle of analysis, which has been called dissection. The two first of these processes of analysis decompose compound light of different degrees of refrangibility; while the two last decompose compound light whose rays have the same refrangibility.
Secr. I.—On the decomposition of light, and the different refrangibility of its rays.
The constituent parts or colours which compose white light, are seven in number, red, orange, yellow, green, blue, indigo, and violet. These colours have been long observed and studied in the rainbow, and in the refractions produced by lenses and prisms, but till the time of Sir Isaac Newton, no satisfactory explanation had been given of their origin and properties. Descartes had found that colours similar to those of the rainbow were produced by prisms; and he endeavoured to explain them by saying, that the particles of the medium, or matter which transmits light, endeavour to revolve with so great force, that they cannot move in a straight line, whence comes refraction; and that those particles which endeavour to revolve more strongly produce a red colour, those that endeavour to move a little more strongly, produce yellow, and so on with the other colours. Now this explanation, as Mr. Whewell has justly remarked, though it contains a gratuitous hypothesis respecting the cause of refraction, yet it proves that Descartes considered the different colours as produced by different degrees of refraction. In like manner Grimaldi, as the same author has observed, explains colours by saying, "that the colour is brighter where the light is dense; and the light is denser on the side from which the refraction turns the ray, because the increments of refraction are greater than the rays that are more inclined;" that is, that the blue rays are more refracted than the red rays. We cannot agree, however, with Mr. Whewell in the opinion, that this explanation of Grimaldi's might give an explanation of most of his facts, but one much more erroneous than a development of Descartes's views would have been." It appears to us quite manifest, that both Descartes and Grimaldi had vague sentiment that the different colours were produced by different degrees of refraction, and that Grimaldi's is the most distinctly expressed of the two; but we cannot for a moment agree with the author above quoted, "that Descartes was led very near the same point with Newton."
The sentiments expressed by Descartes and Grimaldi, were mere notions of the moment, which authors often throw out without much thought, and which are employed in future times to pervert the history of science. If these two authors really thought that colours were produced by different degrees of refraction, why did they not, as they did other opinions, submit them to the test of an experiment, which required neither thought nor labour, and the means of making which were in their hands. Sir Isaac Newton was well acquainted with the writings of Descartes, and so much with Descartes's notions about colours, that the examination of them was the object which he had in view in purchasing his prisms. He never refers to them as anticipatory of his own discoveries, and we must therefore continue to give Sir Isaac the undivided merit of the discovery of the unequal refrangibility of light, as well as its experimental establishment.
We shall now proceed to give our readers some account of this great discovery, and we shall make no apology in doing this in Sir Isaac Newton's own words, abridging his descriptions where they are redundant, or have become unnecessary. We are induced to do this also, because they exhibit the finest model of experimental research, and should be studied by every person who is desirous of investigating truth with diligence and patience.
1. The light of the sun consists of rays which differ in colour and refrangibility.—In a very dark chamber, at a round hole F, fig. 65, about one-third of an inch broad, made in the shutter of a window, I placed a glass prism ABC, whereby the beam of the sun's light, SF, which came in at that hole, might be refracted upwards, toward the opposite wall of the chamber, and there form a coloured image of the sun, represented at PT. The axis of the prism, was, in this and the following experiments, perpendicular to the incident rays. About this axis I turned the prism slowly, and saw the refracted or coloured image of the sun, first to descend, and then to ascend. Between the descent and ascent, when the image seemed stationary, I stopped the prism, and fixed it in that posture, for in that posture the refractions of the light at the two sides of the refracting angles, that is, at the entrance of the rays into the prism, and at their going out of it, are equal to one another.
Then I let the refracted light fall perpendicularly upon a sheet of white paper, MN, placed at the opposite wall of the chamber, and observed the figure and dimensions of the solar image, PT, formed on the paper by that light. This image was oblong, and not oval, but terminated by two rectilinear and parallel sides, and two semicircular ends. On its side it was bounded pretty distinctly; but on its ends very indistinctly, the light there vanishing by degrees. At
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1 Hist. of Inductive Sciences, vol. ii. p. 350. 2 Id. Id. p. 352. Having discovered this fundamental property of light, he contrived the following experiment to prove it by ocular demonstration.
"In the middle of two thin boards DE, de, fig. 67, I made a round hole in each, at G and g, a third part of an inch in diameter; and in the window-shutter a much larger hole being made at F, to let into my darkened chamber a large beam of the sun's light, I placed a prism, ABC, behind the shutter in that beam, to refract it towards the opposite wall; and close behind this prism I fixed one of the boards DE, in such a manner that the middle of the refracted light might pass through the hole made in it at G, and the rest be intercepted by the board. Then, at the distance of about twelve feet from the first board, I fixed the other board de, so that the middle of the refracted light, which came through the hole in the first board, and fell upon the opposite wall, might pass through the hole g in this other board de, and the rest being intercepted by the board, might paint upon it the coloured spectrum of the sun. And close behind this board I fixed another prism abc, to refract the light which came through the hole g. Then I returned speedily to the first prism ABC, and by turning it slowly to and fro about its axis, I caused the image which fell upon the second board de to move up and down upon that board, that all its parts might pass successively through the hole in that board, and fall upon the prism behind it. And in the mean time I noted the places M, N, on the opposite wall, to which that light after its refraction in the second prism did pass; and by the difference of the places at M and N, I found that the light, which, being most refracted in the first prism ABC, did go to the blue end of the image, was again more refracted by the second prism abc, than the light which went to the red end of that image. For when the lower part of the light which fell upon the second board de, was cast through the hole g, it went to a lower place M on the wall; and when the higher part of that light was cast through the same hole g, it went to a higher place N on the wall; and when any intermediate part of the light was cast through that hole, it went to some place in the wall between M and N. The unchanged position of the holes in the boards made the incidence of the rays upon the second prism to be the same in all cases. And yet in that common incidence some of the rays were more refracted and others less; and those were more refracted in this prism, which by a greater refraction in the first prism were more turned out of their way; and therefore, for their constancy of being more refracted, are deservedly called more refrangible."
2. The light of the sky, or the light of the sun reflected from the first surface of bodies, and also the white flames of all combustibles, whether direct or reflected, differ in colour and refrangibility, like the direct light of the sun.
The truth stated in this proposition was established by Newton, by examining all those lights with a prism when they were faint, or transmitting them through the prism, as in section 1, when they were sufficiently intense.
3. The light of the sun consists of rays differing in reflectibility; and those rays that are more refrangible than others are also more reflexible.
A prism ABC, fig. 68, whose two angles at its base BC were equal to one another and half right ones, and the third Let AG, fig. 69, represent the circle which all the most refrangible rays, propagated from the whole disk of the sun, will illuminate and paint upon the opposite wall if they were alone; EL, the circle, which all the least refrangible rays would in like manner illuminate if they were alone; BH, CI, DK, the circles which so many intermediate rays would paint upon the wall, if they were singly propagated from the sun in successive order, the rest being intercepted; and conceive that there are other circles without number, which innumerable other intermediate sorts of rays would successively paint upon the wall, if the sun should successively emit every sort apart. And seeing the sun emits all these sorts at once, they must all together illuminate and paint innumerable equal circles; of all which being, according to their degrees of refrangibility, placed in order in a continual series, that oblong spectrum PT is composed, which was described in the first experiment.
Now, if these circles, whilst their centres keep their distances and positions, could be made less in diameter, their interfering one with another, and consequently the mixture of the heterogeneous rays, would be proportionally diminished. Let the circles AG, BH, CI, &c. remain as before; and let ag, bh, ci, &c. be so many less circles lying in a like continual series, between two parallel right lines ae, and gf, with the same distance between their centres, and illuminated with the same sorts of rays: that is, the circle ag with the same sort by which the corresponding circle AG was illuminated; and the rest of the circles bh, ci, dh, ei, respectively with the same sorts of rays by which the corresponding circles BH, CI, DK, EL, were illuminated. In the figure PT, composed of the great circles, three of those, AG, BH, CI, are so expanded into each other, that three sorts of rays, by which those circles are illuminated, together with innumerable other sorts of intermediate rays, are mixed at QR in the middle of the circle BH. And the like mixture happens throughout almost the whole length of the figure PT. But in the figure pt, composed of the less circles, the three less circles ag, bh, ci, which answer to those three greater, do not extend to one another; nor are there anywhere mingled so much as any two of the three sorts of rays by which those circles are illuminated, and which in the figure PT are all of them intermingled at QR. So then, if we would diminish the mixture of the rays, we are to diminish the diameters of the circles. Now these would be diminished if the sun's diameter, to which they answer, could be made less than it is, or (which comes to the same purpose) if without doors, at great distance from the prism towards the sun, some opaque body were placed with a round hole in the middle of it to intercept all the sun's light, except so much as coming from the middle of his body could pass through that hole to the prism. For so the circles AG, BH, and the rest, would not any longer answer to the whole disk of the sun, but only to that part of it which could be seen from the prism through that hole; that is, to the apparent magnitude of that hole viewed from the prism. But that these circles may answer more distinctly to that hole, a lens is to be placed by the prism to cast the image of the hole (that is, every one of the circles AG, BH, &c.) distinctly upon the paper at PT; after such a manner, as by a lens placed at a window the pictures of objects abroad are cast distinctly upon a paper within the room. If this be done, it will not be necessary to place that hole very far off, no, not beyond the window. And therefore, instead of that hole, I used the hole in the window-shutter as follows.
4. To separate from one another the heterogeneous rays of compound light.
"That the rays which are equally refrangible fall upon a circle answering to the sun's apparent disk, will be proved by experiment by and by." In the sun's light let into my darkened chamber through a small round hole, fig. 70, in my window-shutter, at about ten or twelve feet from the window, I placed a lens MN, by which the image of the hole F might be distinctly cast upon a sheet of white paper placed at L. Then immediately behind the lens I placed a prism ABC, by which the projected light might be refracted either upwards or sidewise, and thereby the round image which the lens alone did cast upon the paper at L, might be drawn out into a long one with parallel sides, as represented at pt. This oblong image I let fall upon another at about the same distance from the prism as the image at L, moving the paper either towards the prism or from it, until I found the just distance where the rectilinear sides of the images pt become most distinct. For in this case the circular images of the hole, which compose that image, after the manner that the circles ag, bh, ci, &c. do the figure pt, were terminated most distinctly, and therefore extended into one another the least that they could, and by consequence the mixture of the heterogeneous rays was now the least of all. The circles ag, bh, ci, &c. which compose the image pt, are each equal to the circle at L; and therefore, by diminishing the hole F, or by removing the lens farther from it, may be diminished at pleasure, whilst their centres keep the same distances from each other. Thus, by diminishing the breadth of the image pt, the circles of heterogeneous rays that compose it may be separated from each other as much as you please. Yet instead of the circular hole F, it is better to substitute an hole shaped like a parallelogram, with its length parallel to the length of the prism. For if this hole be an inch or two long, and but a tenth or twentieth part of an inch broad, or narrower, the light of the image pt will be as simple as before, or simpler; and the image being much broader, is therefore fitter to have experiments tried in its light than before.
5. Homogeneous light is refracted regularly without any dilatation, splitting or shattering of the rays; and the confused vision of objects seen through refracting bodies by heterogeneous light, arises from the different refrangibility of several sorts of rays.
In the middle of a black paper I made a round hole, about a fifth or a sixth part of an inch in diameter. Upon this part I caused the spectrum of homogeneous light, described in the former proposition, so to fall that some part of the light might pass through the hole in the paper. This transmitted part of the light I refracted with a prism placed behind the paper; and letting the refracted light fall perpendicularly upon a white paper, two or three feet distant from the prism, I found that the spectrum formed on the paper by this light was not oblong, as when it is made in the first experiment, by refracting the sun's compound light, but was, so far as I could judge by my eye, perfectly circular, the length being nowhere greater than the breadth; which shews that this light is refracted regularly without any dilatation of the rays, and is an ocular demonstration of the mathematical proposition mentioned above.
In the homogeneous light I placed a paper circle of a quarter of an inch in diameter; and in the sun's unrefracted, heterogeneous, white light, I placed another paper circle of the same bigness; and going from these papers to the distance of some feet, I viewed both circles through a prism. The circle illuminated by the sun's heterogeneous light appeared very oblong, as in the second experiment, the length being many times greater than the breadth. But the other circle, illuminated with homogeneous light, appeared circular, and distinctly defined, as when it is viewed by the naked eye; which proves the whole proposition mentioned above.
In the homogeneous light I placed flies and other minute objects, and viewing them through a prism I saw their parts as distinctly defined as if I had viewed them with the naked eye. The same objects placed in the sun's unrefracted heterogeneous light, which was white, I viewed also through a prism, and saw them most confusedly defined, so that I could not distinguish their smaller parts from one another. I placed also the letters of a small print one while in the homogeneous light, and then in the heterogeneous; and viewing them through a prism, they appeared in the latter case so confused and indistinct that I could not read them; but in the former, they appeared so distinct that I could read readily, and thought I saw them as distinct as when I viewed them with my naked eye: in both cases, I viewed the same object through the same prism, at the same distance from me, and in the same situation. There was no difference but in the lights by which the objects were illuminated, and which in one case was simple, in the other compound; and therefore the distinct vision in the former case, and confused in the latter, could arise from nothing else than from that difference in the lights. Which proves the whole proposition.
In these three experiments, it is farther very remarkable, that the colour of homogeneous light was never changed by the refraction. And as these colours were not changed by refraction, so neither were they by reflexions. For all white, grey, red, yellow, green, blue, violet bodies, as paper, ashes, red lead, orpiment, indigo, bice, gold, silver, copper, grass, blue flowers, violets, bubbles of water tinged with various colours, peacock feathers, the tincture of lignum nephriticum, and such like; in red homogeneous light appeared totally red, in blue light totally blue, in green light totally green, and so of other colours. In the homogeneous light of any colour they all appeared totally of that same colour; with this only difference, that some of them reflected that light more strongly, others more faintly. I never yet found any body which by reflecting homogeneous light could sensibly change its colour.
From all which it is manifest, that if the sun's light consisted of but one sort of rays, there would be but one colour in the world, nor would it be possible to produce any new colour by reflexions and refractions; and, by consequence, that the variety of colours depends upon the composition of light.
6. Every homogeneous ray considered apart is refracted, according to one and the same rule; so that its sine of incidence is to its sine of refraction in a given ratio; that is, every differently coloured ray has a different ratio belonging to it.
Sir Isaac Newton has proved this by experiment; and in other experiments he has determined by what numbers these given ratios are expressed. For instance, if an heterogeneous white ray of the sun emerges out of glass into air; or, which is the same thing, if rays of all colours be supposed to succeed one another in the same line AC, fig. 71, and AD their common sine of incidence in glass be divided into fifty equal parts, then EF and GH, the sines of refraction into air, of the least and most refrangible rays, will be 77 and 78 of such parts respectively. And since every colour has several degrees, the sines of refraction of all the degrees of red will have all intermediate degrees of magnitude from 77° to 77°; of all the degrees of orange from 77° to 77°; of yellow from 77° to 77°; of green from 77° to 77°; of blue from 77° to 77°; of indigo from 77° to 77°; and of violet from 77° to 78.
7. Whiteness, and all grey colours between white and black, may be compounded of colours; and the whiteness of the sun's light is compounded of all the primary colours, mixed in a due proportion.
Let the spectrum or solar image PT, in fig. 72, fall upon the lens MN, above four inches broad, and about six feet distant from the prism ABC, and so figured that it may cause the coloured light which divergeth from the prism to converge and meet again at its focus G, about six or eight feet distant from the lens, and there to fall perpendicularly upon a white paper DE. And if you move this paper to and fro, you will perceive that near the lens, as at d e, the whole solar image (suppose at p r) will appear upon it intensely coloured after the manner above explained, and that by receding from the lens those colours will perpetually come towards one another, and by mixing more and more, dilute one another continually, until at length the paper come to the focus G, where, by a perfect mixture, they will wholly vanish and be converted into whiteness, the whole light appearing now upon the paper like a little white circle. And afterwards by receding farther from the lens, the rays which before converged will now cross one another in the focus G, and diverge from thence, and thereby make the colours to appear again, but yet in a contrary order; suppose at d e, where the red t is now above which before was below, and the violet p is below which before was above.
Let us now stop the paper at the focus G, where the light appears totally white and circular, and let us consider its whiteness. I say, that this is composed of the converging colours. For if any of those colours be intercepted at the lens, the whiteness will cease and degenerate into that colour which arises from the composition of the other colours which are not intercepted. And then if the intercepted colours be let pass and fall upon that compound colour, they mix with it, and by their mixture restore the whiteness. So if the violet, blue, and green be intercepted, the remaining yellow, orange, and red, will compound upon the paper, an orange; and then if the intercepted colours be let pass they will fall upon this compounded orange, and together with it decompound a white. So also if the red and violet be intercepted, the remaining yellow, green, and blue, will compound a green upon the paper, and then be red and violet being let pass, will fall upon this green, and together with it decompound a white. And that in his composition of white the several rays do not suffer any change in their colorific qualities by acting upon one another, but are only mixed, and by a mixture of their colours produce white, may farther appear by these arguments.
If the paper be placed beyond the focus G, suppose at r, and then the red colour at the lens be alternately intercepted, and let pass again, the violet colour on the paper will not suffer any change thereby, as it ought to do if the several sorts of rays acted upon one another in the focus G where they cross. Neither will the red upon the paper Chromatics be changed by any alternate stopping, and letting pass the violet which crosses it.
And if the paper be placed at the focus G, and the white round image at G be viewed through the prism at HIK, and by the refraction of that prism be translated to the place r v, and there appear tinged with various colours, namely, the violet at v and red at r, and others between, and then the red colour at the lens be often stopped and let pass by turns, the red at r will accordingly disappear and return as often, but the violet at v will not thereby suffer any change. And so by stopping and letting pass alternately the blue at the lens, the blue at r will accordingly disappear and return, without any change made in the red at r. The red therefore depends on one sort of rays, and the blue on another sort, which in the focus G, where they are commixed, do not act on one another. And there is the same reason of the other colours.
I considered farther, that when the most refrangible rays Pp, and the least refrangible ones, Tt, are by alternate inclination to one another, the paper, if held very oblique to those rays in the focus G, might reflect one sort of them more copiously than the other sort, and by that means the reflected light would be tinged in that focus with the colour of the predominant rays, provided those rays severally retained their colours or colorific qualities in the composition of white made by them in that focus. But if they did not retain them in that white, but became all of them severally endured there with a disposition to strike the sense with the perception of white, then they could never lose their whiteness by such reflexions. I inclined therefore the paper to the rays very obliquely, as in the second experiment of this book, that the most refrangible rays might be more copiously reflected than the rest, and the whiteness at length changed successively into blue, indigo, and violet. Then I inclined it the contrary way, that the most refrangible rays might be more copious in the reflected light than the rest, and the whiteness turned successively to yellow, orange, and red.
Lastly, I made an instrument XY, in fashion of a comb, whose teeth were about an inch and an half broad, and the intervals of the teeth about two inches wide. Then by interposing successively the teeth of this instrument near the lens, I intercepted part of the colours by the interposed tooth, whilst the rest of them went on through the interval of the teeth to the paper DE, and there painted a round solar image. But the paper I had first placed so, that the image might appear white as often as the comb was taken away; and then the comb being, as was said, interposed, that whiteness by reason of the intercepted part of the colours at the lens did always change into the colour compounded of those colours which were not intercepted, and that colour was by the motion of the comb perpetually varied, so that in the passing of every tooth over the lens all these colours, red, yellow, green, blue, and purple, did always succeed one another. I caused therefore all the teeth to pass successively over the lens, and when the motion was slow, there appeared a perpetual succession of the colours upon the paper: But if I so much accelerated the motion, that the colours by reason of their quick succession could not be distinguished from one another, the appearance of the single colours ceased. There was no red, no yellow, no green, no blue, nor purple, to be seen any longer, but from a confusion of them all there arose one uniform white colour. Of the light which now by the mixture of all the colours appeared white, there was no part really white. One part was red, another yellow, a third green, a fourth blue, a fifth purple, and every part retains its proper colour till it strike the sensorium. If the impressions follow one another slowly, so that they may be severally perceived, there is made a distinct sensation of all the co- Chromatic colours, one after another, in a continual succession. But if the impressions follow one another so quickly that they cannot be severally perceived, there ariseth out of them all one common sensation, which is neither of this colour alone nor of that alone, but hath itself indifferently to them all, and this is a sensation of whiteness."
Such is the account which Newton himself has given of the great discovery of the different refrangibility of light.
In examining the prismatic spectrum it is difficult to discover the terminations or boundaries of the different colours. They pass into one another by insensible shades, and if any person were to lay down their apparent limits by the nicest observations, he would find, what has been very recently discovered, that these limits vary with the state of the atmosphere, and with the altitude of the sun. Sir Isaac Newton, however, did make the attempt, and the following are the results which he obtained, we believe with crown or plate glass. We have added the results obtained long afterwards by Dr. Wollaston and Mr. Fraunhofer with flint glass, which shews the difficulty of this class of observations.
| Newton in Crown Glass | Fraunhofer in Flint Glass | Wollaston in Flint Glass | |-----------------------|--------------------------|-------------------------| | Red | 45 | 56 | 57-6 | | Orange | 27 | 27 | | | Yellow | 40 | 27 | | | Green | 60 | 46 | 82-8 | | Blue | 60 | 48 | 129-6 | | Indigo | 48 | 47 | | | Violet | 80 | 109 | 90 |
The influence of these discoveries on the progress of optical science was very remarkable. They led Sir Isaac to discover that the cause of the imperfections of the refracting telescopes was the different refrangibility of the rays of light. If Ll, for example, is a lens without spherical aberration, upon which parallel rays R, R, R of white light are incident, then it is obvious that the violet, or most refrangible rays will be most refracted in directions Lr, Lr, crossing the axis at r, and there giving a violet focus of light. In like manner the red, or the least refrangible rays, will be refracted in directions Lr, Lr, crossing the axis at r, and there giving a red focus of light. In like manner all the other rays will have foci of their own colour between r and r. If we draw the line ab, meeting the intersection of the extreme violet rays after their convergence with the extreme red rays before their convergence, it will cut the axis at a point e. The line er, is called the longitudinal aberration of refrangibility, or the longitudinal chromatic aberration, and ab is called the lateral aberration of refrangibility, or the diameter of the circle of diffusion, all the coloured rays being diffused over the circle, of which ab is the diameter. The space v a r b, is called the sphere of diffusion; and the section of it shewn in the figure may be regarded as a parallelogram, on account of the smallness of the angles rLe, rLe, which are greatly magnified in the figure. Hence it may be easily shewn, that the longitudinal aberration er, is to the lateral aberration ab, as the focal distance of the lens is to its radius or half its aperture.
In the circle of diffusion ab, the light becomes very faint towards a and b, and very intense in the centre c, so that there is formed at c a sort of general focus, indistinct and coloured. Every part of an object, therefore, will have its image formed in the foci of such a lens similarly indistinct and similarly coloured, and hence we see the reason why refracting telescopes had such great imperfections, that it was necessary to make them of enormous lengths, in order to obtain a sufficient magnifying power.
From these causes, Sir Isaac Newton despaired of the improvement of refracting telescopes, and set himself at an early period of his life to execute reflecting telescopes. His successors, however, Mr. Hall and Mr. Dollond, studied the object of refraction as produced by prisms made of different substances, and found, as we have already fully stated in our history of Optics, that Sir Isaac Newton was mistaken in supposing that all refracting media gave spectra, or separated the colours of white light in the same proportion as their refractive powers, and that different bodies had different dispersive powers, as well as different refractive ones. This grand discovery we shall now proceed to explain.
Sect. II.—On the different Dispersive Powers of Bodies.
The term dispersion has been employed to denote the separation of the different rays of white light into that divergent beam which constitutes the prismatic spectrum, the differently coloured rays having been dispersed or scattered by their different refrangibility. Sir Isaac Newton believed that all bodies whatever, whether water, or crown, or plate, or flint glass, dispersed light in an equal degree, provided the mean refraction, that is, the refraction of the mean or middle ray of the spectrum (the green ray, viz.), of these bodies was equal, or, in other words, that the dispersion, or the angle formed by the extreme red and the extreme violet ray was in different bodies proportional to the mean refraction.
As Sir Isaac Newton submitted to experiment a number of fluid substances, in the form of prisms, it is perhaps one of the most remarkable oversights in the history of science, that he did not think of comparing the length of the spectra which they formed; and it is equally strange that for more than a century he and all his successors should never have thought of forming the spectrum from any other luminous body less in diameter than the sun, or even from any luminous line of small breadth. The consequence of these oversights was, that the most important discoveries relative to light, and to optical instruments, were reserved for another age.
In our History of Optics, and in the article ACHROMATIC GLASSES, we have given a detailed history of the successive labours of Hall, Dollond, Euler, and others, by which the achromatic telescope was invented and perfected.
If we perform the experiment shewn in fig. 65, with two prisms, the one of flint, and the other of crown glass, and measure in each the length of the spectrum PT, or rather the angles which the violet and the red rays PA, TA, make with each other, and the angle which the mean green ray forms with the direction of the coloured rays SY, this last angle will be the mean refraction of the prism, and the first the angular dispersion. We shall then find, that while in crown glass the quotient obtained by dividing the greater by the lesser angle, or the part of the mean refraction to which the dispersion is equal, will be seventeen hundredths ($\frac{1}{17}$), while in flint glass it will be thirty hundredths ($\frac{3}{10}$), this number varying with the nature of the glass.
This result may be exhibited to the eye, by placing behind a prism of crown glass C, another of flint glass F, of such an angle as not to produce any deviation by refraction, the angle of deviation of the green ray produced by the crown glass prism C, being compensated by an equal and opposite deviation produced by the flint glass prism F; that is, the green ray Bq will emerge parallel to the incident ray RA. When this has been effected it will be seen that there is still a spectrum re, which will colour the edges of any object which is viewed through the prism, and in which will have the same position as if they had been produced by a small flint glass prism placed in the same manner as the flint glass prism F.
If we now take two prisms, one of crown, and the other of flint glass, of such angles, that all objects seen through them are colourless, or that a ray of light Bq, when refracted by them, as in fig. 74, shall be white, it will be found that the white pencil Bae will be refracted towards the base of the crown glass prism, the flint glass having corrected the colour produced by the crown glass one, but still left a considerable balance of refraction produced by the latter.
Hence it is manifest that the colours produced by a convex lens of crown glass, as shown in fig. 73, may be corrected by a concave lens of flint glass, while the rays produced by the unbalanced refraction of the convex glass are still converged to a focus. Such a combination of lenses is called an achromatic object-glass, and a telescope in which such an object-glass is used is called an achromatic telescope.
Nothing is easier than to determine by experiment, when we have obtained good glass for the construction of these lenses, the proper radii to which they should be ground, in order to correct the aberration of colour; but it may be readily shewn, and the reader may easily prove it by the methods already described, that the aberration of colour produced by a convex lens of crown glass will be corrected by a concave lens of flint glass, provided the focal lengths of the two lenses are proportional to their dispersive power.
Thus, in fig. 75, if LL is a convex lens of crown glass, whose focus for green rays is at f, and for violet and red rays at v and r, and if F is the virtual focus of a concave lens II of Chromatic flint glass, for the mean green ray, then parallel rays AAA will be refracted to a single focus at RV, where the violet and red rays will be united, provided the focal length of II, viz., EF, is to Ef, the focal length of LL as 0·068, the dispersive power of flint glass, is to 0·033, the dispersive power of crown glass.
The dispersive powers of various glasses, and of some fluids, Dispersive had been measured with considerable care, in reference to powers of the improvement of the telescope, but no attempt was made bodies to investigate it as a branch of physics, exhibiting new and interesting properties of transparent bodies. Dr. Wollaston set the example of beginning this inquiry, and he determined in a very general manner the dispersive qualities of thirty-three substances, which he arranged in the order of their dispersive powers, without giving any numerical estimate of their value. In this state of the subject, Sir David Brewster, by a new method of measuring dispersive powers, which presented considerable facility of observation, made a very extensive series of experiments on the subject, which, in a physical point of view, presented several curious results. In laying the following table of his observations before our readers, we must warn them that they were made, often with the most imperfect specimens of the minerals and fluid substances, from the difficulty of getting any other, and that they were intended only to indicate the general properties of bodies in dispersing light. The want of fixed points in the spectrum, in reference to which the measures could be taken, rendered it necessary to use the extreme points, which varied with the intensity of the light employed, and with the absorbing action of the bodies themselves, when they happened to be coloured or imperfectly transparent. The discovery of the fixed lines, and their use in measuring dispersive powers, introduced by Fraunhofer, has given a new impetus to this subject, and when it is practicable to obtain good prisms of the substance under examination, no other method will or should be adopted.
### Table of the Dispersive Powers of various Solid and Fluid Bodies.
| Names of substances | Part of the whole refraction to which the dispersion is equal. | Dispersion power. | |---------------------|-------------------------------------------------|------------------| | Chromate of lead (greatest refraction), estimated at | 0·770 | 0·400 | | Chromate of lead (greatest refraction), must exceed | 0·570 | 0·296 | | Beadgar, a different kind, melted | 0·394 | 0·267 | | Chromate of lead (least refraction) | 0·388 | 0·262 | | Beadgar, melted | 0·374 | 0·255 | | Oil of cassia | 0·089 | 0·139 | | Sulphur, after fusion | 0·149 | 0·130 | | Phosphorus | 0·156 | 0·128 | | Sulphur of carbon | 0·077 | 0·115 | | Balsam of Tolu | 0·065 | 0·103 | | Balsam of Peru | 0·058 | 0·083 | | Carbonate of lead (greatest refraction) | +0·091 | +0·091 | | Barbadose aloes | 0·058 | 0·085 | | Essential oil of bitter almonds | 0·048 | 0·079 | | Oil of anise seeds | 0·044 | 0·077 | | Acetate of lead, melted | 0·040 | 0·069 | | Balsam of styx | 0·039 | 0·067 |
| Names of substances | Part of the whole refraction to which the dispersion is equal. | Dispersion power. | |---------------------|-------------------------------------------------|------------------| | Guinecum | 0·041 | 0·066 | | Carbonate of lead (least refraction) | 0·056 | 0·066 | | Oil of cumin | 0·033 | 0·065 | | Essential oil of tobacco | 0·035 | 0·064 | | Gum ammoniac | 0·037 | 0·063 | | Oil of Barbadoes tar | 0·032 | 0·062 | | Oil of cloves | 0·033 | 0·062 | | Green coloured glass | 0·037 | 0·061 | | Sulphate of lead | 0·036 | 0·060 | | Deep red glass | 0·044 | 0·060 | | Oil of sassafras | 0·032 | 0·060 | | Opal-coloured glass | 0·038 | 0·060 | | Muriate of antimony (refr. pr. 1·598) | 0·036 | 0·059 | | Rosin | 0·032 | 0·057 | | Oil of sweet fennel seeds | 0·028 | 0·055 | | Oil of spearmint | 0·025 | 0·054 | | Orange-coloured glass | 0·042 | 0·053 | | Rock salt | 0·029 | 0·053 | | Flavouring Boscorich's highest | 0·0527 | | | Caoutchouc | 0·028 | 0·052 | | Oil of pimento | 0·026 | 0·052 | | Flint glass | 0·032 | 0·052 |
| Names of substances | Part of the whole refraction to which the dispersion is equal. | Dispersion power. | |---------------------|-------------------------------------------------|------------------| | Deep purple glass | 0·031 | 0·051 | | Oil of angelica | 0·025 | 0·051 | | Oil of thyme | 0·024 | 0·050 | | Oil of terebenthin | 0·024 | 0·050 | | Oil of wormwood | 0·022 | 0·049 | | Oil of peppery royal | 0·024 | 0·049 | | Oil of caraway seeds | 0·024 | 0·049 | | Oil of dill seeds | 0·023 | 0·049 | | Oil of bergamot | 0·023 | 0·049 | | Fliar glass | 0·029 | 0·048 | | Chio turpentine | 0·028 | 0·048 | | Gum thus | 0·028 | 0·048 | | Oil of lemon | 0·023 | 0·048 | | Flint glass | 0·028 | 0·048 | | Oil of juniper | 0·022 | 0·047 | | Oil of chamomyle | 0·021 | 0·046 | | Gum juniper | 0·025 | 0·046 | | Carbonate of strostites, (greatest refraction) | 0·032 | 0·046 | | Oil of brick | 0·021 | 0·046 | | Flint glass, Boscorich's lowest | 0·0457 | | | Nitric acid | 0·019 | 0·045 | | Oil of lavender | 0·021 | 0·045 | | Balsam of sulphur | 0·023 | 0·045 |
VOL. XVI. ### Chromatics
| Names of Substances | Part of the whole refraction to which the dispersion is equal | Dispersive power | |---------------------|---------------------------------------------------------------|-----------------| | Tortoise shell | 0.027 | 0.045 | | Horn | 0.025 | 0.045 | | Canada balsam | 0.024 | 0.045 | | Oil of myrrh | 0.022 | 0.045 | | Gum olibanum | 0.024 | 0.045 | | Nitric acid | 0.018 | 0.044 | | Cajpeut oil | 0.021 | 0.044 | | Oil of hysop | 0.022 | 0.044 | | Oil of rhodium | 0.022 | 0.044 | | Pink-coloured glass | 0.025 | 0.044 | | Oil of savine | 0.021 | 0.044 | | Oil of poppy | 0.020 | 0.044 | | Zircon (greatest refr.) | 0.045 | 0.044 | | Muriatic acid | 0.016 | 0.043 | | Gum copal | 0.024 | 0.043 | | Nut oil | 0.022 | 0.043 | | Burgundy pitch | 0.024 | 0.043 | | Oil of turpentine | 0.020 | 0.042 | | Oil of rosemary | 0.020 | 0.042 | | Feldspar | 0.022 | 0.042 | | Glue | 0.022 | 0.041 | | Balsam of Capivi | 0.021 | 0.041 | | Amber | 0.029 | 0.041 | | Oil of nutmeg | 0.021 | 0.041 | | Stilbite | 0.021 | 0.041 | | Oil of peppermint | 0.019 | 0.040 | | Spinelle ruby | 0.031 | 0.040 | | Calcareous spar (greatest refraction) | 0.027 | 0.040 | | Oil of rape seed | 0.019 | 0.040 | | Bladder resin | 0.023 | 0.040 | | Tartrate of potash and soda | 0.020 | 0.039 | | Carbonate of potash (greatest refraction) | 0.013 | 0.039 | | Gum elemi | 0.021 | 0.039 | | Sulphate of iron | 0.019 | 0.039 | | Diamond | 0.036 | 0.038 |
The following measures of the dispersive powers of several varieties of glass were taken by Sir John Herschel, by a method which gave him nearly the extreme rays of the spectrum, namely, by viewing the spectrum through a dark blue glass, which stops the green, yellow, and most refrangible red rays, and therefore allows the extreme rays of the spectrum to be seen,—rays which the eye does not recognise in any of the ordinary lights which are used in optical instruments. If we condense the sun's light, as we have done, in order to render visible rays at the extremities of the spectrum, that have not been recognised, we should obtain dispersive powers still higher than those given by Sir John Herschel. By determining, however, the extremities of the spectrum seen by the ordinary light of the sky, it would be easy to accommodate all measures of dispersive power taken in such a light to those taken in the light used by Sir John Herschel, or in the more condensed and consequently elongated spectrum to which we have referred. In order that the measures in the following tables may be correct, it is necessary that they should all have been taken when the sun had the same altitude, because it is quite certain that the violet part of the spectrum diminishes in length very rapidly as the sun approaches the horizon, and some change also takes place at the red extremity.
### Dispersive Powers of different kinds of Glass.
| Names of Substances | Part of the whole refraction to which the dispersion is equal | Dispersive power | |---------------------|---------------------------------------------------------------|-----------------| | Flint glass | 0.03734 | 0.06404 | | Ditto | 0.03705 | 0.06409 | | Crown glass | 0.02139 | 0.04704 | | Plate glass | 0.02616 | 0.05090 |
Sir John Herschel justly remarks, that it ought not to excite surprise that the dispersions deduced by this method should considerably exceed all former estimates.
In the preceding table of dispersive powers we have given two columns of numerical results, the first column containing the part of the whole refraction, or angle of deviation to which the angle of dispersion is equal, and the other the dispersive power itself. The first column is obviously not a measure of the dispersive power, because if the dispersion in that column is \( \frac{1}{2} \)th part of a low refraction in one body, and the 20th part of a high refraction in another body of great refractive power, the dispersive power of the latter must be smaller than that of the former, in the inverse ratio of the index of refraction of the two bodies minus unity. Hence the numbers in the second column, the dispersive powers, are obtained by dividing the first column by the index of refraction minus one.
If we wish to have the intrinsic or absolute dispersive powers of bodies, in reference to the action of their ultimate molecules on the theory of emission, and on the supposition, as Sir John Herschel has remarked, in reference to absolute... refractive powers, of the ultimate atoms of all bodies being equally heavy, we must divide the numbers in the second column of the preceding table by the specific gravities or densities of the bodies. In this way we have computed the results in the following table, containing the substances principally that exercise an extreme action in the dispersion of light.
Table of Absolute Dispersive Powers.
| Substance | Specific gravity used | Absolute dispersive power | |----------------------------|-----------------------|---------------------------| | Sulphate of barytes | 4.48 | 0.00602 | | Strontites | 3.95 | 0.00607 | | Carbonate of barytes | 3.70 | 0.0063 | | Sapphire | 3.50 | 0.0066 | | Chryso-beryl | 3.33 | 0.00675 | | Topaz | 3.50 | 0.00685 | | Fluor spar | 3.17 | 0.0069 | | Cryolite | 2.95 | 0.0074 | | Diamond | 3.50 | 0.0109 | | Plate glass | 2.75 | 0.0112 | | Rock salt | 2.143 | 0.0250 | | Water | 1.00 | 0.032 | | Amber | 1.04 | 0.0400 | | Oil of olives | 0.913 | 0.0415 | | Oil of turpentine | 0.87 | 0.0483 | | Sulphur, fused | 2.00 | 0.065 | | Realgar | 3.50 | 0.0728 | | Phosphorus | 1.75 | 0.0731 | | Oil of anise seeds | 0.987 | 0.078 | | Bi-sulphuret of carbon | 1.27 | 0.081 | | Oil of cassia | 1.044 | 0.131 |
These results present us with several views of considerable interest. The salts of barytes have, in reference to their density, the least dispersive power of all bodies, and next in order are the gems, including even diamond, which occupies so different a place in our table of absolute refractive powers. The inflammable bodies, with the exception of diamond, stand at the head of the table, oil of olive having by far the greater absolute dispersive power of any body yet examined. That this is owing to the hydrogen which it contains is very probable, and has almost been proved by an experiment by Sir John Herschel, who deprived a portion of this oil of most of its hydrogen by making a stream of chlorine pass through it till it refused to act any farther. By this means he converted the oil to a viscous mass, the dispersive power of which was diminished one-half; while its refractive power had hardly suffered any change. This result leads us to conclude that hydrogen has the greatest intrinsic dispersive power of all bodies. Fluorine seems to be the element which has nearly the lowest refractive power.
One of the most interesting results exhibited in the general Table of dispersive powers, relates to the dispersive powers of doubly refracting substances. Dr. Wollaston measured the dispersive power of the ordinary ray in calcareous spar, so far at least as to ascertain that it stood near water, and plate and crown-glass. Sir David Brewster had been led to measure the dispersion of the extraordinary ray, and found it to be much lower than that of water. He was hence induced to examine the dispersive powers of other doubly refracting crystals, and was thus led to the results which we have stated. Similar results have been recently obtained by M. Rudberg and Mr. Cooper, both knowing that the subject had been previously investigated.
Sect. III.—On the Irrationality of the Coloured Spaces Chromatics in the Spectrum, and the existence of a secondary Spectrum.
We are indebted, we believe, to M. Clairaut for the discovery of the irrationality of the coloured spaces in the spectrum. He found that when the flint-glass of an achromatic object glass had its aberration of colour as completely corrected as possible, that is, when the extreme red and violet rays were accurately united in the same focus, still there remained a portion of uncorrected colour, which was of a purple or claret colour on one side of the focus, and of a green colour on the other. If prisms had been used in place of lenses, and the sun's light transmitted through them in the usual manner, there would have been a small residual spectrum, or secondary spectrum as it has been called, consisting of purple and green light. The Abbé Boscovich afterwards observed the same fact, but considered it so extraordinary that he suspected some latent cause of error, and submitted his experiments to the most rigid scrutiny. He at last admitted the irrationality of the coloured spaces in the spectrum as a demonstrated truth, and has shewn how three of the colours of the spectrum may be corrected or united in the same focus in achromatic telescopes. The late Professor Robison obtained similar results, and gave Robison the name of outstanding colours to those which were not united, and form the secondary spectrum.
This subject was more fully investigated by Dr. Blair, Blair in his interesting paper on the unequal refrangibility of light, and he has shewn that the proportions of the coloured spaces vary with the substance of the opposing prisms, so that a complete correction of colour cannot possibly be effected by two media of different dispersive powers. Hence Dr. Blair was led to examine the nature of the dispersive action of different media, and by the most ingenious devices succeeded in producing fluid object-glasses in which the aberration of colour was completely corrected. The telescopes which he made on this principle were so extraordinary, that Professor Robison assures us that one of them, fifteen inches in focal length, equalled in all respects, if it did not surpass, the best of Dollond's forty-two inches long.
Under these circumstances, the scientific world was surprised at the following statement published by Dr. Wollaston in the Phil. Trans. for 1803. "Since the proportions of these colours have been supposed by Dr. Blair to vary according to the medium by which they are produced, I have compared with this appearance, the coloured images caused by prismatic vessels containing substances supposed by him to differ most in this respect, such as strong but colourless nitric acid, rectified oil of turpentine, very pale oil of sassafras, Canada balsam, also nearly colourless. With each of these, I have found the same arrangement of these four colours, and in similar positions of the prisms, as nearly as I could judge, the same proportions of them." Dr. Blair was surprised that Dr. Wollaston should have used such a coarse method of determining a point that required delicate observations, especially with the substances above mentioned; and he remarked to the writer of this article, that if Dr. Wollaston would only make use of lenses, he would see his mistake after a single observation.
There is no doubt, however, that the secondary spectrum can be made very visible by prisms, and if we use a prism of oil of cassia to correct the colour produced by another of sulphuric acid, we shall have a striking ocular proof of the existence of a large secondary spectrum.
The phenomena of a secondary spectrum will be understood from fig. 76, where RR is a ray passing through an aperture in the window-shutter SS, and refracted by a prism P in the direction PM, so as to form the spectrum AB on the palpable colours, red, green, blue, and violet, and if the prism is one of crown-glass, the mean ray PMN which bisects the spectrum will be at the boundary of the blue and green spaces. If we were to take a prism of flint-glass with a much less refracting angle, and form a spectrum CD of the same length as AB; and at the same distance from the prism, the line mn which marks the boundary of the blue and green spaces will no longer be the mean ray of the spectrum, but will be decidedly nearer the red extremity D. The least refrangible half of the spectrum has therefore been more contracted, and the most refrangible half more expanded than in the crown-glass spectrum. If we now take a prism of sulphate of barytes or fluor spar, capable of forming a third spectrum EF of the same length as the other two, the boundary of the blue and green spaces will now be at μ nearer the violet than the red extremity of the spectrum, and the least refrangible half of this spectrum will be more expanded, and the most refrangible half more contracted, than in the crown-glass spectrum.
"If a spectrum," says Sir David Brewster, "formed by flint-glass, had its coloured spaces exactly of the same dimensions with those of an equal spectrum formed by crown-glass, any object such as a window-bar lying parallel to the common section of the refracting planes of the two prisms should appear perfectly colourless when seen through the combined prisms. But if the coloured spaces in the two spectra are not proportional, as shewn in fig. 76, but are irrational, then the window-bar cannot be wholly free from colour, for though the extreme red and violet rays of both the spectra are united, yet the intermediate colours are not rendered coincident. In the spectrum AB, formed by the crown-glass, the first green ray MN, which is here the mean ray, is obviously more refracted than the first green ray mn, in the spectrum CD formed by the flint-glass, and therefore the flint-glass will not be able to refract the green ray, so as to unite it with the red and violet. Hence the green ray will, as it were, be left behind, while the red and violet rays are rendered coincident. Thus, in fig. 76, if a prism p of flint glass is placed behind a crown-glass prism P, so as exactly to correct its dispersion, the spectrum AB will be reduced to a secondary spectrum ab, the upper half of which is green, which is left behind, and the lower half is of a claret colour, formed by the union of the red and violet rays. If the bar of a window had been examined through the combined prisms P, p, the upper side of it would have been tinged with green, and the lower side of it with a claret-coloured fringe.
By comparing, in a similar manner, the spectrum EF, formed by fluor spar, with the spectrum AB, formed by crown glass, it will be found, that the fluor spar having a greater action than the crown glass upon the green ray, will carry it beyond the place of the united red and violet, and will form a secondary spectrum ef, the lower half of which is green, and the upper half of a claret colour, arising from the union of the red and violet light. If the bar of a window were viewed through the combined prisms of crown-glass and rock crystal, it would be tinged with green on its lower side, and with a claret-coloured fringe on its upper side.
When a horizontal window-bar, therefore, is seen through any two prisms which correct each other's dispersion, without uniting all the colours, the green fringe will always be on the same side of the bar with the vertex of the prism which has the least action upon the green light, or which contracts the red and green rays, and expands the blue and violet ones; that is, if the vertex of the flint glass prism is pointing downwards, the uncorrected green fringe will be on the lower side of the bar. By observing, therefore, the position of the green fringe, we can immediately ascertain which of the two prisms has the greatest action upon the green light.
These theoretical deductions from the assumed inequality of the coloured spaces are completely established by observation.
The following Table contains the result of a numerous series of observations made by Sir David Brewster on the secondary spectra of different bodies, the substances being arranged inversely according to their action upon green light. The bodies at the top of the Table form spectra, in which the red and green spaces are most contracted, and the blue and violet ones most expanded. The relative position of some of the substances, particularly the essential oils, is quite empirical; but by a reference to the original experiments, it will be seen whether or not the relative action of any two bodies has been determined.
Table of Transparent Bodies, arranged inversely according to their Action upon Green Light.
| 1 Oil of Cassia. | Balsam of copal. | |-----------------|------------------| | Sulphur. | Oil of fenugreek.| | Sulphuret of Carbon. | Oil of rosemary. | | Balsam of Tolu. | Oil of rhodium. | | 5 Carbonate of lead. | 50 Flint Glass. | | Essential oil of bitter almonds. | Zircon. | | Oil of anise seeds. | Oil of olives. | | Oil of cummin. | Oil of rape seed. | | Oil of sassafras. | Oil of spermaceti. | | 10 Oil of amber. | 55 Oil of Juniper. | | Acetate of lead melted. | Oil of ambergris. | | Opal-coloured glass. | Calcareous spar. | | Orange-coloured glass. | Rock-salt. | | Red-coloured glass. | Gum juniper. | | 15 Oil of sweet fennel seeds. | 60 Tartaric of potash and soda. | | Oil of cloves. | Oil of almonds. | | Muriate of antimony. | Crown Glass. | | Oil of lavender. | Gum Arabic. | | Canada balsam. | Alcohol. | | 20 Oil of Turpentine. | 65 Ether. | | Oil of sage. | Borax, glass of. | | Oil of pennyroyal. | Borax, Tourmaline. | | Oil of peppry. | Leucite. | | Oil of hyssop. | 70 Selenite. | | Oil of spearmint. | Beryl. | | Amber. | Topaz blue. | | Oil of lemon. | Fluor spar. | | Oil of caraway-seeds | Citric Acid. | | Oil of nutmegs. | 75 Malic Acid. | | Oil of thyme. | Ascorbic acid. | | Oil of peppermint. | Nitrous acid. | | Oil of bergamot. | Muriatic acid. | | Oil of marjoram. | Prussic acid. | | Oil of wormwood. | 80 Nitric acid. | | Oil of dill seeds. | Rock crystal. | | Oil of chamomyle. | White of an egg. | | Castor-oil. | Ice. | | Gum copal. | WATER. | | Rosin. | 85 Super-sulphuretted hydrogen. | | Diamond. | Phosphorous acid. | | Nitrate of potash. | Sulphurous acid. | | Oil of beech-nut. | Phosphoric acid. | | Oil of rue. | 89 Sulphuric Acid. | | Oil of Savine. | 45 Nut-oil. |
1 Treatise on New Philosophical Instruments, 1813, p. 316. Finding it impossible to obtain any highly dispersing medium which should refract the rays of the spectrum in the same manner as crown glass, Dr. Blair thought of employing this very imperfection in obtaining a perfect correction of colour. As the green rays removed the outstanding ones, or were not united in the same focus with the red and violet, he considered that if an achromatic concave lens should refract the outstanding green more strongly than the united red and violet, while an achromatic convex lens should also refract the outstanding green more strongly than the united red and violet, then two such achromatic lenses combined might unite the outstanding green with the red and violet, and thus effect a perfect union of all the colours. Hence he took the combination shewn in fig. 77, for a concave lens, composed of a concave lens \(ab\) of crown glass, and a convex lens \(cd\) of a fluid which had its dispersive power of such a character as to unite the red and violet rays as stated in the figures, and leave the green outstanding and most refracted. He then made an achromatic convex lens, fig. 78, composed of a convex lens \(fg\) of an essential oil, the same as that in \(cd\), which disperses the rays in a lesser degree, and of a concave lens \(fg\) of an essential oil which disperses the rays in a much greater degree. This compound lens has its convexity such as to unite at a convenient distance, rays which diverge from the violet focus of the compound concave lens shewn in fig. 77, and therefore its focal length must be much shorter than the other, like the flint lens in a common achromatic. But though the focal lengths of the two compound lenses are thus different, yet, the distance or deviation of the outstanding green, from the united red and violet, is equal in both.
When these two compound lenses are placed in contact, as shewn in fig. 79, it is manifest that the equal and opposite deviations of the green ray will balance each other, and that this ray will therefore be united with the red and violet, and thus form a pencil exempt from secondary colours. The plates of glass shewn by dotted lines at \(ef\), \(cd\), though necessary when the two compound lenses are separate, as in fig. 77, 79, are of course removed, since the two fluids which they separate, as fig. 78, are the same. Hence the compound object-glass, consists of a concave lens of crown glass \(ab\), of a meniscus \(cf\) of a fluid, and a convex glass of another fluid, enclosed in two glasses like watch-glasses. Dr. Blair found it best practice to make all the glasses concave meniscuses in place of having all the concavity Chromatics in one lens \(ab\).
In continuing his experiments, Dr. Blair happened to try the muriatic acid mixed with a metallic solution; he found it best to make his compound convex lens, as shewn in fig. 78, of crown glass, and that fluid which enabled him to correct the colour of the compound concave in fig. 77, and likewise to correct the aberration of figure by a concave which lengthens only by one-third the focal distance of the convex.
When he was trying a compound concave formed only of crown glass and muriatic acid, he observed that this fluid produced an inverted secondary spectrum, and gave a primary spectrum, in which the green rays were among the most refrangible, and hence he was conducted to the idea of forming a compound lens consisting merely of a single concave lens of muriatic acid placed between a plano-convex and a meniscus of crown glass. In this lens, which he actually constructed and used, he observes that the rays of different colours were bent from their rectilinear course with the same equality and regularity as in reflection.
In such telescopes, Dr. Blair found that when the focal length of the object-glass was nine inches, the aperture might be increased as far as three inches; and in order to distinguish such instruments where the aberration is removed from achromatic ones in which it is only partially removed, he proposes the use of the term aplanatic.
**Sect. IV.—On the tertiary spectrum, and the method of correcting the aberration of colour by prisms and lenses of the same kind of glass.**
The existence of the tertiary spectrum was discovered experimentally by Sir David Brewster, who deduced it also spectrum from the constant ratio of the sines. It is produced when the dispersion of a prism of any substance is corrected by another prism of the same substance with a different refracting angle. An irrationality takes place in the coloured spaces, which prevents the correction of colour from being complete. The residuary spectrum was therefore called the tertiary spectrum merely to distinguish it from the secondary one, which is produced by the specific quality in the refracting media, which act in opposition to each other.
In examining the phenomena of this new spectrum, Sir David Brewster was led to a very paradoxical method of exhibiting it.
Having formed a prism of oil of cassia, with a large refracting angle, and viewing through it the broadest horizontal bar of a window, so that the edges of the bar were free of all colours, he inclined the prism so as to make the bar exhibit at its edges the prismatic colours, as shewn in fig. 81, where the edges \(bM\) and \(eN\) had spectra \(baM\), \(efN\), consisting of the usual red and yellow rays, while the edges \(eM\), \(bN\) had spectra \(edM\), \(bcN\) composed of the usual blue and violet rays. These spectra increased from \(b\) and \(e\) towards \(M\) and \(N\), and at the nodes \(b\), \(e\), where the spectra would have vanished, had each face of the prism received the rays symmetrically, the tertiary spectrum was clearly displayed in the form of a green and purple fringe.
In order to produce refraction without colour by two prisms of the same kind of glass, they may be combined, as in fig. 82, where a ray of light \(R\) incident on the first prism \(AB\) is refracted to the axis \(MF\) at \(F\). The prism \(AB\) has a smaller refracting angle than \(CD\), and is placed in an oblique position, so that its dispersion is increased in a greater ratio. Chromatic than its refraction, for the purpose of correcting the dispersion of CD without balancing its refraction; the prism CD having a position in which its refraction and dispersion are a minimum.
The ray R will therefore converge colourless, and meet the axis at F.
If the prisms have the same refracting angle, and are placed in the position shewn in fig. 83, the ray R will emerge colourless in the direction r. This combination of prisms, as well as that in fig. 82, has the property of expanding all objects viewed through them, in a vertical plane passing through their sections BACD, that is, of magnifying them in one plane. Hence, if we place another similar pair of prisms horizontally, this pair also will magnify objects in a horizontal plane, and by combining these two pair of prisms, we obtain an instrument which will expand or magnify objects in all directions.
This instrument was first constructed by Sir David Brewster in 1812, under the name of a teinoscope, for altering the proportions of objects in plans and drawings, by expanding them differently in rectangular directions; and there is reason to think that Dr. Blair was early acquainted with this method of magnifying objects by prisms. Mr. Archibald Blair a few years ago put into Sir David Brewster's hands an instrument of this kind, composed of four prisms which had been executed by his father, but the date of its construction he had no means of discovering. There cannot, therefore, be the shadow of a doubt that both the principles and the invention of an instrument for magnifying objects by means of prisms, were known and published in Scotland long before the celebrated M. Amici of Modena brought forward a contrivance of the same kind. That M. Amici's invention was an independent one will not be questioned.
As we conceive that a telescope of this kind may have many useful applications, we have given in the annexed figure a sketch of the instrument as actually fitted up for use. It consists of two prisms, AB, AC of the same kind of glass, and having a small refracting angle. Their common line of junction at A is horizontal, and their planes of refraction vertical. Other two similar prisms DE, EF, are placed transversely, their common line of junction at E being vertical, and their planes of refraction horizontal. An object M, therefore, seen through the prisms in the direction OM, by an eye placed at O, will be magnified three, four, or five times, or more according to the inclination and angles of the prisms. It is expanded or stretched out in a horizontal plane by the two first prisms ED, EF, and then expanded and stretched out in a vertical plane by the other two prisms, AB, CD.
If we use homogeneous light, we may construct the instrument with only two prisms, as there is no necessity for correcting the colour with a second prism. For solar observations, the two prisms will constitute a telescope, a darkening glass being used as in other instruments. It will be thus equally useful for viewing the lines in the spectrum, where homogeneous light is necessarily used; and by placing two, three, or four instruments in the same tube, we may obtain any magnifying power we desire. The writer of this article is at present occupied with the construction of one of these instruments with rock salt prisms. The length of the instrument which we have drawn is only two inches and three quarters.
Sect. V.—On the Optical Phenomena of the Spectrum.
Although the discovery of the principle, and the actual construction of achromatic and apochromatic telescopes had directed the attention of many observers to the nature of the prismatic spectrum, yet, with the exception of its varying length in different bodies, and the continuity of its coloured spaces, no attempt was made to question the general account of its phenomena given by Sir Isaac Newton.
Owing to his having used the diameter of the sun as the body from which his spectrum was formed, and to the difficulty of procuring in his day good prisms of glass, Sir Isaac never obtained anything like pure homogeneous light, and was therefore unable to determine the exact boundaries of the coloured spaces. Had the spectrum been observed in the same manner on the planet Mercury and on Saturn, the spectrum produced by the same prism would have been very different. On Mercury the rays would have been less pure and homogeneous than that observed on our earth, and the mean refrangible rays of a different colour; while on Saturn the colour would have been more pure and homogeneous.
1. Discoveries of Dr. Wollaston.
The first person, in so far as we know, who proposed to form the spectrum by using a very narrow pencil of light in place of the sun, was Dr. Wollaston, to whom we owe many most valuable observations on the subject.
"I cannot," says he, "conclude these observations on dispersion, without remarking, that the colours into which a beam of white light is separable by refraction, appears to me to be neither seven, as they usually are seen in the rainbow, nor reducible by any means (that I can find) to three, as some persons have conceived; but that by employing a very narrow pencil of light, four primary divisions of the prismatic spectrum may be seen with a degree of distinctness, that I believe has not been described nor observed before. If a beam of daylight be admitted into a dark room by a crevice \( \frac{1}{9} \) of an inch broad, and received by the eye at the distance of ten or twelve feet through a prism of flint glass, free from veins, held near the eye, the beam is seen to be separated into the four following colours only, red, yellowish green, blue, and violet, in the proportions represented in the figure.
"The line A that bounds the red side of the spectrum is somewhat confused, which seems in part owing to the want of power in the eye to converge red light. The line B, between red and green in a certain position of the prism, is perfectly distinct; so also are D, and E, the two limits of violet. But C, the limit of green and blue, is not so clearly marked as the rest; and there are also, on each side of this limit, other distinct dark lines, f and g, either of which, in an imperfect experiment, might be mistaken for the boundary of these colours.
"The position of the prism in which the colours are most clearly divided, is when the incident light makes about equal angles with two of its sides. I then found that the spaces AB, BC, CD, DE, occupied by them, were nearly as the numbers 16, 23, 36, 25." Dr. Wollaston adds, that when the inclination of the prism is altered so as to increase the dispersion of the colours, the proportions of them to each other are then also changed, so that the spaces AC and CE, instead of being as before 39 and 61, may be found altered as far as 42 and 58.
The lines which Dr. Wollaston has described in the preceding observations, are called the fixed lines in the spectrum, and may be considered, as we shall presently have reason to see, as one of the most valuable observations which have been made on this subject. He owed the discovery solely to his having used a narrower line of light, and had he employed a still narrower and brighter line, he would have seen many more lines.
In considering Dr. Wollaston's observations, and comparing the results with those of Sir Isaac Newton, we must carefully attend to the circumstance, that he used a beam of day-light, not one of sun-light, and as this beam emanated from the blue sky, and was light which had been greatly modified by the action of the atmosphere, as we shall soon show, his estimate of the number and nature of the coloured spaces, does not in the least affect or invalidate the observations of preceding authors. The sun's light used by Newton had lost many of its rays, by the absorptive action of the atmosphere, before it fell upon Dr. Wollaston's prism. In consequence of taking it for granted that Dr. Wollaston was analysing the same kind of light that Sir Isaac Newton analysed, both he and Dr Young were misled in the interpretation of the phenomena. Speaking of the observations of Newton and his followers, Dr. Young says, "The observations were however imperfect, and the anatomy was wholly imaginary." Dr Wollaston has determined the division of the coloured image or spectrum, in a much more accurate manner than had been done before: by looking through a prism, at a narrow line of light, he produces a more effectual separation of the colours, than can be obtained by the common method of throwing the sun's image on a wall. The spectrum formed in this manner consists of four colours only, red, green, blue, and violet, which occupy spaces in the proportion of 16, 23, 36, and 5, respectively, making together 100 for the whole length; red being nearly one-sixth, the green and the violet each about one-fourth, and the blue more than one-third of the length. The colours differ scarcely at all in quality within their respective limits, but they vary in brightness; the greatest intensity of light being in that part of the green which is nearest to the red. A narrow line of yellow is merely visible at the limit of the red and green; but its width scarcely exceeds that of the aperture by which the light is admitted, and Dr. Wollaston attributes it to the mixture of the red with the green light. There are also several dark lines crossing the spectrum within the blue portion and in its neighbourhood, in which the continuity of the light seems to be interrupted. This distribution of spectrum Dr. Wollaston has found to be the same whatever refracting substance may have been employed for its formation; and he attributes the difference, which has sometimes been observed in the proportion, to accidental variations of the obliquity of the rays." Hence Dr. Young was led to suppose that the yellow line was the accidental union of the extremity of the red and green spaces,—to regard yellow as a mixture of red and green light; and to suppose that the green space consisted only of homogeneous green without any mixture of yellow. "In consequence," says he, "of Dr. Wollaston's correction of the description of the prismatic spectrum compared with these observations, it became necessary to modify the supposition that I advanced in the last Bakerian lecture respecting the proportions of the sympathetic fibres of the retina; substituting red, green, and violet, for red, yellow, and blue." In this manner the yellow space was struck out of the spectrum on the authority of Dr. Wollaston's observations!
2. Discoveries of Fraunhofer.
Without knowing anything of the discovery of fixed lines in the sky light by Dr. Wollaston, M. Fraunhofer, a celebrated practical optician at Benedictbairn near Munich, made a series of the most beautiful discoveries respecting the spectrum, which he published in 1814 and 1815. By making use of prisms of uniform density, and entirely free of veins, and by excluding all extraneous light, and stopping the rays which formed the coloured spaces which he was not examining, he made the important discovery that the solar Lines in spectrum was covered with a great number of black lines the spectrum was covered with a great number of black lines of different thicknesses parallel to each other, and perpendicular to the length of the spectrum.
All kinds of prisms, fluid or solid, provided they were good, exhibited the same lines, and Fraunhofer found, that these lines had a fixed position in the spectrum, and that they varied with the length of the spectrum, the distance between any two affording a precise measure of the action of the prism on the rays in which these two lines were placed. These lines are darker than the rest of the spectrum, and some of them appear entirely black. The largest lines could scarcely be seen if the aperture exceeded a minute, and the finest lines also disappeared entirely when the aperture was 40°. The aperture used by Fraunhofer was 36" high, and 15" wide, that is, at a distance of 24 feet from the aperture. It was nearly one-fiftieth of an inch wide, and 2-88 inches high. The prism was made of flint glass, had a refracting angle of nearly 60°, and was placed before the object-glass of the telescope, so that the angles of incidence and emergence were equal, or the angle of refraction a minimum. This apparatus is shown in fig. 1, Plate CCCLXXXI, where the prism is seen in front of the object-glass of the telescope of a repeating theodolite resting on a horizontal plane with a steel axis round which it moves. The box on which the axis turns is firmly united with the telescope.
When the prism was turned round, so as to increase the angle of incidence, the lines disappeared, and the same took place when the angle of incidence was diminished. But the lines reappeared at a greater incidence by shortening, and at a smaller incidence by lengthening the telescope.
The solar prismatic spectrum, as seen by Fraunhofer, is represented in fig. 2 of Plate CCCLXXXI which has been Fig. 2 abridged, and many of the lines necessarily omitted, Fraunhofer himself, having been obliged to leave them out of his map. At the line A the red space nearly terminates, and the violet space at B; but when the light of an illuminated cloud fell upon the aperture in the prism, the spectrum appeared to terminate on one side at B, and on the other between G
Dr. Young has given in his Elements of Nat. Phil. vol. I. p. 786, plate 29, a small coloured drawing of the spectrum as seen by Dr. Wollaston and himself, with the yellow line. This line has no existence in the true solar spectrum. At A there is a distinct and well-defined line, the boundary of the red space being a little beyond it. At B there is a mass of lines, forming together a band darker than the adjacent parts. The line at B is very distinct, and of considerable thickness. From B to C may be reckoned nine very delicate and well-defined lines. The line at C is broad, and black like D. Between C and D are found nearly 30 very fine lines, which, however, with the exception of two, cannot be perceived but with a high magnifying power; and with prisms of great dispersion; they are besides well defined. The same is the case with the lines between B and C. The line D consists of two strong lines separated by a bright one. Between D and E we recognize about 84 lines of different sizes. That at E consists of several lines, of which the middle one is the strongest. From E to b there are nearly 24 lines. At b there are three very strong ones, two of which are separated by a fine and clear line. They are among the strongest in the spectrum. The space bF contains nearly 52 lines, of which F is very strong. Between F and G there are about 185 lines of different sizes. At G many lines are accumulated, several of which are remarkable for their size. From G to H there are nearly 190 different lines. The two bands at H are of a very singular nature. They are both nearly equal, and are formed of several lines, in the middle of which there is one very strong and deep. From H to I they likewise occur in great numbers. Hence it follows that, in the space BH, there are 574 lines. The relative distances of the strongest lines were measured with the theodolite, and placed in the figure from observation. The faintest lines only were inserted from estimation by the eye.
The lines F and G in the solar spectrum which we have thus minutely described after Fraunhofer, are not seen in the spectra formed by any white flame, or white light, whether it is generated by ordinary combustion, or produced by the application of intense heat to a solid body. In the flame of a lamp, however, Fraunhofer discovered that there is a double yellow line occupying exactly the same place as the double line D, the two black lines of D corresponding with the two luminous ones of the double yellow line in lamp light. Hence it follows that ordinary white light, produced in the manner already mentioned, has 590 rays of a definite refrangibility which do not exist in solar light, and hence the black lines have been called defective rays or lines.
"Various experiments," says Fraunhofer, "and changes to which I have submitted these lines, convince me that they have their origin in the nature of the light of the sun, and that they cannot be attributed to illusive observations, or any other secondary cause."
By means of the apparatus shewn in fig. 1, Plate CCCLXXXI, Fraunhofer determined in a very accurate manner the distances between the principal fixed lines B, C, D, E, F, G, H, taking those which divided the spectrum most conveniently. The line b for example, would have been better than E for its magnitude and distinctness, but it does not divide the space DF so equally. He repeated these observations with different kinds of flint and crown glass of several fluids, and obtained the results given in the following table.
| Different combinations of Refracting Media | Temperature (°C) | Specific gravity | Angle of the prism | Angle of deviation | |-------------------------------------------|-----------------|-----------------|--------------------|--------------------| | Flint glass, No. 13 | 65 | 3.723 | 26 24 30 17 27 8 | 3 16 | | Crown glass, No. 9 | 63 | 2.535 | 39 20 35 22 38 19 | 2 44-5 | | Water | 65 | 1.000 | 58 5 40 22 36 40 | 3 24 | | Sol. of potash in water | 65 | 1.000 | 58 5 40 22 36 40 | 3 12-4 | | Oil of turpentine | 52 | 1.416 | 58 5 40 27 45 56 | 4 2 10 26 12 54 | | Flint glass, No. 3 | 65 | 0.885 | 58 5 40 39 20 12 | 4 56 | | Flint glass, No. 30 | 65 | 3.512 | 27 41 35 17 35 16-6| 3 8 | | Crown glass, No. 13 | 65 | 3.695 | 21 42 15 14 3 9 | 2 35-6 | | Crown glass | 65 | 2.535 | 42 37 26 25 36 44 | 3 5 | | Flint glass, No. 23 | 65 | 2.756 | 42 56 40 25 39 13 | 3 32-8 | | Flint glass, No. 23 | 65 | 3.724 | 50 15 42 49 55 13-2| 11 12-6 | | Flint glass, No. 23 | 65 | 3.724 | 45 23 14 32 45 12-2| 6 26 |
These valuable data were deduced from measures taken six times for each substance; but as the theodolite was only twenty-four feet distant from the window of his dark room, it became necessary to apply a correction to the angle of deviation μ, arising from the distance 4.25 inches of the centre of the prism from the axis of the theodolite. This correction would have been very great for twenty-four feet, and therefore Fraunhofer, to avoid the uncertainty which
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1. The reader will be desirous of knowing, which of these principal lines were discovered by Dr. Wollaston. The following attempt to do this is given by Sir David Brewster in his Report on Optics, published in the Proceedings of the British Association, vol. i., p. 320, note 2. In the spectrum formed by a narrow beam of day-light, Dr. Wollaston had previously to 1802, discovered seven lines, which he has designated by the letters A, B, f, C, g, D, E, the first line A being, according to his observations, the extreme boundary of the red rays, and the last line E the extreme boundary of the violet rays. The correspondence of these lines with those of Fraunhofer, I have with some difficulty ascertained to be as follows—
A, B, f, C, g, D, E Wollaston. B, D, f, F, G, H, Fraunhofer.
There is no single line in Fraunhofer's drawing in the spectrum, (nor is there any in the real spectrum) coincident with the line C Wollaston; and indeed he himself describes it as not being "so clearly marked as the rest." I have found, however, that this line C corresponds to a number of lines half way between b and F, which, owing to the absorption of the atmosphere, are particularly visible in the light of the sky near the horizon. In order to have seen the lines B and H of Fraunhofer, especially the last, Dr. Wollaston's "beam of day-light" must have come from a part of the sky very near the sun's disc. When the dispersive power of the body under examination is very great, the value of the index of refraction \( m \) given by this last formula will not be rigorously correct, as the equality of the angles of incidence and emergence can only take place for one ray.
Fraunhofer, therefore, measured the distances BC, CD, or when the distance of the two lines B and C, C and D was the smallest, which takes place when the ray or line which bisects these spaces has its angle of incidence and emergence equal. When the substances have a less dispersive power, or the prisms a smaller angle, the same care is not necessary to obtain this degree of accuracy.
If we then call \( E_m \) the index of refraction for the ray E, we have
\[ E_m = \frac{\sin \frac{1}{2} (\mu + \psi + DE)}{\sin \frac{1}{2} \psi} \]
and for the ray F,
\[ F_m = \frac{\sin \frac{1}{2} (\mu + \psi + DE + EF)}{\sin \frac{1}{2} \psi} \]
In this way Fraunhofer obtained the following indices of refraction for the different solids and fluids formerly used:
| Refracting Media | Bm | Cm | Dm | Em | Fm | Gm | Hm | |------------------|----|----|----|----|----|----|----| | Flint glass, No. 13 | 1-627749 | 1-629681 | 1-635036 | 1-642024 | 1-648260 | 1-660285 | 1-671062 | | Crown glass, No. 9 | 1-525832 | 1-526849 | 1-529587 | 1-533005 | 1-536052 | 1-541657 | 1-546556 | | Water | 1-330935 | 1-331712 | 1-333577 | 1-335851 | 1-337818 | 1-341293 | 1-344177 | | Water | 1-330977 | 1-331709 | 1-333577 | 1-335849 | 1-337788 | 1-341261 | 1-344162 | | Solution of potash in water | 1-396929 | 1-400515 | 1-402805 | 1-405632 | 1-408082 | 1-412579 | 1-416368 | | Oil of turpentine | 1-470496 | 1-471530 | 1-474434 | 1-478353 | 1-481736 | 1-488198 | 1-493874 | | Flint glass, No. 3 | 1-602042 | 1-603800 | 1-608494 | 1-614332 | 1-620042 | 1-630772 | 1-640373 | | Flint glass, No. 30 | 1-623570 | 1-626477 | 1-630585 | 1-637356 | 1-643466 | 1-655406 | 1-660672 | | Crown glass, No. 13 | 1-524312 | 1-525299 | 1-527982 | 1-531372 | 1-534337 | 1-539908 | 1-544684 | | Crown glass | 1-554474 | 1-555933 | 1-559075 | 1-563150 | 1-566741 | 1-573335 | 1-579476 | | Flint glass, No. 23, prism of 60° | 1-626596 | 1-628469 | 1-633667 | 1-640495 | 1-646756 | 1-658848 | 1-669686 | | Flint glass, No. 23, prism of 45° | 1-626564 | 1-628451 | 1-633666 | 1-640544 | 1-646780 | 1-658849 | 1-669680 |
Before he proceeded to employ these results to the construction of achromatic telescopes, Fraunhofer endeavoured to determine by exact measurement the illuminating power of the spectrum at different points. In order to do this, he constructed an apparatus represented in figs. 3 and 4, Plate CCCXXXI. To an eye-glass E, made on purpose for the telescope of the theodolite, he applied a small plane metallic mirror \( a \), the edge of which being well defined, cut the field of the telescope in the middle, as shewn in the figure. It was placed at an angle of 45° to the axis of the object-glass A, and in its focus. The eye-glass E is pulled out till the edge of the small speculum \( a \) is distinctly seen. At the side of the eye-glass, and in a direction at right angles to the edge of the speculum \( a \), or to the axis of the telescope, he fixed a tube \( b \), cut at the point \( b \) in the direction of its length, and in this opening he placed a narrow and a shorter tube MN, whose section is seen at \( b \), crossing the larger tube \( cB \) at right angles. A small flame, supplied with oil from an external vessel, was placed in the tube MN, so as to be in the axis of the tube \( cB \). At the point of the narrow tube \( b \) or MN, where it was cut by the axis of the tube \( cB \), was a small round aperture for allowing the light of the flame to fall upon the speculum \( a \). Hence it follows that the eye at E will see in half of the field the speculum \( a \) illuminated by the flame, and in the other half the colours of the spectrum formed by a prism placed, as formerly described, before the object-glass A. By making the tube MN and the flame approach the speculum, we increase its degree of illumination, and can therefore make it equal to the illumination of the part of the spectrum which we wish to determine. In this way he obtained for each coloured space in the spectrum a certain distance of the flame from the speculum, which afforded a measure of the intensity of illumination, the squares of the distances being inversely as the intensities.
"Though at first sight," says Fraunhofer, "it appears difficult to compare the light of two different colours, yet it becomes easy by a little practice. The intensity of the light of the mirror approaches more to that of any colour in the spectrum, if at the same position of the eye-glass its vertical margin is less distinct. If the mirror is adjacent to a part of the spectrum more or less illuminated, the edge of the mirror becomes, in both cases, more distinct; because, in the first case, the mirror appears to be placed in the shadow, and in the second case, it is the colour of the spectrum that is found there. The experiment with the mirror is a little difficult and uncertain, if we perceive clearly the lines of the spectrum, because the brightest and the darkest lines touch one another almost in every colour. On this account the aperture in the window-shutter is made so broad, that only the strongest lines are just visible, and the fine ones not at all. In place of the mirror outside of the shutter by which the light entered, I put a white plane surface illumi- Prismatic rays coming from the edge of the field, cut the axis of the telescope. With the prism of flint glass, No. 13, having an angle of 26° 24' 5', I obtained the following results. Though the experiments were made in clear weather, and at noon, I sometimes perceived, in the course of the observations, a slight change in the density of the light which the prism received. The differences in the four sets of experiments may have been partly owing to this change, and the flame may also have changed its intensity in the course of the observations. If we call the intensity of the light at the brightest part of the spectrum 1, we shall then have:
| Points of the Spectrum where the illumination was measured | Intensity of light | Intensity of light | Intensity of light | Mean Intensity by four experiments | |------------------------------------------------------------|-------------------|-------------------|-------------------|----------------------------------| | At the line B | 0-0100 | 0-044 | 0-053 | 0-020 | | At C | 0-0480 | 0-096 | 0-15 | 0-084 | | At D | 0-6100 | 0-590 | 0-72 | 0-62 | | At 2-7ths of DE from E | 1-0000 | 1-000 | 1-00 | 1-000 | | At F | 0-4400 | 0-38 | 0-61 | 0-49 | | At G | 0-0840 | 0-14 | 0-25 | 0-19 | | At H | 0-0100 | 0-029 | 0-053 | 0-032 | | At | 0-00110 | 0-0072 | 0-009 | 0-005 |
Fraunhofer found the brightest part of the spectrum at the distance of nearly one-third or one-fourth of DE from D.
The results of the preceding experiments are expressed in the curve, fig. 5, which accompanies the spectrum in fig. 2, the preceding values in the last column of the table being the ordinates of the curve, and the angular distances BC, CD in the table in page 416, for flint glass, No. 13, being the abscissa.
If we suppose that the quantity of the light in the differently coloured spaces is represented by the areas of the curves BC, CD, we shall obtain the following results, the area of the space DE being made equal to unity.
M. Fraunhofer next proceeds to apply these interesting results to the construction of achromatic combinations for telescopes. From the results in the table in p. 416; he obtains the following ratios of the different dispersive powers of the differently coloured rays in the different combinations mentioned in the first column.
| Refracting Media | Ca'−Ba' | Da'−Ca' | Ea'−Da' | Fa'−Ef | Ga'−Fa' | Ha'−Ga' | |------------------|---------|---------|---------|--------|---------|---------| | Flint glass, No. 13, and water | 2-562 | 2-871 | 3-073 | 3-193 | 3-460 | 3-726 | | Flint glass, No. 13, and crown glass, No. 9 | 1-900 | 1-956 | 2-044 | 2-047 | 2-145 | 2-195 | | Crown glass, No. 9, and water | 1-349 | 1-468 | 1-503 | 1-560 | 1-613 | 1-697 | | Oil of turpentine and water | 1-371 | 1-557 | 1-733 | 1-732 | 1-860 | 1-963 | | Flint glass, No. 13, and oil of turpentine | 1-868 | 1-844 | 1-783 | 1-843 | 1-861 | 1-899 | | Flint glass, No. 13, and kali | 2-181 | 2-333 | 2-472 | 2-545 | 2-674 | 2-844 | | Kali and water | 1-175 | 1-228 | 1-243 | 1-254 | 1-294 | 1-310 | | Oil of turpentine and kali | 1-167 | 1-268 | 1-386 | 1-381 | 1-437 | 1-498 | | Flint glass, No. 3, and crown glass, No. 9 | 1-729 | 1-714 | 1-767 | 1-808 | 1-914 | 1-956 | | Crown glass, No. 13, and water | 1-309 | 1-436 | 1-492 | 1-518 | 1-604 | 1-651 | | Crown glass and water | 1-337 | 1-682 | 1-794 | 1-839 | 1-956 | 2-052 | | Crown glass, No. 2, and crown glass, No. 13 | 1-174 | 1-171 | 1-202 | 1-211 | 1-220 | 1-243 | | Flint glass, No. 13, and crown glass | 1-667 | 1-704 | 1-715 | 1-737 | 1-770 | 1-816 | | Flint glass, No. 3, and crown glass | 1-517 | 1-494 | 1-482 | 1-534 | 1-579 | 1-618 | | Flint glass, No. 30, and crown glass | 1-932 | 1-904 | 1-997 | 2-061 | 2-143 | 2-233 | | Flint glass, No. 23, and crown glass, No. 13 | 1-904 | 1-940 | 2-022 | 2-107 | 2-168 | 2-268 | The important results embodied in the preceding table completely overturn the opinion of Dr. Wollaston respecting the proportionality of the different colours, and establish beyond all question the irrationality of the coloured spaces. In the very first combination, for example, of flintglass and scotter, the ratio of the dispersion of the rays in the red space BC, is as 1 to 2:56, whereas in the violet space GH, it is as high as one to 3:726. In the combination of flintglass and oil of turpentine, we have a case where the irrationality is very trifling, and what Fraunhofer has not observed, the irrationality is nothing between the orange and blue spaces, and almost nothing between the red and indigo, but very considerable between the green and all the other spaces, and a maximum between the green and the violet.
The differences of the ratios, too, are negative or diminishing in the two first or least refrangible spaces, and positive in all the rest; the negative differences nearly balancing the positive ones; whereas, with very trifling exceptions, the ratios increase towards the most refrangible extremity of the spectrum.
It is obvious that in the construction of achromatic object-glasses of flint and crown glass, there must always be a considerable portion of outstanding colours or aberrations, as Fraunhofer calls it. Hence in place of making the focal lengths of the two lenses in the exact ratio of their dispersive powers, we must alter the ratio so as to make the outstanding colours a minimum for the distinct vision of objects. But in order to do this, we must take into account the intensity of illumination in the coloured spaces; for it is obvious that a given irrationality in the yellow spaces will, from the intensity of its illumination, be much more regarded than a similar irrationality in the violet space. As the outstanding rays, therefore, will be injurious to distinct vision in proportion to their intensity, the distinctness will be a maximum, as Fraunhofer has shewn, when the ratio of dispersion \( x \) is taken, such that
\[ x = \frac{b + c + d + e + f + g}{\beta + \gamma + \delta + \epsilon + \zeta + \eta} \]
in which case we have
\[ (x-b) + (x-c)\delta + (x-d)\epsilon + (x-e)\zeta + (x-f)\eta + (x-g)\eta = 0 \]
where \( b, c, d, e, \) &c. express the quantity of light in the spaces BC, CD, DE, &c. and \( b, c, d, e, \) &c., the quotients or ratios of dispersion
\[ \text{Cm'} - \text{Bm'}, \quad \text{Dm'} - \text{Dm'}, \quad \text{Em'} - \text{Em'}, \quad \text{Fm'} - \text{Fm'}, \quad \text{Gm'} - \text{Gm'} \]
This ratio, or the value of \( x \) calculated for flint glass, No. 30, and crown glass, No. 13, is as 1 to 2:012; but Fraunhofer found that in an object-glass, consisting of these two kinds of glass, the distinctness was a maximum when the ratio was as 1 to 1:98.
To the Chevalier Fraunhofer we owe also the discovery of fixed lines in the light of the planets and fixed stars. As the light of our sun is defective in so many rays, it was to be expected that the light of all the planets which he illuminates, would be equally defective in the same rays. In the brighter colours of the Moon's spectrum, we find the same fixed lines as in the sun's light, and in the same place. The spectra from the light of Mars and Venus contained the lines D, E, b, and F, of solar light, and precisely in the same place. In the spectrum of Sirius he was unable to perceive fixed lines in the orange and yellow colours, but in the green he saw a very strong streak, and in the blue other two very strong ones, having no resemblance to any of the lines in the solar spectrum. Castor gives a spectrum resembling that of Sirius. The streak in the green was so intense, that notwithstanding the weakness of the light, yet he ascertained by measurement that it occupied the same place as the green streak in Sirius. He distinguished also the two Spectrum streaks in the blue, but he could not ascertain their place.
In the spectrum of Polux he found many weak and fixed lines, which resembled those of Venus. The line D he saw distinctly, and occupying the same place as in the pure light. In the spectrum of Capella, he saw the lines D and b, as in solar light. The spectrum of Betelgeus contains numerous fixed lines, which in a favourable atmosphere are sharply defined. There were lines like the solar ones D and b. In the spectrum from Procyon, some lines were perceived with difficulty, but they were not sufficiently distinct to be measured. In the orange space, however, he saw a line at D.
Sect. VI.—On the Physical Properties of the Spectrum.
The physical properties of the spectrum, which have been the subject of experimental investigation, are its heat-producing power, and the chemical and apparently magnetic influence of its rays.
Heating power of the spectrum. That the heat of the Heating coloured rays should be most intense where their light was strongest, was long the general belief of philosophers; and Landriani Rochon, and Sennebier, found by direct experiment that the highest temperature existed in the yellow space. Sir W. Herschel, however, found that the heating Sir W. power increased from the violet to the red space, and that Herschel the thermometer continued to rise when placed beyond the visible red extremity of the spectrum. He therefore drew the conclusion that there were invisible rays in the light of the sun which had the power of producing heat, and which had a less degree of refrangibility than red light. Sir W. Herschel attempted, in vain, to determine the index of refraction of the extreme invisible ray which possesses the power of heating; but he ascertained that at a point one-and-a-half inches distant from the extreme red ray, the invisible rays exerted a considerable heating power, even though the thermometer was placed at the distance of fifty-two inches from the prism.
In 1801, Sir Henry Englefield repeated these experiments; but he does not acquaint us with the kind of glass Englefield of which his prism was made. He obtained the following results, which confirm those of Sir William Herschel.
| Colours of the Spectrum | Temperature | |------------------------|-------------| | Blue | 56 | | Green | 58 | | Yellow | 62 | | Red | 72 | | Beyond Red | 79 |
From our author's own account of the method of making these experiments, we place no confidence in the principal result respecting the invisible rays. "As I had nothing to do with light," says he, "it was not necessary to darken the room; and as I wished to accumulate as large a portion of solar heat as possible, I placed the prism in an open window." As the whole interest of these experiments was concentrated in the determination of invisible heating rays, Sir Henry had a great deal to do with light, as the whole question turned upon an exact appreciation of the termination of the spectrum. In a dark room the spectrum is much longer than in open day, and we have reason to believe from experiment, that Sir Henry Englefield's spectrum did not visibly extend beyond the line C of Fraun-
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1. The Abbé Rochon used a prism of flint glass, and a thermometer containing spirits of wine, and he found the maximum temperature in the yellow-orange rays. See his Observations, 1783.
Dr. Hutton, in his Philosophy of Light and Heat, Edin. 1794, p. 38, remarks that, "the compound light which is white, has a greater power of giving vision in proportion to its power of exciting heat; whereas in the red species, it is the opposite, for here the power of exciting heat is great in proportion to its power of giving vision." Prismatic hofer, so that his maximum temperature of 79° was actually found in the red rays.
With the view of throwing light upon this subject, Sir David Brewster has endeavoured to ascertain the visible extent of the spectrum by various methods of condensing the light, and absorbing by coloured media, the luminous parts of the spectrum. By these means he has traced the visible spectrum, and the fixed lines in it as far beyond the line A as the distance of the group of lines α is from A, and has seen it indistinctly to a distance as great as AB beyond A. Hence there cannot be the least doubt that the experiments beyond the visible red were actually made when the thermometer was placed in the red space. He does not, however, conclude from this that there are no invisible rays beyond the red; but merely that the experiments of Herschel and Englefield were made in a part of the spectrum where rays of light actually exist. On the contrary, our author concludes that there are rays of heat of all degrees of refrangibility, and consequently consisting of waves of all degrees of breadth and velocity. When produced by a slight vibratory movement, the waves of heat are broad and slow; as the temperature rises they become narrower and quicker in their motion. When their velocity is such as to equal that of the extreme red ray, they become faintly visible, and the other colours are successively produced by quicker motion, till white light is radiated. This seems to be the process by which incombustible bodies are gradually raised from the deepest red to the brightest white; and if we examine, by means of a prism, the changes which take place in the gradually increasing light, we shall find that the different rays of the spectrum are successively added to the red light. He conceives, therefore, that the sun emits rays of all degrees of refrangibility, extending probably far beyond the visible extremity of the violet, and though not capable of being rendered sensible, yet exercising powerful influences in the economy of nature.
M. Berard, and Sir Humphry Davy, obtained results analogous to those of Sir W. Herschel; M. Berard finding the maximum heat at the very extremity of the red rays, and Sir H. Davy beyond it.
The most valuable series of experiments on this subject, were made by Professor Wunsch, and Dr. Seebeck of Berlin. So early as 1807, Professor Wunch had made experiments with prisms of various substances, and obtained the following results:
| Substances of which the prisms were made | Place of maximum heat | |------------------------------------------|----------------------| | Alcohol | Yellow space | | Oil of turpentine | Yellow space | | Water | Yellow space | | Green glass | Red | | Yellow glass | Extreme red |
These results were confirmed by Dr. Seebeck, who obtained the following new results:
| Substances of which the prisms are made | Place of maximum heat | |-----------------------------------------|-----------------------| | Sulphuric acid concentrated | Orange | | Solution of sal-ammoniac | Orange | | Solution of corrosive sublimate | Orange | | Crown glass | Middle of red | | Plate glass | Middle of the red | | Flint glass, English | Beyond the red | | Flint glass, Bohemian | Beyond but nearer the red |
The explanation which was given of these results, (which had been ascribed to different prisms of refracting caloric,) by Sir David Brewster, accounts in a very satisfactory manner for all the phenomena. He conceives that transparent Spectrum bodies have the same power of absorbing or stopping certain rays of the thermometric spectrum, as Dr. Robison called it, in the same manner as coloured bodies have the power of stopping certain rays of the luminous spectrum. These last bodies necessarily became coloured by stopping certain rays; but as the eye is not sensible to heat in the same manner as to light, the absorptive power of transparent bodies for heat can only be proved by the thermometer. He considers water as the type of bodies which are uniformly transparent for heat, as its maximum of heat coincides with its maximum of light. A prism of crown glass, on the contrary, is less uniformly transparent for heat; and its maximum of heat is in the red space, because it has absorbed much of the heat in the yellow space. In like manner flint glass has absorbed more of the heating rays in the red than the crown glass, and hence its maximum is about the extremity of the red, or beyond the end of the spectrum as commonly seen. In coloured media the maximum ordinate of their luminous spectrum, shifts along the whole prismatic spectrum; sometimes there are two or more maxima of light, and sometimes narrow and wide spaces entirely defective in light. Hence Sir David Brewster supposes that there are defective spaces, and lines in the thermometric spectrum.
This view of the subject suggests a new mode of investigating the phenomena of the heating rays. If we take a prism of coloured glass to investigate the dark spaces and lines produced by absorbing media, we shall only have a very imperfect approximation to the true result; that is, we never could have absolutely dark spaces in the spectrum, as long as all the rays that went to the formation of the spectrum, passed through all the different thicknesses of the prism. The thinnest parts of the prism allow all the rays to pass, and consequently illuminate the whole spectrum, so that the actions of various thicknesses of the media are confounded, and the real absorptive action at a given thickness concealed. In like manner in the spectrum of heat, the heating rays which pass through the thin parts of the prism, will throw heat into every part of the spectrum; and hence the experiments should be made with the frustums of prisms, where the difference of thickness is small, and the want of area made up by an increased height of the prism. The best way, however, of making the experiments, would be to use a compound prism constructed as in the annexed figure, where AB, MN, is the section of a compound prism consisting of four frusta of prisms, AB, ab, a'b', &c., the frustum AB, being part of a prism ABC, the frustum ab, part of a prism abc, &c., or this compound prism, in place of being ground out of the mass, which would be difficult though practicable, might be composed of a single prism ABC, with parallel plates of the same glass added by cement so as to compose the notched parallelogram ABMN. Very interesting results might also be obtained by using spectra formed by interferences in the manner we shall afterwards describe.
In all experiments with fluid prisms, the results are perplexed with the effects of the plates of glass by which the prisms are confined, in the same manner as we would disturb and indeed nullify the results respecting the absorption of light.
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1 See Professor Powell's valuable Report on Radiant Heat, British Association Reports, vol. I. p. 293. 2 Magazine der Gesellschaft, &c. 3 Turner's Chemistry, 3d Edit. p. 84. 4 Professor Powell's Report, p. 293, 294. the red rays, we cannot attach any definite meaning to these prismatic experiments.
MM. Gay Lussac and Thenard discovered a very energetic chemical action of the solar rays. On exposing to a pencil of solar light, a mixture of hydrogen gas and caloric, in equal volumes, a detonation of the mixed gases took place, and hydrochloric acid (muriatic acid) was formed.
M. Berard repeated the experiments with muriate of silver, and with the preceding mixture of gases which he placed in the different coloured spaces of the spectrum, and he found that the chemical action was in every case more powerful towards the violet extremity and a little beyond it. M. Berard likewise concentrated the least refrangible half of the spectrum by means of a lens, and then the most refrangible half. The latter, though the most intense, produced no effect upon the muriate of silver, but the former blackened it in less than ten minutes.
Mrs. Somerville found that the chemical rays passed as freely through blue glass coloured with cobalt, as through colourless glass. Having dipped a piece of paper in a solution of muriate of silver, and cut it into two parts, one of them was placed under a blue glass, and the other under a white glass at the same instant. The one did not become black more than the other, and there was no difference in the intensity of their colour.
Dr. Thomas Young made a very interesting experiment with the view of determining if the invisible chemical rays interfered with the luminous ones. He produced the Newtonian rings with a thin plate of air, and having formed an image of them by means of the solar microscope, he threw this image upon paper dipped in a solution of nitrate of silver, and placed it at the distance of about nine inches from the microscope. "In the course of an hour," says he, "portions of three dark rings were very distinctly visible, much smaller than the brightest rings of the coloured image, and coinciding very nearly, in their dimensions, with the rings of violet light that appeared upon the interposition of violet glass. I thought the dark rings were a little smaller than the violet rings, but the difference was not sufficiently great to be accurately ascertained. It might be as much as one-thirtieth, or one-fortieth of the diameter, but not greater."
A more decisive experiment was afterwards performed by M. Arago, who formed a set of fringes, by the interference of two solar pencils, proceeding from a common origin, and having kept them very steadily for a long time upon the same part of a piece of paper rubbed with muriate of silver, a series of black lines were traced upon leaving their intervals, smaller than those of the dark and bright fringes formed by violet light.
In the summer of 1831, we had the satisfaction of being Sir John Herschel shown by Sir John Herschel a very interesting experiment on the chemical action of the violet rays. When a solution of platinum in nitro-muriatic acid is mixed with lime water, no precipitation to any considerable extent takes place in the dark, a slight floccy sediment only being formed after long standing. But if a fresh mixture, or an old one cleared by subsidence of this sediment, is exposed to the sun's rays, it instantly becomes milky, and a white precipitate is copiously formed. If the solution of platinum is in excess, the precipitate is of a pale yellow colour. In the common light of a cloudy day, the same effect is produced more slowly. When tubes containing the mixture are exposed within red fluids, or even yellow ones which absorb the violet rays, no precipitation takes place.
The following remarkable facts were published in the Journal de Pharmacie for March 1832, but the abstract
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1 Mem. de la Société d'Arcueil, 1817, tom. iii. A full account of these experiments, occupying a whole chapter of eighteen pages, will be found in Biot's Traité de Physique, tom. iv. p. 600. 2 Traité de l'Air et du Feu, § 66. 3 Phil. Trans. 1826, part ii. p. 136. 4 Biot's Traité de Physique, tom. iv. p. 673, 674. 5 Phil. Trans. 1826, part ii. p. 136. 6 Elements of Nat. Phil. vol. ii. p. 647. 7 See Lond. and Edin. Phil. Mag. No. 1, July 1832, p. 58. Prismatic from which we quote them, does not mention the name of Spectrum, the author. A solution of peroxalate of iron, when kept in the dark or exposed to the heat of boiling water, suffers no change; but if it is exposed to the sun's rays, an infinite number of bubbles of carbonic acid are disengaged, and the solution is like a syrup undergoing strong fermentation.
A solution of muriate of platinum is decomposed by the oxalic acid, and the oxalates by the sun's light, with almost as energetic a disengagement of gas as the solution above mentioned.
Muriate of gold, when reduced by oxalic acid through the agency of light, covers the interior surface of the vessel with an uniform brilliant gilding, which is of a pure green colour by reflected light.
Oxalate of silver under water, is partially decomposed by solar heat (not by culinary heat) into metallic silver and carbonic acid. If the reddish brown solution of muriate of iridium in water, mixed with oxalic acid and heated to ebullition, is exposed to the sun, total discoloration promptly takes place, carbonic acid is disengaged, and grey metallic iridium precipitated.
Magnetic influence of the solar rays.—Though many interesting and apparently accurate experiments have concurred to indicate the existence of this property of light, yet recent inquiries have thrown a considerable doubt upon the conclusions which have been drawn from them. Dr. Morichini, a Roman physician, first succeeded in 1813, in magnetising small needles, by making a focus of the violet rays, collected by a lens, pass repeatedly from the middle to one end of a needle, without touching the other half. By continuing this process for an hour, the needle had acquired distinct polarity. This experiment was repeated successfully by the Marquis of Ridolfi and M. Carpa; and Sir H. Davy and Professor Playfair, and other English philosophers, witnessed the successful performance of this experiment. Professor Configliachi at Rome, with the benefit of an Italian sun, M. Berard of Montpellier, and M. Dhombre Firmas at Alais, failed in obtaining similar effects from the violet rays, and hence the doctrine of the magnetic efficacy of blue light was exploded both by the French and Italian philosophers.
In 1825 Mrs. Somerville repeated, in a different way, the experiments of Morichini. She took a slender sewing needle an inch long, quite devoid of magnetism, and having covered half of it with paper, she fixed it to the panel of the wall with wax, so that its uncovered half should receive the violet rays of a spectrum formed by an equilateral prism of flint glass, whose refracting faces were each 1-4 by 1-1 inches, and which was placed about five feet from the wall. The needle was placed in a vertical plane nearly perpendicular to the magnetic meridian, and inclined to the horizon, and as the sun advanced to the meridian, the needle was moved parallel to itself to keep it in the sun's rays. In less than two hours, when the sun had just passed the meridian, the exposed half of the needle attracted the south and repelled the north pole of the magnetic needle. A second experiment on the same day confirmed this result, and similar effects were obtained by varying the circumstances of the experiment.
In the blue and green rays, the needles were magnetised but less frequently, and always after a longer exposure, but the magnetism was always as strong as in the violet. The indigo rays were nearly as efficacious as the violet. Pieces of clock and watch spring were also magnetised after longer exposure, and magnetic needles had their magnetism increased by the same means. When the violet rays were concentrated with a lens, the magnetism was sooner conveyed to the needle. When the halves of the needles and the pieces of spring were enveloped in violet Prism and green silk, they became magnetic also. Mrs. Somerville does not seem to have tried as a check upon these results, the action of the red rays of the spectrum. The blue glass, we are persuaded, transmitted many of the red rays, and this experiment should have been made by using the common sulphate of copper, which does not transmit, with a proper thickness, any of the less refrangible rays.
In the same year, Professor Christie of Woolwich, made a series of elaborate experiments on the action of the compound rays of the sun on a magnetised needle, and also on needles of copper and glass. The needles were caused to vibrate by the force of torsion in the white light of the sun, and he invariably found that the arc of vibration was more rapidly diminished in the sun's light than in the shade. Professor Christie regards these facts as proving incontestibly the existence of a magnetic influence in the solar rays, and tending to restore the credit of Morichini's experiments. He intended to make his needles vibrate under the separated rays of the sun, and had an apparatus prepared for the purpose; but we are not aware of his having either made these experiments or published any account of them.
M. Baumgartner of Vienna, while repeating the experiments of Mrs. Somerville, found that a steel wire, some parts of which were polished, while the rest were rough or without lustre, were magnetised by the action of the white light of the sun, each polished part exhibiting a north, and each unpolished part a south pole. By concentrating the sun's rays, the effect was produced more rapidly. On a wire eight inches long, he thus obtained eight poles. When the needles were perfectly oxidated or perfectly polished, or had polished lines in the direction of their lengths, he could not succeed in magnetising them.
A less equivocal method of proving the magnetic influence of white light presented itself to Professor Barlocchi. An armed natural loadstone capable of carrying a weight of 1½ Roman pounds (a Roman lb. is equal to 339·179 grams) after three hours exposure to the strong light of the sun, was able to carry 2 oz. or one-sixth of a lb. more, and after an exposure of 24 hours its force was almost doubled. A second loadstone of nearly the same power was put into a dark place of the same temperature as that of the solar rays, but acquired no additional strength. A stronger magnet, which carried 5 lbs. 5 oz. and 2 deniers, was exposed to the light of a cloudy and humid atmosphere, and where it snowed; but it acquired no additional power; while, during the two subsequent days under a clear sky, its energy was more than doubled. The effect was not increased by a lengthened exposure.
Among the most active labourers in this department of science, we must rank M. Zantedeschi of Pavia. He had early obtained results similar to those of Morichini, and about the end of 1825, he had remarked, that suspended wire needles, devoid of all magnetism, and having one of their ends exposed to the white light of the sun under a glass receiver, turned that extremity to the north in the plane of the magnetic meridian. In repeating the experiment of Barlocchi with artificial magnets, M. Zantedeschi found that a horse-shoe magnet, carrying 134 ounces, carried 3½ oz. more after three days exposure to the sun, and by continued exposure, was able to carry 31 oz. He found also that the strength of oxidated magnets was increased by solar light, and that it diminished in those that were not oxidated. A loadstone, for example, not oxidated, lost 2½ oz. of its strength by three hours exposure to solar light, whilst another oxidated and similarly exposed, gained as much and even more strength. When the first was polished like a mirror, its strength suffered no change by a prolonged exposure to the He likewise found that when a loadstone, whether oxidated or not, had its north pole exposed to the sun, it ac- quired strength, but lost strength when its south pole was exposed. In more than sixty experiments, the increase of strength varied from 1 to 2, and to 3½ oz., whereas the cor- responding diminution was from 3½ to 5, and to 5½ oz. In these experiments our author confesses that he often en- countered anomalies of which he could not detect the cause.
In repeating the experiments of Professor Christie, M. Zantedeschi obtained analogous results. Having placed a magnetic needle a foot long in the shade, he drew its place of rest through an arch of 90°, and it performed in 30" four oscillations, the last of which had a semi-amplitude of 70°. When exposed to the solar rays, it performed in the same time, and under the very same circumstances, four oscillations, the last of which had only a semi-amplitude of 60°.
A great degree of doubt has been cast upon the conclu- siveness of all these researches, by a series of well-managed experiments made recently by MM. Ricci and Moser. The following experiments were made after the manner of Mori- chini. The needles were made of soft steel; their mass was very small, and they presented a considerable surface to the action of the light. The prism was placed in the po- sition of minimum deviation, which corresponds to the greatest intensity of the light of the spectrum, the needles were placed on a graduated circle, three or four feet from the prism. The lens used had an aperture of 1-2 inches, and a focal length of 2-3 inches, and by it the focus of the violet rays was made to traverse one-half of the needle 200 times, excepting on the 10th July and the 12th August, when it was done only 100 times, and on the 25th July, when it was done 525 times. The following were the re- sults which they obtained, shewing no change in the number of oscillations.
| Date | Names of the Needles | Duration of Oscillations | |------|----------------------|--------------------------| | April 3 | a, b, c | 22'0" | 22'0" | 10'12 | | 27 | d, e, f | 27'5" | 27'5" | 9'11 | | May 6 | g, h, i | 14'5" | 14'5" | 8'10 | | June 14 | j, k, l | 15'2" | 15'7" | 9'12 | | 16 | m, n, o | 22'0" | 22'0" | 9'11 | | 17 | p, q, r | 22'7" | 22'5" | 8'11 | | 23 | s, t, u | 18'2" | 18'2" | 9'10 | | July 1 | v, w, x | 23'0" | 23'7" | 9'12 | | 11 | y, z, a | 19'5" | 19'5" | 9'12 | | Aug. 10 | b, c, d | 22'4" | 22'2" | 8'10 | | 12 | e, f, g | 22'7" | 22'5" | 8'10 | | 17 | h, i, j | 19'5" | 19'2" | 9'11 |
As reflected light was not used in Morichini's experiments, which our author wished to verify, they employed the direct light of the sun. In the experiments, however, with the needles a, b, c, they fixed the spectrum by means of a helio- state. They remark that the needle had been exposed 17½ hours to the action of the sun without becoming magnetic, whereas Morichini succeeded in magnetising his needles in from fifteen to thirty minutes.
These gentlemen repeated also in a similar manner the experiments of Baumgartner, but it would be unprofitable to detail them here, as they could not reproduce any of the results which he had obtained. They tried also the action of polarised light, but without any better success.
Sect. VII.—Recent Discoveries respecting the Spectrum.
The analysis of white or compound light by the prism was made and perfected by Sir Isaac Newton; but though New analy- sis could not decompose them, he committed a mis-sis of the take in concluding that the colours of the spectrum were spectrum, simple and homogeneous; "that to the same degree of re- frangibility ever belonged the same colour, and to the same colour ever belonged the same degree of refrangibility."
Now, though it is quite true that the green and orange co- lours of the spectrum cannot be decomposed by the prism into more simple ones, the one into blue and yellow, and the other into yellow and red, yet they can be decomposed by other means. This opinion respecting the compound nature of the colours of the spectrum, and the inability of the prism to analyse them, was first maintained by Sir David Brew- ster, who, with the view of placing it beyond a doubt, un- dertook a series of experiments, in which he examined the effects produced on the solar spectrum by viewing it through a great number of coloured media, and reflecting it from co- lored. By these experiments, he not only established the accuracy of his first opinion, that the green and orange co- lours of the spectrum were compound, the one consisting of blue and yellow, and the other of red and yellow, but was led to the more general result, that the whole spectrum was com- pound, consisting of three equal and superposed spectra of red, yellow, and blue light. The following are the general results which he obtained:
1. White light consists of three simple colours, red, yel- low, and blue, by the mixture of which all other colours are formed.
2. The solar spectrum, whether formed by prisms of transparent bodies, or by gratings or grooves in metallic and transparent surfaces, consists of three spectra of equal length beginning and terminating at the same points, viz. a red spec- trum, a yellow spectrum, and a blue spectrum.
3. All the colours in the solar spectrum are compound co- lours, each of them consisting of red, yellow, and blue light in different proportions.
4. A certain quantity of white light, incapable of being decomposed by the prism, in consequence of all its compo- nent rays having the same refrangibility, exists at every
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* See Annales de Chimie, 1829. * Edinburgh Transactions, vol. ix. p. 43. * Id. Id. vol. x. p. 123. point of the spectrum, and may, at some points be exhibited in an insulated state.
"This remarkable structure of the spectrum will be better understood from figs. 88, 89, 90, which represent the three separate spectra, which are shewn in their combined state in fig. 91.
Fig. 91.
"In all these figures, the point M corresponds with the red, or least refrangible extremity of the spectrum, and N with the violet or most refrangible extremity; and the ordinates ax, bx, cx, of the different curves MRN, MYN, MBN, represent the intensity of the red, yellow, and blue ray at any point x of the spectrum.
"If the distance Mx in all these spectra be equal, then, in the combination of them shewn in fig. 91, the ordinates ax, bx, cx, will indicate the nature and intensity of the colour at any point x of the red spectrum. Thus, let
"The ordinate for red light ax = 30, yellow bx = 16, blue cx = 2,
ax + bx + cx = 48 rays,
then the point x will be illuminated with 48 rays of light, viz., 30 of red, 16 of yellow, and 2 of blue light.
"Now, as there must be certain quantities of red and yellow light which will form white, when combined with two blue rays, let us assume these, and suppose that white light, whose intensity is 10, will be formed by 3 red, 5 yellow, and 2 blue rays; hence it follows that the point x is illuminated by
"Red rays...........27 Yellow rays........11 White light........10
48 rays,
or, what is the same thing, the light at x will be orange, rendered brighter by a mixture of white light. The two blue rays, therefore, which enter into the composition of the light at x, will not communicate any blue tinge to the prevailing colour.
"If the point x is taken nearer M, and if, at that point, the blue rays are more numerous in proportion to the yellow than 2 to 5, that is, if they are as 3 to 5, then there will be one blue ray more than what is necessary to make white light with the two yellow and the three red rays, and this blue ray will give a blue tinge to that part of the spectrum, or will modify the peculiar colour of pure red light. In like manner, the blue extremity of the spectrum may have its peculiar colour modified by an excess of red rays, so as to convert it into violet light."
It would occupy too much space to bring forward the body of evidence which supports these views, and we must therefore refer our readers to the original memoir. We shall endeavour, however, in a few words, to prove the general principle of the compound character of the prismatic colours, by facts stated by persons who observed them, without any reference to this opinion. Sir Isaac Newton and Fraunhofer, and many persons besides, have, from long observation of the solar spectrum, concluded that there is a homogeneous Spectrum, unmixed yellow, and a homogeneous unmixed orange space in the spectrum. Newton makes the yellow space 40, and the orange 27, or 67 in all; while Fraunhofer makes the yellow space 27, and the orange space 27, or 54 parts of a spectrum whose length is 360 parts. Now, Dr. Wollaston declares that a beam of daylight is refracted by the prism into five colours only, red, yellowish green, blue, and violet, and he defines their limits with his usual accuracy. Dr. Young, who repeated the same experiments with that exactness which was peculiar to him, declares that the spectrum formed in Dr. Wollaston's manner, consists of four colours only, red, green, blue, and violet, "the colours differing scarcely at all in quality within their respective limits." Now both these accurate observers have rejected the yellow and orange spaces almost entirely, with the exception of the narrow line of yellow light formerly mentioned, thus running counter to all the observations of Newton and Fraunhofer. The cause of such a difference is this: The light analysed by Wollaston was the blue light of the sky, which had been deprived by absorption of many of its rays, having the same refrangibility as those which fell upon the prism. Dr. Young's green space was Sir Isaac Newton's yellow space, deprived of most of its yellow rays, and the red space adjoining the green was Newton's orange space, deprived by the absorption of the atmosphere, of almost all its yellow rays; and the sharp yellow line noticed by Dr. Young, and regarded by Dr. Wollaston as a mixture of red and green light, as if these spaces had overlapped a little, is part of the orange space of Fraunhofer and Newton, deprived of its red rays. This yellow band can be produced artificially upon all kinds of white light, and by the absorption of various media. Hence it is obvious, that by comparing the light reflected and modified by the blue sky with the direct light of the sun, we may obtain irrefragable proof of the compound nature of the yellow and orange spaces. That red light exists at the most refrangible extremity, is obvious from its violet colour; and that blue light exists at the red extremity, may be proved by the following observation of Sir W. Herschel. He had occasion to view the prismatic spectrum, when reflected from clear turned brass, and he observes, "The colour of the brass makes the red rays appear like orange, and the orange colour is likewise different from what it ought to be." Here then, yellow light was seen at the very red end of this spectrum, and it was seen in consequence of blue light having been absorbed by the brass, because blue light mixed with the orange, observed by Sir W. Herschel, would alone recompose the original red. Here, then, there is a proof that blue light, and yellow light, and red light, all exist in the same place, at the least refrangible end of the spectrum. Effects similar to these may be produced by various coloured media, such as chemical solutions, or the coloured juices of plants; and by such means Sir David Brewster has succeeded in insulating white light in the spectrum, incapable of being decomposed by the prism. In a more recent memoir, "On the connexion between the phenomena of the absorption of light and those of thin plates," he has shown that a separation of rays of different colours but equal refrangibility, is effected by the continued production of periodical colours by a number of thin plates acting successively upon the same pencil.
"The existence of fixed lines in the spectrum, as discovered by Fraunhofer, was a fact unexampled in science. Various coloured bodies were known to absorb particular parts of the spectrum, and their peculiar colour was the necessary consequence of this absorption. Some of them, such as smalt blue glass, produced at a certain thickness several dark bands in the spectrum, but these bands shaded off by..." examining the action of all the coloured solid and fluid bodies which he could command, Sir David Brewster was led to observe the action of nitrous acid gas on the spectrum.
With a fine prism of rock salt, having the largest possible refracting angle, he formed a spectrum with the light of a lamp transmitted through a small thickness of the gas, whose colour was a very pale straw yellow, and he was surprised to observe the spectrum crossed with hundreds of lines or bands, much more distinctly pronounced than those of the solar spectrum. In the violet and blue spaces the lines were sharpest and darkest; they were fainter in the green, and almost imperceptible in the yellow and red spaces. By an increase in the thickness of the gas, the lines were better developed in the yellow and red spaces, and became broader in the blue and violet, a general absorption or extinction of the light advancing from the violet extremity, while a specific absorption was going on, on each scale of the lines or bands. It was not easy to obtain a sufficient thickness of gas to bring out the lines at the red extremity, but our author found that heat produced the same effect as an increase of thickness, and by greatly heating a tube half an inch wide, and full of gas, he was able to exhibit the lines and bands to the very extremity of the red space.
"The power of heat alone," says our author, "to render a gas which is almost colourless as red as blood, without decomposing it, is in itself a most singular result, and my surprise was greatly increased when I afterwards succeeded in rendering the same pale nitrous acid gas so absolutely black by heat, that not a ray of the brightest summer's sun was capable of penetrating it. In making this experiment the tubes frequently exploded, but by using a mask of mica and thick gloves, and placing the tubes in cylinders of tinned iron, with narrow slits to admit the light, there is little danger of any serious accident."
It is a remarkable fact that the liquefied gas in the very same tube, produces none of the fixed lines, and exercises no other action on the spectrum than any ordinary fluid of the same orange colour.
As the points of maximum absorption in coloured bodies were distinctly coincident with some of the principal lines in the solar spectrum, our author suspected that the same might be true with regard to the nitrous gas lines, and he therefore formed the solar and the gaseous spectrum with light passing through the same aperture, so that the lines in the one stood opposite to those in the other, and their coincidence became a matter of simple observation. He then superimposed the two spectra, when both were formed by solar light, and thus exhibited at once the two series of lines and bands, with all their coincidences and deviations. This phenomenon is one of the most splendid in physical optics.
"The general coincidence," continues our author, "thus recognizable by the eye, requires to be more particularly explained. Though some of the larger lines in the gaseous spectrum coincide with some of the larger ones in the solar spectrum, yet, in many cases, faint and narrow lines in the one coincided with strong and broad lines in the other; and there were some strong gaseous lines, and even broad bands, to which I could discover no counterpart in Fraunhofer's map of the spectrum, which at this stage of my inquiry was the standard to which I appealed. This discrepancy at first embarrassed me, and as I observed it in parts of the spectrum where Fraunhofer had drawn every line which he had seen with his finest instruments, I abandoned all hopes of being able to establish the general principle of their identity. I was therefore obliged either to renounce this principle as
one contradicted, or rather not confirmed by observation, or to consider Fraunhofer's delineations as in fault, and to enter upon the Herculean task of making a better map of the spectrum.
"After a little practice in the observation of the solar spectrum, I discovered most of the lines which I had in vain sought for in Fraunhofer's map, as the counterpart of those in the gaseous spectrum. I saw well marked groups, of which he had only given one of the lines, and shaded bands, and well defined lines, which his method of observation had not permitted him to discover.
"After I had laid down all the principal features in the New map spectrum, I was able to examine the two classes of lines of the solar spectrum, part passu. The action of the gas upon invisible lines in the spectrum rendered them visible by slightly enlarging them, and this enlargement of a solar line indicated the existence of a corresponding line in the gaseous spectrum.
"By this double process, and by methods of observation which I believe have never before been used in optical researches, I have been able to execute three different maps of the spectrum: first, a map of the lines in the solar spectrum; secondly, a map of the same spectrum, exhibiting at the same time the action of nitrous acid gas upon solar light, previously deprived of a number of its definite rays; and, thirdly, a map shewing the action of the gas upon a continuous and uninterrupted spectrum of artificial white light. The general scale of these delineations is four times greater than that of Fraunhofer, but some portions of them are drawn on a scale twelve times greater, which became necessary from the impossibility of representing in narrower limits the numerous lines and bands which I have discovered. The length of Fraunhofer's spectrum is 15½ inches; mine, upon the same scale, is nearly 17 inches. The length of the general spectrum, which I have delineated, is about five feet eight inches, and the length of a spectrum, corresponding to the scale on which I have delineated parts of it, is seventeen feet.
"Fraunhofer has laid down in his map 354 lines, but in the delineations which I have executed, the spectrum is divided into more than 2000 visible and easily recognised portions, separated from each other by lines more or less marked, according as we use the simple solar spectrum, or the solar and gaseous spectra combined, or the gaseous spectrum itself, in which any breadth can be given to the dark spaces."
In this minute examination of the spectrum, our author was led to the discovery of a system of lines and bands, particularly in the red and green spaces, which at other times wholly disappeared; but by a diligent comparison of these observations, he found that these lines and bands depended on the proximity of the sun to the horizon, and were produced by the absorptive action of the earth's atmosphere.
Sir David Brewster is about to publish his map of the lines of the spectrum, as produced by the light of the sun itself; and another map of the spectrum, as modified by the action of the earth's atmosphere. He has, however, made the following general observations on the subject, in reference to the fixed lines of Fraunhofer:
"The atmosphere," he remarks, "acts very powerfully round the line D, and in the space immediately on the least refrangible side of it. It develops a beautiful line in the middle of the double line D, and by enlarging a group of small lines on the red side of D, it creates a band almost as dark as the triple line D itself. It widens generally all the lines, but especially the darkest one, which I call m, between C and D. It develops a band on the least refrangible side of m, and it acts especially upon several lines, and develops a separate band
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1 A remarkable example of a very definite action on a part of the spectrum was discovered by Sir D. Brewster in the triple oxalate of chromium and potash. It absorbs a very definite band on the least refrangible side of B, a part of the spectrum over which it exercises no general absorptive action. This band lies in the space Ba of Fraunhofer's map, so that if x is its place, Bx will be 1 Ba, or its index of refraction in the water spectrum is almost exactly 1:33070-1.—See Phil. Trans. 1833, p. 93.
Phil. Trans. 1837, part ii. p. 242.—Edin. Trans. vol. xii. Periodical on the most refrangible side of C. The lines A, B, and C Colours are greatly widened, and lines and bands are particularly developed between A and B, and generally throughout all the red space.
Most of the lines thus widened by the atmosphere are faint lines previously existing in the spectrum, and I have no doubt that they would be seen in the spectrum of the lime ball light condensed by a polyzonal lens, and acted upon by thirty miles of atmosphere.
The absorptive action of the atmosphere shews itself in a less precise manner in the production of dark bands, whose limits are not distinctly defined. A very remarkable narrow one, corresponding to one produced by the nitrous acid gas, is situated on the most refrangible side of C. Another very broad one lies on the most refrangible side of D, close to a sharp and broad band of yellow light, displayed by the general absorption of the corresponding part of the superimposed blue spectrum. There is also an imperfectly defined atmospheric action, corresponding to a group of lines where Dr. Wollaston placed his line C.
This general description of the atmospheric lines, while it indicates the remarkable fact, that the same absorptive elements which exist in nitrous acid gas, exist also in the atmospheres of the sun and of the earth, leads us to anticipate very interesting results from the examination of the spectra of the planets. Fraunhofer had observed in the spectra of Venus and Mars some of the principal lines of the solar spectrum. This, indeed, is a necessary consequence of their being illuminated by the sun, for no change which the light of that luminary can undergo is capable of replacing the rays which it has lost. But while we must find in the spectra of the planets and their satellites, all the defective lines in the solar spectrum, we may confidently look for others arising from the double transit of the sun's light through the atmospheres which surround them.
PART V. ON PERIODICAL COLOURS.
The phenomena of periodical, or recurrent colours, as Dr. Young has very appropriately called them, are among the most interesting in optics, and have been treated of by him with great ability, under our article on CHROMATICS, though not in that popular and descriptive manner, which is required in a work like this. We shall therefore endeavour to give as perspicuous an account as we can of this interesting portion of physical optics.
SEC. I.—On the Interference of Light.
The discovery of the interference of light in its simplest form, is due to Grimaldi, as we have already seen. He admitted the sun's light into a dark room, through two small and equal apertures of a circular form. Two cones of diverging light were thus formed, and by receiving them on a screen, held beyond the place where the cones intersected each other, two overlapping luminous circles were seen on the screen. A partially illuminated penumbra surrounded each of these cones, and at the place where the rays from each aperture met, the screen was, generally speaking, more strongly illuminated by the union of the two lights; but the boundaries of the penumbral portions which overlap are much darker than the corresponding portions of the penumbra which do not overlap, as if the one light had at this part put out the other. Upon intercepting the light from one of the apertures, this dark part became brighter, and upon restoring the light it again became darker. The result, therefore, was here unambiguous, and justified the observation of Grimaldi, "that an illuminated surface may be rendered darker by the addition of light."
Dr. Hooke made a similar experiment, and observed the darkness produced at the overlapping part of the two cones; but this result, remarkable as it was, seems to have excited Colours no interest during nearly a century and a half, till Dr. Young, who was unacquainted with the experiment of Grimaldi, obtained the same result in a different manner, and thus laid the foundation of the most interesting department of physical optics. The following is the experiment which he gave as "An Experimental Demonstration of the Interference of Light." I made a small hole in a window shutter, and covered it with a piece of thick paper, which I perforated with a fine needle. For greater convenience of observation I placed a small looking-glass without the window shutter in such a position as to reflect the sun's light in a direction nearly horizontal upon the opposite wall, and to cause the cone of diverging light to pass over a table on which were several little screens of card paper. I brought into a sunbeam a slip of card about \( \frac{1}{35} \)th of an inch in breadth, and observed its shadow either on the wall, or on other cards held at different distances. Beside the fringe of colour on each side of the shadow, the shadow itself was divided by similar parallel fringes of smaller dimensions, differing in number according to the distance at which the shadow was observed, but leaving the middle of the shadow always white. Now these fringes were the joint effects of the portions of light passing on each side of the slip of card, and inflected or rather diffracted into the shadow. For a little screen being placed either before the card, or a few inches behind it, so as either to throw the edge of its shadow on the margin of the card, or to receive on its own margin the extremity of the shadow of the card, all the fringes which had before been observed in the shadow on the wall, immediately disappeared, although the light inflected on the other side was allowed to retain its course, and although this light must have undergone any modification that the proximity of the other edge of the slip of card might have been capable of occasioning. Nor was it for want of a sufficient intensity of light that one of the two portions was incapable of producing the fringes alone; for when they were both uninterrupted the lines appeared, even if the intensity was reduced to one-tenth or one-twentieth.
Although this experiment is a very decisive one, yet M. Fresnel made one still more instructive and general, and free from any of the objections that might have been urged against that of Dr. Young. He took two plane mirrors M, N, which were inclined at a very great angle, a little less than 180°, and having allowed a beam of light Ra, Rb, proceeding from a luminous point R such as the focus of a small lens, he received the reflected rays on a piece of paper PQ. If the light was homogeneous, there was seen upon the paper a succession of bright and dark bands alternating. These bands are parallel to the line of intersection of the two mirrors, and they are placed symmetrically on both sides of a plane passing through the line of intersection of the mirrors, and through a point A bisecting the distance of the points D, E, the virtual points of divergence of the two reflected pencils aG, bG. That these parallel bands are produced by the mutual interference of the two beams is at once proved by intercepting one of them, by covering one of the mirrors, when the whole series disappears. It is found also by measuring distances of the same bands from the line of intersection of the mirrors, and when the paper PQ is placed at different distances from the mirrors, that their different
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1 Phil. Trans. 1804; or Elements of Nat. Phil. vol. ii. p. 639. A still more simple and elegant method of exhibiting the phenomena of interference, has been given by Professor Lloyd of Dublin, which any person may repeat with a single piece of plate glass. Having placed horizontally a piece of black glass QP, with his eye behind it at QM, he viewed by very oblique reflexion, when the angle of incidence was nearly 90°, a horizontal narrow aperture placed at A, a distance of three feet from the reflector QP. The proper degree of obliquity was easily found by bringing the reflected image of the aperture A to coincide very nearly with the direct image, in which case the direction of the reflected plane BPQ bisected the distance AA'. When the ray AM, which fell directly upon the eye at M, interfered with the reflected rays CM, which reached it by a longer path, they produced a system of fringes or bands, which were distinctly visible when received upon an eye-piece placed at a short distance from the reflector. This system of bands was exactly similar to one-half of the system seen in Fresnel's experiment. With compound white light the first band was a bright one and colourless. This was followed by a very sharply defined black band; then came a coloured, and so on alternately, seven alternations being easily counted, and the breadth of the bands being, as near as the eye could judge, the same throughout the series, and increasing with the obliquity of the reflected beam. "The first dark band was of intense blackness," but the darkness of the succeeding bands were less intense, as they were of higher orders, and after three or four orders they were completely obliterated by the closing in of the bright bands. At the same time the coloration of the bright bands increased with the order of the band, until after six or seven alternations, the colours of different orders became superimposed, and the bands were thus lost in a diffused light of nearly uniform intensity." These bands," continues Professor Lloyd, "are most perfectly defined when the eye-piece is close to the reflector. Their breadth and coloration increased with the distance of the eye-piece, but remained of a finite and very sensible magnitude, when the latter was brought into actual contact with the edge, a circumstance which distinguished them altogether from the diffracted fringes formed on the boundary of the shadow."
When homogeneous light is used, the bands are alternately bright and dark, and varying in magnitude with the refrangibility of the light, as will be afterwards more fully explained. If the light of the sun is used, the bands may be distinctly seen upon a white screen placed at MQ. That they are produced by interference may be easily proved, either by stopping the direct ray AM, or the reflected ones CM, when the whole system of bands disappears.
The leading phenomena of interference may be likewise exhibited by transmitting the light emanating from a luminous point through the two faces of a prism, the inclination of which is about 180°. The pencil or ray passing through one of the faces, will be slightly inclined to the ray passing through the other, at a small angle, and will interfere with it at their point of concourse, and produce the usual fringes. This form of the experiment is described by Sir Isaac Newton, who considered the fringes as produced by inflexion.
In all the preceding experiments two pencils of light issuing from the same point or luminous origin, are made again to meet, the one having arrived at the point of concourse Periodical by a different and a longer path. Now it is obvious from the experiments, that when the two portions of light thus intervening reach the spot where they interfere by paths exactly equal, they form a bright fringe, having the intensity of its light greater than that of either portion. It is also evident that other bright fringes are produced, when their paths differ in length; and if we suppose d to be the difference of paths by which the second bright fringe is produced, similar bright fringes will be produced when the differences in the lengths of the paths are 2d, 3d, 4d, 5d, 6d, &c. But it is manifest from the preceding experiments, that if the two portions of light interfere at intermediate points, or when the difference in the length of their paths is \( \frac{1}{2}d \); \( d + \frac{1}{2}d \); \( 2d + \frac{1}{2}d \); \( 3d + \frac{1}{2}d \); \( 4d + \frac{1}{2}d \), &c., the two interfering portions shade each other and produce blackness, as appears from the dark fringes lying between the bright ones. Here, then, we have a remarkable property of light established by direct experiment, and well fitted to guide us in our inquiries into the physical cause of the various phenomena of light. We shall find the same beautiful property shewing itself under various aspects in a succession of interesting phenomena, which we shall now proceed to describe.
**Sect. II.—On the Colours of Thin Plates.**
The colours of thin plates were first observed by Mr. Boyle, who remarks that all chemical essential oils, as also good spirits of wine, by shaking till they rise in bubbles, appear of various colours, which immediately vanish when the bubbles burst, so that a colourless liquor may be immediately made to exhibit a variety of colours, and lose them in a moment, without any change in its essential principles. Mr. Boyle also noticed these colours in soap bubbles and in turpentine, and he succeeded in blowing glass sufficiently thin to exhibit them. In 1666 Lord Brereton observed similar colours produced by the thin plates which are formed on the surface of glass by the action of the weather. In the year 1672 Dr. Hooke exhibited to the Royal Society a soap bubble with all its colours, in fulfilment of a promise which he had made at a previous meeting, "to exhibit something which had neither refraction nor reflexion, and yet was diaphanous." "By means of a glass pipe he blew several small bubbles out of a mixture of soap and water, when it was observable that at first they appeared white and clear, but that after some time the film growing thinner, there appeared upon it all the colours of the rainbow, first a pale yellow, then orange, red, purple, blue, green, with the same series of colours repeated." Dr. Hooke made considerable progress in the investigation of this class of phenomena, and made experiments with thin plates of Muscovy glass, (Mica). He found that a faint yellow plate of this substance laid upon a blue one constituted a very dark purple; and Sir Isaac Newton, in a private letter to Dr. Hooke, a copy of which is now before us, acknowledges that Hooke had observed previous to him, "the dilatation of the coloured rings by the obliquation of the eye, and the apparition of a black spot at the contact of two convex lenses, and at the top of a water bubble."
Sir Isaac Newton, whose investigations we shall presently give in his own words, made great progress in discovering the law of the phenomena, and it is a curious fact not to be overlooked by physical inquirers, that his theory of the phenomena, elaborated with the utmost care and generalising an extensive series of facts, is now exploded, while the theoretical views of Dr. Hooke are almost universally admitted.
Mr. Melville of Edinburgh proposed to make a permanent soap bubble by freezing, but we believe the experiment... Periodical has never yet succeeded. Dr. Joseph Reade has however been more fortunate in making what may be called a permanent soap bubble, for illustrating the colours of thin plates, which we saw him exhibit at the meeting of the Physical Section of the British Association at Liverpool in 1837. The following is his own account of the method of making it: "Having put two ounces of distilled water into an eight-ounce phial, and having added about the size of a large pea of Castile soap, I placed the bottle in a saucepan of boiling water on the fire; the bottle was speedily filled with a dense volume of vapour, which expelled all the air. I now corked it, and after cooling and thus condensing the vapour, had perhaps as perfect a vacuum as could be formed, even by the best air-pump. I now held the bottle laterally between my hands, and by means of a circular and brisk motion formed a circular film, on which by resting the bottle on an inclined plane, were formed after a short time, all the parallel bands or series of colours in the following order: 1. a white or silvery segment at top; 2. a snuff-coloured brown, inclined at bottom to a deep red; 3. blue; 4. yellow; 5. red; 6. blue; 7. green; 8. red; 9. green; 10. red; 11. green. (As in fig. A.)
"After some time a black segment was seen to form at the top of the white, and continually to increase in size. (See fig. B.) After a few minutes the parallel bands increased in breadth, and running into one another, only three or four distinct bands were seen. Nothing can exceed the beauty of these colours, equal to those of the rainbow, or the plumage of the tropics: whilst writing this description, I have these bands in a bottle before me, feasting my eyes on their beauty. In a few minutes more this black segment or aqueous film occupies, perhaps, half the circular film, and the lower half becomes yellow tinged with orange. (See fig. C.)
"If we now incline the bottle towards the experimenter's breast, the saponaceous atoms producing these colours, are seen to float in the region of the black or aqueous; when placed again on the inclined plane, they fall to the bottom of the film. In some time more the entire film becomes black, and all the colours disappear.
"Having now placed the bottle in a basin of boiling water, the evaporation was increased, and the black film soon became clothed with saponaceous atoms, which being variously condensed, produced all the colours of the clouds when the sun is setting on a summer's evening. On again placing the bottle on the inclined plane, the parallel bands were again formed by the attraction of cohesion, and the colours afterwards gave place to the black film. I held the bottle laterally between my hands, and by means of a circular motion washed it, and thus clothed it with saponaceous atoms, which went through the same process on placing the bottle on the inclined plane. By means of washing the film every morning, I preserved it for more than three weeks."
The colours of thin plates are often exhibited in nature in the most beautiful manner. On the surface of little pools of mossy water, and especially in the proximity of springs containing iron, we observe thin bright films generally whitish and often yellow and reddish. Between the plates of a mass of mica, and sulphate of lime, and talc, we observe thousands of open spaces where the rings are sometimes circular, consisting only of the first tint above blackness, viz., the white of the first order, sometimes two, three, or more colours according to their size, while at other times the rings or fringes are extremely numerous and often irregular. These colours are all produced by thin plates of air or of vacancy in these fissile minerals, and the colours may be all changed by admitting water or fluids of different refractive powers. In some specimens of Labrador felspar, Sir David Brewster has found crystallised cavities so thin or with so little depth, as to give the most splendid colours of thin plates, and to afford one of the finest subjects of popular display in the microscope. The colours produced by heat on highly polished steel, are all the colours of a thin plate of oxide; and they are often beautifully displayed on the sides and on the bars of grates.
When glass is exposed to the action of the weather, its surface acquires a thin film, which at first can only be rendered visible by examining the faint light reflected from it when it is in contact with a fluid of nearly the same refractive power. It forms most rapidly on the panes of glass in stable windows, but it is seen in the highest perfection in the specimens of decomposed glass found among the remains of Roman buildings. The glass is to a certain depth entirely decomposed into thin films of extreme beauty, reflecting the most brilliant colours to the eye, and transmitting tints of the most exquisite brilliancy, and far surpassing any of the colours produced by art. Coloured films of the richest tints are also seen upon both the faces of cleavage of a sort of artificial mother of pearl, which has been called macrite, and described by Mr. Horner in the Phil. Trans. These films are all thin plates of extreme tenacity.
When we breathe upon glass at a proper temperature, and examine with a magnifier the margin of the film while evaporating, or when we observe the evaporation of different volatile fluids, we shall perceive many interesting examples of the colour of thin plates.
One of the finest exhibitions of this kind with which we are acquainted, is that which is produced by the ammonio-sulphate of copper, and which was observed by Sir David Brewster. A solution of the ammonio-sulphate of copper in water, is spread upon a clear plate of glass or any other surface. In the course of an hour or two, a similarly coloured film is formed upon its surface, exhibiting the colours of thin plates from the white of the first order upwards. When the solution is strong, or the stratum of fluid deep, the thickness of the film increases, and the colours rise to higher orders, of a beautiful green and pink colour. When the colours are such as we would wish to preserve, an aperture must be made in the film, and by inclining the plate the fluid must be allowed to run out slowly, leaving the film on the surface of the glass. This film will become hard and permanent after the aqueous part of it has been evaporated. The fringes of colour take the shape of the mass of fluid, or of the piece of glass whose surface is covered with it.
One of the most extraordinary examples of the colours of thin plates, or rather of the blackness that immediately precedes these colours, and one which almost requires the evidence of ocular demonstration to credit, is the existence of filaments, or of a down of quartz so exceedingly minute as to be incapable of reflecting light. The very remarkable specimen of quartz in which this was discovered by Sir David Brewster belongs to the cabinet of the Duchess of Gordon. The original crystal was $2\frac{1}{4}$ inches in diameter, and of a light smoky colour, but impervious to light except in small pieces. Mr. Sanderson, lapidary in Edinburgh, had broken up the crystal for the purposes of his profession, but the apparent foulness of the fracture induced him to lay it aside. The following is an account given by Sir David Brewster, of the principal fracture.
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1 Edin. Trans. vol. xi. p. 322. 2 Phil. Trans. 1837, part ii. 3 Id. Id. 1836, p. 49, and 1837, part ii. "At first sight, the absolute blackness of the separated surfaces seemed to me as it did to every one, to be owing to a thin film of opaque and minutely divided matter that had insinuated itself into a fissure of the crystal; but this opinion was immediately overturned when I observed that both surfaces were equally and uniformly black, and that they were also perfectly transparent by transmitted light.
"Although I had now no doubt that the phenomenon was entirely of an optical nature, and that the blackness of the surfaces arose from their being composed of short and slender filaments of quartz, whose diameter was so exceedingly small that they were incapable of reflecting a single ray of the strongest light, yet it became desirable to establish this curious fact by experimental evidence.
"Having found that no detergent substances either removed or diminished the superficial blackness, I subjected the fragment to the action of cold and hot acids; but the surface continued unaltered by these operations. I now immersed the fragment in oil of anise seed, which approaches to quartz in its refractive power, and upon examining the light reflected at the separating surfaces of the oil and the quartz, I found that the blackness disappeared, and that the fragment, whether seen by reflected or transmitted light, comported itself like any other piece of quartz of the same translucency. Upon removing the oil from the surfaces, it resumed its original blackness, and the filamentous or velvety nature of the surface was rendered evident to the eye by the slight change of tint which was produced by pressing the filaments to one side.
"As the preceding experiments are sufficient to prove that the size of the minute filaments was less than the thickness of a thin plate of quartz which lost the power of reflecting light, it became interesting to obtain an approximate measure of their magnitude. The thinnest substance ever observed, is the aqueous films of the soap bubble previous to its bursting. Newton, however, remarked that the films still reflected a faint image of the candle or of the sun. Hence its thickness must have corresponded nearly with what Newton calls the beginning of black, which appears in water at a thickness of the 750,000th part of an inch. The tint, however, of the quartz surface is of a much deeper character, and cannot exceed the very black of Newton's scale which corresponds in quartz to one-third of the one-millionth part of an inch, or to one-fourth of the thinnest part of the soap bubble.
"If the two surfaces of quartz had separated by filaments of a larger size, the colour of the filamentous surface might have been red, or blue, or yellow, or green; but though such a structure would have been more dazzling to the eye, it yet would have been less wonderful than the one which has now been described."
The colours of thin plates may also be exhibited by pressing together two glass prisms that have moderate refracting angles. Various coloured fringes or portions of coloured rings will be seen by viewing the light reflected from the surfaces in contact, and in a much fainter degree, by examining the transmitted light. The same phenomena may be seen with unusual brilliancy, by taking a thick piece of glass, and having made a scratch on one side of it with a file, apply a heated wire to the scratch, so as to produce a crack in the glass, which may be extended at pleasure by a second and third application of the hot wire. If we now examine the surface of this crack in different directions, we shall see it covered with coloured fringes, which may be made to vary in breadth and position, by opening or closing the crack with the force of the hand.
When we wish to examine and measure the coloured rings with care, the method used first by Hooke, and subsequently by Newton, should be adopted. Two convex lenses of very long focal length are placed the one above the other, so as to touch at their vertex; or a plano-convex lens may have its plane side AB laid upon the convex side CD, fig. 97, of another lens. Sir Isaac Newton used for the uppermost lens a plano-convex one, whose focal length was fourteen feet, and for the lowermost a double convex lens, whose focal length was fifty feet. These lenses must then be held together, and pressed if necessary by three clamp screws, as shown in fig. 95. The following is the general account of the phenomenon given by Sir Isaac Newton, though somewhat abridged:
"Next to the pellucid central spot, made by the contact of the glasses, succeeded blue, white, yellow, and red. The blue was so little in quantity, that I could not distinguish any violet in it, but the yellow and red were as copious as the white, though four or five times more than the blue. The next order of colours round those in the second, was violet, blue, green, yellow, and red, all of them copious and vivid except the green. The third order was purple, blue, green, yellow, and red, the green being more vivid than in the last order. The fourth order was only green and red, the green being copious and lively, being bluish on one side, and yellowish on the other. The red was very imperfect: the succeeding rings or orders of colours were very faint; and after three or four orders, they ended in perfect whiteness. The form of the whole system of rings, when the lenses were most compressed, so as to produce the black spot in the centre, as shown in fig. 96, where a, b, c, d, e, f, g, h, i, k, l, m, n, o, p, q, r, s, t, v, x, y, z, indicate the different colours beginning at the centre, viz. 1. black, blue, white, yellow, red; 2. violet, blue, green, yellow, red; 3. purple, blue, green, yellow, red; 4. green, red; 5. greenish blue, red; 6. greenish blue, pale red; 7. greenish blue, reddish white.
In order to find the interval between the glasses, or the thickness of the plate of included air (or space) at which each colour was produced, Sir Isaac measured the diameter of the first six rings at their brightest part, and found their squares to be in the arithmetical progression of the odd numbers 1, 3, 5, 7, 9, &c., and the intervals between the glasses are obviously in the same progression, one of the surfaces being plane, and the other spherical. He then measured the diameter of the rings at their darkest points, and found their squares to be in the arithmetical progression of the even numbers, 2, 4, 6, 8, 10, &c.
In order to find the absolute thickness of the plate of air or space at which these different rings were produced, he measured the diameter of the fifth ring at its darkest point as produced by the different object-glasses.
| Diameter of Sphericity of the object-glass | Diameter of fifth dark ring | |-------------------------------------------|-----------------------------| | 182 inches | 100 | | | 1774784 | | 184 inches | 5 | | | 88850 |
and dividing these diameters by 5, we obtain the diameter of the first ring he obtained, \( \frac{1}{88739} \) and \( \frac{1}{88850} \), but as these measurements were taken at an angle of incidence of 4°, the results must be diminished in the ratio of the secant of 4°, or 10029; so that we have \( \frac{1}{88952} \) and \( \frac{1}{89063} \); the mean of which, \( \frac{1}{8900} \) nearly expresses in parts of an inch, the thickness of the air at the darkest part of the first dark ring at a perpendicular incidence.
By multiplying this interval by the series of odd and even numbers, 1, 3, 5, 7, &c., and 2, 4, 6, and 8, &c., we obtain the following measures of all the rings:
| Thickness of the air at the brightest part | Thickness of the air at the darkest part | |------------------------------------------|----------------------------------------| | First Ring | \( \frac{1}{178000} \) | | Second Ring | \( \frac{3}{178000} \) | | Third Ring | \( \frac{5}{178000} \) | | Fourth Ring | \( \frac{7}{178000} \) |
After measuring the diameters of the rings at different angles of incidence, Sir Isaac obtained the following results:
From the results given in the preceding table, Sir Isaac Newton has deduced the following rule:—“That the thickness of the air is proportional to the secant of an angle, whose sine is a certain mean proportional between the sines of incidence and refraction; and that mean proportional, so far as, by these measures, I can determine it, is the first of an 106 arithmetical mean proportionals between these sines counted from the higher sine, that is, from the sine of refraction when the refraction is made out of the glass into the plate of air, or from the sine of incidence when the refraction is made out of the plate of air into the glass.”
Sir Isaac next proceeds to describe the rings formed by the light transmitted through the two glasses. In this system of transmitted rings, the order of the colours was yellowish red; black, violet, blue, white, yellow, red; violet, blue, green, yellow, red. The colours in these rings are very faint at a perpendicular incidence, but become brighter as the incidence increases.
In fig. 97 Sir Isaac has represented the different colours reflected and transmitted, AB and CD being the surfaces of the glasses which touch at E, and the lines uniting them representing their distances in arithmetical progression. The words above the straight line AB are the colours of the reflected rings, and those below the circular arch CD those of the transmitted rings.
When water was introduced between the lenses, the colours became fainter, and the rings less, and Sir Isaac found that the intervals were inversely as the indices of refraction in water and air.
The rings were always larger in the homogeneous red light of the spectrum than in the violet light, in the ratio of 14½ to 9, and he concluded from more detailed observations that the thicknesses of the air between the glasses when the rays were successively formed by the limits of the seven different colours, red, orange, yellow, green, blue, indigo, and violet, are to one another as the cube roots of the squares of the eight lengths of a cord which sound the notes in an eighth, sol, la, fa, sol, la, mi, fa, sol, that is, in the cube roots of the squares of the numbers 1, \( \frac{5}{8} \), \( \frac{3}{4} \), \( \frac{3}{2} \), \( \frac{7}{8} \), \( \frac{1}{2} \), \( \frac{1}{4} \), \( \frac{1}{8} \); or 1; 0·924; 0·885; 0·825; 0·763; 0·711; 0·681; 0·630; that is, if 1 be the interior diameter of any such red ring formed by the extreme red rays, the cube root of \( \frac{1}{8} \) will be the interior diameter of a ring at the boundary of the red and orange, and so on.
The colours of thin plates of fluid or solid bodies are not so easily studied as those of air, from the difficulty of procuring, and working with, such evanescent films. Sir Isaac Newton, however, studied them in soap bubbles which, as soon as they were blown, he covered with a clear glass. In this way he observed the colours to emerge like so many concentric rings surrounding the summit of the bubble. As the bubble became thinner, and the subsidence and evaporation of the water, the rings dilated slowly, till they covered the whole bubble, descending in order to the bottom of it, till where they vanished successively. After all the colours were emerged at the top, there grew in the centre of the rings a small round black spot, which continually dilated itself till it became sometimes more than a-half or three quarters of an inch in breadth before the bubble broke. Some light was still reflected from the water at this spot, and Sir Isaac saw within it several smaller round spots much blacker and
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1 It has been noticed by M. Le Blanc, that if the two extreme terms of the preceding numbers, or any other two equidistant from the extremes be multiplied together, their product will always be equal to \( \frac{1}{4} \). By observing how much the colours at the same places of the bubble, or at divers places of equal thicknesses, were varied by the several obliquities of the rings, Sir Isaac obtained the following thicknesses of the water requisite to exhibit one and the same colour at several obliquities:
| Incidence on the water | Refraction into the water | Thickness of the water | |------------------------|--------------------------|-----------------------| | 0° | 0° | 10 | | 15° | 11 | 11 | | 30° | 22 | 1 | | 45° | 32 | 2 | | 60° | 40 | 30 | | 75° | 46 | 25 | | 90° | 48 | 35 |
These results harmonise entirely with the rule already given for thin plates of air.
When the system of rings seen by reflection was examined by a prism, and perhaps only eight rings visible, Sir Isaac counted sometimes more than forty on the side of the system to which the refraction was made, though he reckoned by estimation more than a hundred. Soap bubbles also, before they exhibited any colours to the naked eye, have appeared through a prism girded about with many parallel and horizontal rings, to produce which effect it was necessary to hold the prism parallel, or very nearly parallel to the horizon, and to dispose it so that the rings might be refracted upwards.
Sir Isaac then proceeds to a very important part of the subject, namely, to explain the composition of the colours of thin plates, a topic of great interest and extensive application. "Let there be taken," says he, "on any right line from the point Y, fig. 98, the lengths YA, YB, YC, YD,
Fig. 98.
Now, if A2 be supposed to represent the thickness of any thin transparent body, of which the internal violet is most copiously reflected on the first ring, then HK will represent its thickness, at which the outermost red is most copiously reflected in the same series. Also A6 and HN will denote the thicknesses at which those extreme colours are most copiously reflected in the second series, and A10 and HQ the thicknesses at which they are most copiously reflected in the third series, and so on. And the thickness at which any of the intermediate colours are reflected most copiously, will be defined by the distance of the line AH from the intermediate parts of the lines 2 K, 6 N, 10 Q, &c., against which the names of those colours are written below.
But, farther, to define the latitude of these colours in each ring or series, let A1 be the least thickness and A3 the greatest thickness, at which the extreme violet in the first series is reflected, and let HI, and HL, be the like limits for the extreme red, and let the intermediate colours be limited by the intermediate parts of the lines 1 I, 3 L, against which the names of those colours are written, and so on. But yet with this caution, that the reflections be supposed strongest at the intermediate spaces, 2 K, 6 N, 10 Q, &c., and from thence to decrease gradually towards these limits, 1 I, 3 L, 5 M, 7 O, &c. on either side; where you must not conceive them to be precisely limited, but to decay indefinitely. And whereas I have assigned the same latitude to every series; I did it, because although the colours in the first series seem to be a little broader than the rest, by reason of a stronger reflexion there, yet that inequality is so insensible as scarcely to be determined by observation.
Now, according to this description, conceiving that the rays, originally of several colours, are by turns reflected at the spaces 1 I L 3, 5 M O 7, 9 P R 11, &c. and transmitted at the spaces AHI 1, 3 LM 5, 7 OP 9, &c. it is easy to know what colour must in the open air be exhibited at any thickness of a transparent thin body. For, if a ruler be applied parallel to AH, at that distance from it by which the thickness of the body is represented, the alternate spaces 1 II 3, 5 MO 7, &c. which it crosseth, will denote the reflected original colours, of which the colour exhibited in the open air is compounded. Thus, if the constitution of the green in the third series of colours be desired, apply the ruler as you see at π ρ σ φ, and by its passing through some of the blue at π, and yellow at σ, as well as through the green at ρ, you may conclude that the green exhibited at that thickness of the body is principally constituted of original green, but not without a mixture of some blue and yellow.
By this means you may know how the colours from the centre of the outward rings ought to succeed in order as they were described. For, if you move the ruler gradually from AH through all distances, having passed over the first space which denotes little or no reflexion to be made by the thinnest substances, it will first arrive at 1 the violet, and then very quickly at the blue and green, which, together with that violet compound blue, and then at the yellow and red, by whose farther addition that blue is converted into whiteness, which whiteness continues during the transit of the edge of the ruler from 1 to 3, and after that by the successive deficiency of its component colours, turns first to compound yellow, and then to red, and last of all the red ceaseth at L. Then begin the colours of the second series, which succeed in order during the transit of the edge of the ruler from 5 to O, and are more lively than before, because more expanded and severed. And for the same reason, instead of the former white there intercedes between the blue and yellow a mixture of orange, yellow, green, blue, and indigo, all which together ought Periodical to exhibit a dilute and imperfect green. So the colours of the third series all succeed in order; first, the violet, which a little interferes with the red of the second order, and is thereby inclined to a reddish purple; then the blue and green, which are less mixed with other colours, and consequently more lively than before, especially the green: Then follows the yellow, some of which towards the green is distinct and good, but that part of it towards the succeeding red, as also that red, is mixed with the violet and blue of the fourth series, whereby various degrees of red, very much inclining to purple, are compounded. This violet and blue, which should succeed this red, being mixed with, and hidden in it, there succeeds a green. And this at first is much inclined to blue, but soon becomes a good green, the only unmixed and lively colour in this fourth series. For as it verges towards the yellow, it begins to interfere with the colours of the fifth series, by whose mixture the succeeding yellow and red are very much diluted and made dirty, especially the yellow, which, being the weaker colour, is scarce able to show itself. After this the several series interfere more and more, and their colours become more and more intermixed, till, after three or four more revolutions, (in which the red and blue predominate by turns) all sorts of colours are in all places pretty equally blended, and compound an even whiteness.
And since the rays ended with one colour are transmitted, where those of another colour are reflected, the reason of the colours made by the transmitted light is from hence evident.
If not only the order and species of these colours, but also the precise thickness of the plate, or thin body at which they are exhibited, be desired in parts of an inch, that may be also obtained by the assistance of the preceding observations. For according to these observations, the thickness of the thinned air, which between two glasses exhibited the most luminous parts of the first six rings were
\[ \frac{1}{178000} \quad \frac{3}{178000} \quad \frac{5}{178000} \quad \frac{7}{178000} \quad \frac{9}{178000} \]
Suppose the light reflected most copiously at these thicknesses be the bright citrine yellow, or confine of yellow and orange, and these thicknesses will be \(F_\lambda, F_\mu, F_\sigma, F_\tau\). And this being known, it is easy to determine what thickness of air is represented by \(G_\phi\), or by any other distance of the ruler from \(AH\).
But farther, since the thickness of air was to the thickness of water, which between the same glasses exhibited the same colour, as 4 to 3, and the colours of thin bodies are not varied by varying the ambient medium; the thickness of a bubble of water, exhibiting any colour, will be \(\frac{4}{3}\) of the thickness of air producing the same colour. And so according to the same observations, the thickness of a plate of glass, whose refraction of the mean refrangible ray, is measured by the proportion of the sines 31 to 20, may be \(\frac{20}{31}\) of the thickness of air producing the same colours; and the like of other mediums. I do not affirm that this proportion of 20 to 31 holds in all the rays; for the sines of other sorts of rays have other proportions. But the differences of these proportions are so little that I do not here consider them. On these grounds I have composed the following table, wherein the thickness of air, water, and glass, at which each colour is most intense and specific, is expressed in parts of an inch divided into ten hundred thousand equal parts.
| Reflected Tints | Transmitted Tints | Air | Water | Glass | |-----------------|------------------|-----|-------|-------| | Very black | White | 1 | | | | Black | | | | | | Beginning of | | | | | | black | | | | | | Blue | Yellowish red | 2 | | | | White | Black | 4 | | | | Yellow | Violet | 7 | | | | Orange | Blue | 8 | | | | Red | | 9 | | | | Violet | White | 11 | | | | Indigo | | 12 | | | | Blue | Yellow | 14 | | | | Green | Red | 15 | | | | Yellow | Violet | 16 | | | | Orange | | 17 | | | | Bright red | Blue | 18 | | | | Scarlet | | 19 | | | | Purple | Green | 21 | | | | Indigo | | 22 | | | | Blue | Yellow | 23 | | | | Green | Red | 25 | | | | Yellow | Bluish green | 27 | | | | Red | Bluish green | 29 | | | | Bluish red | | 32 | | | | Bluish green | Green | 24 | | | | Green | Red | 35 | | | | Yellowish green | | 36 | | | | Red | Bluish green | 40 | | | | Bluish green | Greenish blue | 46 | | | | Red | | 52 | | | | Bluish green | Greenish blue | 58 | | | | Red | | 65 | | | | Bluish green | Greenish blue | 71 | | | | Ruddy white | | 71 | | |
Although these experiments of Newton were repeated by many of his successors, particularly by Mariotte, Mazzea, and Dutoir, their observations were generally made on the colours of thin plates as exhibited by a plate of air between two plane glasses. Mazzea observed that a diminution of the breadth, and a change in the position of the coloured rings, were produced by an increase of temperature; and that the same system of rings was produced in vacuo. He made a great number of experiments with salt, vapour, wax, and rosin pressed between the plane glasses, but he was perplexed with the phenomena, not being aware that he was actually making experiments with mixed plates in which the colours follow an entirely different law.
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1 Mem. Acad. Berlín, 1792, and Memoires Presentées, tom. ii. p. 28. 2 Memoires Presentées, tom. iv. p. 724. 3 He made the lenses almost red hot, and by pressing them with a rod of iron, he saw the rings produced as before. medical Dutour and Sir William Herschel observed the reflected and transmitted tints at the same time; the last series being reflected to the eye by the lower surface of the glass, and M. Dutour observed these reflected ones more distinctly by making the shadow of an opaque body pass over the upper surface. Sir W. Herschel observed additional sets of rings by increasing the number of reflecting faces; and in producing the primary reflected system by a lens pressed against a metallic reflector, he remarks that the transmitted system must in this case be conceived to have been absorbed by the metallic surface.
In an interesting paper on the colours of thin plates published in the Mémoires D'Arnaud, M. Arago has given an account of some important discoveries on this subject. In viewing the reflected rings through a rhomb of Iceland spar, having its principal section parallel or perpendicular to the plane of incidence, he observed, that the intensity of light in one of the images varied with the angle of incidence, and that this image vanished at an incidence of 35° the maximum polarising angle for glass. He discovered the very same property in the transmitted rings. This eminent philosopher has also shewn, that when the reflected and the transmitted systems of rings are superposed, they completely neutralise each other, forming white light; and hence he concluded, that their colours were complementary, and the intensities of their illumination exactly the same.
M. Arago next examined the system of rings when formed between a lens and a metallic reflector. When he observed them with the thumb of spar above mentioned, one of the images vanished as formerly at 35° of incidence, the angle of maximum polarisation of glass; but above and below that angle M. Arago observed the most singular phenomena. At incidences below 35°, the two images formed by the doubly refracting rhomb differed only in intensity, the colours and the diameters of the rings being exactly the same in both. Above the polarising angle, however, the rings in the two images were of complementary colours, the orders of colours in the one beginning from a black centre, and in the other from a white centre. M. Arago also observed that the rings of the same order of colours in the two images had different sizes. When a plate of any substance of intermediate density between the two substances between which it was contained was used, similar phenomena were produced. When the surface of the metallic reflector was slightly tarnished, M. Arago observed a second system of rings with complementary colours, arising from the light irregularly scattered at the metallic surface, as they could be seen in every position of the eye.
Without knowing of these discoveries of M. Arago, Mr. Airy, about twenty years afterwards, published similar results respecting the modification of the rings above and below the maximum polarising angle; and Professor Lloyd ingeniously observed, that an analogous result may be obtained by combining (as in Fresnel's experiment, fig. 92) a metallic reflector with one of glass. The light being polarised perpendicularly to the plane of reflexion, the central band will be white, when the angle of incidence is below the polarising angle of the glass; at the polarising angle, the interference bars will vanish altogether; and beyond that incidence they will reappear with a dark centre in place of a white one. This method of observation would seem to be peculiarly adapted to the investigation of the change of phase produced by metallic reflexion at various incidences.
A consideration of Fresnel's expressions, which had led Mr. Airy to make his experiments with a metallic surface, led him also to expect that when the rings were formed between two transparent surfaces of different refractive powers, and when the light was polarised perpendicularly to the plane of incidence, the rings should be black centred at incidences below the maximum polarising angle of the least refractive surface, or greater than that of the highest refractive surface, and should be white-centred, when the angle of incidence was between these angles. By forming the rings between plate glass and diamond, Mr. Airy found his anticipations correct. In the course of these experiments Mr. Airy observed that the rings did not disappear at the polarising angle of diamond, but that the first black ring contracted as the incidence was gradually increased, and at last took the place of the central white spot. Hence he concluded that there is still some light reflected at the maximum polarising angle of diamond, and that this body has no angle of complete polarisation. See page 376, col. 2.
Sect. III.—On the Diffraction or Inflexion of Light.
M. Grimaldi, to whom we owe the discovery of the interference of light, likewise made some important experiments on what is called the diffraction or inflexion of light. Having admitted a ray of solar light into a dark room, through a small hole AB, he placed in the conical beam ABCD an opaque body EF. The shadow of this body was not bounded by the straight lines AEH, BFG, nor by the penumbra without the IL formed by the lines BEI, AFL, but was enlarged to shadow MN, and was much greater than it should have been, if formed by rays passing in straight lines past the edges of the body. Without the shadow of the body there were three fringes of coloured light, the broadest and most luminous of which, next to the shadow X, fig. 100, was MNO. There was no colour in the middle at M, but it was blue at the side NN, and red at the other side OO. The second fringe
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1 Phil. Trans. 1807, 1809, 1810. 2 Tom. iii. 3 Cambridge Transactions, 1832.—The following note upon this paper we confess ourselves unable to understand, "I have carefully verified," say Mr. Airy, "this assertion, (that it is indifferent whether the light is polarised before or after reflexion) because I think that it leads to important theoretical conclusions. If polarisation were a modification of light (as Dr. Brewster and others have supposed,) it might be conceived that polarisation before incidence, might destroy its power of producing rings at a certain angle, or might change the tints; but when the reflexion is performed, and the rings are actually visible to the eye with a dark centre, it seems quite impossible that any modification or physical change in the light should make that centre appear white. The satisfactory explanation is, that polarisation is a resolution of the light into two sets at right angles to each other, performed in such a manner that the two sets can in general be separately exhibited, and that in this instance only one is transmitted to the eye." The authors criticised in the preceding extract, viz. Malus, Young, Biot, Herschel, and we believe MM. Arago and Fresnel, all considered polarisation as a modification of common light, and if a modification certainly a physical change, with no reference to theory. In every point of view, indeed, the term is philosophical and unexceptionable. If polarisation is a resolution, as Mr. Airy affirms, of vibrations in an infinite number of planes, into two sets at right angles to each other, which we do not question, this is certainly a pretty considerable modification and a very remarkable physical change. As we have nothing to do with theories in describing phenomena, we must continue to use the terms which have been regarded as appropriate by our contemporaries.
4 Report on Optics, British Association Reports, Rep. 4, 1833, p. 366. Periodical QPR was narrower than MNO. It was colourless in the middle at P, faintly blue at QQ, and faintly red at RR. The third fringe STV resembled the other two, but was the narrowest and the faintest in its colours. They were bent round the edges of the body, as shewn in fig. 101.
Grimaldi likewise discovered fringes within the shadow, which were best seen when the body was long, the light great, and its distance from the aperture considerable. These internal fringes increased with the breadth of the body, and they became narrower when they increased in number. They were bent round the angles of the body, as shewn in fig. 102, where ADBC is the shadow, and a, b, c, d the internal fringes. Short lucid streaks were seen proceeding, as it were, from the angle D, and returning to it, as shewn in the figure.
What Dr. Young has called the crested fringes of Grimaldi, are shewn in figure 103. These fringes are formed by any body that has a rectangular termination. At the line which bisects the right angle there is a white central fringe, bounded by hyperbolic curves, whose asymptotes are the diagonal line, and on each side of this are two or three other bands, disposed in hyperbolic curves, which are convex to the diagonal, and converging in some degree, as they recede from the angular point.
In repeating the experiments of Grimaldi, Sir Isaac Newton admitted the sun's light through a small hole in a piece of lead the forty-second part of an inch in diameter; he found the breadth of the shadow of a human hair, which was the 280th of an inch in diameter, to be as follows:
| Distance of the hair from the aperture | Distance of the paper receiving the shadow from the hair | Breadth of the shadow | |----------------------------------------|----------------------------------------------------------|----------------------| | 12 feet | 4 inches | 0.01666 inches | | 12 | 24 | 0.03572 | | 12 | 128 | 0.125 |
Upon comparing the breadths of the fringes without the shadow, and their intervals at different distances, he found them to be nearly in the same proportion, the breadths of the fringes being as the numbers $1, \sqrt{\frac{1}{3}}, \sqrt{\frac{1}{9}}, \ldots$ &c., and their intervals to be in the same progression. Hence the Periodical fringes and their intervals together was as the number 1, $\sqrt{\frac{1}{3}}, \sqrt{\frac{1}{9}}, \ldots$ &c.
When the hair was surrounded with water, the very same phenomena were seen, and metals, stones, glass, wood, horn, ice, &c., produced the very same fringes. The following was the order of the colours, reckoning from the shadow:
First fringe,... Violet, indigo, pale blue, green, yellow, red. Second fringe,... Blue, yellow, red. Third fringe,... Pale blue, pale yellow, red.
When homogeneous light was used, Sir Isaac found that the fringes were largest in red light, least in violet light, and of an intermediate size in green light. In one case, the distance between the middle of the first fringe on each side of the shadow was $\frac{1}{37.5}$ of an inch in red light, and $\frac{1}{40}$ in violet light.
From experiments made by Sir Isaac Newton on the knife light which passed by the edge of a knife, and on that which passed between two knife edges parallel to each other, he concluded that the light of the first fringe passed by the edge of the knife at a distance greater than the 100th part of an inch; the light of the second at a greater distance than that of the first; and the light of the third fringe at a greater distance than that of the second.
Sir Isaac then stuck into a board the points of two knives with straight edges, so that their edges formed an angle of 1° 47' 26", and from the observations which he made on the light which passed between them, he concluded that the light which forms the fringes is not the same light at all distances of the paper from the knives, obviously considering each fringe as produced like caustic curves, by the intersection of the inflected rays. When the fringes formed by these inclined knife edges were received on paper held at a great distance, the fringes formed by the one knife edge were bent into the shadow of the other knife, and formed cubical hyperbolas, whose asymptotes were for one set the knife edge which produced the fringes, and a line perpendicular to the line bisecting the angle formed by the knives.
Although many attempts were made during the last century to complete the unfinished labours of Newton on this subject, yet no decided discovery was made till the time of Dr. Thomas Young. This distinguished natural philosopher, in endeavouring to explain the origin of the fringes which surrounded the shadow of the margin of a small circular aperture, conceived that the light nearest its centre was least inflected, and that nearest its edges most; and that another portion of light reflected from the margin of the aperture, and coinciding either exactly or nearly with the direct light, after a circuitous path, would interfere with that light, and produce colour. In November 1803 he confirmed this supposition to a certain extent, in so far as the production of the colours by interference was concerned, by his discovery of the interference of light, as already described.
The fringes formed by inflexion, as observed by Dr. Young, are shewn in fig. 104, where ABCD is the shadow of the inflecting body with its internal fringes, which he considered as produced by the light passing on each side of the inflecting body, and bent into the shadow, so as to interfere in the manner already described. This he clearly proved to be the case; but he was less successful in explaining the external fringes between AC and GH, and between BD and EF. He ascribed them to the interference of rays reflected from the margin of the inflecting body, with rays which passed by it directly.
Dr. Young examined the crested fringes of Grimaldi in the same manner as he did the internal fringes ABCD. He found that when a screen was placed within a few inches of the inflecting angle of the body, so as to receive only one of the edges of the shadow, all the crested fringes disappeared; but if the rectangular point of the screen was opposed to the point of the shadow, so as barely to receive the angle of the shadow, or its extremity, the fringes were in no way affected.
M. Fresnel in France, and M. Fraunhofer at Munich, were simultaneously occupied in studying the inflexion of light, and each of them published the results of their labours, without any knowledge apparently that the other had been similarly occupied. We shall begin by giving an account of Fresnel's experiments.
In place of a small hole, M. Fresnel adopted the ingenious idea of substituting a lens of short focal length, which collected the solar rays into its focus, from which they diverged as from a small luminous aperture. When bodies were placed in this bright light, they gave distinct fringes, the magnitude of which he was able to measure at various distances behind the inflecting body, and at various distances of the inflecting body from the lens, simply by viewing the fringes with an eye-glass furnished with a micrometer, instead of rendering them dull by receiving them upon paper. In this manner he measured their breadths within the one or two-hundredth part of a millimeter. He traced the external fringes up to their very origin, and by the aid of a lens of a short focus, he saw the third fringe at the distance of less than the one-hundredth part of a millimeter from the edge of the inflecting body. M. Fresnel made also the important discovery that the phenomena varied with the distance of the inflecting body from the radiating or luminous point, as will be seen from the following measures:
| Distance of the inflecting body from the focus of the lens | Distance behind the body where the inflection was measured | Angular inflexion of the red rays first fringe | |-----------------------------------------------------------|----------------------------------------------------------|-----------------------------------------------| | 0 feet 4 inches | 3-281 feet | 12° 6' | | 19—48 | 3-231 | 3° 55 |
from which it appears that the rays experience a less degree of inflexion in proportion to the distance of the inflecting body from the radiant point.
When the inflecting body was kept at a fixed distance from the lens, M. Fresnel measured the inflexion of the same fringe at different distances behind the inflecting body, and the result of these measurements was, that the successive positions of the same fringe did not lie in a straight line, but formed a curve, whose concavity is turned towards the inflecting body. The successive positions of the same fringe in all the orders of colours he found to be hyperbolas, having the radiating point and the edge of the inflecting body for their common foci. In some of these curves the maximum deviation from a straight line was the twenty-fifth part of an inch, or nearly fifty times greater than the error of observation.
In explaining the exterior fringes, M. Fresnel agreed with Dr. Young in supposing that rays reflected from the margin of the inflecting body interfered with the direct rays; but he was afterwards led, both by theory and experiment, to renounce this opinion.
He found that the fringes were absolutely independent of the curvature of the margin of the inflecting body, and that Periodical in cases where the margin was made extremely narrow or Colours sharp, the small quantity of light which it could reflect would be incapable of producing, by its interference with the direct light, such bright fringes as are actually observed. To assure himself of this, he took two plates of steel, the edge of each of which was rounded in one half of its length, and sharp in the remaining half; he placed the rounded portion of one edge opposite the sharp portion of the other, and vice versa. Hence, if the position of the fringes depended on the form of the surface, the effect would thus be doubled, and the fringes appear broken in the middle. They were on the contrary perfectly straight throughout their whole length. M. Fresnel was therefore obliged to suppose, and the supposition he found to be perfectly conformable to the undulatory theory, that rays that pass at a sensible distance deviate from their primitive direction, and interfere with those which pass directly by the edge of the body.
In order to settle this question, M. Fresnel compared the results of Dr. Young's hypothesis with those of his own, and he found that the breadth of any fringe of homogeneous light should be on the two hypotheses as 2 to 1-8726, and having measured the diameter of such a fringe, he found his own hypothesis more consistent with observation than Dr. Young's.
In order to exhibit to the eye the hyperbolic form of the fringes, we have given a representation of them in fig. A, where LL is a lens of short focus, by which the rays of the sun entering a dark chamber, are refracted to a focus F, from which they again diverge, forming the cone Fma. Or the lens may be fixed in a large diaphragm DD', which may stand on the table before the window at which the sun's light enters. By means of a coloured glass VV' placed on either side of the diaphragm, or on either side of F, the light may be rendered homogeneous. If we now place a screen of any kind, EC, at some distance from F, having its edge somewhat smaller, and free of dust, and receive its shadow GT' upon a sheet of paper, or any white ground, TT', or on a glass plate roughened with emery, we shall obtain the section of the fringes formed by diffraction. The line FEG, which is the geometrical shadow, is not the real shadow. On the side of it towards GT', the paper will not appear black, but illuminated with a visible shade, which goes on decreasing nearly uniformly for a considerable distance. On the other side of EG there are several fringes or alternations of light and darkness. The first fringe B parallel to the shadow is bright, then a band S almost entirely black, which is the black fringe of the first order, then a second bright fringe B', which is followed by the second black fringe S'. These alternations continue to a great distance from G, so that even the sixteenth or seventeenth order may be observed, the bright fringes becoming less coloured, and the black ones more luminous, till they are no longer visible.
By varying the distance of the paper TT' from the screen EC, or of the screen from the focus F, the same fringes are produced with certain variations depending on their distances, the fringes being propagated in hyperbolas, as shewn in the figure. The fringes are largest in red R, smallest in violet V, and of an intermediate size in green light G, as re-
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1 Mémoire sur la Diffraction, p. 370. Prof. Lloyd's Report, ut antec. Periodical presented in fig. 106, where they are shewn as on each side of the shadow of a human hair HH.
Fig. 106.
The following table shews the angular distances of the seven first fringes from the inflecting body EC, and the geometrical shadow:
| For Red Light. | Angles, FE = 1000 | Angles, FE = 100 | Angles, FE = 10 | |---------------|------------------|-----------------|----------------| | First order...| 3° 35" | 11° 20" | 35° 51" | | Second order..| 5° 15" | 16° 35" | 52° 25" | | Third order...| 6° 30" | 20° 32" | 1° 4° 56" | | Fourth order..| 7° 32" | 23° 50" | 1° 15° 21" | | Fifth order...| 8° 27" | 26° 44" | 1° 24° 31" | | Sixth order...| 9° 17" | 29° 20" | 1° 32° 44" | | Seventh order.| 10° 3" | 31° 45" | 1° 41° 21" |
| For Violet Light. | Angles, FE = 1000 | Angles, FE = 100 | Angles, FE = 10 | |-------------------|------------------|-----------------|----------------| | First order... | 2° 58" | 9° 22" | 29° 36" | | Second order... | 4° 20" | 13° 42" | 43° 18" | | Third order... | 5° 21" | 16° 55" | 53° 36" | | Fourth order... | 6° 13" | 19° 40" | 1° 2° 15" | | Fifth order... | 6° 59" | 22° 5" | 1° 9° 48" | | Sixth order... | 7° 40" | 24° 15" | 1° 16° 38" | | Seventh order... | 8° 18" | 26° 14" | 1° 22° 53" |
When the diverging light passes through a small circular aperture, very beautiful phenomena are observed, which were studied about the same time by M. Fresnel and Sir John Herschel. M. Fresnel had deduced the phenomena from theory, and subsequently confirmed his deductions by experiment. The following is Sir John Herschel's account of the phenomena:—“Suppose,” says he, “we place a sheet of lead, having a small pin-hole pierced through it, in the diverging cone of rays from the image of the sun, formed by a lens of short focus, and in the line joining the centres of the hole and focus prolonged, place a convex lens or eye-glass, behind which the eye is applied. The image of the hole will be seen through the lens as a brilliant spot, encircled by rings of colours of great vividness, which contract and dilate, and undergo a singular and beautiful alternation of tints, as the distance of the hole from the luminous point, on the one hand, or on the eye-glass, on the other, is changed. When the latter distance is considerable, the central spot is white, and the rings follow nearly the order of the colours of thin plates. Thus, when the diameter of the hole was about 1-30th of an inch, its distance (a) from the luminous point about six feet six inches, and its distance (b) from the eye-lens twenty-four inches, the series of colours was observed to be,
1st order. White, pale yellow, yellow, orange, dull red. 2nd order. Violet, blue (broad and pure), whitish, greenish yellow, fine yellow, orange red, very full and brilliant. 3rd order. Purple, indigo blue, greenish blue, pure brilliant green, yellow green, red. 4th order. Good green, but rather sombre and bluish, bluish white, red.
Sir John Herschel's experiment was made on the 12th July 1819, but was not published till 1825.
We owe also to Sir John Herschel the following beautiful experiment with two equal apertures placed near each other. The rings are formed about each as in the case of
“The series of tints exhibited by the central spot is evidently, so far as it goes, that of the reflected rings in the colours of thin plates; the surrounding colours are very capricious, and appear subject to no law.”
Sir John Herschel, Treatise on Light, sect. 729, 730. Periodical one aperture, but these are accompanied with a set of straight parallel fringes, bisecting the interval between their centres, and perpendicular to the line joining their centres. Two other sets of similar fringes appear in the form of a St. Andrew's cross, forming equal angles with the first set, as shown in Plate CCCLXXXI fig. 5. When the apertures are unequal, as in fig. 6, these fringes assume the form of hyperbolae, having the aperture in their common focus. By varying the number and shape of the apertures, the phenomena became exceedingly beautiful.
M. Poisson deduced from theory that the centre of the shadow of a small circular opaque disc, exposed to light emanating from a single luminous point, would be precisely as much illuminated by the diffracted light as it would be by the direct light, if the disc were removed. By using a small disc of metal, cemented to a clear and homogeneous plate of glass, M. Arago confirmed this very remarkable result.
We owe to MM. Arago a series of beautiful discoveries respecting the influence of transparent screens in the phenomena of inflexion. When a thick piece of glass was used as a screen on one side of the inflecting body, the rings wholly disappear, as if the screen were opaque. If the screen is very thin, like a film of sulphate of lime or mica, the fringes still remain visible, but shift their places, and are moved from the side where the screen is interposed.
If we make this experiment on the fringes produced by two apertures, we have only to cover one of the apertures with the screen. The same effect, however, will be produced, if we cover both apertures with screens of different thicknesses. In this case the fringes will shift their places from the thicker plate, without suffering any other change.
This beautiful property has been most ingeniously employed by MM. Arago and Fresnel, in measuring the refractive powers of different gases. For as the displacement of the coloured fringes depends on the refractive power, as well as thickness of the plate, its refractive power may be computed from the displacement. In the same manner, if one of the interfering rays are made to pass through tubes filled with different gases, while the other does not, the displacement produced by the gas will give a measure of the refractive power of the gaseous medium.
We come now to give some account of the experiments of Joseph Fraunhofer, made with instruments of extreme accuracy, and furnishing data of the highest importance in physical optics.
The apparatus employed by Fraunhofer was a repeating theodolite, whose vernier read off to 4″. In the centre of the circle, but above it, this instrument carries a flat circular plate six inches in diameter, having its axis coincident with that of the theodolite, and graduated separately to 10″. In the middle of this disc is placed a metallic screen, in which the necessary apertures are made, and which is in the axis of the theodolite. The divisions of this disc serve to measure if necessary the angle of incidence of the rays. A telescope, having an object-glass of twenty lines in aperture, and 16½ inches in focal length, is placed three-and-a-half inches from the centre of this disc. This telescope is placed firmly on the alidade of the divided circle, whose diameter is twelve inches, and the whole is counterpoised. The axis of the telescope is exactly parallel to the horizon, as well as to the plane of this circle. The magnifying power which he employed was from thirty to fifty times. The instrument did not communicate with the floor of the room, from which it was wholly insulated. The heliostate was placed in the prolongation of the optical axis, at a distance of 38 feet 7½ inches from the centre of the theodolite. In order to make the heliostate follow the sun in his hourly motion, the observer could move the screw of the mirror by means of a long rod of iron, which extends from the heliostate to the theodolite, and with this apparatus he could also vary at pleasure the intensity of the solar light.
The opening of the heliostate is vertical, being two inches high, and commonly from the 50th to the 100th of an inch.
By means of an achromatic microscope, magnifying 110 times, Fraunhofer measured the aperture in the metallic screen, which he did to the fifty-thousandth, and sometimes to the hundred-thousandth part of an inch, provided the body was very fine at its edge.
Fraunhofer's first observations were made with a single slit, which was placed before the object-glass of the telescope, which had been previously directed to the aperture in the heliostate, so that the aperture was bisected by the wire of the micrometer. He then saw the fringes shown in fig. 7, Plate CCCLXXXI. The middle fringe or band L₁ bisected by the micrometric wire K, was white, becoming yellow towards L₂ and L₃, where it was red. In the space L₄ there is a spectrum with very lively colours, viz. indigo near L₁, then blue, green, yellow, and red, near L₃. The spectrum in L₅ is much less intense, viz. blue near L₆, and yellow, green, and red near L₇. The spectrum in the space L₈, L₉, is still fainter, being green on the side L₈, and red on the side L₉. A great number of spectra follow these, becoming fainter and fainter, and losing themselves in a band of light, which is spread over a great space. All these spectra on both sides of K are perfectly equal, and consequently symmetrical. Both the colours and the spectra shade into one another imperceptibly. The following table contains the average of the distances L₁, L₂, L₃, and L₄, from the central line, all of them being equal, the measures being taken from the red extremity of each spectrum, so that if we wish to have the angle of deviation of L₁ from KK we have only to multiply the value of L₁ in the table by 2, and so on.
| Width of the aperture in parts of a Paris inch | Breadth of each spectrum or values of KL₁, KL₂, &c. | Product of the aperture by the deviation L₁ | |-----------------------------------------------|-------------------------------------------------|------------------------------------------| | 0-11545 | 37″-66″ | 0-0000210 | | 0-065098 | 1—11—17 | 0-0000210 | | 0-03630 | 1—56—6 | 0-0000209 | | 0-02346 | 3—4—43 | 0-0000210 | | 0-01237 | 5—48—7 | 0-0000209 | | 0-01210 | 6—1—84 | 0-0000212 | | 0-01020 | 6—57—3 | 0-0000206 | | 0-00671 | 11—6—4 | 0-0000217 | | 0-00642 | 11—12—2 | 0-0000209 | | 0-00397 | 21—10—3 | 0-0000207 | | 0-00308 | 23—32—7 | 0-0000211 | | 0-00218 | 33—40 | 0-0000213 | | 0-00215 | 35—17 | 0-0000220 | | 0-00114 | 1—4—53 | 0-0000215 |
From these observations, M. Fraunhofer deduces the following conclusions:
1. That the angles of deviation of the luminous rays which pass through a single aperture, are in the inverse ratio of the width of that aperture. 2. That when a ray is diffracted in passing through a narrow aperture, the distance of similar rays from the middle in the several spectra, form in each case an arithmetical progression, whose difference is equal to the first term. 3. That if γ is the aperture, the arches L₁, L₂, or the deviation of the inflected rays, are in general for the radius of a circle equal to 1, \( L₁ = \frac{0-0000211}{γ} \), \( L₂ = \frac{0-0000211}{γ} \), \( L₃ = \frac{0-0000211}{γ} \).
By observing whether the micrometer were appeared or disappeared in the different spectra, Fraunhofer ascertained that the spectra nearest KK, are not composed of homo-
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1 Neue Modification des Lichtes durch gegenseitige Einwirkung und Bewegung der Strahlen, und Gesetze derselben. Von Jos. Fraunhofer in München. Without a date. Periodical genous light, but that it becomes more and more homo- Colours. genous at greater distances from the axis.
Our author next proceeds to describe the phenomena observed when the two edges which form the narrow ap- erture, are at different distances from the object-glass. When the effective width of the aperture thus formed, is from the 25th to the 50th of an inch, the spectra are the same as those before described; but when the opening becomes less, the spectra on one side of the axis become wider, horizontally than those on the other side. When the ap- parent aperture was extremely narrow, the spectra on one was two or three times wider than those on the other. By continuing to close the distant edges, the longest spectra begin to disappear successively, so that the fifth spectrum for example, fills almost suddenly the whole field of the telescope, till it ceases to be visible, then the fourth spec- trum presents the same phenomena, then the third, and so on. During these changes the spectra on the other side remain unchanged, but when all the former have vanished, they also disappear in their turn, not successively, but all at once, which happens when no light passes between the edges. The large spectra are always on the side of the screen, which is nearest the object-glass.
When the apertures, both in the heliostate, and in front of the object-glass, were small circular ones, a system of rings are produced, absolutely the same as those of thin plates, with this difference only, that the centre is white in place of black. The rays increase in size as the apertures diminish. By varying the apertures he obtained the fol- lowing results:
1. That the diameter of the coloured rings are in the inverse ratio of the diameter of the apertures. 2. That the distances of the extreme rings (of any given refrangibility) from the centre form an arithmetical pro- gression, whose difference is smaller than the first term. 3. That if \( \gamma \) is the diameter of the aperture in Paris inches, we shall have
\[ L = \frac{0.0000214}{\gamma} = L_1 - L_2 = L_3 - L_4, \text{ &c.} \]
\[ L_1 = \frac{0.0000257}{\gamma} \]
\[ L_2 = \frac{0.0000257}{\gamma} + L_1, \text{ and so on.} \]
The most important and interesting of Fraunhofer's re- searches, are those which relate to the spectra produced by gratings, consisting of a number of parallel wires placed parallel also to the narrow linear aperture in the heliostate. He formed these gratings of fine wires stretched across a rectangular frame; the two shorter ends of the frame con- sisted of two fine screws made with the same die, and hav- ing 250 threads in an inch. By placing a wire in each thread he ensured their exact parallelism. The diameter of each wire was 0.002021 of a French inch, and the edge of each wire was distant from the adjacent edge 0.003682 of an inch. The aperture of the heliostate was two inches long, and the 100th of an inch wide, and the frame of wires was placed before the object-glass, so that no other light could be admitted but through them. Fraunhofer was as- tonished at the phenomena which he saw. He saw the colourless line of light A, Plate CCCXLXXXI, fig. 8, in the aperture of the heliostate, exactly as if no wires had been interposed, and at some distance from it on both sides a great number of coloured spectra exactly similar to those produced by a good prism. They were larger in propor- tion to their distance from the central bright line, and they diminished in intensity in the same proportion. A part of these spectra are shewn in figure 8, where A is the aperture of the heliostate, absolutely without co- lours. On each side of A the spectra are perfectly sym- metrical. When the apparatus is well made, the space AH' is absolutely black. The first spectrum occupies the Period space H', C'; H' being the usual limit of the violet, and Col- C' that of the red. Between H'' and C'', there is a second spectrum twice as long as the first; the order of the co- lours being the same, but their intensity a little less. The third spectrum occupies the space between C'' and F'', but a part of its violet rays are mixed with the red of the se- cond spectrum, and also a part of the red rays of the third with the blue rays of the fourth spectrum. The fourth spectrum is seen between F'' and D'', its blue extremity losing itself in the third, and its red extremity in the fifth spectrum. Many other spectra succeed these, and when the apparatus is good, thirteen may be easily reckoned on each side of A.
But what is the most interesting fact, when the appar- atus is good, and the adjustments correct,—the fixed dark lines in the prismatic spectrum are seen at C', D', these lines being the same as those similarly marked in Plate CCCXLXXXI, fig. 1. It is remarkable, however, that A is not seen, a fact which M. Fraunhofer neither notices nor explains. The lines both great and small, are absolutely the same, both in this and the prismatic spectrum, though their distances are widely different.
In a grating in which \( \gamma \), the distance between the wires is 0.000028, and \( \delta \), or the diameter of the wires 0.001324, the following are the distances:
| Distance between the lines B' and C' | Diffused Water Spectrum. | |-----------------------------------|--------------------------| | \( \gamma \) | \( \delta \) | | \( \gamma \) | \( \delta \) |
Total length from B' to H' \( = 19 \) 10 19 1 3
The diffracted spectrum is therefore a very extraordinary one. The green space from E to F, is almost exactly the same in both, but all the less refrangible spaces are greatly expanded in the diffracted spectrum, and the more refran- gible spaces greatly contracted, the red space CD in the diffracted spectrum being twice as great as in the water spectrum; and the violet space in the water spectrum twice as small.
M. Fraunhofer repeated the above experiments with ten different gratings; in which the breadth of the wires and the spaces between them were varied, and he deduces from them the following laws:
1. For two different gratings in which the parallel wires are of the same size and placed at equal distances, the magnitude of the spectra which arise from the reciprocal influence of a great number of rays diffracted by narrow apertures, and their distances from the axis are in the in- verse ratio of the intervals \( \gamma + \delta \), that is the space from the middle of one of the openings to the middle of the other.
2. In all the perfect mean spectra, or those in which the fixed lines are seen, the distances between the coloured rays of the same nature in the different spectra, or between the same fixed lines in them, form an arithmetical progression, whose distance is equal to the first term.
3. In gratings in which the diameter \( \delta \) of the wires and the distance \( \gamma \) between them is expressed in Paris inches, the first term of the progression for the fixed lines B, C, D, E, or those rays which have the corresponding degrees of re- frangibility is represented by the following numbers:
\[ B = \frac{0.00002541}{\gamma + \delta}; \quad C = \frac{0.00002425}{\gamma + \delta}; \quad D = \frac{0.00002175}{\gamma + \delta}; \]
\[ F = \frac{0.00001943}{\gamma + \delta}; \quad F = \frac{0.00001789}{\gamma + \delta}; \quad G = \frac{0.00001585}{\gamma + \delta}; \]
\[ H = \frac{0.00001451}{\gamma + \delta}. \] If we represent the numerator of each of these expressions by \(a\), the angle of deviation of one of the same coloured rays in the first spectrum by \(y'\), in the second by \(y''\), in the third by \(y'''\), we have generally
\[ y' = \frac{a}{y + b}, \quad y'' = \frac{a}{y + c}, \quad \text{etc.} \]
If \(s\) stands for the number of the spectrum, \(v\) being \(=0\) for the axis, \(=1\) for the first spectrum, \(2\) for the second spectrum, and if we put \(E = y + b\) we shall have in general,
\[ S(v) = \frac{a}{v}. \]
The preceding results having been obtained with angles so small that the arcs and their sines and tangents are nearly in the same proportion, M. Fraunhofer began a new series of experiments with the view of obtaining spectra, in which the angles should be larger and by which he might determine whether it was the arcs or their sines or tangents which had the proportions assigned by the experiments.
This inquiry rendered it necessary to obtain much finer gratings than those he had used. He therefore coated a plate of glass with two or three folds of gold leaf in order to have the interstices filled up, and by a peculiar arrangement be traced upon the glass parallel lines in which \(t\) was \(=0.00114\) of an inch. When the lines were drawn closer, no gold remained upon the glass. With this system the spectra were larger, and the fixed lines distinctly seen, but they did not answer his purpose. He therefore thought of spun glass which answered as well as wires; and having covered a plate of glass with a thin coat of fat, so thin that it could scarcely be seen, he traced parallel lines upon it, the intervals of which were only half the size of those on the gold leaf. The spectra produced by this system of lines gave the fixed lines very distinctly, so that their distances from the axis could be accurately measured; but he could not succeed in tracing either upon a layer of fat or black varnish lines closer than this. He at last succeeded in his object of tracing a finer system of lines by using a diamond, with which, by the aid of a machine, he traced lines so fine upon the surface of glass, that they could not be seen by the most powerful compound microscope. In this way he obtained a set of several thousand lines in which \(t = 0.0001223\) of an inch, and which are at distances so very equal, that the fixed lines in the first and second spectrum are clearly seen.
With the system when \(t = 0.0001223\), and the number of lines 3601, the fixed line \(D\) is seen double in the first spectrum.
When the light fell vertically on this grating, Fraunhofer obtained the following measures:
| Names of fixed lines | Distance of fixed lines from the axis A. | Distance of lines in first spectrum. | |----------------------|----------------------------------------|-------------------------------------| | C | 11° 25' 20" | C' from D' | | C'' | 23° 19' 42" | C'' from D'' | | D' | 10° 14' 31" | D' from E' | | D'' | 20° 49' 44" | D'' from F' | | E' | 9° 9' 0" | E' from F' | | E'' | 18° 32' 34" | E'' from G' | | F' | 8° 26' 6" | F' from G' | | F'' | 17° 3' 34" | F'' from H' | | G' | 7° 27' 19" | G' from H' | | G'' | 15° 3' 9" | G'' from I' | | H' | 6° 52' 36" | H' from J' |
With this grating, the third, fourth, and following spectra were well seen, but the fixed lines could not be seen with sufficient distinctness for accurate measurement in those beyond the first and second.
He therefore used another grating in which \(t = 0.005919\) of an inch; and when the light fell upon it vertically, he obtained the following results for the first five spectra, with the lines \(D, E, F, G, H\) for the first four, with \(E\) for the first three with \(F\) and \(G\), and the first two for \(H\).
| Names of fixed lines | Distance of fixed lines from the axis. | |----------------------|---------------------------------------| | C' | 2° 20' 57" | | D' | 2° 6' 30" | | D'' | 4° 13' 7" | | D''' | 6° 20' 7" | | D'''' | 8° 27' 43" | | D''''' | 10° 35' 53" | | E' | 1° 53' 7" | | E'' | 3° 46' 17" | | E''' | 5° 39' 50" | | E'''' | 7° 33' 41" |
| Names of fixed lines | Distance of fixed lines from the axis. | |----------------------|---------------------------------------| | E' | 9° 28' 3" | | F' | 1° 44' 19" | | F'' | 3° 28' 45" | | F''' | 5° 13' 23" | | F'''' | 6° 58' 18" | | G' | 3° 45' 57" | | G'' | 4° 37' 30" | | G''' | 1° 27' 0" | | G'''' | 2° 50' 11" |
all the observations made with both the systems of lines are represented by the expression
\[ \sin S(v) = \frac{a}{v}. \]
That is with rays falling vertically, the sines of the angle of deviation of any fixed line or ray of definite refrangibility from the axis in the different spectra which succeed others, are as the numbers 1, 2, 3, 4, 5.
The last of these systems of lines has the remarkable property of having all the spectra on one side of the axis twice as luminous as those on the other. Fraunhofer supposed that one of the sides of each line had been sharper than the other, and confirmed this opinion by tracing lines on a layer of fat, so that one line was less sharp than the other, and it produced the same inequality in the intensity of the light of the spectra on each side of the axis.
If the ray does not fall vertically upon the system of grooves or lines, but is inclined to it in a plane which intersects the parallel lines vertically, the same effect is produced as if the distance between the middle of the lines or \(t\) were diminished in the ratio of the radius to the cosine of the angle of incidence. Hence the distance of the spectra from the axis increases as the cosine of the angle of incidence. If \(\sigma\) therefore is the angle of incidence, then we have \(\sin S(v) = \frac{pa}{\cos \sigma}\). This, however, is only true when the system of lines is coarse and \(\sigma\) not very large. But in fine systems of lines it is otherwise, the spectra on both sides of the axis are no longer symmetrical, and in the system where \(t = 0.0001223\), when \(\sigma\) is \(= 55^\circ\) we have the deviation of \(D'\) on one side of the axis is \(= 15^\circ 16'\), and on the other side of the same axis \(50^\circ 33'\).
Hitherto we have treated of spectra formed by the light transmitted through the gratings, or through plates of glass with systems of lines etched upon one of its surfaces. But M. Fraunhofer examined also the spectra produced by reflection from the etched surfaces of the glass plates. For this purpose, he coated the surface of the glass with a black resinous varnish of the same refractive power as the glass. Then when light reflected from the system itself reading the lines which are etched. In this way knowing the number of lines, viz. 3601, and the distance between the first and last, he obtained \(t\) or the distance between the middle of any two lines.
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1 M. Fraunhofer remarks that it requires much good fortune even with \(t = 0.0001223\) to find a diamond point which shall trace several thousands of such lines without being altered, and he had succeeded only in obtaining one system. It is only by trial that an useful diamond point can be obtained. As every line requires to be drawn singly with great care, the labour of drawing two thousand is enormous. Fraunhofer has drawn lines so close that there are 32,000 of them in a Paris inch. By etching the first and last line of the system somewhat stronger than the rest, our author, by a microscopic apparatus, measured the distance between these two lines, the etching machine itself reading the lines which are etched. In this way knowing the number of lines, viz. 3601, and the distance between the first and last, Periodical of lines fell on the object-glass of the telescope, the very same phenomena appeared as when the light passed through the same system of lines at the same angle of inclination, spectra not symmetrical being seen. The intensity of the spectra was still such that the distances of the various lines can be determined with great accuracy.
M. Fraunhofer has noticed it as very remarkable that under a certain angle of incidence a portion of a spectrum produced by reflexion, consists of entirely polarised light. This angle of incidence varies greatly for the different spectra, and even still very perceptibly for the different colours of one and the same spectrum. Thus with the glass system of lines where $\epsilon = 0.0001223$, the ray $E'$ in the green part of the first spectrum is polarised when $\sigma = 49^\circ$, but the same green part of the second spectrum on the same side of the axis is only polarised when $\sigma = 40^\circ$, and the green part of the first spectrum lying on the opposite side of the axis, is not polarised till $\sigma = 69^\circ$. In this last case the remaining colours of the spectrum are imperfectly polarised. This was less the case in the second spectrum above mentioned where the colour still remained polarised when the angle of incidence was perceptibly altered. In the spectrum where the green light was polarised at $69^\circ$, the light was at no angle of incidence so completely polarised as in the first spectrum at $49^\circ$. With a system of lines in which $\epsilon$ is larger than the one above mentioned, the green rays in the spectra already referred to, are polarised at totally different angles of incidence.
A very singular consequence arises from the formula deduced from theory by M. Fraunhofer, and representing his experiments. If the distance $\epsilon$ between the lines is less than the length of an undulation, and the light falls vertically on the grating, so that $\sigma = \epsilon$, it follows that no coloured ray remains visible, however the light may fall, and only the white light becomes visible in the axis. Hence all scratches or inequalities on polished surfaces can produce no spectra, or no disturbance in the light which the surface refracts or reflects, and consequently no imperfection in the images which they form. M. Fraunhofer likewise draws the conclusion "that it would be impossible by any means to render such inequalities (those less than $\epsilon$) visible," and that a microscopic object, the diameter of which is $\epsilon$, and consists of two parts, cannot be recognised as consisting of two parts. "This," he adds, "shows us the limits which are set to vision through microscopes." This result, if clearly established, would be a very remarkable one.
It is very obvious, that if the distances of the etched lines or the wires in gratings are unequal, the larger distances will give smaller spectra, and the smaller distances larger spectra, which will be mixed with each other. Fraunhofer, however, conceived it would be interesting to know what would happen if the intervals $\epsilon$ were regularly unequal, that is, if the inequality in the distances were regularly repeated in equal parts. With this view he etched parallel lines in various ways, regularly unequal upon plates of glass covered with gold leaf. If the distances between the lines are expressed by $\epsilon', \epsilon'', \epsilon'''$, and if one of the equal parts, which consists of unequal $\epsilon$s is expressed by $\epsilon' + \epsilon'' + \epsilon''' + \ldots + \epsilon_n$, then the distances of the various spectra were found by experiment to be
$$\sin \theta = \frac{\epsilon'}{\epsilon'} + \frac{\epsilon''}{\epsilon''} + \frac{\epsilon'''}{\epsilon'''} + \ldots + \frac{\epsilon_n}{\epsilon_n}$$
The phenomena of spectra thus produced are chiefly remarkable on account of their different intensity. With some systems of lines of this kind, several spectra or parts of them may be wholly wanting, or have so slight an intensity that they are not easily observed, whilst the succeeding ones again become very intense. Owing to this cause, the fixed lines in these spectra may be observed. In the usual systems consisting of equal spaces, the lines $C^{III}, F^{III}$ or the fixed lines $C, F$, in the twelfth spectrum can be seen; but with a regularly unequal system of lines, where every division consists of three shades different among themselves, and are as $25 : 33 : 42$, the lines $C^{III}, D^{III}, E^{III}$, and $F^{III}$ are seen with such distinctness that their distances from the axis can be accurately measured. The reason of this is, that with such systems of lines, the tenth and the eleventh spectra are almost wholly wanting. With this system of lines, indeed, Fraunhofer saw $E^{XIV}$ or the line $E$ in the 24th spectrum so distinctly that its distance could be measured.
M. Fraunhofer, by means of concentric circular lines etched upon glass at equal distances, and by making the aperture in the heliostate circular, has produced circular spectra with circular fixed lines. Their distances from the axis are in the same proportion as in the spectra produced by parallel lines.
When the gratings and systems of lines are immersed in fluids, the same phenomena are produced, but the distances of all the spectra from the axis are diminished in the inverse ratio of the indices of refraction.
Fraunhofer has given also some fine drawings of a beautiful class of phenomena produced by the diffraction of light passing through round and quadrangular apertures either singly or arranged regularly. When a plate of brass perforated with two equal apertures $0.02297$ of an inch in diameter, and $0.03831$ distant, is placed in front of the object glass, and the aperture of the heliostate round, the extraordinary appearance shewn in fig. 9, plate CCCLXXXII, was seen. It consisted of 65 elliptical spectral spectra distributed in concentric rings, the outermost of which contains 28 spectra, the next 20, the next 12, and the central one five. When the circular apertures are arranged so as to correspond with the four angles of a square, the effect produced is similar to fig. 10.
One of the most splendid figures of this kind is produced by crossing two gratings with the wires at right angles to each other, a circular image is covered with narrow spectra radiating from the centre, but occupying only parts of different radii. The spectra are rectangular of about a line wide, and from five lines to five inches long. The violet end of each is towards the centre. In some places the spectra touch and overlap each other, but the greater number are insulated.
Our limits will not permit us to pursue this most curious subject any farther, and we must refer our readers to Fraunhofer's own work, and to another recently published by Schwedt of Spire in 1835, in which he has given drawings of an immense variety of beautiful phenomena, all of which are capable of being explained by the undulatory theory.
As the various phenomena of diffraction observed by Arago, Fresnel, Young, Fraunhofer, and Schwedt, are susceptible of being explained by the undulatory theory, even facts of the same class, and having a similar origin, cannot possess much interest in our inquiries into the physical causes to which they must be ultimately referred.
We shall now proceed, however, to give an account of a new series of facts discovered by Sir David Brewster, of which the undulatory theory is not capable in its present form of giving the least explanation, and we must there- In all the phenomena of gratings and systems of lines observed by Fraunhofer, the central image of the luminous aperture in the heliostate is white, a result that might have been expected, as that light is reflected from the original surface of the glass, and cannot interfere with any other light. "If the lines," says Fraunhofer, "were so thick that one touched another, and consequently had no space between them, no light could be regularly reflected from the etched surface, and would, as from every other polished surface, be dispersed. Were the intermediate spaces equally wide as the lines, the etched surface could only regularly reflect half as much light as an equal surface of glass that was not etched, therefore the quantity of regularly reflected light from an etched surface of glass, is in proportion to the quantity of light which is reflected from a surface of glass of the same size not etched, or as the width of the spaces between any two neighbouring lines is to the width of these lines."
These conclusions however irresistible they seem to be, are very far from having any foundation. For upon examining a series of several systems of lines or grooves cut on steel for him by the late Sir John Barton, Sir David Brewster observed, that in several of them the central image hitherto described as white or colourless, had a distinct colour which was the same in every part of the system. In one of the systems on which there were 1000 lines in an inch, the central image had its tint a greenish blue at a perpendicular incidence, which suffered no change by turning round the plate, nor by reflecting the light from different parts of the system. He found the same colours on various other systems of lines, and upon examining them at different angles of incidence, he found that the tints varied with the incidence, being a maximum at a vertical incidence, diminishing as the incidence increased, and disappearing at an angle of 90°. The following were the general results with the grooves on steel:
| Number of grooves in the inch | Orders and portions of orders of colours from 0° to 90° of incidence | |-------------------------------|---------------------------------------------------------------| | 500 Citron yellow of the first order shading to white. | | | 625 One complete order of colours, together with the reddish yellow of the second order. The colours very faint. | | | 1000 Four complete orders of colours. | | | 1000 One complete order, with blue, green, and yellowish green of the second order. | | | 1250 One complete order, with blue and bluish green of the second order. The colours very faint. | | | 2000 One complete order, together with blue, green, and greenish yellow of the second order. | | | 2500 One complete order, together with the full blue of the second order. | | | 3333 Gamboge yellow of the first order. | | | 5000 One complete order, together with bluish white of the second order. | | | 10,000 One complete order with blue and fainter blue of the second order. | |
In the third specimen, with 1000 grooves, mentioned in this table, the following were the four orders of colours:
| Colours | Angles of incidence | |---------|--------------------| | White | 90° 0' | | Yellow | 80° | | Reddish orange | 77° 3° | | Pink | 76° 20° | | Junction of Pink and blue | 75° 40° |
The property established by the preceding experiment is certainly one of a very remarkable kind. That a pure and highly polished metallic surface, which reflects light perfectly white, should actually decompose it when the surface is reduced to narrow lines, is inconsistent with every doctrine respecting light. Here there are no rays to interfere,
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1 Fraunhofer, Edin. Journal of Science, No. xiii. p.109. Periodical no doubly refracted pencils, and in short, none of the ordinary conditions on which the decomposition of light depends.
That the colour does not arise from light that has entered a certain way into the body interfering with that which is regularly reflected, is obvious from the fact, that in two specimens of 2000 grooves in an inch, impressed on black wax, the new colours were very distinct, the vertical tint being a greenish yellow of the second order, in one specimen, and a gamboge yellow in the other, in addition to one complete order of colours at greater incidences.
The following experiments, intended to show the effect of a variable refracting power in the reflecting surface, are calculated to give us some insight into the nature of this new property of light. In the following table Sir David Brewster has described the changes produced upon the colours, by placing different fluids on the reflecting surface:
| No. of grooves | Maximum vertical tint without a fluid | Maximum tint with three different fluids | |----------------|--------------------------------------|----------------------------------------| | 3124 | No colour | 1. Water; tinge of yellow. | | | | 2. Alcohol; tinge of yellow. | | | | 3. Oil of cassia; faint reddish yellow.| | 500 | Citron yellow of first order | 1. Water; tinge of red. | | | | 2. Alcohol; diluted pink. | | | | 3. Oil of cassia; a bluer pink. | | 625 | Reddish yellow of second order | 1. Water; faint pink of second order | | | | 2. Alcohol; ditto more pink. | | | | 3. Oil of cassia; bluish pink of second order. | | 1,000 | Yellowish green of second order | 1. Water; pinkish red, second order. | | | | 2. Alcohol; brilliant pink, ditto. | | | | 3. Oil of cassia; greenish blue, third order. | | 1,250 | Bluish green, faint | 1. Water; yellow, second order. | | | | 2. Alcohol; yellower. | | | | 3. Oil of cassia; yellowish pink. | | 2,000 | Greenish yellow, second order | 1. Water; brownish red, second order. | | | | 2. Alcohol; pinkish red, ditto. | | | | 3. Oil of cassia; greenish blue. | | 2,500 | Blue, second order | 1. Water; dilute green. | | | | 2. Alcohol; greenish white, second order. | | | | 3. Oil of cassia; bright gamboge, yellow. | | 3,333 | Gamboge yellow, first order | 1. Water; pinkish red, first order. | | | | 2. Alcohol; reddish pink. | | | | 3. Oil of cassia; bright blue, second order. | | 5,000 | Bluish white, second order | 1. Water; pale yellow. | | | | 2. Alcohol; yellow, with tinge of orange. | | | | 3. Oil of cassia; yellowish pink, second order. | | 10,000 | Fine blue, second order | 1. Water; greenish white, second order. | | | | 2. Alcohol; yellowish white. | | | | 3. Oil of cassia; brilliant gamboge yellow. |
Similar results were obtained with grooves impressed upon wax, so that it follows that more orders of colours, and higher tints, at a given incidence, are developed by diminishing the refractive power of the grooved surface. But one of the most interesting results in this table is the part in which the colours are entirely developed by the fluids applied to the surface; and hence if we had transparent fluids of much higher refractive powers, the colours would be produced when the intervals were much larger.
Similar phenomena were developed when the grooves were impressed on the fusible metal, on tin, and on isinglass. In those on isinglass, the new colours were seen also in the transmitted central image, and were extremely brilliant; but they were not decidedly complementary to those in the reflected image. The following were the colours of the reflected and transmitted image in isinglass, beginning from 90° of incidence:
| Colour of the reflected central image | Colour of the same image, seen by transmission | |---------------------------------------|-------------------------------------------------| | Yellow | Deep blue | | Orange | Paler blue | | Pink | Blue | | First limit of pink and blue | Blue | | Blue | Pink | | Green | Orange pink | | Yellow | Orange | | Orange | Yellow | | Pink | Yellow | | Second limit of pink and blue | Yellow | | Blue | Yellow |
Our author was now desirous of observing what took place in the prismatic images, when the colours appeared in the principal or central image.
Let Ab, fig. 107, be the reflected image of a long rectangular aperture from the spaces between the grooves, and ab', a'b', a''b'', a'''b''', the prismatic images of it, rr', rr'', rr''', &c. being the violet sides, and rr, rr', &c. the red sides of these spectra.
Then in the
First spectrum ab, the violet rays are obliterated at m at an incidence of 74°, and the red rays at n at an incidence of 66°, the intermediate colours, blue, green, being obliterated at intermediate points between m and n, and at angles of incidence intermediate between 74° and 66°. In the
Second spectrum a'b', the violet rays are obliterated at m' at an incidence of 66° 20', and the red at n' at 54° 45'. In the
Third spectrum a''b'', the violet rays are obliterated at m'' at 57°, and the red at n'' at 41° 35'. And in the
Fourth spectrum a'''b''', the violet rays are obliterated at m''' at 48°, and the red rays at n''' at 23° 30'. Another similar succession of obliterated tints takes place on all the prismatic images at a lesser incidence, as shown at \( p \), \( p' \), the violet being obliterated at \( p \) and the red at \( r \), and the intermediate colours at intermediate points. In this second succession the line \( p' \) begins and ends at the same angle of incidence, as the line \( m''n'' \) in the third prismatic image \( a''b'' \); and the line \( p'q' \) on the second prismatic image corresponds with \( m''n'' \) on the fourth prismatic image.
This singular obliteration of the colours is shown more clearly in fig. 108, where \( r \) is a part of one of the prismatic images, \( rr \) the red space, \( gg \) the green space, \( bb \) the blue, and \( vv \) the violet space. The line of obliteration \( mn \) begins at \( m \), the extreme violet being obliterated there, so that the curve of illumination \( abm \), fig. 109, is just affected at one extremity \( m \). The line advances into the spectrum, and at the point corresponding to \( d \), fig. 109, a portion of the blue and violet is obliterated, as shown by the notch in the curve; at \( e \) a portion of the green and blue; at \( h \) a portion of the red and green, and at \( n \) the extreme red.
A similar obliteration of tints takes place on the ordinary image \( AB \).
The first obliteration, viz., that of the violet, takes place at \( q \), fig. 107, and that of the red at \( p \); while the intermediate colours disappear at intermediate points. This first space of obliteration has no corresponding one at the same incidence in any of the prismatic images.
The second obliteration of the violet in \( AB \) takes place at \( q \), and that of the red at \( r \), and this corresponds in incidence with the obliterations \( m'n', m''n'' \) on the second prismatic image.
The third obliteration of the violet takes place at \( s \), and that of the red at \( t \), and this corresponds in incidence with the four obliterations on the second and fourth prismatic images, viz., \( p, p', m''n'', m''n'' \).
In all these phenomena the points \( m, n, p, r, s, t \) are only the points of minimum intensity, or of maximum obliteration; for the tints never entirely disappear, and those obliterated at each line \( mn \) form an oblique spectrum containing all the prismatic colours.
The analysis of these curious and apparently complicated phenomena becomes very simple when they are examined under homogeneous illumination. The effect produced in red light is represented in fig. 110, where \( AB \) is the image of the rectangular aperture reflected from the faces \( n \) of the steel, and the four images on each side of it correspond with the prismatic images. All these nine images, however, consist of homogeneous red light, which is obliterated at the fifteen shaded rectangles, which are the minima of the Periodical Colours.
The centres \( p, r, t, n, s, c. \) of these rectangles correspond with the points marked with the same letters in fig. 107; and if we had drawn the same figure for violet light, the centres of the rectangles would have corresponded with \( o, g, s, m, p, c. \) in fig. 107. The rectangles should have been shaded off to represent the phenomena accurately, but the only object of the figure is to show to the eye the position and relations of the minima of the periods.
If it should be practicable to remove a still greater portion of the faces \( n \), the first minimum \( p \), fig. 110, would commence at a greater angle of incidence; and other two rows of minima, namely, rows of five and six, would be found extending to the fifth and sixth prismatic images. The arrangement and succession of these is easily deducible from fig. 110, where the law of the phenomenon is obvious to the eye.
The following table contains the angles of incidence reckoned from the perpendicular at which these minima occur in the extreme rays.
**Position of the minima in red light.**
| Ord. | Im. | 1st Prism | 2nd Prism | 3rd Prism | 4th Prism | |------|-----|-----------|-----------|-----------|-----------| | First minima | ...76° 0' | 66° 0' | 55° 45' | 41° 35' | 23° 30' | | Second minima | ...55° 45' | 41° 35' | 23° 30' | 23° 30' | 23° 30' | | Third minima | ...23° 30' | 23° 30' | 23° 30' | 23° 30' | 23° 30' |
**Position of the minima in violet light.**
| Ord. | Im. | 1st Prism | 2nd Prism | 3rd Prism | 4th Prism | |------|-----|-----------|-----------|-----------|-----------| | First minima | 81° 30' | 74° 60' | 66° 20' | 57° 48' | 48° | | Second minima | 66° 20' | 57° 48' | 48° | 48° | 48° | | Third minima | 48° | 48° | 48° | 48° | 48° |
When the steel with 1000 grooves is exposed to common light, and the incident ray is very near the perpendicular, the 5th, 6th, 7th, and 8th prismatic images are combined into a mass of whitish light, terminated externally by a black space. As the angle of incidence increases, the 6th, 7th, 8th, and 9th images are combined into this mass, then the 7th, 8th, 9th, and 10th images, and so on; the black space which terminates this mass receding from the axis or image \( AB \), fig. 107, as the obliquity of the incident ray increases.
Having covered the steel plate with water and oil of cassia in succession, I found the angular distances of the black space to be as follows at the same incidence:
- Air: 12° 23' - Water: 17° 15' - Oil of cassia: 21° 22'
The sines of which are inversely as the indices of refraction of the fluids.
Phenomena analogous to those above described take place on the grooved surfaces of gold, silver, and calcareous spar, &c.
In order to study this subject under a more general aspect, I was desirous of examining the phenomena exhibited by grooved surfaces of different refractive powers. It was Periodical obviously impossible to procure systems of lines upon transparent bodies in which the grooves should have exactly the same distance and magnitude; but I conceived it practicable to impress upon different substances the very grooves which produced the preceding phenomena, and I succeeded in impressing the system of 1000 grooves upon tin, realgar, and isinglass.
The following results were obtained with tin, the colours being those upon AB, fig. 107:
- White ........................................... 90° 0' - Yellow ........................................... 80 - Pink .................................................. 75 30 - First junction of pink and blue ............ 73 10 - Blue .................................................. 72 - Bluish green ....................................... 70 15 - Yellow .................................................. 63 - Bright pink .......................................... 54 - Second junction of pink and blue .......... 47 - Bluish green ....................................... 41 - Yellow .................................................. 36 - Pink .................................................... 32 - More and more pink
First minimum of red .................................. 72° 0' Second .................................................. 61 15
The following results were obtained with realgar:
- White ........................................... 90° 0' - Yellow ........................................... 80 - Pink .................................................. 75 30 - First junction of pink and blue ............ 73 10 - Blue .................................................. 72 - Bluish green ....................................... 70 15 - Yellow .................................................. 63 - Bright pink .......................................... 54 - Second junction of pink and blue .......... 47 - Bluish green ....................................... 41 - Yellow .................................................. 36 - Pink .................................................... 32 - More and more pink
First minimum of red .................................. 72° 0' Second .................................................. 61 15
The following results were obtained with isinglass. The colours were generally the same as in the steel:
- The first limit of pink and blue was at .... 75° 45' - The blue of second order ...................... 73 45 - The second limit of pink and blue was at .. 54 30
In these experiments the tin gave nearly the same results as the steel; but in the realgar and the isinglass similar tints were produced at a less angle of incidence than in the steel. The minima of the periods were exhibited very finely on the isinglass, and were produced at smaller angles of incidence.
In a specimen with 1000 grooves upon isinglass, the third pink, or that seen upon steel at 36°, was the highest; but after drying, the pink descended to yellow, and subsequently to green.
If the isinglass is removed from the steel when it is still soft, the edges of the grooves get rounded and lose their sharpness, and only one prismatic image is seen on each side of the ordinary image, as in mother-of-pearl.
The mass of white light is finely seen in the impressions taken upon tin, but never appears upon isinglass.
The prismatic colours seen on mother-of-pearl are exactly of the same kind as the prismatic images of grooved surfaces, with this difference, that a single prismatic image only is seen on each side of the common colourless image. Periodical. The following account of these colours has been given by Sir David Brewster, who first analysed them, and discovered their communicability to wax, the fusible metals, &c.
Mother-of-pearl, which constitutes the interior lining of the shell of the pearl oyster, and of various other shells, has been long employed in the arts for the purposes of use and ornament. Every one must have observed the play of prismatic tints, from which this substance derives much of its value as an ornament; but the nature and origin of these tints were never made the subject of investigation, till Sir David Brewster took up the subject, and published the results of his observations in the Phil. Trans. for 1814.
In order to study well the properties of this substance, we must select a regularly formed piece or plate of mother-of-pearl, which is known by the uniformity of its colour in day-light, and scarcely exhibits in that light any of the prismatic tints. Let this plate be now ground flat on both sides, (but not polished), upon a hone or a piece of slate, or upon a bit of glass, with the powder of schistus, or with fine emery. When this is done, hold the plate close to the eye, and view it by reflexion the flame of a candle, or of an argand lamp, or the flame of two or three candles, so placed as to appear like one, and we shall see a dull and reddish image, free from all prismatic colours, its dulness arising from the imperfect polish of the surface. On one side of, or above or below, this image, will be seen a brighter image with the colours of the spectrum, nearly as if it had been formed by a prism.
When the plate of mother-of-pearl is turned round in its own plane, the prismatic image will follow the motion of the plate, and revolve round the common image, the blue rays being nearest the common image, and the red rays farthest from it. If the plate is so placed that the prismatic image is in the plane of reflexion, and between the common image and the observer, it will be found that the distance between the two images increases with the angle of incidence, being about 2° 7' at an incidence not far from the perpendicular, and 9° 14' at a very great obliquity. This distance between the images varies more rapidly when the plate is turned round 180° in azimuth, so that the common image is between the prismatic image and the observer; but in this case we cannot measure the angle accurately much beyond 60°, when it is nearly 4° 30'.
Beyond the prismatic image, and in the same line with it and the common image, will be observed a mass of coloured light, nearly as far beyond the prismatic image as the prismatic image is from the common image. The distance of this patch of coloured light varies according to a different law from that of the prismatic image, as the rays which form it have previously suffered refraction. This mass of light has a beautiful crimson colour at great angles of obliquity. At 37° of incidence it is green, and at less angles it has a yellow hue, approaching to white, and becomes very luminous. These colours become more brilliant when the plate is polished, and have an origin quite different from those of the prismatic image.
Hitherto we have considered the phenomena only when the surface has that degree of polish which accompanies smooth grinding. If a greater degree of polish, however, is communicated to the plate, the common image becomes more brilliant, and a new prismatic image starts up, diametrically opposite to the first prismatic image, and at the same distance from the common image. This second prismatic image resembles in every respect the first. Its brilliancy increases with the polish, and when this polish is very high, the second prismatic image is nearly as bright as
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1 See Phil. Trans. 1886, p. 55. The first, which has its brilliancy a little impaired by polishing. This second image is never accompanied, like the first, by a mass of coloured light. If the polish of the surface is removed by grinding, the second prismatic image vanishes, and the first resumes its primitive brilliancy.
When the preceding experiments are repeated on the opposite surface of the plate of mother-of-pearl, the same phenomena are observed, but in a reverse order, the first prismatic image and the mass of coloured light being now seen on the opposite side of the plate.
In examining the light transmitted through the mother-of-pearl, we shall perceive analogous phenomena. A coloured image will be seen on each side of the common image, having the same angular distance from it as those seen by reflection, and resembling them in every particular; the blue light being nearest the common image, and the red light farthest from it. These two images, however, are generally fainter than those seen by reflection. When the second prismatic image is extinguished by removing the polish, it is then the most brilliant when seen by transmission; and, in general, the image which is brightest by reflection is faintest by transmission, and vice versa.
In measuring the angular distances of the prismatic image from the common image seen by reflection, Sir David Brewster had occasion to fix the mother-of-pearl to a goniometer by means of a cement made of rosin and bees' wax. Upon removing it from the cement, by insinuating the edge of a knife, and making it spring off, the plate of mother-of-pearl left a clean impression of its own surface; and he was surprised to observe, that the cement had actually received the property of producing the colours which were exhibited by the mother-of-pearl. This was at first attributed by our author, and others who saw the experiments, to a very thin film of mother-of-pearl detached from the plate, and left upon the cement; but subsequent experiments convinced him that the mother-of-pearl communicated to the cement its own properties.
In order to show this remarkable property to advantage, the plate of mother-of-pearl should be fixed to a handle like a seal, and its surface carefully cleaned. The sealing-wax should be black, and the impression taken by an impulse of the plate against the black sealing-wax, when rendered as fluid as possible by heat.
The properties of mother-of-pearl may also be communicated in this way to balsam of tolu, gum-arabic, gold leaf placed upon wax, tinfoil, the fusible metal composed of bismuth and mercury, and to lead, by hard pressure, or the blow of a hammer. When the impression is first made upon the fusible metal, the play of colours is singularly fine; but the metallic surface soon loses its polish, and the colours gradually decay.
If dissolved isinglass or gum-arabic, &c., are placed upon the plate of mother-of-pearl, and allowed to harden upon it, they will exhibit in the most splendid manner the colours of the substance; or if we imitate these gums between two plates of mother-of-pearl, we shall have transparent films, exhibiting on both sides the play of the prismatic tints.
In order to show that in these cases no part of the mother-of-pearl is left on the surface, Sir David Brewster plunged a piece of wax, with the impression, into nitric acid, which would instantly have destroyed the carbonate of lime, of which the mother-of-pearl is chiefly composed, but it had no effect on the colorific property of the surface. In soft cements, made of bees' wax and rosin, the slightest degree of heat destroys the superficial configuration, by which the colour is produced. In sealing-wax, isinglass, gum-arabic, and realgar, a much greater heat is necessary to destroy it; but in tinfoil and lead, its destruction can only be effected by the temperature at which they cease to become solid.
If we now examine the prismatic images reflected from the wax which has received the impression from an unpolished piece of mother-of-pearl, we shall find that the single prismatic image which is thus produced is on the right hand Colours of the common image, whereas it is on the left hand of the common image in the mother-of-pearl itself.
At different angles of incidence, the two coloured images formed by the wax, follow the same laws as those produced by the mother-of-pearl; but the mass of green and crimson tints never appears in the impressions taken from mother-of-pearl, because they are produced by light which has penetrated the mother-of-pearl, and has after refraction been reflected from one or more thin plates which lie between the strata of which the mother-of-pearl is composed.
In communicating to isinglass or gum-arabic the superficial structure of mother-of-pearl, their transparency enables us to observe the phenomena of the transmitted colours. The two prismatic images were both visible—the primary one being remarkably brilliant, and the second one scarcely perceptible; but when the light was transmitted through the gum, the primary image was nearly extinct, while the secondary one was unusually brilliant and highly coloured, far surpassing in splendour those which are formed by transmission through the mother-of-pearl itself. When both the surfaces of isinglass or gum-arabic have received the superficial structure of mother-of-pearl, four images are seen. The transmitted colours are more brilliant in gum-arabic than in balsam of tolu, as the latter reflects more light; but the prismatic images have the same degree of colour or dispersion when formed by isinglass or gum-arabic, as when they are formed upon metallic surfaces.
From these facts, it is obvious that the principal phenomena of mother-of-pearl have their origin in a particular configuration of its surface. By the use of the microscope Sir David Brewster discovered in every specimen of mother-of-pearl that gave the prismatic images, a grooved structure upon its surface, resembling the delicate texture of the skin at the tip of an infant's finger, or the lines which mark out islands and coasts upon a map.
In many specimens of mother-of-pearl, the grooves are parallel, but they are often arranged in all possible directions like the veins of agate, and in this case the common reflected image is surrounded with a number of prismatic images sometimes arranged in a circular or oval form, more or less regular. Sometimes the spaces between the grooves, or rather the edges of the strata of the shell, can be seen by the naked eye, or by a magnifying power of six or eight times, in which case the prismatic images are less highly coloured, having whitish light in their centre, and are placed close to the common image. At other parts of the same plate, more than 3000 grooves may be counted in an inch, and in some places they cannot be detected by ordinary magnifying powers. When the spaces between the grooves are wide, a new groove often begins, and there is often a sudden change from a series of wide grooves to a series of close ones. When the mother-of-pearl is scratched, the bottom and sides of the scratches are grooved exactly like the parts that are polished.
The direction of the grooves is always at right angles to the line joining the common image and the prismatic image. Had the grooved structure appeared only upon its external surface, the phenomena and the communicable colours would have disappeared when the surface was ground down; but the surprising part of the phenomenon is, that if we grind down the external surface with the finest powders, and polish it to the utmost degree, we never can grind out the grooved structure, and replace it by a flat surface. The edges of the shell break off by the action of the finest powders, so that the termination of one stratum cannot pass into the subjacent stratum without being separated by a distinct line or edge, formed by the fracture of its thin marginal parts. As all the strata have thus a prismatic termination, the mass of green and crimson light is reflect- Sir John Herschel discovered in very thin plates of mother-of-pearl a pair of nebulous prismatic images more distant from the central image than the two prismatic ones above described, and also a pair of fainter nebulous images, the line joining which is perpendicular to the line joining the first pair. He saw them by looking through thin pieces between the 70th and 300th of an inch thick. They are produced by a veined structure, in which there were 3700 veins in an inch. They cross the common grooves at all angles, and are parallel to the plane passing through the centres of the two systems of the coloured rings.
The spurious discs of stars first noticed by Sir William Herschel, have been justly considered as arising from diffraction, and have been described as supplementary facts to those discovered by Fraunhofer, in Sir John Herschel's Treatise on Light; and we believe every thing distinctly known on the subject, is due to the latter. A good telescope with a small power, shews fixed stars as a bright mass of light with some extraneous rays. A magnifying power of from 200 to 400, in favourable weather, exhibits the star as a well defined disc surrounded by several alternately dark and bright equidistant rings slightly coloured at their margin. These rings are best seen in refracting telescopes, with which the central disc is larger than in reflectors.
In applying apertures of various figures to the mirrors and object-glasses of telescopes, Sir John Herschel obtained the following results:
With the largest circular diaphragm, either near to, or distant from the spectrum or object-glass, the disc and rings increase inversely as the diameter of the aperture. When the aperture was only one inch in a telescope of seven feet in focal length, the disc of the star was well defined and surrounded with one ring only faintly tinged with white, faint red, black, very faint blue, white, extremely faint red, and black, reckoning from the centre. With a half inch aperture, the rings were invisible, the disc greatly enlarged, the light shading off to the circumference like some comets. This is shewn in fig. 11, Plate CCCLXXXII.
With annular apertures the phenomena were highly beautiful. When the outside of the annulus was three inches and the inside one and a quarter, Capella appeared as in fig. 12, and the double star Castor as in fig. 13. When the breadth of the annular aperture is diminished, the disc and the breadth of the rings also diminish, while the number of visible rings increases. The appearance of Capella with annular apertures of 5·5 inches exterior, 5·5 interior, of 0·7 exterior, of 0·2 interior, and of 2·2 exterior, of 2·0 interior, is represented in figs. 14, 15, and 16. In the last figure the disc was reduced to a point, and the rings were so numerous and close that they could scarcely be counted. When the breadth of this annulus was reduced one-half, the rings were invisible.
When two annuli, as shewn in the annexed figure, were used, large halos or rings were seen by Sir John Herschel, as in fig. 17, plate CCCLXXXI.
With an aperture of the shape of an equilateral triangle, or the opening between two concentric equilateral triangles, the figure was that shewn in fig. 18, in which the small central disc was extremely bright, and the field of the telescope black.
When the preceding figure is seen with the telescope out of focus, it changes into fig. 19.
When three circular apertures are placed at the angles of an equilateral triangle, the effect shewn in fig. 20, is produced.
When three equal and similar annular apertures were arranged in the same manner, the effect was as in fig. 12. When this was thrown out of focus, it had the appearance in fig. 21; when brought better into focus, it changed into fig. 22; and when in focus, into fig. 12.
Sect. IV.—On the Colours of concave Mirrors or thick Plates.
The colours produced by thick plates were discovered by Sir Isaac Newton, who has given the following account of them. "There is no glass or speculum however well polished but besides the light which it refracts or reflects regularly, scatters every way irregularly a faint light by means of which the polished surface, when illuminated in a dark room by a beam of the sun's light, may be easily seen in all positions of the eye. There are certain phenomena of this scattered light, which, when I first observed them, seemed very strange and surprising to me. The sun shining into my darkened chamber through a hole in the shutter AB, one-third of an inch wide, I let the intromitted beam of light RR fall perpendicularly upon a glass speculum M, ground concave on one side, and convex on the other, to a sphere of five feet and eleven inches radius, and quick-silvered over on the convex side. And holding a white opake chart, or a quire of paper, at the centre of the sphere to which the speculum was ground, that is, at the distance of about five feet and eleven inches from the speculum, in such a manner that the beam of light might pass through a little hole made in the middle of the chart to the speculum, and thence be reflected back to the same hole; I observed upon the chart four or five concentric irises or rings of colours like rainbows, after the manner that those in the fourth and following observations of the first part of this third book encompassed the black spot, but yet larger and fainter than those. If the distance of the chart from the speculum was much greater or much less than that of six feet, the rings became dilute and vanished,
Fig. 112.
The colours of these rainbows succeeded one another from the centre outwards, in the same form and order with those transmitted through the two object glasses.
Measuring the diameters of these rings upon the chart, I found them also in the same proportion to one another with the rings between the two object glasses. For the diameters of the four first of the bright rings measured between the brightest parts of their orbits at the distance of six feet from the speculum, were 1\(\frac{1}{2}\), 2\(\frac{1}{2}\), 3\(\frac{1}{2}\), 3 inches, whose squares are in arithmetical progression of the numbers 1, 2, 3, 4. If the white circular spot in the middle be reckoned amongst the rings, and its central light, where it seems to be most luminous, be put equipollent to an infinitely little ring; the squares of the diameter of the rings will be in the progression 0, 1, 2, 3, 4, &c. I measured also the diameters of the dark circles between these luminous ones,
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1 Sir John Herschel's Treatise on Light, § 769, &c. in which the reader will find the subject more fully treated. 2 This account is abridged from Newton's Optics, book ii. part iv. p. 204. Sir David Brewster has remarked, that the method which he has found the most simple for exhibiting these colours, is to place the eye immediately behind a small flame, from a minute wick fed with oil or wax, so that we can examine them even at a perpendicular incidence. The colours of thick plates may be seen even with a common candle held before the eye at the distance of ten or twelve feet from a common pane of crown glass in a window that has accumulated a little fine dust upon its surface, or that has on its surface a deposition of fine moisture. Under these circumstances they are so very bright, that they may be seen even when the pane of glass is clean.
Sect. V.—On the Colours produced by Double Plates of Glass of equal thickness.
In 1815, Sir David Brewster published in the Edinburgh Transactions an account of a new species of coloured double fringes produced by the reflection of light between two plates of parallel glass of equal thickness.
In these experiments he cut the plates of glass AB, CD, fig. 112, out of the same piece, and having placed between them a bit of soft bees' wax, he pressed them together till they were at the distance of nearly the tenth of an inch, and slightly inclined to each other as in the figure, till one or more of the reflected images of a circular luminous disc seen in the direction VR by an eye at V, were reflected from the bright and direct image formed by transmitted light. When this was done the reflected image was crossed with about fifteen or sixteen beautiful parallel fringes. The three central fringes consist of blackish and whitish stripes, and the exterior ones of brilliant stripes of red and green light; and the central fringes have the same appearance in relation to the external fringes, as the internal have to the external rings formed by thin plates. If the two plates of glass are turned round in a plane at right angles to the incident ray, the reflected images will move round the bright image, and the parallel fringes will always preserve a direction at right angles to a line joining the centres of the bright and reflected images. Hence it follows, that the direction of the fringes is always parallel to the common section of the four reflecting surfaces, which exercise an action upon the incident light.
The position of the plates remaining as before, let the inclination of the plates, or, what is the same thing, the distance of the bright and the reflected image, be varied by a gentle motion of one of the plates, the coloured fringes will be found to increase in breadth as the inclination of the plates is diminished, and to diminish as the inclination of the plates is increased.
If the light of the circular object, instead of falling perpendicularly upon the plates, is incident at different obliquities, so that the plane of incidence is at right angles to the common section of the plates, no fringes are visible across any of the images. But if the plane of incidence is parallel to the common section of the plates, the reflected images increase in brightness with the obliquity of incidence, and the coloured fringes become more vivid. When
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1 Memoir de l'Acad. de Paris, 1705, p. 136. 2 Phil. Trans., 1807, Part II. 3 See our Article Chromatics, vol. vi. p. 639, sect. ix. 4 Treatise on Optics, Lardner's Cyclop. § 77. 5 Vol. vii. p. 435. Periodical the angle of incidence increases from $0^\circ$ to $90^\circ$, the images that have suffered the greatest number of reflections are crossed by other fringes, inclined to them at a small angle. At an angle of about $44^\circ$, the image formed by four reflexions is covered with interfering fringes; but it is not till the angle of incidence is greater, that this is distinctly seen on the image formed by two reflexions.
Hitherto he had observed no fringes upon the first or bright image, which is composed of light that has not suffered reflexion from the second plate of glass. By concealing, however, the bright light of the first image, so as to perceive the image formed by a second reflexion, within the first plate, and by viewing this image through a small aperture, which he found of great service in giving distinctness to all the phenomena, he observed fringes across the first image, far surpassing in precision of outline, and in richness of colouring, every analogous phenomenon which he had seen. When these fringes were concealed, he also observed other fringes on the image immediately behind them, and formed by a third reflexion, from the interior of the first plate. He concealed the second image, upon which the fringes were extremely bright, and very faint stripes were seen upon the one immediately behind it.
In examining these phenomena a little more attentively, he observed that the size of the fringes in the first image varied with the distance of the eye from the plates, while those on the second and fourth image diminished with that distance.
In pursuing this inquiry, our author found that the production of the fringes depends upon the action of all the four surfaces of the two plates of parallel glass; and that the magnitudes of the fringes are inversely as the thickness of the plates that produce them at a given inclination.
When the eye is placed between the plates and the luminous object, so as to see the first, third, fifth, seventh, &c. reflected images, the coloured fringes are also seen, and have the same character as those already described.
All these phenomena are seen without any variation when polarised light is used in place of common light.
In order to explain the changes which the light undergoes in its passage through the plates of glass, let AB, CD, fig. 113, be a section of two plates at right angles to the common section of their surfaces, and let RS be a ray of light incident nearly in a vertical direction. This ray, after passing through the first plate AB, will suffer a small refraction at P and Q, and emerge in the direction QV parallel to RS. At the point P, in the second plate CD, the ray TP will be reflected to a, again reflected to b, and, after suffering a refraction at b and c, will emerge in the direction cd, forming with RV an angle equal to twice the inclination of the plates. A portion of the reflected ray Pa, will enter the first plate at a, and having suffered reflection and refraction at b, the reflected portion $\beta\gamma$ will reach the eye at $\delta$. The ray P ab c will likewise suffer a reflection at c and e, and will reach the eye at g. In like manner, a part of the ray PQ will be reflected at Q, and move in the direction Qrstuv, and another part of it in the direction swxyz, and these rays will suffer several other reflexions; but the images which they form will be so faint, that the eye will not be capable of perceiving them. When the observer, therefore, looks at a luminous body, in the direction SR, through the glass plates, he will perceive two images, one of which is a bright image, seen by the transmitted light QV, and the other is a faint image, seen principally by the reflected light P ab c d and composed of several images, formed by the pencils c d, u v, e f, z s, and e g. The bright image is not crossed by coloured
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1 Vol. vi. p. 636, sect. vi. 2 Treatise on Light, § 688—606. See also Biot's Traité de Physique, tom. iv. p. 246.
Then the next ring, viz. the dark one, m, m, m, forming the boundary of the remote external system, m, m, m, A, and of the near central system m', m', m'; B. The four groups of rings thus developed, assume at greater incidences the character shewn in fig. 25, but they are not seen all at once; and in tracing their form it is necessary to cause the image on which they are produced to be reflected successively from different parts of the lens. The rings are so closely packed together, at a distance from the white centres x'x, to which they are all related, that it is extremely difficult to perceive them in the present object-glass. At a still greater angle of incidence, the rays close in upon the centres x'x, and become exceedingly close and nearer as the points x'x approach to the circumference of the lens, and the rays become brighter from the increase of the light at greater obliquities.
In some double object-glasses the rays can only be seen by looking through the convex crown glass lens. In one object-glass the four bounding fringes at x'x, fig. 25, united and formed a black cross, as shewn in fig. 26. From a series of experiments Sir David Brewster has found that in the object-glass shewn in fig. 113, the action of the two surfaces 1, 2 of the convex lens AB, and the inner surface 3 of the concave one CD, are necessary to the production of these fringes, and hence he concludes that the rings arise from the interference of two pencils of light, one of which has suffered three reflections within the convex lens AB, and has passed four times through its thickness, with another pencil which has suffered two reflexions within the convex lens, and one reflection from the inner surface of the concave lens, and has passed four times through the thickness of the convex lens, and twice through the thickness of the meniscus of air.
In a triple object-glass, which gave a system of rings similar to that in fig. 26, they were covered with another system of very minute fringes, parallel to one another, and to the line joining the centres x and x'.
SECT. VI.—On the Colours of Double Plates of Glass of Unequal Thickness, and other Analogous Phenomena.
Mr. W. Nicholson observed the colours of thick plates in the glasses employed for the sights of sextants, and he considered them analogous to those of thin plates. They have been ascribed, however, by Dr. Young, "to the rays twice reflected in the second plate only."
Mr. Knox of Belfast has described some interesting phenomena already briefly noticed by Dr. Young, in the article CHROMATICS. Having formed a system of the rings of thin plates, by placing a convex lens on a piece of silvered glass, he observed the common system of reflected rings, and also the transmitted system reflected to his eye by the silvered glass. This is shewn in fig. 27, Plate CCCCLXXXIII., where A and B are the two systems; but he was surprised to observe between them a system of parallel fringes CD EF, passing through the intersection of the two circular systems. These fringes were equal in number to the number of the rings in A and B. They were equidistant, reached the edge of the lens on both sides, and were formed at right angles to the direction of the light, and to a line forming the centres of the systems A and B.
Our author then tried the effect of combining two primary systems of reflected rings in the same manner. With this view he placed a double-convex lens about thirty-six inches in focal length, on a piece of plate glass, with its under side painted black, and upon the lens he placed a piece of plate glass. By these means two sets of primary rings were produced, whose relative positions could be altered at pleasure. By using the shadow of a black card, he found that, instead of parallel fringes, as in fig. 27, he had a new species of rings of a circular form, from two to three times the diameter of the primary rings from which they originated. These rings passed, as before, through the intersections of the primary ones; and the ring which divided the two classes passed through a point, whose distance from the centre of each primary set were in proportion to their longest diameters.
These rings, which Mr. Knox calls intersectionary ones, may be made to vary infinitely in their dimensions, according as the diameters of the primary sets differ more or less, being least where that difference is greatest, and increasing in size as the two primary sets approach to equality, until at last they become straight lines, when the two primary sets are equal. The dimensions of the intersectionary rings will also, as Mr. Knox remarks, ceteris paribus, diminish as the two primary sets approach, and increase as they recede from one another.
As these intersectionary rings are almost always accompanied by a second, and sometimes by a third set of equal or unequal dimensions, Mr. Knox supposes that they may be produced by primary sets, combined with either transmitted or reflected sets, provided the two between which they are formed are of unequal dimensions.
Considering the intersectionary fringes as diagonals to the angles at which they were formed, Mr. Knox conjectured that if he could form rectilineal fringes by flat plates, and combine them at different angles, he would produce a third or diagonal set placed between the other two. He accordingly took a pair of slips of glass, and by applying two of their ends together, and using some friction, and a considerable degree of pressure, he formed a fine set of rectilineal fringes. By applying a third slip of glass longitudinally to the upper one of the first two, he formed a similar set of rectilineal fringes at right angles to the first; and he immediately observed the diagonal fringes, which he had anticipated, appear in the angle between the two primary sets, as shewn in fig. 28, where B and C are the primary fringes, and D the intersectionary set, divided into two classes, as shewn by the dotted line. By forming the second set of fringes at different angles with the first, the central band of the intersectionary fringes always bisected the angle. It is a curious circumstance, that though the diagonal fringes are formed by the crossing of the two primary sets, yet they never appear at the opposite angle A, nor could they be made to appear in any angle formed by primary fringes, unless these fringes were so disposed as to have their red sides turned towards each other.
Dr. Young has given an explanation of these curious phenomena in the article CHROMATICS, to which we must refer the reader.
Mr. Henry Fox Talbot upon superposing two films of blown glass and viewing through them a homogeneous yellow flame, and even the light of the sky, observed bright and dark stripes, or coloured bands and fringes, which were not produced by either of them separately. These phenomena, as Sir John Herschel remarks, are obviously referable to the same principle as the fringes discovered by Sir David Brewster, "the interference taking place here between rays respectively twice reflected within the upper lamina, and once reflected at the upper surface of the lower lamina, or else between rays, one of which is thrice reflected in the mode represented by $AaB'a'BaA$, and the other in that represented by $AaB'a'A'aA$, the interval between the glasses being supposed to be exactly equal to the..." Periodical thickness of the upper one in both cases, a condition which is sure to obtain somewhere when the laminae are curved.
**Sect. VII.—On the Colours of Fibres formed by Reflection and Transmission.**
Every person must have observed the fine thread or line of the spider's web glittering with the brightest colours, when the light of the sun is reflected from it to the eye. By examining these colours attentively, they are found to vary with the angle of incidence. The only attempt that we know of to explain this phenomenon, that deserves notice, is that of Sir John Herschel—“These colours,” says he, “may arise either from a similar cause, (namely, that which produces colour in a single scratch or fissure, or the interference of light reflected from its opposite edges,) or from the thread itself as spun by the animal, consisting of several agglutinated together, and thus presenting not a cylindrical but a furrowed surface.” That the structure thus assumed would produce colours cannot be doubted, but they would be the colours of grooved surfaces such as we have already described, which we conceive to be exceedingly different from those of the spider's line.
It appears from a preceding section, that when a reflecting surface has its breadth reduced to an extreme degree of narrowness, whether it be a metallic or a transparent one, it is no longer capable of reflecting white light, but decomposes it in the manner already described, the coloured bands being transverse to the direction of the reflecting line. Sir David Brewster has therefore applied this new property of reflecting surfaces to account for the colour of the spider's line, which presents analogous phenomena, and which from its extreme minuteness, must necessarily decompose white light. If this is not the true cause of the phenomenon, it will not be easy to obtain any rational explanation from the doctrine of interference, without taking for granted the existence of a structure of a very peculiar kind.
**On the Colours of Fibres by transmitted Light.**
If we take a number of fibres of wool, and fix them in parallel directions, as in Fraunhofer's gratings, they would of course produce parallel spectra or coloured bands similar to those already described. If we take a mass of the same wool, in which the fibres have every possible direction, they will then exhibit spectra or fringes lying in every possible direction, or circular ones. As both these results would be equally obtained, if the fibres were cut down into particles as long as they are broad, we should have parallel fringes when the particles are arranged in straight lines, and circular ones when the particles are scattered like dust upon a plate of glass in all directions. Hence fibres of minute particles will produce circular fringes, which increase in diameter as the particles are smaller in diameter.
When we therefore look at a candle placed at a little distance, through wool, or cotton, or vapour lying upon glass, or diluted blood, or milk, or the seed and farina of plants, &c., &c., we observe round the image of the candle a light area terminating in a dark reddish margin. This is followed by a ring of bluish green light, and then a red ring, and when the fibres or particles have an uniform size, the green and red rings are frequently repeated. It is to Dr. Young that we owe the discovery that the diameter of these rings is always the same when the size of the particles or fibres is uniform and equal, and that they vary inversely as the size of the fibres or particles. On this principle he constructed his eriometer for measuring the diameter of minute particles and fibres, such as wool, &c. It is composed of a plate of brass, or copper, or even card, with an aperture in the centre about the fortieth of an inch in diameter, and surrounded by a circle of perforations about half an inch in diameter, the number of perforations in the circle being about ten or twelve, and as minute as possible. When the instrument is constructed on such a small scale, the eye requires the aid of a lens. The wool, or plate of particles, are then attached to the end of a slider, and when the light of an argand lamp, or two or three candles placed in a line, so as to unite their flames, is transmitted through the wool or particles, the slider is drawn out till the first dark red coloured circle coincides with the circle of particles, and the index then shews on the scale upon the slider the number which indicates the size of the fibres or particles. The basis of this scale, rather an imperfect one, Dr. Young took from Dr. Wollaston's measure of the magnitude of the seeds of the puff ball or Lycoperdon bovista, which he found to be the 8500ths of an inch in diameter. Dr. Young found the rings formed by these minute seeds to be three-and-a-half on the scale of his instrument, and hence he assumes the thirty thousandths part of an inch, (more accurately the 29750th), as the value of an unit on his scale. The following results were obtained by Dr. Young:
| Table of the diameter of Fibres and minute particles. | |------------------------------------------------------| | Milk diluted, very indistinct, about..................3 | | Lycoperdon bovista, dust of, very distinct.............3½ | | Blood of a bullock, from beef..........................4½ | | Human blood, diluted with water.......................5 | | Smut of barley (male ear)..............................6½ | | Blood of a mare.......................................6½ | | Human blood diluted with water after standing five days...6½ | | Blood recently diluted with serum, only...............8 | | Pus....................................................7½ | | Silk, very irregular..................................12 | | Wool of the beaver, jointed, very uniform.............13 | | Angola wool, about...................................14 | | Vigonia wool..........................................15 | | Siberian hare's wool, Scotch hare's wool, foreign coney wool, yellow rabbit's wool, about........15½ | | Mole's fur............................................16 | | Skate's blood.........................................16 | | British coney wool, American rabbit's wool, about.....16½ | | Saxon wool, a few fine fibres..........................17 | | Buffaloo's wool.......................................18 | | Wool of the mountain sheep, ovis montana..............18 | | Seal wool, finest, mixed, about.......................18½ | | Shawl wool............................................18 or 19 | | Goat's wool...........................................19 | | Cotton, very unequal..................................19 | | Peruvian wool, mixed, the finest locks...............20 | | Welch wool, a small lock of...........................20 | | Saxon wool............................................23 or 24 | | Wool of an Escorial ram...............................23 to 24 | | Southdown wool.......................................24 | | Lionceza wool, 24 to 29...............................generally 25 | | Paular wool, 24 to 29...............................generally 25½ | | Alpaca wool, about..................................26 | | Laurestinus, farina of...............................26 | | Ryeland merino wool.................................27 | | Merino Southdown wool...............................28 | | Lycopodium seed, beautifully distinct...............32 | | Southdown ewe wool.................................39 | | Coarse wool, Sussex.................................46 | | Coarse wool, from same, worsted......................60 |
The diameter of the fibres or particles in the preceding table may be obtained in parts of an inch, by dividing the 1/30,000 of an inch by the numbers opposite them. The diameter of the particles of the human blood will be
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1 Treatise on Light, §693. See also Riot, Traité de Physique, tom. iv. p. 246. Young has ingeniously observed, that if we square the number belonging to the pound of wool, and subtract 325, the remainder will be nearly the number of pounds of wool that are worth 100 guineas. In the case of good Lionessa wool for example, whose number is 25, we shall have $25 \times 25 - 325 = 300$ for 100 guineas, or seven shillings per pound.
In experiments of this kind it would be better to express the magnitude of the first ring in parts of the radius, or by the angle which the whole ring subtends at the eye. Those who have any of Mr. Barton's scales, with the number of lines in an inch marked, can easily compare the diameter of the first ring produced either by fibres or particles, with the distance of the red spaces in the two first prismatic images; and by the rule of proportion he will find the magnitude of the fibres or particles.
**Sect. VIII.—On the colours of mixed plates.**
The phenomena of colour observed by M. Mazeas, when he pressed between two glasses, suet, Spanish wax, resin, common wax, and the sediment of urine, are clearly those of mixed plates, though they have not hitherto been recognised as such. He put between his glass plates a little ball of suet about one-fourth of a line in diameter, and pressed it between the two surfaces, warming them at the same time in order to disperse the suet. He then rubbed them violently together in a circular manner, and was surprised at looking at a candle through them to see it surrounded with two or three concentric rings, very broad, and with very lively and delicate colours, namely, a red inclining to yellow, and a green like that of an emerald. By continuing the friction, the rings assumed the colours of blue, yellow, and violet, especially when he looked through the glasses on bodies directly opposed to the sun.
M. Mazeas shows very clearly that these were not the colours of thin plates on account of the distance between the glasses, and also because the colours disappeared by melting the suet, but a new species of colours which he tried in vain to explain.
M. Dutoir repeated and varied the experiments of Mazeas, but did not succeed in explaining them.
Dr. Thomas Young, apparently without knowing of the experiments of Mazeas, though they are fully detailed by Priestley, has described the very same colours under the name of the colours of mixed plates, and the merit of discovery of these colours has been ascribed to him by almost all modern writers. The method of producing the colours by suet, as given above by Mazeas, is exceedingly simple, while many persons have failed in repeating the experiments described by Dr. Young. It is to Dr. Young, however, that we owe the higher obligation of having discovered the general principle to which these colours must be referred, though he has not examined the phenomena with that attention which they merited. The following is the account which Dr. Young has given of the colours of mixed plates.
"I first noticed the colours of mixed plates, in looking at a candle through two pieces of plate glass, with a little moisture between them. I observed an appearance of fringes resembling the common colours of thin plates; and, upon looking for the fringes by reflection, I found that these new fringes were always in the same direction as the other fringes, but many times larger. By examining the glasses with a magnifier, I perceived that wherever these fringes were visible, the moisture was intermixed with portions of air, producing an appearance similar to dew. I then supposed that the origin of the colours was the same as that of the colours of haloes; but, on a more minute examination, I found that the magnitude of the portions of air and water was by no means uniform, and that the explanation was therefore inadmissible. It was, however, easy to find two portions of light sufficient for the production of these fringes; for the light transmitted through the water, moving in it with a velocity different from that of the light passing through the interstices filled only with air, the two portions would interfere with each other, and produce effects of colour according to the general law. The ratio of the velocities, in water and in air, is that of 3 to 4; the fringes ought therefore to appear where the thickness is 6 times as great as that which corresponds to the same colour in the common case of thin plates; and, upon making the experiment with a plain glass and a lens slightly convex, I found the sixth dark circle actually of the same diameter as the first in the new fringes. The colours are also very easily produced, when butter or tallow is substituted for water; and the rings then become smaller, on account of the greater refractive density of the oils; but, when water is added, so as to fill up the interstices of the oil, the rings are very much enlarged; for here the difference only of the velocities in water and in oil is to be considered, and this is much smaller than the difference between air and water. It appears to be necessary for the production of these colours, that the glasses he held nearly in a right line between the eye and the common termination of a dark and luminous object; the portion of the rings, seen on the dark ground, is then more distinct than the remaining portion; and, instead of being continuations of the rings, they exhibit everywhere opposite colours, so as to resemble the colours of common thin plates seen by reflection, and not by transmission. In order to understand this circumstance, we must consider that where a dark object as A is placed behind the glasses, the whole of the light which comes to the eye, is either refracted through the edges of the drops, (as the rays B, C,) or reflected from the internal surface (as D, E,) while the light which passes through those parts of the glasses which are on the side opposite to the dark object, consists of rays refracted as before through the edges, (as F, G,) or simply passing through the fluid, (as H, I.) The respective combinations of these portions of light exhibit a series of colours of different orders, since the internal reflection modifies the interference of the rays on the dark side of the object, in the same manner as in the common colours of thin plates seen by reflection. When no dark object is near, both these series of colours are produced at once; and since they are always of an opposite nature at any given thickness of a plate, they neutralize each other, and constitute white light.
In applying the general law of interference to these colours, as well as to those of thin plates already known, it is impossible to avoid a supposition which is a part of the undulatory theory, that is, that the velocity of light is the greater, the rarer the medium; and there is also a condition annexed to this explanation of the colours of mixed plates, as well as to that of the colours of simple thin plates, which involves another part of the same theory; that is,
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1 See Dr. Young's Introduction to Medical Literature, p. 552. 2 Memoires i rescripts, tom. ii. p. 43. 3 Id. Id. tom. iv. p. 288, 289; or Priestley On Vision, vol. ii. p. 505. Periodical that where one of the portions of light has been reflected at the surface of a rarer medium, it must be supposed to be retarded one half of the appropriate interval; for instance, in the central black spot of a soap bubble, where the actual lengths of the paths very nearly coincide, but the effect is the same as if one of the portions had been so retarded as to destroy the other. From considering the nature of this circumstance, I ventured to predict, that if the two reflections were of the same kind, made at the surfaces of a thin plate of a density intermediate between the densities of the mediums containing it, the effect would be reversed, and the central spot, instead of black would become white; and I have now the pleasure of stating, that I have fully verified this conclusion by interposing a drop of oil of sassafras between a prism of flint-glass and a lens of crown-glass; the central spot seen by reflected light was white, and surrounded by a dark ring. It was however necessary to use some force, in order to produce a contact sufficiently intimate; and the white spot differed, even at last, in the same degree from perfect whiteness, as the black spot usually does from perfect blackness. There are also some irregularities attending the phenomena exhibited in this manner by different refracting substances, especially when the reflexion is total, which deserves farther investigation.
We are not aware that these interesting experiments of Mazzea and Dr. Young, have been repeated by any modern writer, our treatises on optics containing merely an abstract of Dr. Young's results. The subject has, however, been recently taken up by Sir David Brewster, who had found that the colours of mixed plates, in place of being merely the result of an experiment, was a natural phenomenon of a very interesting kind, which frequently presented itself in the examination of minerals. He was therefore led to attach a greater interest to the phenomena which they present. As the paper which contains the results of his inquiries has not been published, we can only give a brief notice of them.
In order to produce these colours so as to be permanent, he found that the froth of albumen, (the white of an egg beat up into froth) ground circularly as it were between two thick plates of glass pressed firmly together, when the circular motion is stopped, exhibits the colours very splendidly. The glasses may then be held together by wax or by screws. If we desire to have a circular system of rings, we must use a convex lens in place of one of the plates. Whipped cream answers also very well; but paste that has become very smooth by age he found preferable to any other substance which he employed. When the experiment is successfully made, the colours are extremely splendid, the flame of a candle or other luminous body being of a bright colour complementary to that of the coloured light which surrounds it.
Upon looking with a microscope at the albumen or paste thus pressed into a film, it is found to resemble accurately the strata of cavities containing the new fluids, and sometimes water, as in sulphate of lime, &c., the paste being sometimes found in separate ramified patches of all shapes surrounded with air, while on some occasions numerous air cavities are included in the paste.
Although Dr. Young was correct in ascribing the colours of mixed plates to the interference of rays moving with different velocities in passing through the two contiguous media, yet his analysis of the phenomena is imperfect, and his determination of the interfering pencils incorrect. In his 39th lecture he ascribes "the colours seen in the dark part beyond the object, to the light scattered irregularly from the surfaces of the fluid," and in his description of fig. 115, above quoted, he specifies as the interfering portions which produce the colours, viz. rays reflected from the internal surface of the cavities, with rays refracted through the edges of the drops; and rays refracted through the edge with rays simply passing through the fluid. That the colours of mixed plates are not produced either by refracted or reflected pencils, is at once proved by the fact, that the same colours may be produced when the most refracting medium is terminated by a fine edge, and is itself a plate with perfectly parallel surfaces, so that there is no edge to reflect light, and no inclined faces to refract it.
Having obtained this result, Sir David Brewster viewed the subject in another aspect, and has shewn that the phenomenon is entirely one of inflexion caused by the edge of a transparent inflecting body sufficiently thin to produce colours by the interference of the retarded light passing through the body close to its edge, with the light passing within the body close to its edge. The same pencils which interfere in the common case of a diffusing body interfere in the present case; but the pencils that pass through the transparent plate are modified by the retardation which they experience, so as to produce the phenomena of colours. The oppositely coloured pencils in mixed plates form part of the system of diffracted fringes, the colour seen upon the luminous body occupying the shadow of the diffusing edge, and the opposite colour seen around or beside the candle occupying the first fringe on each side of the shadow. The colours in the fringe on the left hand of the shadow of the diffusing edge, are seen by the eye on the right hand side of the candle, and those in the fringe on the right hand of the shadow are seen by the eye on the left hand side of the candle.
All the preceding phenomena may be produced by breaking down a thin transparent plate into minute portions, and when these portions are made to float in a fluid, or are placed between two plates of glass containing a fluid of nearly the same refractive power as the solid portions. The solid portions will thus act like much thinner plates placed in air, and we shall observe the identical phenomena which take place with the cavities in paste or albumen. When we examine with the microscope a very narrow portion of the solid substance bounded by lines nearly parallel, we perceive the phenomena of mixed plates under a new aspect. The space between the shadow of the two edges is filled up with a bright band of the colour which occupies the two first external fringes, and as the innermost fringe of each edge overlap each other, the colour has double the intensity. The same phenomenon is often seen in the parallel ramifications, or in the long cavities produced by the paste or albumen when the last act of friction was to move one of the plates of glass in a straight line.
The phenomena of mixed plates have been discovered by Sir David Brewster in sulphate of lime, in which there are shallow crystallised cavities in great numbers forming regular strata in the mineral. These cavities are filled with water, and in this case the diffusing edge is the edge of the cavity, and the light which passes through the water has its velocity greater than that which passes through the solid mineral.
In one remarkable circular stratum of cavities, all the cavities in the centre were the deepest, and gradually diminished in depth towards the circumference, till the microscope could scarcely resolve them. The highest orders of colours were therefore in the middle of the stratum, and they gradually descended to a white of the first order, at the circumference of the circular stratum.
An interesting experiment lately published by Henry Fox Talbot, Esq., is a phenomenon of mixed plates, in which the densest of the plates is too thick to produce the colours at its edge in common light. "Make," says he, "a circular hole in a piece of card, of the size of the pupil..." of the eye. Cover one half of this opening with an extremely thin film of glass (probably mica would answer the purpose as well or better.) Then view through this aperture a perfect spectrum formed by a prism of moderate dispersive power, and the spectrum will appear covered throughout its length with parallel obscure bands resembling the absorptions produced by iodine vapour. The cause of this phenomenon probably is, that one half of the light which passes through the glass film has its undulations thereby retarded by a certain quantity.
This phenomenon, which we have observed under various modifications, arises from the diffraction of the light passing on each side of and very close to the edge of the film; for if we cover this edge even with a fine wire, we obstruct the diffracted fringes modified by the retardation of the denser plate, and the phenomenon vanishes. Hence in order to see the fringes in perfection, we must make the aperture no wider than to include the space on each side of the edge of the film within which the light passes that is concerned in the phenomenon. As the light here used is perfectly homogeneous, the bands are alternately coloured and obscure.
Sir David Brewster has observed the fringes described by Mr. Talbot with thick plates having fine edges, and immersed in fluids, and even with plates of glass the 15th part of an inch thick, by looking through their edge upon a highly dispersed spectrum formed by a large refracting angle of a prism, and magnified by a powerful telescope. The perfection of the edge of the film, and an equality of thickness, are essential to the production of the fringes in their most interesting form. Sir David Brewster obtained them by looking through plates of sulphate of lime, where there were not properly speaking plates of different refractive powers, but where the plate suddenly became thicker, as shown in the annexed figure, where AB, EC is the plate of sulphate of lime, which becomes thicker at E. When the eye looks through it at the spectrum in the direction ED, it sees the fringes which have the same magnitude as if the plate ADB were removed, and the eye looked only through the plate CE, in which case it is clearly a phenomenon of mixed plates.
When these dark lines were produced in great perfection, they had a sort of granular structure resembling fine screws at their edges, and sometimes appearing to enclose minute specks of light. Several other remarkable phenomena accompanying these fringes were communicated by the author to the British Association at Liverpool, and will appear in the next volume of the Philosophical Transactions.
PART VI.—ON THE DOUBLE REFRACTION OF LIGHT.
In the preceding part of this article we have supposed that when light is transmitted vertically through the surfaces or through the mass of transparent bodies, or when it passes obliquely and is refracted by the same bodies, it leaves the surface or emerges from the body in a single pencil, either perfectly white, or decomposed into a diverging beam of coloured light. This supposition is perfectly correct, under certain circumstances, for gaseous and fluid bodies, for glass slowly and equally cooled, and for a numerous class of crystallized bodies, whose primitive form is either the cube, the regular octahedron, or the rhombohedral dodecahedron. When these bodies have the same temperature and density throughout their mass, and are not exposed to any pressure, a pencil of light will be transmitted through them single, according to the laws already explained.
When a pencil of light falls upon all other bodies, such as artificial crystals, or salts on crystallized minerals, which have not the above primitive forms; upon animal substances, such as bone, horn, shell, hair, crystalline lenses of animal and elastic integuments; upon vegetable bodies, such as particular leaves, stalks and seeds; upon artificial bodies such as gums, resins, jellies, and solid bodies that have a variable density, during the transient passage of heat, or from rapid cooling, or unequal temperature and pressure;—when a pencil of light falls upon such bodies, it will be refracted into two distinct pencils, more or less inclined to each other according to the mechanical condition of the body, or to the direction in which the pencil passes through it. In some minerals and artificial crystals, this refraction of the two pencils is very great, and, generally speaking, is in such crystals, easily observed and measured; but in some cases it is not visible unless by transmitting the light through prisms of the body with large refracting angles; and in very many cases, the existence of two pencils is inferred only from certain other properties, which always accompany the property of giving double pencils. This separation of a single pencil into two, is called double refraction, and the bodies which have such a property are called doubly refracting crystals or bodies.
In many regular crystals there is one line through which, if a single pencil of light is transmitted, it does not experience double refraction. This line is called the axis of the crystal, or the axis of double refraction, and such crystals are denominated crystals with one axis of double refraction. In other crystals there are two lines through which, if the single pencil of light is transmitted, it does not experience double refraction. Such crystals are denominated crystals with two axes of double refraction. We shall treat of these two classes in separate sections.
Sect. I.—On the Law of Double Refraction in Crystals with one Axis.
As Bartholinus discovered the property of double refraction in the mineral called Iceland spar, calcareous spar, or by chemists, carbonate of lime; and it is well fitted for exhibiting the phenomena, owing to its perfect transparency, and the great separation of the two pencils, this mineral consists, according to Stromeyer, of 56·15 parts of lime, and 43·7 of carbonic acid. It crystallises in the form of an obtuse rhombohedron, as shewn in fig. 117, where ABCB, ABCB', &c. are the faces of the rhombohedron, and AA', the axis of the rhombohedron, on the two joining its two obtuse angles. The following are the dimensions of the crystal, as given by Malus, who observes that the first angle, from which all the rest are derived, is within ten seconds of the truth.
| Inclination of the faces ABCB', AB'C'B'',... | 105° 5' | | Inclination of the faces ABCB', A'C'B'C'... | 74° 55' | | Plane angle between the edges AB, AB'... | 101° 55' | | AB, BC... | 78° 4° 59' | | Inclination of the faces to the axis AA'... | 45° 23° 26' | | Angle between the edges AB, AB', &c. and the axis AA'... | 66° 44° 45' | | Inclination of the edges AB, AB', to the opposite faces AB'C'B'', &c... | 109° 8° 12' | | A'C, A'C, to the opposite faces ABCB'... | 70° 51° 48' | | Length of the diagonal AA', the sides AB A'C being unity... | 1·2598 |
Fig. 117. As Iceland spar cleaves equally in planes parallel to all its six faces, it is easy to cut out of any mass of it an accurate rhombohedron in which all the three sides \(AB, A'B', A''B''\) are equal; but for the purposes of experiment it is sufficient to have a piece with two smooth and parallel faces formed by cleavage, or ground and polished parallel to the cleavage planes.
Let \(AX\) be such a piece of Iceland spar, and \(ABDC\) its upper surface. If we place its lower surface upon a piece of paper having a black line \(MN\), drawn upon it, and if we place the eye above the upper surface, we shall see the line \(MN\) distinctly doubled; or if it should appear single, the two images will separate, by turning the spar a little round, and the line will appear double as at \(MN\), and \(mn\). The one line will be found to coincide with the other, when \(MN\) is parallel to \(AD\), the short diagonal of the rhomboidal face \(ABDC\), and the lines will appear most separated, when \(MN\) is parallel, as in the figure, to the long diagonal \(BC\). If a black spot is used in place of a line placed at \(O\), it will appear double in every position, and in turning the crystal round, the other image \(E\) will seem to revolve round \(O\).
The best way, however, of shewing the double refraction of the Iceland spar, is to make a ray of light \(Rr\), fall upon the crystal at \(r\). This ray will be refracted into two pencils or rays, \(rO, rE\), and these will be again refracted at the second surface of the spar in directions \(Ee, Oo\), parallel to the original ray \(Rr\). If we measure the angles of refraction of the fixed ray \(Oo\), corresponding to several angles of incidence from \(0^\circ\) to \(90^\circ\), we shall find that these angles are always to one another in the constant ratio of \(1:1:654\), and that the refracted ray \(rO\) is always in the plane of incidence. It follows, therefore, that this ray is refracted as in water and glass, according to the ordinary law of refraction discovered by Snellius. Hence it is called the ordinary ray. But if we make the same experiments on the other ray \(rE\), we shall find that at an incidence of \(0^\circ\), in place of passing on unrefracted, it is actually refracted \(6^\circ 12'\); that at other incidences its angles of refraction are not regulated by the law of Snellius, and that the ray \(rE\) is not in the plane of incidence. We conclude, therefore, that the ray \(rE\) is refracted according to some new or extraordinary law which remains to be investigated, and hence the ray \(rE\) is called the extraordinary ray.
If we grind down the two solid obtuse angles \(A, A'\), fig. 116, so that the ground surfaces are perpendicular to the axis \(AA'\), and if we polish their surfaces, and transmit a ray perpendicularly through them, so as to be parallel to the axis \(AA\), we shall find that there is only one refracted ray, and hence there is no double refraction along the axis. Now this line \(AA'\) is not a fixed axis like that of the earth; it is merely a fixed direction; for every line parallel to \(AA'\) enjoys the property of an axis, as there is no double refraction in lines parallel to \(AA'\).
In order to obtain an accurate idea of the law of extraordinary refraction, or that by which the extraordinary ray \(rE\) is regulated, let us take a rhomb, such as that shewn in fig. 117, and grind it into an accurate sphere, and then polish it. Let \(ACBD\) be such a sphere, \(AB\) being its axis of double refraction, corresponding with \(AA'\) in fig. 117, and let \(O\) be its centre. If we now bend a piece of sheet lead or sheet copper into an arch \(ADB\), and making a small hole in it opposite to \(A\), and another opposite to \(B\), cement a handle to it about \(D\), it will enable us, in the following manner, to detect the general law of extraordinary refraction. The use of the two holes in this experiment is to ensure that the light admitted through one of them, and emerging at the other, shall pass through the centre \(O\) of the sphere. Let one of the holes be now placed in the surface of the sphere at \(A\), and the eye at the other hole at \(B\); the hole will appear single, shewing that the double refraction is there nothing. Let the hole be shifted gradually from \(A\) to \(C\), and the other hole will move from \(B\) to \(D\). In the different positions from \(A\) to \(C\), it will be seen that the hole begins to become double on leaving \(A\), and that the distance of the two images of it gradually increases from \(A\) to \(C\), where it becomes a maximum. The same result will be obtained by making the hole move from \(A\) to \(D\); or in any quadrant of the sphere passing through the poles \(A\) and \(B\). The very same result will be obtained if the hole moves from \(B\) to \(C\), or from \(B\) to \(D\). If we now make one of the holes move along the equator \(COD\), we shall find that by placing the eye at the other hole, the distance of the images will be exactly the same, or the double refraction the same in every part of the equator. Hence the general law of double refraction in crystals with one axis, is a very simple one: it is nothing at the poles; it increases gradually from the poles to the equator, where it is a maximum; it is the same in the same parallels of latitude; and the preceding experiments will also shew that the line joining the centres of the two images is in the plane of the meridian.
Let the index of extraordinary refraction be now measured at the pole \(A\), where it is the same as the ordinary index, and at the equator \(C\), and it will be found to be \(1:654\), or \(m\), in the former case, and \(1:488\), or \(m'\), in the latter. Now Huygens was led by a theory which has been explained under Chromatics, to the following law, which he verified by experiment, and which has been confirmed by the experiments of Wollaston, Malus, and other philosophers. Upon the axis \(AB\) of the sphere, fig. 118, describe an ellipse \(Ac, Bd\), whose lesser axis \(AB\) is to its greater axis \(cd\) as \(\frac{1}{m}\) is to \(\frac{1}{m'}\), or as \(\frac{1}{1:654}\) is to \(\frac{1}{1:488}\), or as \(0:604\) to \(0:674\); and if an oblate spheroid is supposed to be generated by the revolution of this ellipse round its lesser axis \(AB\), the reciprocal of the index of refraction of the extraordinary ray at any point of the spheroid will be measured by its radius at that point; that is, if \(RabO\) is a ray incident at \(b\), the radius \(\frac{1}{Oa}\) will be the extraordinary index of refraction for that ray. If we therefore make
\[ \phi = ROA, \text{ or the inclination of the incident ray to the axis.} \]
\(R = cd\), the major axis of the spheroid.
\(r = AB\), the lesser axis.
Then it may be shewn, as Malus has done,\(^1\) that
\[ Oa = \sqrt{(r^2 \sin^2 \phi + R^2 \cos^2 \phi)} \]
---
\(^1\) Théorie de la Double Refraction, p. 143. and as the index of refraction of the extraordinary ray is the reciprocal of this radius, we have
\[ m' = \frac{\sqrt{(r^2 \sin^2 \phi + R^2 \cos^2 \phi)}}{R r}, \text{ or} \]
\[ m'^2 = \frac{1}{r^2} - \left( \frac{R^2}{r^2} \right) \sin^2 \phi. \]
In the case of Iceland spar, we have
\[ m'^2 = 2735693 - 0.535510 \sin^2 \phi, \text{ or} \]
\[ m' = \sqrt{2735693 - 0.535510 \sin^2 \phi}. \]
As the index of extraordinary refraction thus found, is always equal to the index of the ordinary refraction minus another quality which depends on the difference between the radius of the sphere and that of the spheroid, the crystals in which this happens may be called negative doubly refracting crystals.
The preceding law of double refraction was believed by Malus to be universal, and applicable to all crystals that had this property. M. Biot, however, discovered that in quartz or rock crystal the extraordinary ray had its index of refraction \( m' \) greater than the ordinary index \( m \). This mineral crystallises in six-sided prisms, as shown in the annexed figure, terminated by six-sided pyramids, A and B. If we grind down and polish the summits A and B of a large crystal, perpendicular to the axis AB, and if we determine the index of refraction when the rays pass along AB, we shall find it to be as Malus found it, 1.5484, and without any double refraction, and 1.5544 in a direction perpendicular to the axis.
If we now grind the crystal into a sphere ACBD, fig. 121, and if we perform the very same experiments with it as we did with Iceland spar, we shall obtain analogous results. The double refraction will be found to increase from the poles A, B, to the equator CD, and to be the same in every part of the equator, and in each parallel of latitude; the only differences between it and calcareous spar being that the double refraction is less, and that the index of refraction of the extraordinary ray is always greater than that of the ordinary ray.
The extraordinary refraction of quartz will, therefore, as M. Biot has shewn, be represented by a prolate spheroid generated by the revolution of the ellipse AcBa', whose greater axis AB is to the lesser cd as \( \frac{1}{1.5484} \) is to \( \frac{1}{1.5544} \), or as 6458 is to 6418. Hence, if \( RbaO \) is a ray incident on the sphere at b, the radius \( \frac{1}{Oa} \) will be the index of extraordinary refraction for that ray. Hence we shall obtain
\[ Oa = \frac{Rr}{\sqrt{(r^2 \sin^2 \phi + R^2 \cos^2 \phi)}}, \text{ and} \]
\[ m'^2 = \frac{1}{r^2} + \left( \frac{R^2}{r^2} \right) \sin^2 \phi. \]
Hence, as the index of extraordinary refraction is equal to the index of ordinary refraction plus another quantity depending on the difference between the radius of the sphere and that of the prolate spheroid, crystals which have this property may be called positive doubly refracting crystals.
The following geometrical rule of finding the direction of the extraordinary ray, when the incident ray forms different angles with the axis given by Huygens, may be interesting to some of our readers. Let CGHF be what is called the principal section of a crystal of calcareous spar, &c., or a plane passing through the axis, which will be in the direction CH. Let SK, VK, be rays incident on that plane upon the surface CG, and equally inclined to IKL, perpendicular to the surface at the point of incidence K; then if KM is the refracted ray, corresponding to a ray IK, incident perpendicularly, the other rays VK, SK, will be refracted in directions KT, KX, so that TM is equal to XM.
Although the determination of the extraordinarily refracted ray is very simple, when the crystal is supposed to be spherical, yet the formula becomes complicated when we suppose the ray incident in any given direction upon a natural surface of Iceland spar. Malus has investigated such a formula, but our limits will not allow us to give more than the resulting expression.
Making \( \theta \) = the angle of incidence,
\( \theta' \) = the angle of extraordinary refraction,
\( \pi \) = the azimuth of incidence, or the angle which the plane of incidence forms with the axis.
\( \pi' \) = the same angle for the extraordinary ray.
\( \lambda \) = the inclination of a line perpendicular to the face of incidence to the axis of the crystal.
\[ A = R^2 \sin^2 \lambda + r^2 \cos^2 \pi, \]
\[ C = \sin \lambda \cos \lambda (R^2 - r^2). \]
Then
\[ \tan \theta' \cos \pi' = \frac{A}{\sqrt{A - R^2 \sin^2 \theta}} \left( r^2 \cos^2 \pi + A \sin^2 \pi \right) + C, \]
and
\[ \tan \theta' \sin \pi' = \frac{R^2 \sin \theta \sin \pi}{\sqrt{A - R^2 \sin^2 \theta}} \left( r^2 \cos^2 \pi + A \sin^2 \pi \right). \]
When the refracting surface is parallel to the axis, (as in the six-sided prism of calcareous spar), we shall have \( \lambda = 90^\circ \), in which case \( A = r^2 \) and \( C = 0 \). The formula then becomes
\[ \tan \theta' \cos \pi' = \frac{r^2 \sin \theta \cos \pi}{R \sqrt{1 - \sin^2 \theta}} \left( r^2 \cos^2 \pi + R^2 \sin^2 \pi \right), \]
\[ \tan \theta' \sin \pi' = \frac{R \sin \theta \sin \pi}{\sqrt{1 - \sin^2 \theta}} \left( r^2 \cos^2 \pi + R^2 \sin^2 \pi \right), \]
and dividing the one equation by the other, we have
\[ \tan \pi' = \frac{R^2}{r^2} \tan \pi. \]
When the refracting surface is parallel to the axis, and the plane of incidence perpendicular to the axis, then \( \pi' = 90^\circ \), or \( \lambda = 90^\circ \); consequently \( \cos \pi' = 0 \), and the first equation becomes
\[ \tan \theta' = \frac{R \sin \theta}{\sqrt{1 - R^2 \sin^2 \theta}} \text{ and } \sin \theta' = R \sin \theta. \]
When the refracting surface is parallel to the axis, and the plane of incidence passes through the axis, in which case the refraction is in the plane of a principal section, then \( \pi' = 0^\circ \), the second equation becomes
\[ \tan \theta' = \frac{R \sin \theta}{\sqrt{1 - R^2 \sin^2 \theta}}. \]
1 Traité de la Lumière, sect. 17. Double Refraction.
\[ \text{Tang. } \theta' = \frac{r}{R} + \frac{r \sin \theta}{\sqrt{1 - r^2 \sin^2 \theta}} \]
or, by substituting \( r \sin \theta \) for its equal sin. \( \theta \), we have
\[ \text{Tang. } \theta' = \frac{r \sin \theta}{\sqrt{1 - r^2 \sin^2 \theta}} \]
which substituted in the general formula, gives
\[ \text{Tang. } \theta' = \frac{2}{R} \text{Tang. } \theta. \]
When the refracting surface is perpendicular to the axis of the crystal, as in the chaux carbonatée basée de Haury, then \( \lambda = 0^\circ \), and \( A = r^2 \), and \( C = 0 \). Hence
\[ \text{Tang. } \theta' \sin \pi' = \frac{R^2 \sin \theta \sin \pi}{\sqrt{1 - R^2 \sin^2 \theta}} \]
\[ \text{Tang. } \theta' \cos \pi' = \frac{R^2 \sin \theta \cos \pi}{\sqrt{1 - R^2 \sin^2 \theta}} \]
and dividing the one equation by the other we have
\[ \text{Tang. } \theta' = \text{Tang. } \pi, \quad \text{and } \pi' = \pi, \]
which shows that the refracted ray is in the plane of the incident ray. Hence we obtain from the preceding equation,
\[ \text{Tang. } \theta' = \frac{R^2 \sin \theta}{\sqrt{1 - r^2 \sin^2 \theta}} \]
In order to find the path of the extraordinarily refracted ray, Huygens has given the following elegant geometrical construction. Let EBH be the elliptical section of the oblate spheroid which regulates the double refraction of the crystal, formed by the surface upon which the ray is incident. Let the ray RC fall upon its centre, and let BCK be the intersection of the plane of incidence with the face of the crystal. Let EMH be a part of the oblate spheroid within the crystal or below its surface, the axis of the spheroid passing through and having any inclination to the surface. Then draw in the plane KCR a line CO perpendicular to RC, and having drawn OK perpendicular to OC, or parallel to CR, make OK equal to \( \frac{1}{3m} \) incid., or the reciprocal of the sine of the angle of incidence. Through K draw KT perpendicular to the plane of incidence BRCK, and through KT draw a plane which shall touch the spheroid. Let I be the point where this plane touches the spheroid, then drawing the line CI, this line will be the extraordinarily refracted ray.
As there are several researches and instruments in which it may be required to find the focus of the extraordinary pencil when the doubly refracting substance has the form of a lens, we shall here give the formula obtained by Malus. Calling \( r, r' \) the radii of the anterior and posterior surfaces of a convex lens,
\( d \) = the distance of the radiant point,
\( a \) = the larger semi-axis of the spheroid of double refraction.
\( b \) = the shorter semi-axis of \( da \).
\( F \) = the focal length for the ordinary ray.
\( f \) = the focal length of the extraordinary ray, when \( d \) is infinite, or the rays parallel, and \( \varphi \) = the focal length of the extraordinary ray required.
Thus Malus has shown that
\[ \varphi = \frac{a^2 bdr'}{(r+r') (2b^2-a^2-a^2b)-a^2br'} \]
\[ F = \frac{-br'}{(r+r') (1-b)} \]
When the radii \( r, r' \), are equal, or the lens equally convex, we have
\[ \varphi = \frac{a^2 bdr}{2d (2b^2-a^2-a^2b)-a^2br'} \]
\[ f = \frac{-2 (2b^2-a^2-a^2b)}{a^2 br} \]
If we suppose \( a \) to be equal to \( b \), or the spheroid to become a sphere, by which the ordinary refraction is regulated, we obtain for the ordinary ray,
\[ F = \frac{br}{2(1-b)} \]
Hence, the difference between the ordinary and extraordinary focal lengths will be
\[ f-F = 2F \frac{a^2-b^2}{2b^2-a^2-a^2b} \]
If we change the signs of \( r \) and \( r' \) in these expressions, they will apply to concave lenses.
If we now take Iceland spar, and suppose the lens to be formed of that substance, we have only to substitute in the preceding equation, the following values of \( a \) and \( b \); viz.:—
| \( a \) | \( b \) | |------|------| | 0.6741717 | 0.6044871 |
whence
| \( m \) | \( m' \) | |------|------| | 1.65429 | 1.48330 |
we obtain
\[ \varphi = \frac{r}{8.8228502} \]
\[ F = \frac{r}{0.764180} \]
\[ f-F = \frac{F}{1.1144546} \]
In the case of quartz or rock crystal, where
| \( a \) | \( b \) | |------|------| | 0.645813 | 0.641776 |
\[ m = 1.558176 \]
\[ m' = 1.548435 \]
we obtain
\[ \varphi = \frac{r}{0.962824} \]
\[ F = \frac{r}{0.895775} \]
\[ f-F = \frac{F}{0.074846} \]
In all lenses of the same substance, the ratio of \( f \) to \( F \) will be constant, whatever be the form of the lenses, provided the incident rays are parallel. If, in the general expression of \( \varphi \), we make \( d \) infinite, we have
\[ \varphi = \frac{a^2 bdr'}{(r+r') (2b^2-a^2-a^2b)} \]
and making \( a = 0 \), we have for the ordinary ray,
\[ F = \frac{-br'}{(r+r') (1-b)} \]
whence
\[ \frac{\varphi}{F} = \frac{a^2 (1-b)}{2b^2-a^2-a^2b} \]
a result independent of the values of \( r \) and \( r' \).
The focus of the rays refracted extraordinarily by a doubly refracting lens, becomes more and more distant, as List of the Primitive Forms and Crystals which have one axis of Double Refraction.
From a very extensive series of experiments on the double refraction of crystallised bodies, Sir David Brewster was led to the general law, that all crystals whose primitive form has only one axis of figure, or one pre-eminent line, round which the matter of the crystals is symmetrically arranged, had also one axis of double refraction, and that their axis of figure was likewise the axis of double refraction.
The following are the primitive forms which possess this geometrical and optical property:
1. Rhombohedron with an obtuse summit. 2. Rhombohedron with an acute summit. 3. Regular six-sided prism. 4. Octohedron with a square base. 5. Right prism with a square base.
Table of Crystallised Minerals and other bodies which have one axis of double refraction.
In the following table the crystals are arranged under their primitive forms, so far as these forms have been determined by crystallographers. The sign + indicates that the crystals have positive double refraction, like quartz, and — that they have negative double refraction, like Iceland spar.
1. Rhombohedron with an obtuse summit.
- Carbonate of lime (Ice-land spar). - Carbonate of lime and iron. - Carbonate of lime and magnesia. - Phosphato-arseniate of lead. - Carbonate of zinc. - Nitrate of soda.
2. Rhombohedron with acute summit.
- Corundum. - Sapphire. - Ruby.
- Phosphate of lead. - Ruby silver. - Levyne. - Tourmaline. - Rubellite. - Alum stone. - Chlorate of soda. - Dioprase. - Quartz.
- Emerald. - Beryl. - Apatite. - Nepheline. - Arseniate of lead. - Hydrate of magnesia. - Zircon. - Oxide of tin. - Tungstate of lime. - Mellite. - Molybdate of lead. - Octohedrite. - Muriate of potash. - Cyanide of mercury. - Idiocease. - Wernerite. - Paranthine or scapolite. - Meoinite. - Somervillite. - Edingtonite. - Arseniate of potash. - Subphosphate of potash. - Phosphate of ammonia and magnesia. - Sulphate of nickel and copper. - Hydrate of strontites. - Apophyllite of Uton. - Oxalivite. - Superacetate of copper and lime. - Titanite. - Ico, certain crystals. - Murio-carbonate of lead.
The following crystals and organised bodies have one axis of double refraction, but their primitive form has not been accurately determined.
Position of the axis.
- Muriate of lime........Axis of six-sided prism. - Muriate of strontian......Do. do. - Hyposulphate of lime......Axis of hexagonal tables. - Mica from................Perpendicular to laminae. - Mica, with amiantus......Perpendicular to the laminae. - Nacre,(See Phil.Trans.)1836,p.)........Perpendicular to laminae. - Boracite................Axis of rhomb of 90°. - Apophyllite,(see compo-see of Haug),........Perpendicular to the table. - Sulphate of potash and iron........Axis of six-sided prism. - Tortoise shell, (Sir. J. Herschel),...........
1 Sir John Herschel. When M. Malus published his theory of double refraction, and even so late as 1816, all crystals were believed to have only one axis of double refraction, one of the rays being refracted by the ordinary law, and the other by the extraordinary law, above explained. During the examination of an extensive class of minerals and artificial salts, Sir David Brewster was led to the discovery of crystals with two axes of double refraction.
The general character of the phenomena presented by such crystals will be understood from fig. 129, where we may suppose, as before, the crystal to have the form of a sphere. In place of there being one line along which there is no double refraction, there are two such as POP, P'OP', in which the incident pencil is not divided into two. The double refraction increases on each side of these axes, from P to C and A, and from P' to D and B. The double refraction increases in the very same manner from p to C and B, and from p' to D and B, according to a law which will afterwards be explained.
In continuing his investigations, Sir David Brewster found crystals in which the axes POP, P'OP' formed all possible angles with each other from 0° up to 90°; and he was led also to the important result, that these two axes were not real axes of double refraction like those in uniaxial crystals, but were only resultant axes, as he called them, or axes of compensation. The grounds on which he formed this opinion were, that these lines POP, P'OP' had no relation whatever to any fixed or permanent lines in the primitive form of the crystals, like the axes of uniaxial crystals, and that the double refraction did not altogether vanish along these lines, as in Iceland spar and other minerals with no axis. He was led to these results, not by measuring the double refraction itself, but by the phenomena which will be presently explained. At this time it was the opinion of our author, and afterwards that of M. Biot and other distinguished philosophers, that in biaxal as in uniaxial crystals, one of the rays was refracted according to the ordinary law of Snellius, and the other according to an extraordinary law, and hence the investigation of the extraordinary law occupied the attention of our author.
In commencing this inquiry, he assumed as the two real axes of double refraction, the line AB bisecting the angle formed by the apparent or resultant axes POP, P'OP', and another line at right angles to it, viz. either the line CD, or the line perpendicular to it passing through O.
If the principal axis AB is positive, then if we assume O as the second axis, it must be taken positive also; but if CD is assumed as the second axis, it must be taken negative. Now, it is obvious that if we take AB as a positive, and CD as a negative axis, the double refraction in the direction AB is the maximum double refraction of the axis CD, because the effect of the axis AB is here nothing. In like manner, the double refraction along CD is a measure of the maximum double refraction of the axis AB. Hence we can easily ascertain the relative intensities of the doubly refracting force of each axis AB and CD. Having done this, the next step was to compute the double refraction at the point P produced by the positive axis AB acting alone as a single axis, and also the double refraction produced at the same point P, by the negative axis CD. When this was done, the two double refractions were found to be equal and opposite, and hence they compensated each other, and produced an axis PP, in which there was no double refraction, and which was the resultant of the actions of AB and CD.
In this way, and by experiments which will be related in a subsequent chapter, our author was led to the following method of finding the general law of extraordinary refraction in biaxial crystals.
Make \( b \) = axis of revolution of the two spheroids. \( a, a' \) = the other axis of the spheroid. \( \beta, \beta' \) = the inclination of the incident ray to the axes of the crystal. \( \psi \) = the angle of the doubly refracting forces emanating from each axis. \( \xi \) = half the difference of the angle at the base of the parallelogram of forces.
Then, as the velocity of the ray is inversely as the variable radius of the spheroid, \( \frac{1}{b^2} \) will be the square of the velocity of the ordinary ray, and \( \frac{1}{a^2}, \frac{1}{a'^2} \) the square of the minimum velocity of the extraordinary ray, in virtue of the separate action of each axis, AB and CD. Hence the difference between the squares of the velocities of the ordinary and extraordinary rays will be
\[ \left( \frac{a^2 + b^2}{a^2 b^2} \right) \sin^2 \beta - \left( \frac{a'^2 + b'^2}{a'^2 b'^2} \right) \sin^2 \beta' \]
the sign being positive when the axis is positive, and vice versa. But as these expressions represent the sides of the parallelogram of forces, we have
\[ \tan \xi = \frac{\left( \frac{a^2 + b^2}{a^2 b^2} \sin^2 \beta \right) - \left( \frac{a'^2 + b'^2}{a'^2 b'^2} \sin^2 \beta' \right)}{\left( \frac{a^2 + b^2}{a^2 b^2} \sin^2 \beta \right) + \left( \frac{a'^2 + b'^2}{a'^2 b'^2} \sin^2 \beta' \right)} \]
Consequently the difference between the squares of the velocities of the ordinary and extraordinary ray produced by the combined action of the two axes, will be
\[ \left( \frac{a^2 + b^2}{a^2 b^2} \sin^2 \beta \right) (\sin \psi) \]
Hence, calling V the velocity required, we have
\[ V^2 = \frac{1}{b^2} + \frac{\left( \frac{a^2 + b^2}{a^2 b^2} \sin^2 \beta \right) (\sin \psi)}{\sin (\xi + \psi)} \]
and
\[ V = \frac{1}{b^2} + \frac{\left( \frac{a^2 + b^2}{a^2 b^2} \sin^2 \beta \right) (\sin \psi)}{\sin (\xi + \psi)} \]
The form of the compound, or irregular spheroid, may therefore be computed for all doubly refracting crystals.
The general law of extraordinary refraction which has now been explained, may be thus expressed.
The increment of the square of the velocity of the extraordinary ray produced by the action of two axes of double refraction, is equal to the diagonal of a parallelogram whose sides are the increments of the square of the velocity produced by each axis separately, and calculated by the law of Huygens, and whose angle is double of the angle formed by the two planes passing through the ray and the respective axes.
When the two axes are of equal intensity, and of the same character, the preceding law gives the very same results as the law of Huygens does for one axis placed at right angles to the other two.
From these views it follows as a necessary consequence, a consequence first deduced from them by M. Biot, that the difference of the squares of the velocities of the two rays are proportional to the product of the sines of the angles which each of them make with the two resultant axes \( P \) and \( P' \), and hence making these angles \( \phi \) and \( \phi' \), and \( V \) the velocity of the extraordinary, and \( v \) that of the ordinary ray, we shall have \( V = (v^2 + a \sin \phi \times \sin \phi')^{1/2} \), \( a \) being a constant coefficient.
M. Fresnel was led by theoretical considerations to suppose that in crystals with two axes, the law of double refraction was still more complicated; and he soon confirmed the accuracy of his views by direct experiment, and was thus able to establish a general law which embraced all crystals with one axis.
Our limits will not allow us to do more than give the following brief abstract of his labours, which we owe to M. Pouillet.
When the light is incident in the plane perpendicular to \( AB \), fig. 129, that is in the plane of \( CD \), one of the two rays is regulated by the ordinary law of refraction, whereas when the ray is incident in the plane \( AB \), perpendicular to the axis \( CD \), the other of the two rays is regulated by the ordinary law of refraction. The formulae by which Fresnel expressed his law, are as follow:
making \( V \) = the velocity of the ordinary ray, \( v \) = that of the extraordinary ray, \( A \) = the angle of the ray with the one axis, \( a \) = the angle of the ray with the other axis, \( D \) = ordinary velocity in uniaxial crystals, or the constant velocity in the section perpendicular to \( CD \), fig. 129, in biaxial crystals, \( d' \) = the extraordinary velocity in uniaxial crystals, or the constant velocity in the section perpendicular to \( AB \), fig. 129, then
\[ V^2 = D^2 + (d'^2 - D^2) \sin^2 \frac{A}{2} \]
In uniaxial crystals, when the two axes are reduced to one, we have \( A = a \), so that
\[ V^2 = D^2 + (d'^2 - D^2) \sin^2 \frac{A}{2} \]
that is, the ordinary velocity is constant in all directions and equal to \( D \), and, as the second equation indicates, the extraordinary velocity \( v \) depends on the angle \( A \), which the extraordinary ray makes with the axis.
When this ray is in the section perpendicular to the axis, we have \( A = 90^\circ \), and \( \sin^2 \frac{A}{2} = 1 \), hence \( V = d \).
When the ray is parallel to the axis \( A = 0^\circ \), and \( \sin^2 \frac{A}{2} = 0 \), consequently \( V = D^2 \), or \( V = D \). In this way \( D \) is the expression of the velocity in this case, and it is on this account that the term ordinary velocity is applied to all those which are given by the different values of \( V \).
When the ray on the contrary, is in the section perpendicular to \( AB \), the sum of the angles \( A \) and \( a \) is always equal to two right angles, and \( \sin^2 \frac{A}{2} + \sin^2 \frac{a}{2} = 1 \), whence it follows that \( V^2 = d^2 \), and \( V = d \), and it is on this account that the term extraordinary velocity is applied to all those that are given by the values of \( v \).
When \( d \) is greater than \( D \), the minimum of the ordinary velocity takes place when \( a = A \), or when \( V = D \), and the maximum takes place when \( a - A \) is the greatest possible, which happens in the plane of the axes \( APCBDP' \). The minimum becomes the maximum, and vice versa when \( D \) is greater than \( d \). The maximum and minimum for the extraordinary ray take place also, when \( v = d \), and consequently for the case when the ray is in the plane of the axes, but they in like manner change their part when \( d \) is greater or less than \( D \).
In every case the difference of the squares of the velocities is expressed by the formula
\[ v^2 - V^2 = (d^2 - D^2) \sin a \sin A. \]
That is to say, the two ordinary and extraordinary rays having a common direction, the difference of the squares of their velocities are proportional to the product of the sines of the angles which each of them makes with the two axes.
This remark, adds M. Pouillet, had been made by Sir David Brewster and M. Biot before Fresnel had pointed out the simple law which embraces the phenomena in all its extent.
List of the primitive forms of Crystals that have two axes of double refraction.
From a great number of experiments, Sir David Brewster found that the property of possessing two axes of double refraction belonged to all the crystals that are included axes in the prismatic system of Mohs, or which have the following primitive forms of Haug:
- A right prism, Base a rectangle. - Oblique prism Base an oblique parallelogram. - Octohedron Base a rhomb.
In these solids there is no single line or axis of symmetry.
The following table, which we could have enlarged considerably, contains most of the crystals with two axes, whose primitive forms have been determined by crystallographers:
1. List of Crystals of known Primitive Forms, and having Two Axes of Double Refraction.
| Crystal Name | Position of Principal Axis | Position of Second Axis | |--------------|---------------------------|-------------------------| | Cynophane, Young | Axis of right prism. | Perp. to the sides. | | Fersud, ditto. | Perp. to axis. | Axis of right prism. | | Prehnite, ditto. | Parallel to longest side of prism, or perp. to best cleavage planes. | Axis of right prism, or perp. to longest faces. | | Stilbite, ditto. | Axis of right prism. | Axis of that prism, or other axis. | | Comptonite, Brooke | Perp. to axis of right prism. | Axis of the prism. | | Thomsonite, ditto. | Perp. to axis of prism. | Perp. to sides of prism. | | Anhydrite, Bouron | Axis of right prism. | Parallel to a side of rhomb. prism. | | Tartrate of potash, ditto. | Perp. to the flat rhomb. faces. | Greater diagonal of its rhomb. base. | | Staurolide, Haug | Axis of right prism. | Diagonal of its rhomb base. | | Daphnite, do. | Axis of right prism. | In plane of laminae. | | Mica, do. | Axis of right prism. | Long diagonal, or axis of prism. | | Tale, do. | Axis of right prism. | Perp. to axis of prism. | | Spodumene, do. | Axis of prism. | Perpendicular to axis. | | Sulphate of barytes, ditto. | Short diagonal of rhomb. base. | In plane of laminae. | | Sulphate of strontian, | Axis of the prism. | Long diagonal, or axis of prism. | | Sulphate of soda, Bouron | Citric acid, do. | Perp. to axis of prism. | | Tartarate of potash and soda, do. | Chromate of lead, do. | Mesotype of Aurergne, do. | | Stillbite, Brooke | Perp. to laminae. | Axis of prism. |
1 Elements de Physique Expériment. liv. viii. cap. 1. ### Double Refraction
#### Names of Minerals
| Name of Mineral | Position of Principal Axis | Position of Second Axis | |-----------------|----------------------------|-------------------------| | Needlestone of Faroe | Axis of prism | Perpendicular to axis | | Sulphate of lime, Haury | In plane of the laminae | Axis of prism | | Epidote, do | In base of prism | Axis of prism | | Axinite, do | Perp. to laminae | In plane of laminae | | Acetate of strontian | Perp. to laminae | In plane of laminae |
#### Oblique Quadrangular Prism—Base a Rectangle
| Name of Mineral | Position of Principal Axis | Position of Second Axis | |-----------------|----------------------------|-------------------------| | Borax, Haury | Perp. to axis of prism | Perp. to axis of prism | | Euclase, do | Perp. to plates | Plane of axis perpendicular to the flat tables |
#### Oblique Quadrangular Prism—Base a Rhomb
| Name of Mineral | Position of Principal Axis | Position of Second Axis | |-----------------|----------------------------|-------------------------| | Diopside, Haury | Perpendicular to sides of the prism | Axis of prism, or other axis | | Augite, do | Perp. to axis of prism | Axis of prism | | Glauconite | Perp. to axis of prism | Axis of prism | | Grammatite | Perp. to axis of prism | Axis of prism | | Sulphate of iron, Wollaston | Perp. to axis of prism | Axis of prism | | Super-sulphate of potash, Bowerman | Perp. to axis of prism | Axis of prism | | Acetate of copper, do | Perp. to axis of prism | Axis of prism | | Tartaric acid, do | Perp. to axis of prism | Axis of prism | | Oxalic acid, do | Perp. to axis of prism | Axis of prism | | Soda, do | Perp. to axis of prism | Axis of prism | | Hydrate of strontian | Perp. to axis of prism | Axis of prism | | Bitartrate of potash | Perp. to axis of prism | Axis of prism | | Sulphate of soda | Perp. to axis of prism | Axis of prism |
#### Oblique Quadrangular Prism—Base an oblique Parallelogram
| Name of Mineral | Position of Principal Axis | Position of Second Axis | |-----------------|----------------------------|-------------------------| | Feldspar, Haury | Perp. to broad sides of prism | Axis of prism, or other axis | | Kyanite, do | Perp. to broad sides of prism | Axis of prism, or other axis | | Sulphate of copper, do | Perp. to broad sides of prism | Axis of prism, or other axis |
#### Octahedron with a Rectangular Base
| Name of Mineral | Position of Principal Axis | Position of Second Axis | |-----------------|----------------------------|-------------------------| | Nitrate of potash | Axis of octahedron | Perp. to sides of base | | Arragonite | Axis of prism or perpendicular to axis of octahedron | Perp. to sides of base | | Carbonate of lead | In plane of base of two pyramids | Axis of the octahedron | | Sulphate of lead | Axis octahedron | Perp. to sides of rect. base | | Topaz | Long diagonal of rhomb. base | Short diagonal of the rhomb. base |
#### Octahedron with a Rhombic Base
| Name of Mineral | Position of Principal Axis | Position of Second Axis | |-----------------|----------------------------|-------------------------| | Sulphur, Haury | Diagonal of base through o of Haury, Pl. xxxix | Axis of octahedron or other axis | | Sphene, do | Diagonal of base through o of Haury, Pl. xxxix | Axis of octahedron or other axis |
#### Minerals belonging to the Prismatic System of Mohs, and having two axes, but not included in any of the above divisions of Haury
| Name of Mineral | Position of Principal Axis | Position of Second Axis | |-----------------|----------------------------|-------------------------| | Sulphate of magnesia, Mohs | Cleavable diagonal of square base | Perpendicular to axis, or the other diagonal | | Sulphate of manganese | Axis of rhomboideal prism | Short diagonal of rhomboideal base | | Sulphate of zinc | Axis of hexab. prism | Perp. to axis | | Sulphate of ammonia | Axis of prism | Perp. to axis | | Sulphate of cobalt | Axis of prism | Perp. to axis | | Carbonate of strontian, do | Axis of prism | Perp. to axis | | Carbonate of barytes, do | Perp. to plates | Plane of plates | | Disilicate of ammonia | Perp. to plates | Plane of plates |
#### Names of Minerals
| Name of Mineral | Position of Principal Axis | Position of Second Axis | |-----------------|----------------------------|-------------------------| | Muriate of barytes | Realgar | In plane of laminae | | Orpiment | Lepidolite | Perp. to laminae | | Mesotype | Mesolite | Axis of prism | | Serpentine | Natrolite | Ditto | | Sulphato-bi-carbonate of lead | Axes of acute rhomb. | In plane of laminae | | Phosphate of iron | In plane of laminae | Perp. to laminae | | Harmotome, Petalite, Chalasite | Perp. to laminae | In plane of laminae |
#### Crystals whose primitive form has not been determined, but which have been found to have two axes, and which must belong to the Prismatic System of Mohs
| Name of Mineral | Position of Principal Axis | Position of Second Axis | |-----------------|----------------------------|-------------------------| | Jolite | Indurated talc | Axis of prism | | Carbonate of potash | Perp. to plates | Plane of axis perpendicular to the flat tables | | Carbonate of copper | Shorter diagonal of the rhomboideal base | Axis of hexagonal prism, or long diag. of rhomb. base | | Sulphate of ammonia and magnesia | Short diagonal of the rhomboideal base | Axis of prism, or other axis | | Sulphate of soda and magnesia | Perp. to rhomb plates | Long diagonal of rhomboideal plates | | Sulphate of nickel | Oxy-nitrate of silver | Axis of prism, or other axis | | Nitrate of ammonia | Nitrate of strontian, with water of crystallization | Axis of prism, or other axis | | Nitrate of mercury | Nitrate of bismuth | Axis of prism, or other axis | | Nitrate of lead, certain specimens | Mr. Herschel, Edin. Phil. Jour. vol. ii. p. 184 | Axis of prism, or other axis |
#### Minerals belonging to the Prismatic System of Mohs, and having two axes, but not included in any of the above divisions of Haury
| Name of Mineral | Position of Principal Axis | Position of Second Axis | |-----------------|----------------------------|-------------------------| | Crystalized Cheltenham salts | Perp. to axis | Axis of prism, or other axis | | Muri-sulphate of magnesia and iron | Perp. to laminae | Perpendicular to hexag. plates | | Benzoinate of ammonia | Perp. to plates | Axis of prism | | Benzoic acid | Perp. to plates | Plane of axis in that diag. | | Chromic acid | Perp. to plates | Axis of prism | | Spermaceti | Perp. to plates | Plane of plates | | Boracic acid | Perp. to plates | In plane of plates | | Succinic acid | Perp. to plates | In plane of these faces | | Super-tartarate of potash | Perp. to plates | Axis of prism | | Tartrate of potash and antimony | Perp. to plates | Axis of prism | | Camphor | Perp. to plates | Axis of prism | | Hydrate of barytes | Perp. to plates | Axis of prism | | Prussiate of potash | Perp. to plates | Axis of prism | | Mother-of-pearl | Perp. to plates | Axis of prism | | Carbonate of ammonia | Perp. to plates | Axis of prism | | Hyposulphite of lime | Perp. to plates | Axis of prism | | Sir John Herschel, Edin. Phil. Jour. vol. i. p. 15 | Perp. to plates | Axis of prism | | Sir John Herschel | Perp. to plates | Axis of prism |
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**Note:** The table provides detailed information about various minerals, their positions of principal and second axes, and their relationships to the prismatic system of Mohs. Sect. III.—On Crystals with three Axes of Double Refraction.
Having determined the primitive form to which those crystals belong which have one and two axes of double refraction, Sir David Brewster found that all those crystals which have no resultant axes, or, properly speaking, no double refraction, belonged to that class of primitive forms which have three rectangular axes of form, namely, the cube, the regular octahedron, and the rhomboïdal dodecahedron, or to the tessular system of Mohs. Since, however, every real axis of double refraction, coincides with a prominent line in the primitive form of the crystal, he conceived that those crystals which had no apparent double refraction had actually three equal rectangular axes, the effect of which was to compensate each other at every point of the crystal, or, in other words, to have an infinite number of resultant axes.
In confirmation of these views, our author found various indications of positive and negative doubly refracting structures in alum, diamond, &c., as if these equal axes had not exactly compensated each other, either from the three not being perfectly equal, or from their not being placed accurately at right angles to each other.
The following is a list, which might be considerably extended, of the primitive forms of the crystals that have no double refraction.
Primitive form a cube.—Muriate of soda, muriate of potash, muriate of silver.
Primitive form an octahedron.—Diamond, fluor spar, muriate of ammonia, pleonaste, nitrate of lead, sulphate of alumina, soda, alum, ruby copper, spinelle, nitrate of strontian octahedron, nitrate of lead, nitrate of barytes, sulphate of ammonia and chromium, sulphate of ammonia and iron, sulphate of alumina and ammonia.
Primitive form a rhomboïdal dodecahedron.—Garnet, bende, sodalite, essonite, helvin, lazulite.
There are some crystals, such as arseniate of iron, amphiogene, analcime, boracite, apolome, all of which have double refraction, and therefore cannot belong to the tessular system.
Sect. IV.—On Crystals with Planes of Double Refraction.
In all the crystals to which we have hitherto referred, the double refraction is related to one or more axes; but Sir David Brewster has found, that in analcime, a mineral ranked in the tessular system, there are several planes or sections of the crystal in which there is no double refraction, the double refraction increasing with the distance from these planes, according to a law which will be afterwards mentioned. When the ray is incident in any direction that does not lie in one of these planes, it is separated into two images by double refraction. This is the only substance which is known to possess this remarkable property.
Part VII. On the Polarisation of Light.
The light emitted from the sun, from a candle, or from any self-luminous body, before it has suffered reflection from, or refraction by, any body, is called common light. If we allow a beam of such light to fall upon any refracting or reflecting body, whether transparent or metallic, or to pass by any diffracting body, it will suffer precisely the same changes, whether its upper, its under, its right or its left side, or any other side of the beam, is turned towards the refracting, reflecting, or diffusing body. Hence it follows, that this beam of light has the same properties in all its sides.
Now this is not true of all light. If the preceding beam of common light is reflected at a particular angle, from transparent bodies, or passes obliquely through a number of refracting surfaces, or is transmitted through certain crystals, or suffers total reflexion from the second surfaces of transparent bodies, or from the surface of metals—in all these cases it has suffered such a change, that it no longer has the same properties in all its sides, but, on the contrary, exhibits distinct and remarkable properties in its different sides, or, what is the same thing, has polarity. This beam of common light is therefore said to be polarised. The different kinds of polarisation which may thus be impressed upon common light are three, viz., plane polarisation, circular polarisation, and elliptical polarisation; or the whole of these three kinds of polarisation may be included in the general name of elliptical polarisation, which becomes circular when the two axes of the ellipse are equal, and rectilineal or plane when the minor axis of the ellipse is infinitely small. We shall now proceed to explain the phenomena of these three kinds of polarisation, in their order.
Chap. I.—On Plane Polarisation.
There are four ways by which common light may be plane polarised.
1. By double refraction. 2. By one reflection from transparent bodies. 3. By several refractions through transparent surfaces. 4. By the absorption or dispersion of part of the light.
These various processes exhibit many interesting phenomena and laws, which we shall proceed to explain.
Sect. I.—On the Polarisation of Light by Double Refraction.
The polarisation of light by the double refraction of Iceland spar was discovered by Huygens. Upon examining by double refraction, Fig. 130.
Having taken two pieces of Iceland spar, he placed them symmetrically, as in fig. 130, with all the faces of the one parallel to all the faces of the other, A r X, A' G X' being the principal sections of two rhombs. A ray of common light R r, incident upon the first crystal at r, is divided into two pencils r C, r D, O being the ordinary and E the extraordinary ray, as formerly explained. Now the ordinary ray D G, falling upon the second crystal at G, and the extraordinary one CF at F, should have been each subdivided, by double refraction, into two pencils, by that crystal; but they are not, the ordinary ray D G being only refracted ordinarily, and the extraordinary ray C F only extraordinarily, as seen in the figure, where these rays F H, GK, emerge singly at H and K, the one an ordinary and the other an extraordinary ray. If the upper rhomb remains fixed while the under one is turned round 90°, so that its principal section is perpendicular to that of the upper one, as shown in fig. 131, the same phenomena will take place, with this difference only, that the ray D G refracted ordinarily by the first crystal, is refracted extraordinarily by the second, and the ray C F, refracted extraordinarily by the first crystal, is refracted ordinarily by the second.
Hence it is manifest, that the ray of common light R r, and the two doubly refracted pencils C F, D G, have all Polarisation different properties. For if R r were to fall upon the second rhomb, it would be divided into two pencils; whereas C F and D G refuse to be so divided, and are each refracted in different ways by the second crystal.
Now, in every other position of the four rhombs, between the two where their principal sections are parallel, or perpendicular to one another, the two pencils C F, D G are divided into two pencils, and four separate pencils emerge from the second rhomb.
In order to understand the phenomena presented by these four pencils, when the second rhomb performs a complete revolution behind the first one, let us suppose that the lower rhomb begins to revolve from the position in fig. 129, which we shall call 0° of azimuth, and in which case we shall have two horizontal pencils H E, K O, fig. 131, whose sections are shown in the annexed figure at B, opposite 0°; A representing the appearance of the aperture through the first rhomb.
When the second rhomb has just begun to move out of its position of parallelism to the first, two extremely faint images begin to appear between the other two; and at 22½° of azimuth they will appear as at C. At 45° of azimuth their intensity will be equal, as at D; at 67½° the two most distant ones will have become the faintest; and at 90° the four images will be reduced to two, this being the position shown in fig. 130. By continuing to turn the second rhomb, other two faint images start up, which at 112½° appear as at G; at 135° the four images are equally bright, as at H; at 157½° the two outermost are the faintest; and at 180° they all coalesce into one bright image, as at K, having twice the brightness of either of those at A or F, and four times the brightness of any one of the four at D or H.
In making the preceding experiment, it will be seen that two of the images gradually increase in brightness, while other two gradually diminish. Malus investigated the law of the intensity for these images, both when the pencil of common light is incident perpendicularly and obliquely. Our limits will permit us to give only the simplest case, making o = ray refracted ordinarily by the first rhomb.
e = the ray refracted extraordinarily by the first rhomb.
oo = the ray refracted ordinarily by the first rhomb, and ordinarily by the second.
oe = the ray refracted ordinarily by the first rhomb, and extraordinarily by the second.
ee = the ray refracted extraordinarily by the first rhomb, and extraordinarily by the second.
eo = the ray refracted extraordinarily by the first rhomb, and ordinarily by the second.
Q = the quantity of light contained in the incident ray.
(1 - m) Q = the quantity of light absorbed by the first rhomb.
P = the intensity of any of the pencils.
P_o = the intensity of the ordinary emergent ray.
P_e = the intensity of the extraordinary emergent ray.
Then, since the quantity of light contained in the two emergent rays is equal to the incident light diminished by the quantity absorbed, we shall have Q - (1 - m) Q = m Q; and since the light is equally divided between the two pencils, we obtain P_o = \(\frac{1}{2}\) m Q, and P_e = \(\frac{1}{2}\) m Q.
But the quantity of light m Q which falls upon the second rhomb will be reduced by absorption to m Q - (1 - m) m Q = m^2 Q; consequently, if a is the angle formed by the principal sections of the two rhombs, we shall have
First pencil, \(P_{oo} = \frac{1}{2} (m^2 Q \cos^2 a)\).
Third pencil, \(P_{oe} = \frac{1}{2} (m^2 Q \cos^2 a)\).
Second pencil, \(P_{ee} = \frac{1}{2} (m^2 Q \sin^2 a)\).
Fourth pencil, \(P_{eo} = \frac{1}{2} (m^2 Q \sin^2 a)\).
When the principal sections of the two rhombs are parallel, then \(a = 0\), and \(\sin^2 a = 0\), consequently \(P_{oo} = 0\), and \(P_{ee} = 0\); that is, the third and fourth pencils will disappear.
When, on the contrary, the principal sections are at right angles to one another, \(a = 90^\circ\), and \(\cos^2 a = 0\), consequently \(P_{oo} = 0\), and \(P_{ee} = 0\); that is, the first and second pencils will disappear.
When the incident ray is not perpendicular to the surface of the first rhomb, the intensities of the pencils are functions of the angles of incidence and the angle which the ray forms with the principal sections.
All the phenomena above described may be produced by combining any two positive and any two negative crystals; but if a positive is combined with a negative crystal, the same effects will be produced when the principal sections are at right angles to each other, as when they are parallel in the other cases.
The difference between common and polarised light, as evinced by the phenomena of double refraction, is, that the former may always be divided into two pencils by a doubly-refracting crystal, whereas the latter is not capable of being so divided under certain circumstances.
As the four polarised pencils, when united, as at K, fig. 132, produce a pencil of common light, or rather a pencil which cannot be distinguished from common light, it is highly probable that the Iceland spar, in converting common into polarised light, by refracting it into two pencils, has not communicated to it any new property, but has merely separated it into two elements, just as a prism separates a pencil of white light into its seven elementary colours by refraction, these colours again forming white light by their reunion.
Sect. II.—On the Polarisation of Light by Reflection.
We have already stated, in the history of Optics, the manner in which the celebrated French philosopher Malus discovered the polarisation of light by reflection. Upon repeating his experiments with a variety of opaque and transparent bodies, not metallic, such as glass, water, &c., he found that, when light was reflected at a particular angle from such bodies, it was polarised exactly like one of the pencils formed by double refraction, the pencil polarised by reflection having all its properties identically the same with that of the doubly-refracting crystal. Like the latter, it was no longer capable of being divided into two pencils by a rhomb of Iceland spar; and as, in the polarisation of light by the double refraction of one crystal, that property depends on the angle formed between the principal sections of the two crystals, as shown in figs. 130 and 131, so, in the present case, the polarisation depends on the angle formed between the plane of reflection and that of the principal section of the crystal which polarises the light. In all such phenomena, indeed, as Malus remarks, the plane of reflection replaces the plane of the principal section of the crystal.
If we receive the ray polarised by reflection from water at an angle of 52° 45', upon any crystal having double refraction, it will not be divided into two pencils when the plane of reflection is parallel to the principal section, as if it had been a pencil of common light; but it will be refracted entirely, according to the ordinary law, as if the crystal had lost the power of double refraction. If, on the other hand, hand, the principal section of the crystal is perpendicular to the plane of reflection, the reflected ray will be refracted wholly, according to the law of extraordinary refraction. In all intermediate positions it will be divided into two rays, according to the same law, and in the same proportions, as if it had acquired its new character by the influence of double refraction.
In order to analyse this phenomenon completely, he placed the principal section of a crystal vertically; and after having divided a ray into two by it, he made these two rays fall on the surface of water, at an angle of $52^\circ 45'$. The ordinary ray was partially reflected, like common light, but the extraordinary ray penetrated the water wholly; and not a single particle of it was reflected. When the principal section of the crystal was, on the contrary, perpendicular to the plane of incidence, the extraordinary ray was partially reflected, and the ordinary ray was wholly refracted.
Malus found the phenomena to be the same for all other transparent bodies, whether solid or fluid; but the angle at which light experienced this modification was in general greater in bodies which refracted light most. Below and above this limit the rays were more or less modified.
This property of reflected light takes place at a different angle for pencils reflected at the second surfaces of bodies, and the sine of the angle at the first surface is to the sine of the angle at the second, as the sine of incidence is to the sine of refraction. Hence, in parallel plates, either of glass or other bodies, the two pencils which are reflected in the same directions from both surfaces have equally received this new property, and the light which has received it is said to have been polarised by reflection. M. Malus found the same property in black bodies, such as black marble, ebony, &c.
Malus next proceeded to study the phenomena when the light R, polarised by one plate of glass A, was reflected from a second plate C, fig. 138, the ray RA being incident on the first plate, and the polarised ray AC on the second plate at an angle of $56^\circ$, the polarising angle of glass. In the case shown in the figure, the plane of reflection ACE, in the plate C, is at right angles to the plane of reflection RAC from the first plate, and in this case the reflected pencil CE wholly vanished, all the reflected and polarised light AC having penetrated the glass at C. If we now turn round the plate C from this point, or zero, into different azimuths, so that it is always equally inclined to the polarised ray, a small portion of the ray AC will be reflected from C, and this portion will increase till it becomes a maximum, when the plane ACE is parallel to RAC, or in the azimuth of $90^\circ$. By continuing to turn the plate C, the reflected ray CE will gradually diminish, and when C has reached the azimuth of $180^\circ$ its plane of reflection will be perpendicular to that of A, and the reflected ray CE will wholly disappear. In advancing from $180^\circ$ to $270^\circ$, CE will again reach its maximum, and at $360^\circ$, when it has returned to its position, as in the figure, it will again return to its minimum.
While the reflected pencil CE passes from its minimum intensity at $0^\circ$ and $180^\circ$, to its maximum at $90^\circ$ and $270^\circ$, Malus supposes the intensity to vary as the square of the cosine of the angle of azimuth, or of any even power of the cosine. Calling $\alpha$ the angle of azimuth which the plane of the second reflection makes with a plane perpendicular to RAC, I the maximum intensity of the reflected pencil, and P the intensity corresponding to any azimuth $\alpha$, then $P = I \cos^2 \alpha$. If we make $\alpha$ equal to $0^\circ$, $90^\circ$, $180^\circ$, and $270^\circ$, we shall have, when $\alpha = 0^\circ$, $\cos \alpha = 1$, $\cos^2 \alpha = 1$, and consequently $P = I$, or the reflected pencil is a maximum. When $\alpha = 90^\circ$, $\cos \alpha = 0$, $\cos^2 \alpha = 0$, and $F = 0$, that is, the reflected pencil wholly disappears.
It is obvious, from the arrangement of glasses in fig. 132, that, if the light R proceeds from the sky, an observer with his eye placed at E will see a black spot in the part of the sky from which the light R comes, as the whole of the light penetrates the plate C. If the light R comes from a house, the house will disappear if it is at a considerable distance; and by turning round C, the house will have its greatest brightness when the two planes of reflection are parallel. If, in the position when the house was invisible, we breathe upon the plate C, the house will suddenly become visible, and will again disappear when the breath has evaporated. If we now place the plate C at an angle of $52^\circ 45'$ to the ray AC, the house will be seen; but if we again breathe upon CC, the house will disappear. The cause of these phenomena is, that by breathing upon C we make the reflecting surface an aqueous one, which refuses to reflect light at an angle of $52^\circ 45'$, but reflects it at $56^\circ$.
If we place beside each other two sets of reflectors arranged in the manner shown in the figure, C being inclined in the one set $56^\circ$ to A, and in the other set $52^\circ 45'$, and the plates C, C being near each other, we may, by breathing upon each at the same time, exhibit the paradoxical phenomenon of reviving and extinguishing a luminous image by the same breath, or we may appear to breathe at the same time light and darkness.
1. On the Law of the Polarisation of Light by Reflection.
After determining the angle at which different bodies polarised light, Malus concluded that "this angle followed neither the order of refractive powers nor that of the dispersive forces, and that it was a property of bodies independent of the other modes of action, which they exercised over light."
In repeating the experiments of Malus, Sir David Brewster measured the polarising angles of a greater number of substances, but experienced many difficulties in connecting them together by a simple law. In some substances the light was not completely polarised at any angle. In others purple and blue light was left at the polarising angles; and in various specimens of glass, different parts of the same surface gave different polarising angles. The first of these phenomena he ascribed to the circumstance that the differently coloured rays of white light were polarised at different angles; and the second he found to arise from changes that had taken place on the surfaces of glass by partial decomposition, owing to the action of the atmosphere. By rejecting those substances where the action of the surface was thus masked or disturbed, he was led to the following general law, that the index of refraction of any body is the tangent of its angle of polarisation.
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1 Théorie de la Double Refraction, sect. 48. 2 This experiment was described by Sir David Brewster, in the Edin. Phil. Journal, vol. vii. p. 146. See also his Letters on Natural Magic, p. 125. 3 Bull. de Sciences de la Soc. Philos. Juin 1811, No. 45, tom. ii. p. 294. The following were the experiments on which this law was founded:
| Names of the Bodies | Observed Polarising Angles | Calculated Polarising Angles | |---------------------|---------------------------|-----------------------------| | Air | 45° or 47° | 45° 0' 32" | | Water | 52 | 59° | | Fluor spar | 54 | 50 | | Obsidian | 56 | 3 | | Sulphate of lime | 56 | 28 | | Rock crystal | 57 | 22 | | Sulphate of barytes | 58 | 29 | | Opal-coloured glass | 58 | 1 | | Topaz | 58 | 40 | | Mother-of-pearl | 58 | 47 | | Iceland spar | 58 | 23 | | Orange-coloured glass | 59 | 12 | | Spinelle ruby | 59 | 6 | | Zircon | 53 | 8 | | Glass of antimony | 54 | 43 | | Sulphur | 64 | 10 | | Diamond | 58 | 2 | | Chromate of lead | 67 | 42 |
Upon repeating these experiments with homogeneous light, our author also found that the angle of polarisation varied with the refrangibility of the light, and that the tangent of the polarising angle was equal to the index of refraction of the light employed.
Hence we are able to explain why, at the maximum polarising angle, a portion of unpolarised light must always remain, and why this portion increases with the refractive and dispersive power of the body. This will be understood from the following table:
**Water**
| Index of Refraction | Colour of the Light | Polarising Angle | Variation | |--------------------|--------------------|------------------|-----------| | 1:330 | Red | 53° 4' | 0° 15' | | 1:336 | Green, or mean ray | 53 11 | | | 1:342 | Violet | 53 19 | |
**Plate Glass**
| Index of Refraction | Colour of the Light | Polarising Angle | Variation | |--------------------|--------------------|------------------|-----------| | 1:515 | Red | 56 34 | | | 1:525 | Green | 56 45 | 0° 21 | | 1:535 | Violet | 56 55 | |
**Oil of Cassia**
| Index of Refraction | Colour of the Light | Polarising Angle | Variation | |--------------------|--------------------|------------------|-----------| | 1:597 | Red | 57 57 | | | 1:642 | Green | 58 39 | 1° 24 | | 1:687 | Violet | 59 21 | |
Now it is obvious, that when the green or mean, or most luminous ray, is polarised, and therefore vanishes, neither the red nor the violet has wholly vanished, and consequently a portion of unpolarised light, composed of a portion of these two colours, will still be visible. In oil of cassia the quantity of light is considerable, and is of a fine blue colour.
Dr. A. Seebeck's experiments.
So recently as 1830, Dr. A. Seebeck of Berlin has published a series of very accurate and valuable experiments made by an instrument constructed for the purpose, which, if any doubts had existed about the accuracy of the preceding law, were sufficient to remove them. Dr. Seebeck's principal object seems to have been to obtain accurate measures of the polarising angle of different glasses, when the surfaces were newly polished, in order to reconcile the law to that class of bodies in which the deviations had been found to arise from some chemical or mechanical changes produced upon their surface. The following table contains Dr. Seebeck's experiments:
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1 Mean of four observations by Malus, Biot, Arago, and Brewster. 2 See Poggendorf's Annalen, 1830, No. 9, p. 27; or Edin. Journ. of Science, N. S. vol. v. p. 99. Let \( AB \), fig. 135, be a ray incident at the first surface \( MN \), \( AD \) the ray polarised at that surface, \( AC \) the ray incident at the second surface \( PQ \), and \( CM \) the ray polarised at that surface; then, if \( CF \) be the refracted ray, the angle \( MCF \) is a right angle. But \( DAC \) is a right angle, and on account of the parallelism of \( MN \), \( PC \), and \( BA \), \( CF \), the angle \( FCP \) is equal to \( DAM \); but \( MCP \) is equal to \( MAC \), hence the whole \( MCF \) is equal to the whole \( DAC \), or a right angle.
Cor. 1. The ray \( CM \), reflected by the second surface, is at right angles to the ray \( AB \) incident on the first surface.
Cor. 2. The internal reflected ray \( CM \) forms with the external reflected ray \( AD \), an angle equal to the angle of deviation \( CAO \).
Cor. 3. The ray \( CF \), emerging from the second surface, forms with the first reflected ray \( AD \), an angle equal to the complement of the angle of deviation.
When a pencil of light is incident upon the separating surface of two media having different indices of refraction \( m, m' \), it will be polarised at an angle whose tangent is equal to the quotient of the greater index of refraction divided by the lesser, or \( \frac{m}{m'} \), if \( m \) exceeds \( m' \).
This truth is a necessary consequence of the general law, and was also deduced from direct experiment. If the uppermost of the two media is a parallel plate, such as water lying upon a horizontal surface, &c., the separating surface of the two media cannot, at any angle of incidence upon the first surface, completely polarise the incident light, unless the sine of the angle whose tangent is \( \frac{m}{m'} \), is, when multiplied by \( m \), less than unity. Thus, in the case of water and glass, the polarising angle is \( 68^\circ 47' \); but no ray incident upon the water, even at \( 90^\circ \), can fall at such an oblique incidence upon the glass as \( 48^\circ 47' \). For sin \( 48^\circ 47' \times m' \) (or the angle of refraction at an incidence of \( 90^\circ \)), is \( = 1 - 0048 \). When the upper medium has a higher refractive power than the lower, and lies in a parallel plate upon it, the same law is applicable, with this difference, that the ray is now polarised at the second surface of the denser medium, and the angle of polarisation is that whose cotangent is equal to the index of refraction \( \frac{m}{m'} \) of the separating surfaces.
In the preceding observations, we have considered only the light which is incident at the polarising angle. It becomes interesting to inquire, what is the condition of the light which is incident at angles above and below the polarising angle. Malus, Arago, Biot, Fresnel, and other distinguished philosophers, considered the light thus reflected as consisting of two pencils, one of which preserved its state of common light, while the other pencil was polarised in the plane of incidence. In the year 1815, however, Sir David Brewster was led, by direct experiments, to a very different opinion, namely, that the pencil of light which was supposed to have preserved its character of common light had suffered a physical change in its condition, or had acquired in various degrees a character approaching to complete polarisation. He found, for example, that a pencil of light reflected from glass, either at \( 62^\circ 30' \) or \( 50^\circ 20' \), was so far polarised, that it was wholly polarised by a second reflection at either of these angles; whereas, had the unpolarised part been common light, it could not have been polarised at any angle but \( 56^\circ 45' \). In like manner, he found that three reflections at \( 65^\circ 33' \) or \( 46^\circ 30' \), and four at \( 67^\circ 33' \) or \( 43^\circ 51' \), polarised the whole pencil; and in general he found that a ray of light partly polarised by reflection at any angle, will be more and more polarised by every successive reflection in the same plane till its polarisation is complete, whether the reflections are made at angles all above or all below the polarising angle.
These views were not acceded to by philosophers, though founded on direct experiment; and, so late as 1825, Sir John Herschel, in discussing the question, gives his decision in favour of the opinion held by the French philosophers. Sir David Brewster was therefore induced to repeat and extend his experiments, and succeeded in establishing his original view of the subject on an irrefragable basis. A brief account of these experiments will form the subject of the next section.
2. On the Motion of the Plane of Polarisation by Reflection.
MM. Fresnel and Arago, and Sir David Brewster, were engaged about the same time in inquiries upon this subject. If we suppose a pencil of polarised light polarised in a plane inclined \( 45^\circ \) to a vertical line, and if we reflect it at different angles from a transparent surface in which the plane of reflection is perpendicular to the horizon, the plane of polarisation will be gradually reduced from \( 45^\circ \) to \( 40^\circ, 35^\circ, 30^\circ, 25^\circ, \ldots \) as we diminish the angle of incidence from \( 90^\circ \) till we reach the polarising angle, when the plane of polarisation will be inclined \( 0^\circ \), or will be brought into the plane of reflection. At angles less than the polarising angle the plane continues to turn in the same direction, till at \( 0^\circ \) it is again inclined \( 45^\circ \) to a vertical plane, or to the plane of reflection, having performed a revolution of \( 90^\circ \), the first \( 45^\circ \) during the change of incidence from \( 90^\circ \) to \( 56^\circ 45' \), and the other \( 45^\circ \) from \( 56^\circ 45' \) to \( 0^\circ \).
M. Fresnel represented these changes by the following law: \( i \) being the angle of incidence, \( r \) the angle of refraction, \( x \) the primitive inclination of the plane of the polarised ray to the plane of reflection, and \( \phi \) the inclination to which that plane is brought by reflection.
\[ \tan \phi = \tan x \frac{\cos (i + r)}{\cos (i - r)} \]
When \( x = 45^\circ \), \( \tan x = 1 \), and
\[ \tan \phi = \frac{\cos (i + r)}{\cos (i - r)} \]
In these formulae founded on the law of the tangents, \( i + r \) is the supplement of the angle which the reflected ray forms with the refracted ray, while \( i - r \) is the deviation produced by refraction.
The following are the observations with glass and water, with which M. Arago verified the last of these formulae, in which \( x = 45^\circ \).
### Glass
| Angles of Incidence | Inclination of Plane of Polarisation to Plane of Reflection | Difference | |--------------------|-------------------------------------------------------------|------------| | 24 | 88° 55' | -1° 1' | | 39 | 24° 38' | +0° 3 | | 49 | 11° 45' | -0° 53 | | 60 | 5° 15' | -0° 14 | | 70 | 19° 52' | -0° 32 | | 80 | 32° 45' | -0° 40 | | 85 | 38° 55' | -0° 24 | | 87 | 40° 55' | -0° 41 | | 88 | 41° 15' | -1° 29 | | 89 | 44° 35' | +0° 43 |
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1 Treatise on Light, sect. 366, 367. Water.
| Angles of Incidence | Inclination of Plane of Polarisation to Plane of Reflection | Difference | |---------------------|-------------------------------------------------------------|------------| | Observed | Calculated | | | 60° | 10° 20' | -0° 31' | | 70° | 25 20 | +0 32 | | 80° | 36 20 | +0 21 | | 85° | 40 50 | +0 18 |
The following observations were made by Sir David Brewster on glass.
Glass.
| Angles of Incidence | Inclination of Plane of Polarisation to Plane of Reflection | Difference | |---------------------|-------------------------------------------------------------|------------| | Observed | Calculated | | | 90° | 45° 0' | 0° 0' | | 88° | 43 4 | +0 35 | | 86° | 40 43 | +0 7 | | 84° | 38 47 | +0 25 | | 80° | 33 13 | -0 33 | | 75° | 28 45 | +1 4 | | 70° | 22 6 | +1 3 | | 65° | 14 40 | +0 47 | | 60° | 6 10 | -0 6 | | 56° | 0 0 | 0 | | 50° | 9 0 | 0 | | 45° | 16 55 | +0 24 | | 40° | 22 37 | -0 24 | | 30° | 32 25 | -0 54 | | 20° | 39 0 | -1 4 | | 10° | 44 0 | +0 11 |
Diamond.
| Angles of Incidence | Inclination of Plane of Polarisation to Plane of Reflection | Difference | |---------------------|-------------------------------------------------------------|------------| | Observed | Calculated | | | 90° | 45° 0' | 0° 0' | | 85° | 34 30 | +0 34 | | 80° | 24 0 | +0 48 | | 75° | 14 30 | +1 22 | | 70° | 4 30 | +0 36 | | 67° | 0 0 | 0 | | 60° | 12 30 | +0 49 | | 50° | 24 0 | +0 30 |
Our author also made another series of experiments, which confirms the general formula. As \( x = 45^\circ \) in the preceding experiments, he wished to observe the law of variation for \( \phi \) when \( x \) varied from \( 0^\circ \) to \( 90^\circ \). He took a crystal of quartz with a fine natural face parallel to the axis, and he found, that at an angle of incidence of \( 75^\circ \), and when \( x = 45^\circ \), the inclination of the plane of polarisation to the plane of reflection was \( 26^\circ 20' \). We have therefore
\[ \frac{\cos(i + r)}{\cos(i - r)} = \tan. 26^\circ 20', \]
and consequently the general formula becomes \( \tan. \phi = \tan. x \cdot \tan. 26^\circ 20' \), by which the third column in the following table has been calculated.
Values of \( x \).
| Inclination of Plane of Polarisation | Difference | |-------------------------------------|------------| | Observed | Calculated | | 0° | 0° 0' | | 10° | 4 54 | | 20° | 10 0 | | 30° | 15 50 | | 35° | 20 0 | | 40° | 23 30 | | 45° | 26 20 | | 50° | 30 0 | | 55° | 35 30 | | 60° | 40 0 | | 70° | 53 0 | | 80° | 70 0 | | 90° | 90 0 |
It is a curious circumstance, that at an incidence of \( 45^\circ \) the deviation produced by refraction, or \( i - r \), is, in every substance, the complement of the angle of refraction \( r \) to \( 45^\circ \); and in the action of all substances in turning round the planes of polarisation, at an incidence of \( 45^\circ \), the angle of rotation, when the plane of the polarised ray is \( = 45^\circ \), is equal to the angle of refraction, while the new inclination of the plane of polarisation to the plane of reflection, or \( \phi \), is equal to the deviation \( i - r \).
These phenomena may be represented to the eye as in fig. 136, where MN represents the plane of incidence divided into ninety equal parts, and \( ab, ab, ab \) the planes of polarisation of the same pencil of light incident at the angles marked upon the curve line. At \( 90^\circ \) of incidence, for example, the pencil A has its plane of polarisation inclined \( 45^\circ \) to the plane of reflection M; but at \( 70^\circ \) the same plane is inclined only \( 21^\circ \), and at \( 56^\circ \) it is inclined \( 0^\circ \). At \( 40^\circ \) it is inclined \( 23^\circ \) in another direction; at \( 23^\circ \) about \( 38^\circ \), and at \( 0^\circ \) it is inclined \( 45^\circ \).
3. On the partial Polarisation of Light, and the Law of its Intensity.
In order to apply these results, Sir David Brewster conceives common light to be composed of two pencils A, B, fig. 136, having their planes of polarisation \( ab, cd \) at right angles to each other, and of equal intensity. Two such pencils united comports itself under all circumstances exactly like common light. We are as much entitled to consider a beam of common light as composed in this manner, as we are to regard white light as composed of seven differently refrangible rays. The prism analyses the one, and doubly-refracting crystals, and the action of transparent surfaces, analyse the other, and common light is recomposed by the two oppositely polarised pencils, as much as white light is recomposed by the union of the seven coloured rays. Considering common light in this manner, Sir David Brewster was led to obtain a complete explanation of the phenomena of polarisation produced by reflection and refraction.
A beam of common light will be represented as at AB, composed of the two beams A, B, \( ab \) and \( cd \) being the planes of polarisation of each of them. In order, however, to analyse the action of a reflecting surface in changing the physical condition of the beam of common light, we shall represent it as in fig. 136, where the planes of polarisation \( ab, cd \) are each inclined \( 45^\circ \) to the plane of incidence MN. These two beams are obtained from a rhomb of Iceland spar, upon one of whose surfaces is placed an aperture of the size of A or B, and the rhomb is turned round till its principal section is inclined \( 45^\circ \) to the plane of incidence. MN. In this position the double beam or pencil A, B will turn its planes of polarisation as in fig 136.
At an incidence of 90°, or as near it as possible, no change is produced in the pencils A, B, the angle ace being still 90°. At an incidence of 80° the angle ace is reduced from 90° to 66°; at 70° it has been reduced to 40°; and at 56°, the maximum polarising angle, it has been reduced to 0°, or the planes of polarization ab, cd being now parallel, or, what is the same thing, the whole of the reflected pencil being polarised in the plane of incidence. Below the polarising angle, the planes ab, cd continuing to turn in the same direction, are again inclined to each other. At 40° they are inclined 50°; at 23° they are inclined 35°; and at 0°, or a perpendicular incidence, they are again brought back to their primitive inclination of 90°, or the state of common light. The two curves in the figure show the progressive change which takes place in the planes of polarisation, these planes being a tangent to the curve at the incidence which corresponds to any particular part of it.
"Such," says Sir David Brewster, "being the action of the reflecting forces upon A and B taken separately, let us now consider them as superposed and forming natural light. At 90° and 0° of incidence, the reflecting force produces no change in the inclination of their axes or planes of polarisation; but at 56° in the case of glass, and 67° 43' in the case of diamond, the axes of all the particles are brought into a state of parallelism with the plane of reflection; and consequently when the image which they form is viewed by the rhomb of calcareous spar, they will all pass into the ordinary image, and thus prove that they are wholly polarised in the plane of reflection.
Hence we see that the total polarisation of the reflected pencil at an angle whose tangent is the index of refraction, is effected by turning round the planes of polarisation of one half of the light from right to left, and of the other half from left to right, each through an angle of 45°. Let us now consider what takes place at those angles where the pencil is only partially polarised. At 80°, for example, the angle of the planes a b, c d is 66°, that is, each plane of polarisation has been turned round in opposite directions from an inclination of 45° to one of 33° with the plane of reflection. The light has therefore suffered a physical change of a very marked kind, constituting now neither natural nor polarised light. It is not natural light, because its planes of polarisation are not rectangular; it is not polarised light, because they are not parallel. It is a pencil of light having the physical character of one half of its rays being polarised at an angle of 66° to the other half. It will now be asked how a pencil thus characterised can exhibit the properties of a partially polarised pencil, that is, of a pencil part of whose light is polarised in the plane of reflection, while the rest retains its condition of natural light. This will be understood by placing the analysing rhomb, with its principal section, in the plane of reflection, and viewing through it the images A and B at 80° of incidence. As the axis of A is inclined 33° to MN or the section of the rhomb, the ordinary image of it will be much brighter than the extraordinary image, the intensity of each being in the ratio of $\cos^2 \phi$ to $\sin^2 \phi$, $\phi$ being the angle of inclination, or 33°, in the present case. In like manner, the ordinary image of B will be in the same ratio brighter than its extraordinary image, that is, by considering A and B in a state of superposition, the extraordinary image of a pencil of light reflected at 80° will be fainter than the ordinary image in the ratio of $\sin^2 33°$ to $\cos^2 33°$. But this inequality in the intensity of the two pencils is precisely what would be produced by a compound pencil, part of which is polarised in the plane of reflection, and part of which is common light.
When Malus, therefore, and his successors analysed the Polarisation pencil reflected at 80°, they could not do otherwise than conclude that it was partially polarised, consisting partly of light polarised in the plane of reflection, and partly of natural light. The action of successive reflections, however, afforded a more precise means of analysis, in so far as it proved that the portion of what was deemed natural light had in reality suffered a physical change, which approximated it to the state of polarised light; and we now see that the portion of what was called polarised light was only what may be called apparently polarised; for though it disappears, like polarised light, from the extraordinary image of the analysing prism, yet there is not a single particle of it polarised in the plane of reflection.
"These results lead to conclusions of general importance. The quantity of light which disappears from the extraordinary image, is obviously the quantity of light which is really or apparently polarised at a given angle of incidence; and if we admit the truth of the law of repartition discovered by Malus, and represented by $P_{oo} = P_0 \cos^2 \phi$, and $P_{oe} = P_0 \sin^2 \phi$, and as we can determine $\phi$ for substances of every refractive power, and for all angles of incidence, we may consider as established the mathematical law which determines the intensity of the polarised pencil, whatever be the nature of the body which reflects it, whatever be the angle at which it is incident, whatever be the number of reflections which it suffers, and whether these reflections are all made from one substance, or partly from one substance and partly from another.
Let us suppose that a beam of common light composed of two portions A, B (fig. 138) polarised + 45° and − 45° to the plane of reflection, is incident on a plate of glass at such an angle that the reflected pencil composed of C and D has its planes of polarisation inclined at an angle $\phi$ to the plane MN. When a rhomb of calcareous spar has its principal section in the plane MN, it will divide the image C into an extraordinary pencil E and an ordinary one F; and the same will take place with D, G being its extraordinary, and H its ordinary image. If we represent the whole of the reflected pencil, or C + D, by I, then C = $\frac{1}{2}$, D = $\frac{1}{2}$, E + F = 1, and G + H = 1. But since the planes of polarisation of C and D are each inclined $\phi$ degrees to the principal section of the rhomb, the intensity of the light of the doubly-refracted pencils will be as $\sin^2 \phi$, $\cos^2 \phi$; that is, the intensity of E will be $\frac{1}{2} \sin^2 \phi$, and that of F, $\frac{1}{2} \cos^2 \phi$. Hence it follows that the difference of these pencils, or $\frac{1}{2} \sin^2 \phi - \frac{1}{2} \cos^2 \phi$, will express the quantity of light which has passed from the extraordinary image E into the ordinary one F; that is, the quantity of light apparently polarised in the plane of reflection MN. But as the same is true of the pencil D, we have $\frac{1}{2} (\sin^2 \phi - \frac{1}{2} \cos^2 \phi)$ or $\sin^2 \phi - \cos^2 \phi$ for the whole of the polarised light in a pencil of common light C + D. Hence, since $\sin^2 \phi + \cos^2 \phi = 1$, and $\cos^2 \phi = 1 - \sin^2 \phi$, we have for the whole quantity of polarised light
$$Q = 1 - 2 \sin^2 \phi.$$
But tan. $\phi = \tan x \frac{\cos (i + r)}{\cos (i - r)}$
And as tan.$^2 \phi = \frac{\sin^2 \phi}{\cos^2 \phi}$ and $\sin^2 \phi + \cos^2 \phi = 1$, we have the quotient and the sum of the quantities $\sin^2 \phi$ and $\cos^2 \phi$, by which we obtain
---
1 Phil. Trans. 1830, p. 71. \[ \sin^2 \phi = \frac{1}{(\tan x \cos (i + r))^2 + 1} \]
That is, \( Q = 1 - 2 \frac{\tan x \cos (i + r)}{\cos (i - r)} \)
As tan \( x = 1 \) in common light, it is omitted in the preceding formula.
This formula may be adapted to partially polarised rays, that is, to light reflected at any angle different from the angle of maximum polarisation, provided we can obtain an expression for the quantity of reflected light.
M. Fresnel's general formula has been adapted to this species of rays, by considering them as consisting of a quantity \( a \) of light completely polarised in a plane making the angle \( x \) with that of incidence, and of another quantity \( 1 - a \) in the state of natural light. Upon this principle it becomes
\[ 1 = \frac{\sin^2 (i - r)}{\sin^2 (i + r)} \cdot \frac{1 + a \cos^2 x}{2} \\ + \frac{\tan^2 (i - r)}{\tan^2 (i + r)} \cdot \frac{1 - a \cos^2 x}{2} \]
But as we have proved that partially polarised rays are rays whose planes of polarisation form an angle of \( 2x \) with one another as already explained, \( x \) being greater or less than \( 45^\circ \), we obtain a simpler expression for the intensity of the reflected pencil, viz. the very same as that for polarised light.
\[ 1 = \frac{\sin^2 (i - r)}{\sin^2 (i + r)} \cos^2 x + \frac{\tan^2 (i - r)}{\tan^2 (i + r)} \sin^2 x. \]
Hence we have
\[ Q = \left( \frac{\sin^2 (i - r)}{\sin^2 (i + r)} \cos^2 x + \frac{\tan^2 (i - r)}{\tan^2 (i + r)} \sin^2 x \right), \]
\[ \left( 1 - 2 \frac{\tan x \cos (i + r)}{\cos (i - r)} \right)^2. \]
This formula is equally applicable to a single pencil of polarised light of the same intensity as the pencil of partially polarised light. In all these cases it expresses the quantity of light really or apparently polarised in the plane of reflection.
In order to show the quantity of light polarised at different angles of incidence, I have computed the following table for common light, and suited to glass in which \( m = 1.525 \).
| Angle of Incidence | Angle of Refraction | Inclination of Planes of Polarisation and Plane of Reflection | Quantity of Light Reflected out of 1000 Rays | Quantity of Polarised Light Q | Ratio of Polarised to Reflected Light | |--------------------|-------------------|---------------------------------------------------------------|---------------------------------------------|-------------------------------|-------------------------------------| | 0° 0' | 0° 0' | 45° 0' | 43-23 | 0 | 0 | | 10° | 6° 32 | 43 51 | 43-39 | 1-74 | 0-0408 | | 20° | 12° 58 | 40 13 | 43-41 | 7-22 | 0-16618 | | 25° | 16° 5 | 37 21 | 43-64 | 11-6 | 0-26388 | | 30° | 19° 8½ | 33 40 | 44-73 | 17-25 | 0-3853 | | 35° | 22° 6 | 29 8 | 46-33 | 24-37 | 0-5260 | | 40° | 24° 56 | 23 41 | 49-10 | 33-25 | 0-6773 | | 45° | 27° 37½ | 17 22½ | 53-66 | 44-09 | 0-82167 | | 50° | 30° 9 | 10 18 | 61-36 | 57-36 | 0-9360 | | 55° | 33° 15 | 0 0 | 79-5 | 79-5 | 1-000 | | 60° | 34° 36 | 3 44 | 95-31 | 91-6 | 0-9628 | | 65° | 36° 28 | 12 45 | 124-86 | 112-7 | 0-9024 | | 70° | 38° 2 | 15 32 | 160-67 | 139-0 | 0-7794 | | 75° | 39° 18 | 20 52 | 257-26 | 152-94 | 0-6014 | | 78° | 39° 54 | 30 44 | 329-95 | 457-67 | 0-4718 | | 79° | 40° 1 | 31 59 | 359-27 | 157-69 | 0-3769 | | 80° | 40° 13 | 33 13 | 391-7 | 156-6 | 0-4000 | | 82° | 40° 35 | 36 22 | 429-44 | 145-4 | 0-29112 | | 84° | 40° 42 | 36 2 | 560-32 | 134-93 | 0-2498 | | 85° | 40° 47 | 39 12 | 616-38 | 123-75 | 0-2008 | | 86° | 40° 51 | 40 22½ | 676-26 | 108-67 | 0-1696 | | 87° | 40° 54 | 41 32 | 744-11 | 99-33 | 0-12972 | | 88° | 40° 57½ | 42 42 | 819-9 | 65-9 | 0-0864 | | 89° | 40° 58 | 43 51 | 904-81 | 36-32 | 0-04014 | | 90° | 40° 58 | 45 0 | 1000-0 | 0 | 0-0000 |
As the preceding formula is deduced from principles which have been either established by experiment or confirmed by it, it may be expected to harmonise with the results of observation. At all the limits where the pencil is either wholly polarised or not polarised at all, it of course corresponds with experiment; but though, in so far as I know, there have been no absolute measures taken of the quantity of polarised light at different incidences, yet we are fortunately in possession of a set of experiments by M. Arago, who has ascertained the angles above and below the polarising angle at which glass and water polarise the same proportion of light. In no case has he measured the absolute quantity of the polarised rays; but the comparison of the values of \( Q \) at those angles at which he found them in equal proportions, will afford a test of the accuracy of the formula. This comparison is shown in the following table, in which column 1 contains the angles at which the reflecting surface polarises equal proportions of light; column 2 the values of \( Q \); or the inclination of the planes of polarisation; and column 3 the intensities of the polarised light computed from the formula.
Glass: No. 1. - 82° 48'...37° 33'...2572 - 24° 18'...37° 21'...2637 - 82° 5'...36° 47'...2838 - 26° 6'...36°...3090 - 78° 20'...32° 38'...4186 - 29° 42'...33'...4064 - 86° 31'...41° 54'...1080 - 16° 12'...41° 27'...1236
Water: No. 4. - 82° 48'...37° 33'...2572 - 24° 18'...37° 21'...2637 - 82° 5'...36° 47'...2838 - 26° 6'...36°...3090 - 78° 20'...32° 38'...4186 - 29° 42'...33'...4064 - 86° 31'...41° 54'...1080 - 16° 12'...41° 27'...1236
The agreement of the formula with experiments made with as great accuracy as the subject will admit, must be allowed to be very satisfactory. The differences are within the limits of the errors of observation, as appears from the following table:
Glass: No. 1. - Deviations from Experiment: 0-0065 - Part of the whole Light: 1/3
No. 2. - Deviations from Experiment: 0-0262 - Part of the whole Light: 3/8
No. 3. - Deviations from Experiment: 0-0122 - Part of the whole Light: 1/2
Water: No. 4. - Deviations from Experiment: 0-0156 - Part of the whole Light: 7/4
M. Arago has concluded, from the experiments above stated, that equal proportions of light are polarised at equal angular distances from the angle of complete polarisation. Thus, in glass No. 1, the mean of $89^\circ 48'$ and $24^\circ 18'$ is $53^\circ 33'$, which does not differ widely from the maximum polarising angle, or $55^\circ$, which M. Arago considers as the maximum polarising angle of the glass. In order to compare this principle with the formula, I found, that in water No. 4, the angle which polarises almost exactly the same proportion of light as the angle of $86^\circ 31'$, is $15^\circ 10'$, the value of $\phi$ being $41^\circ 54'$ at both these angles; but the mean of these is $50^\circ 50'$ in place of $53^\circ 11'$, so that the rule of M. Arago cannot be regarded as correct, and cannot therefore be employed, as he proposes, to determine the angle of complete polarisation.
4. On the Law of the Polarisation of Light by successive Reflections.
We come now to show the application of the preceding law of intensity to the phenomena of the polarisation of light by successive reflections.
"When a pencil of common light," says Sir David Brewster, "has been reflected from a transparent surface, at an angle of $61^\circ 3'$, for example, it has experienced such a physical change that its planes of polarisation form an angle of $6^\circ 45'$ each with the plane of reflection. When it is incident on another similar surface at the same angle, it is no longer common light, in which $x = 45^\circ$, but it is partially polarised light, in which $x = 6^\circ 45'$. In computing, therefore, the effect of the second reflection, we must take the general formula $\tan \phi = \tan x \frac{\cos (i + r)}{\cos (i - r)}$; but as the value of $x$ is always in the same ratio to the value of $\phi$, however great be the number of reflections, we have $\tan \phi = \tan^n \phi$ for the inclination $\theta$ to the plane of reflection produced by any number of reflections $n$, $\phi$ being the inclination for one reflection. Hence, when $\theta$ is given by observation, we have $\tan \phi = \sqrt[n]{\tan \theta}$. The formula for any number $n$ of reflections is therefore $\tan \phi = \left( \frac{\cos (i + r)}{\cos (i - r)} \right)^n$. It is evident that $\phi$ never can become equal to $0^\circ$; that is, that the pencil cannot be so completely polarised by any number of reflections at angles different from the polarising angle, as it is by a single reflection at the polarising angle; but we shall see that the polarisation is sensibly complete, in consequence of the near approximation of $\theta$ to $0^\circ$.
"I found, for example, that light was polarised by two reflections from glass at an angle of $61^\circ 3'$, and $60^\circ 29'$ by another observation. Now, in these cases, we have
| Quantity of Unpolarised Light | |-----------------------------| | after 1st Reflection | | after 2nd Reflection |
two reflections at $61^\circ 3'$...$6^\circ 45'$...$0^\circ 47'$...$0^\circ 00037$ $60^\circ 29'$...$5^\circ 38'$...$0^\circ 33'$...$0^\circ 00018$
The quantity of unpolarised light is here so small as to be quite unappreciable with ordinary lights.
"In like manner, I found that light was completely polarised by five reflections at $70^\circ$. Hence, by the formula, we have
Values of $\theta$. Unpolarised Light.
| Values of $\theta$ | Unpolarised Light | |-------------------|-------------------| | 1 reflection at $70^\circ$...$20^\circ$...$0^\circ$...$0^\circ 23392$ | | 2...$7^\circ$...$32^\circ$...$0^\circ 03432$ | | 3...$2^\circ$...$45^\circ$...$0^\circ 00460$ | | 4...$1^\circ$...$0^\circ$...$0^\circ 00060$ | | 5...$0^\circ$...$22^\circ$...$0^\circ 00008$ |
The quantity of unpolarised light is here also unappreciable after the fifth reflection.
In another experiment it was found that light was wholly polarised by the separating surface of glass and water at the following angles:
Values of $\theta$. Unpolarised Light.
By 2 reflections at $44^\circ 51'$...$0^\circ 56'$...$0^\circ 0005$ By 3...$42^\circ 27'$...$0^\circ 26'$...$0^\circ 0001$
"In all these cases the successive reflections were made at the same angle; but the formula is equally applicable to reflections at different angles,
"1. When both the angles are greater than the polarising angle,
Unpolarised Light.
1 reflection at $58^\circ 2'$, and 1 at $67^\circ 2'$...$0^\circ 34'$...$0^\circ 0002$
"2. When one of the angles is above and the other below the polarising angle,
Unpolarised Light.
1 reflection at $53^\circ$, and 1 at $58^\circ 2'$...$0^\circ 12'$...$0^\circ 000024$
This experiment requires a very intense light; for I find in my journal that the light of a candle is polarised at $53^\circ$ and $78^\circ$.
"In reflections at different angles, the formula becomes
$$\tan \phi = \frac{\cos (i + r)}{\cos (i - r)} \times \frac{\cos (I + l')}{\cos (I - l')}$$
and $i$ being the angles of incidence. In like manner, if $a$, $b$, $c$, $d$, $e$, &c. are the values of $\phi$ or $\theta$ for each reflection, or rather for each angle of incidence, we shall have the final angle, or $\tan \phi = \tan a \times \tan b \times \tan c \times \tan d$, &c.
"It is scarcely necessary to inform the reader, that when a pencil of light reflected at $58^\circ 2'$ is said to be polarized by another reflection at $67^\circ 2'$, it only means, that this is the angle at which complete polarisation takes place in diminishing the angle gradually from $90^\circ$ to $67^\circ 2'$, and that even this angle of $67^\circ 2'$ will vary with the intensity of the original pencil, with the opening of the pupil, and with the sensibility of the retina. But when it shall be determined experimentally at what value of $\phi$, or rather at what value of $Q$, the light entirely disappears from the extraordinary image, we shall be able, by inverting the formula, to ascertain the exact number of reflections by which a given pencil of light shall be wholly polarised.
"As the value of $Q$ depends on the relation of $i$ and $r$, that is, on the index of refraction, and as this index varies for the different colours of the spectrum, it is obvious that $Q$ will have different values for these different colours. The consequence of this must be, that in bodies of high dispersive powers, the unpolarised light which remains in the extraordinary image, and also the light which forms the ordinary image, must be coloured at all incidences; the colours being most distinct near the maximum polarising angle. This necessary result of the formula was found to be experimentally true in oil of cassia, and various highly dispersive bodies. In realgar, for example, $\phi$ is $= 0$ at an angle of $65^\circ 0'$ for blue light, at $68^\circ 37'$ for green light, and at $66^\circ 49'$ for red light. Hence there can be no angle of complete polarisation for white light, which was also found to be the case by experiment; and as $Q$ must at different angles of incidence have different values for the different rays, the unpolarised light must be composed of a certain portion of each different colour, which may be easily determined by the formula.
"Such are the laws which regulate the polarisation of..." light by reflection from the first surfaces of bodies that are not metallic. The very same laws are applicable to their second surfaces, provided that the incident light has not suffered previous or subsequent refraction from the first surface. The sine of the angle at which $\phi$ or $Q$ has a certain value by reflection from the second surface, is to the sine of the angle at which they have the same value at the first surface, as unity is to the index of refraction. Hence $\phi$ and $Q$ may be determined by the preceding formulæ after any number of reflections, even if some of the reflections are made from the first surface of one body and the second surface of another."
**Sect. III.—On the Polarisation of Light by Refraction.**
We have already seen that the polarisation of light by refraction was discovered by independent observations by Malus, Biot, and Brewster; but the priority of discovery belongs to Malus, who, from various observations, arrived at the following conclusion: "When a ray of light falls upon a plate of glass at an angle of $54^\circ 35'$, all the light which it reflects is polarised in one direction. The light which traverses the glass is composed, 1st, of a quantity of light polarised in a direction opposite to that which is reflected, and proportional to it; and, 2dly, of another portion not modified, and which preserves the character of direct light."
Sir David Brewster made analogous experiments with thin films of glass, films of mica, folds of gold-beaters' skin, and films of gold leaf; but he occupied himself chiefly in determining the law of the phenomena depending on the number of plates through which the light was transmitted, and found that the number of plates which polarise a maximum of light by transmission at different angles of incidence, are to one another as the co-tangents of the angles of incidence.
In determining this law, our author employed forty-seven plates of crown-glass, each about three inches long and one broad, and with these he obtained the following results:
| Number of Plates in each Parcel | Angles of Incidence at which Light is Polarised, by Calculation. | Angles of Incidence at which Light is Polarised, by Experiment. | Difference between the Calculated and Observed Angles. | |-------------------------------|---------------------------------------------------------------|---------------------------------------------------------------|-----------------------------------------------------| | 8 | $79^\circ 11'$ | $78^\circ 52'$ | $0^\circ 19'$ | | 10 | $76^\circ 33'$ | $76^\circ 24'$ | $0^\circ 9'$ | | 12 | $74^\circ 9'$ | $74^\circ 2'$ | $0^\circ 2'$ | | 14 | $71^\circ 30'$ | $72^\circ 15'$ | $0^\circ 45'$ | | 16 | $69^\circ 4'$ | $69^\circ 40'$ | $0^\circ 36'$ | | 18 | $66^\circ 43'$ | $66^\circ 43'$ | $0^\circ 0'$ | | 21 | $63^\circ 21'$ | $63^\circ 39'$ | $0^\circ 18'$ | | 24 | $60^\circ 8'$ | $61^\circ 0'$ | $0^\circ 52'$ | | 27 | $57^\circ 10'$ | $56^\circ 53'$ | $0^\circ 12'$ | | 29 | $55^\circ 16'$ | $54^\circ 50'$ | $0^\circ 26'$ | | 31 | $53^\circ 28'$ | $53^\circ 16'$ | $0^\circ 12'$ | | 33 | $51^\circ 44'$ | $51^\circ 0'$ | $0^\circ 44'$ | | 35 | $50^\circ 5'$ | $50^\circ 23'$ | $0^\circ 17'$ | | 39 | $47^\circ 1'$ | $46^\circ 50'$ | $0^\circ 11'$ | | 41 | $45^\circ 36'$ | $45^\circ 49'$ | $0^\circ 14'$ | | 43 | $43^\circ 34'$ | $44^\circ 0'$ | $0^\circ 26'$ | | 47 | $41^\circ 41'$ | $42^\circ 0'$ | $0^\circ 19'$ | | 100 | $22^\circ 42'$ | $22^\circ 42'$ | $0^\circ 0'$ | | 200 | $11^\circ 49'$ | $11^\circ 49'$ | $0^\circ 0'$ | | 500 | $4^\circ 47'$ | $4^\circ 47'$ | $0^\circ 0'$ | | 1,000 | $2^\circ 24'$ | $2^\circ 24'$ | $0^\circ 0'$ | | 2,000 | $1^\circ 12'$ | $1^\circ 12'$ | $0^\circ 0'$ | | 4,000 | $0^\circ 36'$ | $0^\circ 36'$ | $0^\circ 0'$ | | 14,000 | $0^\circ 1'$ | $0^\circ 1'$ | $0^\circ 0'$ | | 6,640,000 | $0^\circ 0''$ | $0^\circ 0''$ | $0^\circ 0'$ |
If $n$, $n'$, therefore, represent the number of plates in any two parcels, and $\phi$, $\phi'$ the angles at which the pencil was polarised, we have
$$ n : n' = \cotan. \phi : \cotan. \phi', $$
and
$$ n \times \tan. \phi = n' \times \tan. \phi'. $$
That is, the number of plates in any parcel, multiplied by the tangent of the angle at which it polarises light, is a constant quantity. From a great number of observations made with a parcel of eighteen plates, our author found the constant quantity for crown-glass to be $41:84$ when the light was that of a good wax-candle placed at the distance of about twelve feet, so that we have
$$ \tan. \phi = \frac{41:84}{n}; $$
that is, divide the constant quantity by any given number of plates, and the quotient will be the natural tangent of the angle at which that number will polarise a pencil of light.
In this way the second column of the table, which differs very little from the observed column, was computed.
From these experiments our author drew the same conclusion as in the case of reflection, respecting the state of what Malus calls the unpolarised light, namely, that it had suffered a physical change; and he also showed that light could be polarised by successive refractions, each refraction bringing it gradually nearer and nearer to the state of complete polarisation.
1. **On the Motion of the Planes of Polarisation by Refraction.**
The first, and we believe the only, experiments that have been made on this subject, were those of Sir David Brewster; and we shall therefore give an account of his researches in his own words, abridging it as much as possible.
"If we take a plate of glass, deviating so slightly from parallelism between its faces as to throw aside from the direct transmitted image of a luminous body the faint images formed by reflection between its inner surfaces, we shall obtain, even at the greatest obliquities, a pencil of light free of all admixture of reflected light.
"Let this plate be placed upon a divided circle, so that we can observe through it two luminous discs of polarised light A, B (fig. 139) formed by double refraction, and having their planes of polarisation inclined $+45^\circ$ and $-45^\circ$ to the plane of refraction. At a perpendicular incidence, the inclination of the planes of polarisation will suffer no change; but at an incidence of $30^\circ$ they will be turned round $40'$, so that their inclination to MN or the angle $\alpha\epsilon\epsilon$ will be $45^\circ 40'$. At $45^\circ$ their inclination will be $46^\circ 47'$. At $60^\circ$ it will be $50^\circ 7'$; and it will increase gradually to $90^\circ$, where it becomes $66^\circ 19'$. Hence the maximum change produced by a single plate of glass upon the planes of polarisation is $66^\circ 19' - 45^\circ = 21^\circ 19'$, an effect exactly equal to
---
1 Phil. Trans. 1830, p. 133. what is produced by reflection at angles of 39° or 70°. It is remarkable, however, that this change is made in the opposite direction, the planes of polarisation now approaching to coincidence in a plane at right angles to that of reflection; a difference which might have been expected from the opposite character of the resulting polarisation.
"In this experiment the action of the two surfaces is developed in succession, so that we cannot deduce from the maximum rotation of 21° 19', the real action of the first, or of a single surface, which must be obviously more than half of the action of the two surfaces, because the planes of polarisation have been widened before they undergo the action of the second surface.
"In order to obtain the rotation due to a single surface, I took a prism of glass ABC, having such an angle BAC, that a ray RR, incident as obliquely as possible, should emerge in a direction Rr perpendicular to the surface AC. I took care that this prism was well annealed, and I caused the refraction to be performed as near as possible to the vertex A, where the glass was thinnest, and consequently most free from the influence of any polarising structure. In this way I obtained the following measures:
Glass.
| Angles of Incidence | Inclination of Planes ab, cd, to the Plane of Reflection | Rotation of Plane for one Surface | |---------------------|----------------------------------------------------------|----------------------------------| | 87° 38' | 51° 15' | 9° 15' | | 54° 50' | 47° 25' | 2° 25' | | 32° 20' | 45° 92' | 0° 22' |
"I next made the following experiments with two kinds of glass, the one a piece of parallel plate-glass, and the other a piece of very thin crown. The latter had the advantage of separating the reflected from the transmitted light.
Plate Glass. Crown Glass.
| Incidence | Inclination | Rotation for two Surfaces | Inclination | Rotation for two Surfaces | |-----------|-------------|---------------------------|-------------|---------------------------| | 0° | 0° | 45° | 0° | 0° | | 10° | 6° 30' | 0° 13 | 45° 13 | 45° 6 | | 20° | 13° 5 | 0° 27 | 45° 27 | 45° 28 | | 30° | 16° 15 | 0° 32 | 45° 32 | 45° 40 | | 40° | 19° 20 | 0° 40 | 45° 40 | 46° 0 | | 50° | 22° 19 | 1° 12 | 46° 12 | 46° 25 | | 60° | 25° 10 | 1° 30 | 46° 30 | 46° 56 | | 70° | 27° 55 | 1° 42 | 46° 47 | 47° 34 | | 80° | 30° 29 | 2° 48 | 47° 42 | 48° 24 | | 90° | 33° 52 | 3° 54 | 48° 54 | 48° 59 | | 100° | 35° 0 | 5° 7 | 50° 7 | 50° 36 | | 110° | 36° 53 | 6° 48 | 51° 43 | 52° 7 | | 120° | 38° 29 | 8° 7 | 53° 7 | 53° 59 | | 130° | 39° 45 | 9° 55 | 54° 55 | 56° 18 | | 140° | 40° 42 | 12° 10 | 57° 10 | 59° 5 | | 150° | 41° 19 | 15° 45 | 60° 45 | 62° 24 | | 160° | 41° 21 | 16° 39 | 61° 39 | 63° 9 | | 170° | 41° 28 | | 66° 19 | |
"The last column but one of the table was calculated by the formula
\[ \cot \phi = \cos (i - r) \]
\( n \) being in this case 2. The errors, however, being almost all negative, I suspected that there was an error of adjustment in the apparatus; and upon repeating the experiment at 80°, the point of maximum error, I found that the inclination was fully 58° 40', giving a difference only of 25 in place of 1° 55'.
"In these experiments \( x = 45° \) and \( \cot x = 1 \); but in order to try the general formula when \( x \) varied from 0° to 90°, I took the case where the angle of incidence was 80°, Polarisation and $\phi = 58^\circ 40'$ when $x = 45^\circ$. The following were the results.
| Values of $x$ | Inclination observed | Inclination calculated | Difference | |--------------|----------------------|------------------------|------------| | 0° | 0° | 0° | 0° | | 2° | 7° 10 | 7° 20 | -1° 10 | | 5° | 9° 40 | 8° 19 | +1° 21 | | 10° | 17° 10 | 16° 25 | +0° 45 | | 15° | 24° 42 | 24° 6 | +0° 36 | | 20° | 32° 30 | 31° 19 | +1° 11 | | 25° | 39° 15 | 37° 54 | +1° 21 | | 30° | 44° 10 | 43° 57 | +0° 13 | | 35° | 49° 38 | 49° 28 | +0° 10 | | 40° | 54° 36 | 54° 31 | +0° 5 | | 45° | 58° 40 | 59° 5 | -0° 25 | | 50° | 63° 10 | 63° 19 | -0° 9 | | 55° | 66° 58 | 67° 15 | -0° 17 | | 60° | 70° 18 | 70° 56 | -0° 38 | | 65° | 74° 8 | 74° 24 | -0° 16 | | 70° | 76° 56 | 77° 42 | -0° 46 | | 75° | 79° 20 | 80° 53 | -1° 33 | | 80° | 83° 23 | 83° 58 | -0° 35 | | 85° | 86° 23 | 86° 0 | +0° 23 | | 90° | 90° 0 | 90° 0 | 0° |
"The last column but one was calculated by the formula $\cot \phi = \cot x \cdot (\cot 58^\circ 40')^2$.
"In determining the quantity of polarised light in the refracted pencil, we must follow the method already explained for the reflected ray, mutatis mutandis. The principal section of the analysing rhomb being now supposed to be placed in a plane perpendicular to the plane of reflection, the quantity of light $Q'$ polarised in that plane will be
$$Q' = 1 - 2 \cos^2 \phi,$$
the quantity of transmitted light being unity. But
$$\cot \phi = \cot x \cos (i - r),$$
and as $\cot \phi = \frac{\cos^2 \phi}{\sin^2 \phi}$, and $\sin^2 \phi + \cos^2 \phi = 1$, we have the quotient and the sum of $\sin^2 \phi$ and $\cos^2 \phi$ to find them.
Hence $\cos^2 \phi = \frac{(\cot x \cos (i - r))^2}{1 + (\cot x \cos (i - r))^2};$
and by substituting this for $\cos^2 \phi$ in the former equation, it becomes
$$Q' = 1 - 2 \frac{(\cot x \cos (i - r))^2}{1 + (\cot x \cos (i - r))^2}.$$
"Now since, by Fresnel's formula, the quantity of reflected light is
$$R = \frac{1}{2} \left( \frac{\sin^2 (i - r)}{\sin^2 (i + r)} + \frac{\tan^2 (i - r)}{\tan^2 (i + r)} \right),$$
the quantity of transmitted light $T$ will be
$$T = 1 - \frac{1}{2} \left( \frac{\sin^2 (i - r)}{\sin^2 (i + r)} + \frac{\tan^2 (i - r)}{\tan^2 (i + r)} \right).$$
Hence $Q' = \left( 1 - \frac{1}{2} \left( \frac{\sin^2 (i - r)}{\sin^2 (i + r)} + \frac{\tan^2 (i - r)}{\tan^2 (i + r)} \right) \right)$
$$\left( 1 - 2 \frac{(\cos (i - r))^2}{1 + (\cos (i - r))^2} \right).$$
"This formula is applicable to common light, in which $\cot x = 1$ disappears from the equation; but, on the same principles which we have explained in a preceding paper, it becomes for partially polarised rays and for polarised light,
$$Q' = \left( 1 - \frac{1}{2} \left( \frac{\sin^2 (i - r)}{\sin^2 (i + r)} \cos^2 x + \frac{\tan^2 (i - r)}{\tan^2 (i + r)} \sin^2 x \right) \right)$$
$$\left( 1 - 2 \frac{(\cot x \cos (i - r))^2}{1 + (\cot x \cos (i - r))^2} \right).$$
"In all these cases the formula expresses the quantity of light really or apparently polarised in the plane of refraction.
"As the planes of polarisation of a pencil polarised at $45^\circ$ and $-45^\circ$ cannot be brought into a state of coincidence by refraction, the quantity of light polarised by refraction can never be mathematically equal to the whole of the transmitted pencil, however numerous be the refractions which it undergoes; or, what is the same thing, refraction cannot produce rays truly polarised, that is, with their planes of polarisation parallel."
2. On the partial Polarisation of Light by one or more Refractions.
"The analysis given in the preceding paragraphs, of the changes produced on common light, considered as represented by two oppositely polarised pencils, furnishes us with the same conclusions respecting the partial polarisation of light by refraction which we have already deduced respecting the partial polarisation of light by reflection. Each refracting surface produces a change in the position of the planes of polarisation, and consequently a physical change upon the transmitted pencil by which it has approached to the state of complete polarisation.
"This position I shall illustrate by applying the formula to the experiments in a preceding page.
"According to the first of these, the light of a wax-candle at the distance of ten or twelve feet is wholly polarised by eight plates or sixteen surfaces of parallel plate-glass at an angle of $78^\circ 52'$. Now I have ascertained that a pencil of light of this intensity will disappear from the extraordinary image, or appear to be completely polarised, provided its planes of polarisation do not form an angle of less than $88^\circ 49'$ with the plane of refraction for a moderate number of plates, or $88^\circ 19'$ for a considerable number of plates, the difference arising from the great diminution of the light in passing through the substance of the glass. In the present case the formula gives
$$\cot \phi = (\cos (i - r))^{16} \text{ and } \phi = 88^\circ 50';$$
so that the light should appear to be completely polarised, as it was found to be.
"At an angle of $61^\circ 0'$ the pencil was polarised by twenty-four plates or forty-eight surfaces. Here
$$\cot \phi = (\cos (i - r))^{16} \text{ and } \phi = 88^\circ 36'.$$
"At an angle of $48^\circ 34'$ the light was polarised by forty-seven plates or ninety-four surfaces. Here
$$\cot \phi = (\cos (i - r))^{99} \text{ and } \phi = 88^\circ 27'.$$
"It is needless to carry this comparison any further; but it may be interesting to ascertain by the formula the smallest number of refractions which will produce complete polarisation. In this case the angle of incidence must be $90^\circ$.
"Hence, $\phi = 56^\circ 29'$ and $(\cos (i - r))^{99}$ gives $88^\circ 36'$ and $(\cos (i - r))^{100} 88^\circ 4'$; that is, the polarisation will be nearly complete by the most oblique transmission through four and a half plates or nine surfaces, and will be perfectly complete through five plates or ten surfaces."
Sect. IV.—Comparison of the Laws of Intensity for Light polarised by Reflection and Refraction.
"Having obtained formulæ for the quantity of light polarised by refraction and reflection, it becomes a point of great importance to compare the results which they furnish. Calling $R$ the reflected light, these formulæ become..."
\[ Q = R \left( 1 - 2 \frac{\cos(i + r)}{\cos(i - r)} \right) \]
\[ Q' = 1 - R \left( 1 - 2 \frac{\cos(i - r)}{\cos(i + r)} \right). \]
But these two quantities are equal, and hence we obtain the important general law, that—At the first surface of all bodies, and at all angles of incidence, the quantity of light polarised by refraction is equal to the quantity polarised by reflection. I have said 'of all bodies,' because the law is equally applicable to the surfaces of crystallized and metallic bodies, though the action of their first surface is marked or modified by other causes.
It is obvious from the formula that there must be some angle of incidence where \( R = 1 - R \), or the reflected equal to the transmitted light. When this takes place, we have \( \sin^2 \varphi = \cos^2 \varphi \); that is,
The reflected is equal to the transmitted light, when the inclination of the planes of polarisation of the reflected pencil to the plane of reflection, is the complement of the inclination of the planes of polarisation of the refracted pencil to the same plane; or, if we refer the inclination of the planes to the two rectangular planes into which the planes of polarisation are brought, — The reflected will be equal to the transmitted light when the inclination of the planes of polarisation of the reflected pencil to the plane of reflection is equal to the inclination of the plane of polarisation of the refracted pencil to a plane perpendicular to the plane of reflection.
In the following table, the inclination of the planes of polarisation of the reflected and the refracted pencil, and the quantities of light reflected, transmitted, and polarised, at all angles of incidence upon glass, \( m \) being equal to 1525, and the incident light = 1000, are given:
| Angle of Incidence | Angle of Refraction | Quantity of Reflected Light | Quantity of Transmitted Light | Quantity of Polarised Light | |-------------------|--------------------|----------------------------|----------------------------|---------------------------| | 0° 0' | 0° 0' | 45° 0' | 45° 0' | 43-23 | | 1° 0' | 1° 84' | 44° 57' | 45° 0' | 43-26 | | 2° 0' | 2° 32' | 43° 51' | 45° 0' | 43-39 | | 3° 0' | 3° 12' | 42° 43' | 45° 0' | 43-41 | | 4° 0' | 4° 06' | 41° 35' | 45° 0' | 43-44 | | 5° 0' | 5° 00' | 40° 27' | 45° 0' | 43-46 | | 6° 0' | 6° 00' | 39° 19' | 45° 0' | 43-47 | | 7° 0' | 7° 00' | 38° 11' | 45° 0' | 43-48 | | 8° 0' | 8° 00' | 37° 03' | 45° 0' | 43-49 | | 9° 0' | 9° 00' | 35° 55' | 45° 0' | 43-50 | | 10° 0' | 10° 00' | 34° 47' | 45° 0' | 43-51 | | 11° 0' | 11° 00' | 33° 39' | 45° 0' | 43-52 | | 12° 0' | 12° 00' | 32° 31' | 45° 0' | 43-53 | | 13° 0' | 13° 00' | 31° 23' | 45° 0' | 43-54 | | 14° 0' | 14° 00' | 30° 15' | 45° 0' | 43-55 | | 15° 0' | 15° 00' | 29° 07' | 45° 0' | 43-56 | | 16° 0' | 16° 00' | 27° 59' | 45° 0' | 43-57 | | 17° 0' | 17° 00' | 26° 51' | 45° 0' | 43-58 | | 18° 0' | 18° 00' | 25° 43' | 45° 0' | 43-59 | | 19° 0' | 19° 00' | 24° 35' | 45° 0' | 43-60 | | 20° 0' | 20° 00' | 23° 27' | 45° 0' | 43-61 | | 21° 0' | 21° 00' | 22° 19' | 45° 0' | 43-62 | | 22° 0' | 22° 00' | 21° 11' | 45° 0' | 43-63 | | 23° 0' | 23° 00' | 20° 03' | 45° 0' | 43-64 | | 24° 0' | 24° 00' | 18° 55' | 45° 0' | 43-65 | | 25° 0' | 25° 00' | 17° 47' | 45° 0' | 43-66 | | 26° 0' | 26° 00' | 16° 39' | 45° 0' | 43-67 | | 27° 0' | 27° 00' | 15° 31' | 45° 0' | 43-68 | | 28° 0' | 28° 00' | 14° 23' | 45° 0' | 43-69 | | 29° 0' | 29° 00' | 13° 15' | 45° 0' | 43-70 | | 30° 0' | 30° 00' | 12° 07' | 45° 0' | 43-71 | | 31° 0' | 31° 00' | 10° 59' | 45° 0' | 43-72 | | 32° 0' | 32° 00' | 9° 51' | 45° 0' | 43-73 | | 33° 0' | 33° 00' | 8° 43' | 45° 0' | 43-74 | | 34° 0' | 34° 00' | 7° 35' | 45° 0' | 43-75 | | 35° 0' | 35° 00' | 6° 27' | 45° 0' | 43-76 | | 36° 0' | 36° 00' | 5° 19' | 45° 0' | 43-77 | | 37° 0' | 37° 00' | 4° 11' | 45° 0' | 43-78 | | 38° 0' | 38° 00' | 3° 03' | 45° 0' | 43-79 | | 39° 0' | 39° 00' | 1° 55' | 45° 0' | 43-80 | | 40° 0' | 40° 00' | 0° 47' | 45° 0' | 43-81 |
It is obvious, from a consideration of the principle of the formula for reflected light, that the quantity of polarised light is nothing at 0°, because the force which polarises it is there a minimum. At the maximum polarising light, \( Q \) is only 79-5, because the glass is incapable of reflecting more light at that angle, otherwise more would have been polarised. The value of \( Q \), then, rises to its maximum at 78° 7', and descends to its minimum at 90°; but the polarising force has not increased from 56° 45' to 78° 7', as the value of \( \varphi \) shows. It is only the quantity of reflected light that has increased which occasions a greater quantity of light to disappear from the extraordinary image of the analysing rhomb.
The case, however, is different with the refracted light. The value of \( Q' \) has one minimum at 0°, and another at 90°, while its maximum is at 78° 7'; while the force has its minimum at 0°, and its minimum at 90°, where its effect is a minimum only because there is no light to polarise. At the incidence of 78° 7', where the quantities \( Q, Q' \) reach their maxima, the reflected light is exactly one half of the transmitted light; \( \sin^2 \varphi' = \cos^2 \varphi \) and tan. \( \varphi' = \cos \varphi \).
At 85° 50' 40", where the transmitted light is one half of the reflected light, the deviation \( (i - r') = 45° \), and the quantity of polarised light is one third of the transmitted light, one sixth of the reflected light, and one ninth of the incident light. \( \sin^2 \varphi' = \cos^2 \varphi \) reflected light : transmitted light, and cot. \( \varphi = \sin(i - r') \).
At 45° we have \( (i + r') + (i - r') = 90° \) and \( \varphi = (i - r') \), tan. \( (i - r') = \frac{\cos(i + r')}{\cos(i - r')} \), and tan. \( (i - r') = \frac{\sin(i - r')}{\sin(i + r')} \).
At 56° 45', the polarising angle, the formula for reflected light becomes \( R = \frac{1}{2} (\sin^2 (i - r')) \); but at this angle we have \( r' = 90° - i \). Hence we obtain the following simple expression in terms of the angle of incidence, for the quantity of light reflected by all bodies at the polarising angle:
\[ R = \frac{1}{2} (\cos(2i))^2. \]
Sect. V.—On the Action of Single Plates and Single Surfaces in polarising Light by Reflection and Refraction.
In the article on the polarisation of light in this work, M. Arago has described an elegant experiment, from which he deduced the conclusion, that the quantity of polarised light contained in the pencil transmitted by a transparent plate is exactly equal to the quantity of light polarised at right angles, which is found in the pencil reflected by the same plate.
Now this law is true only at the maximum polarising angle, the two pencils being unequal at all other angles of incidence. The apparent equality observed by M. Arago seems to have arisen from other light being blended with the pencils. In order, therefore, to obtain the true law for single plates, we must determine it for a single surface.
In order to do this, Sir David Brewster employed a well-annealed prism of colourless glass, in place of a plate of glass, and he made the ray BI from a sheet of whitepaper BA enter the surface ED perpendicularly at I, while another ray AS was reflected to the eye from the surface ED of the prism. He then made the experiment with a doubly refracting prism C in the manner described by M. Arago, and obtained the law for a single surface, viz.
That the quantity of polarised light in the pencil re-
 Polarisation by a transparent surface is exactly equal to the quantity in the pencil reflected by it.
That is, what was supposed to be true of plates is true only of surfaces.
The action of a single plate on light, involving the combined action of three refractions and two reflections, without following the light beyond one internal reflection, is sufficiently complex, and has been analysed in the following manner by Sir David Brewster.
He took a plate of glass of the form MN, fig. 142, having an oblique face Md cut upon one of its ends.
Fig. 142.
"A ray of light RA, polarised + 45° and − 45°, was made to fall upon it at A, at an angle of incidence of nearly 83°, so that the inclination of the planes of polarisation of the reflected ray AP was about 364°. Now the ray AC, after reflection in the direction CS, without any refraction at B, where it emerges perpendicularly to Md, would also have had the inclination of its planes of polarisation equal to 364° if there had been no intermediate refraction at A; but this refraction alone being capable of producing an inclination of 53°, or a rotation of 53° − 45° = 8°, and this rotation being in an opposite direction from that produced by the second reflection at C, the inclination of the planes of polarisation for the ray CS is nearly 444°, the reflection of C having brought back the ray AC almost exactly into the state of natural light.
Without changing either the light or the angle, I cemented a prism Mcd on the face Md, so that cd was parallel to dN, and I found that the second refraction at b, equal to that at A, changed the inclination of the planes of polarisation to 53°; that is, the two refractive actions at A and b had overcome the action of reflection at C, and the pencil bs actually contained light polarised perpendicular to the plane of reflection.
In order to put this result to another test, I took a plate Mc NQ (fig. 142) of the same glass, which separated the pencil bs reflected at the second surface, from the parallel pencil AP reflected from the first surface, and I found, that at an angle of 83°, the value of the inclination I or ϕ for the ray was about 37°, while the value of I for the ray bs was nearly 53°, an effect almost equal to the refractive action of a plate at 83° of incidence.
When the pencil RA is incident on the first surface at the polarising angle or 56° 45°, the rotation produced by refraction at A is about 2°, or the inclination I = 45° + 2° = 47°; but the maximum action of the polarising force at C is sufficient to make I = 0° whether x is 45° or 47°. Hence CB is completely polarised in the plane of reflection, and the refractive action at b is incapable of changing the plane of polarisation when I = 0°; the reason is, therefore, obvious why the two rotations at A and b, of 2° each, produce no effect at the maximum polarising angle.
If we now call
ϕ = inclination to the plane of reflection produced by the first refraction at A,
ϕ' = inclination produced by the reflection at C,
ϕ'' = inclination produced by the second refraction at b,
We shall have
\[ \cot \phi = \cos(i - r), \text{ or } \tan \phi = \frac{1}{\cos(i - r)} \]
\[ \tan \phi' = \tan x \frac{\cos(i + r)}{\cos(i - r)} = \frac{\cos(i + r)}{\cos(i - r)} \]
\[ \cot \phi'' = \cot x (\cos(i - r)) = \frac{\cos(i - r)}{\cos(i + r)} \]
These formulae are suited to common light, where z = 45°, but when x varies they become
\[ \cot \phi = \cot x (\cos(i - r)), \]
\[ \tan \phi' = \tan x \left( \frac{\cos(i + r)}{\cos(i - r)} \right), \]
\[ \cot \phi'' = (\cot x \left( \frac{\cos(i - r)}{\cos(i + r)} \right)). \]
Resuming the formula for common light, viz. cot ϕ =
\[ \frac{\cos(i - r)}{\cos(i + r)}, \]
it is obvious that when \( \cos(i - r)^2 = \cos(i + r)^2 \),
\[ \cos(i + r), \cot \phi' = 1, \text{ and } \phi'' = 45°; \text{ that is, the light is restored to common light.} \]
In glass where m = 1-525 this effect takes place at 78° 7', a little below 78° in diamond, and a little above 80° in water.
At an angle below this ϕ becomes less than 45°, and the pencil contains light polarised in the plane of reflection; while at all greater angles ϕ is above 45°, and the pencil contains light polarised perpendicular to the plane of reflection. Hence we obtain the following curious law:
"A pencil of light reflected from the second surfaces of transparent plates, and reaching the eye after two refractions and an intermediate reflection, contains, at all angles of incidence, from 0° to the maximum polarising angle, a portion of light polarised in the plane of reflection. Above the polarising angle the part of the pencil polarised in the plane of reflection diminishes till \( \cos(i + r) = \cos(i - r)^2 \), when it disappears, and the whole pencil has the character of common light. Above this last angle the pencil contains a quantity of light polarised perpendicular to the plane of reflection, which increases to a maximum and then diminishes to zero at 90°."
Let us now examine the state of the pencil CS after only one refraction and one reflection. Resuming the formula tan ϕ' =
\[ \frac{\cos(i + r)}{\cos(i - r)} \]
it is evident that when \( \cos(i - r)^2 = \cos(i + r)^2 \), ϕ' = 45°, and the light restored to common light. This takes place in glass at an angle of 82° 44'. At all angles beneath this the pencil contains light polarised in the plane of reflection; but at all angles above it, the pencil contains light polarised perpendicular to the plane of reflection, the quantity increasing from 82° 44' to its maximum, and returning to its minimum at 90°.
Let us now apply the results of the preceding analysis to M. Arago's experiment. Suppose the angle of incidence to be 78° 7', and let the light polarised by reflection at A (fig. 142) be = m, and that polarised by one reflection also = m. Then, since the pencil bs is common light, the polarised light in the whole reflected pencil AP, bs, is = m, whereas the light polarised by the two refractions is = 2m; so that the experiment makes two quantities appear equal when the one is double that of the other. If the angle exceeds 78° 7', the oppositely polarised light in the pencil bs will neutralise a portion of the polarised light in the pencil AP, and the ratio of the oppositely polarised rays which seem to be compensated in the experiment may be that of 3m or even 4m to 1.
We may now obtain formulae for computing the exact quantities of polarised light at any angle of incidence, either in the pencil CBS or bs. The primitive ray RA being common light, AC will not be in that state, but will have its planes of polarisation turned round a quantity \( x \) by the refraction at A; so that \( \cot x = \cos (i - r) \). Hence we must adopt for the measure of the light reflected at C the formula of Fresnel for polarised light whose plane of incidence forms an angle \( x \) with the plane of reflection. The intensity of AC being known from the formula for common light, we shall call it unity, then the intensity I of the two pencils polarised \( -x \) and \( +x \) to the plane of reflection will be
\[ I = \frac{\sin^2(i - r)}{\sin^2(i + r)} \cos^2 x + \tan^2(i - r) \sin^2 x, \]
and
\[ Q = I \left( 1 - 2 \frac{(\cos(i + r))^2}{(\cos(i - r))^2} \right) \]
In like manner, if the intensity of CB = 1, we have
\[ \tan x = \frac{\cos(i + r)}{\cos(i - r)} \]
and the intensity I of the transmitted pencil bs
\[ I = 1 - \frac{\sin^2(i - r)}{\sin^2(i + r)} \cos^2 x + \tan^2(i - r) \sin^2 x \]
and
\[ Q = \left( I 1 - 2 \frac{(\cos(i - r))^2}{(\cos(i + r))^2} \right) \]
The following table, computed from the formulae in the preceding page, shows the state of the planes of polarisation of the three rays AC, CS, and bs.
| Angle of Incidence on the First Surface | Angle of Refraction at First Surface and Angle of Incidence on Second Surface | Inclination of Plane of Polarisation of AC | Inclination of Plane of Polarisation of CS | Inclination of Plane of Polarisation of bs | |----------------------------------------|-------------------------------------------------|------------------------------------------|------------------------------------------|------------------------------------------| | 0° 0' | 0° 0' | 45° 0' | 45° 0' | 45° 0' | | 32° 0' | 20° 33' | 45° 34' | 32° 29' | 32° 51' | | 40° 0' | 25° 10' | 45° 58' | 24° 12' | 24° 56' | | 45° 0' | 27° 55' | 46° 17' | 17° 49' | 18° 38' | | 56° 30' | 33° 30' | 47° 22' | 0° 0' | 0° 0' | | 67° 0' | 37° 34' | 48° 57' | 18° 29' | 20° 50' | | 70° 0' | 39° 30' | 49° 31' | 23° 34' | 27° 6' | | 75° 0' | 39° 46' | 50° 45' | 32° 22' | 37° 48' | | 78° 57' | 40° 29' | 51° 49' | 33° 10' | 44° 59' | | 79° 0' | 40° 33' | 51° 56' | 33° 19' | 45° 46' | | 80° 0' | 40° 42' | 52° 16' | 40° 27' | 47° 46' | | 83° 0' | 41° 5' | 53° 21' | 44° 39' | 53° 40' | | 85° 30' | 41° 23' | 54° 47' | 59° 58' | 60° 13' | | 90° 0' | 41° 58' | 56° 29' | 56° 29' | 66° 19' |
Sect. VI.—On the Polarisation of Light by Absorption and Dispersion.
In the preceding section we have considered common light as consisting of two pencils of polarised light, having their planes of polarisation at right angles to each other. The very same results would take place if we considered a beam of light as consisting of two sets of polarised rays, all those of one set having their planes of polarisation in every possible direction, and of another set having their planes of polarisation at right angles to those of the former. In this view of the subject, common light is polarised when it issues from the luminous body, and when we polarise it or decompose it by double refraction, or polarise it completely by reflection or refraction, we merely separate the one half of it polarised from the half polarised. This effect is analogous to the decomposition of white light into its colours. All the colours exist in the sun's light, and they are merely separated by prismatic refraction, or by interference, or by absorption.
Now common light may also be decomposed by dispersion and absorption; that is, if we can contrive any method of dispersing or absorbing one of the two polarised pencils of common light, we shall exhibit the other pencil in its state of natural polarisation.
Crystals in which this effect is produced are called singly polarising or singly refracting crystals. They were first observed and described by Sir David Brewster in the year 1812. The first mineral in which he discovered this property was the agate, in which one of the pencils is dispersed into a nebulous mass of light, sometimes of the form of a crescent, so that the bright image was all polarised in one plane, like one of the pencils of Iceland spar. The mass of nebulous light, too, was always polarised in a plane perpendicular to that of the bright image.
The same author discovered a similar property in certain specimens of the carbonate of barytes, which exhibited several interesting phenomena, in thick crystals of mica, and in mother of pearl, and very curiously in oil of mace, and other substances. The same property he found in various artificial crystals, but particularly in nitre.
Another very beautiful example of polarised nebulous images was observed by the same author in an artificial kind of nacre already referred to, in which there are three nebulous polarised images, with a bright image enclosed in the middle one of the three nebulous images.
A similar property was discovered in 1815 in the tourmaline, nearly about the same time, by M. Biot and M. Seebeck, the priority belonging to the former. This crystal has double refraction, like all other crystals of the same class; but when it is cut by planes passing through the axis of the crystal, and has a certain thickness (about the twenty-fifth of an inch, but which varies in different specimens), it transmits only one of the pencils formed by double refraction. The pencil polarised in the plane of the principal section is absorbed, or somehow or other lost, while the one depolarised at right angles to that section is transmitted. If we take two such plates of tourmaline, and place them with their axes parallel, the unabsorbed pencil will be freely transmitted through both; but if we begin to turn one of them round, the light will become fainter and fainter, and the luminous object will vanish entirely when the axes are at right angles to each other. By continuing to turn, the image again appears, reaches its maximum when the axes are parallel, and then vanishes when the axes are at right angles. The same phenomenon is exhibited by the agate in reference to its distinct image.
In all the singly refracting and polarising crystals above enumerated, the effects described arise from a certain degree of imperfection in the structure and combination of the elementary crystals, by which one of the polarised pencils is reflected or absorbed; but Sir David Brewster found that the same property may be communicated to any crystal, merely by altering its superficial conformation.
If we take a hexahedral prism of nitrate of potash, and... observe a luminous object through two of its inclined surfaces that have a good natural or artificial polish, we shall perceive two distinct and perfectly formed images. If we now roughen these two surfaces, and cement upon each of them a plate of glass by means of balsam of capivi, the character of the two images will be greatly changed. The image that has suffered the greatest refraction will be as distinct as before, but the other image will be either of a faint reddish colour, or wholly invisible, according to the degree of roughness induced upon the refracting surfaces. When oil of cassia is used instead of the balsam, the least refracted image, if it was visible before, will now be completely extinguished.
By substituting pure alcohol, or the white of an egg, instead of the balsam, the least refracted image will become distinct, and the most refracted image will be either a mass of nebulous light, or almost invisible.
In order to explain these phenomena, we must recollect that the index of refraction for the ordinary image of nitre is 1·511, and that of the extraordinary image 1·328. When the rough surface of the nitre is covered with balsam of capivi, which has nearly the same index of refraction as the ordinary image, the same effect is produced as if the rough surface had been polished for the ordinary rays. All the little pits or depressions in the rough surface being filled up with balsam, the ordinary rays suffer little or no refraction in penetrating the crystal, and therefore the image which they form will be as clear and distinct as in the first experiment. But since the index of refraction for the extraordinary image is much less than that of the balsam, the rays of which it is composed will not enter the crystal undisturbed, but will be scattered in the same manner as if its surface was rough, and had a refractive power corresponding to the difference between the index of refraction for the extraordinary ray, and the index of refraction for the balsam. When water or alcohol is substituted in room of the balsam, the effects now described are interchanged, the roughness being removed for the extraordinary rays by the application of a fluid of the same refractive density, while the rays that form the ordinary image are dispersed by the refractions which still exist at the rough surface of the crystal.
These effects will be better understood by supposing the crystal to consist of an extraordinary and an ordinary medium, arranged in alternate strata, or as water exists in wet hydrophanous opal. When the superficial polish of both these media is removed, the application of the balsam restores, as it were, the polish of the ordinary medium, without restoring that of the extraordinary medium; while the application of the alcohol restores the polish of the extraordinary medium without restoring that of the ordinary medium.
These results were repeatedly obtained with calcareous spar, arragonite, nitre, carbonate of potash, and other crystals; and we have now before us a singly-polarising prism of Iceland spar, made nearly twenty years ago, which answers all the common purposes of a plate of tourmaline or a parcel of glass plates.
Sect. VII.—On the Depolarisation of Light, and the Colours of thin Crystallized Plates in Polarised Light.
The phenomena which we are about to describe are among the most splendid in optics. They were discovered by independent observation by M. Arago and Sir David Brewster; the priority of discovery belonging to M. Arago. The very same colours, indeed, as we shall presently see, had been observed by Huygens, Robison, Malus, and others, in Iceland spar; but they were not aware of their nature and origin.
There are four methods of exhibiting these colours, which may be used at pleasure, and each of which has its advantages.
1. If we take a plate of agate or tourmaline, or any other of the artificial singly-polarising crystals already described, and having cut it into two parts, each of which is at least equal to the diameter of the pupil of the eye, though this is not absolutely requisite, fix each of them above an aperture in a piece of card or brass, so that no light passes at their edges. Let them be now placed at the distance of an inch or more, the one near the eye being capable of turning round the axis of vision; and let this last be turned into such a position that the light of the sky, or that of a flame enlarged by a lens placed near it, is no longer able to penetrate the second singly-refracting plate.
If we now introduce between the two plates of tourmaline, for example, a plate, either thick or thin, of any doubly-refracting substance, we shall observe very curious effects. If the plate is thick, such as one of sulphate of lime, the 30th of an inch and upwards, we shall find that the insertion of the plate has revived, as it were, the light which refused to pass through the second plate, and this light will be white. If we turn the sulphate of lime round, we shall find four positions 90° from each other, in which the revived light is a maximum; and other four bisecting these, and also 90° from each other, in which the light entirely vanishes, as before the sulphate of lime had been introduced.
This property of reviving the light has been called the depolarisation of light, because the sulphate of lime deprives the rays polarised by the first plate, of that kind of polarisation which prevented them from penetrating the second plate.
But if the plate is thin, between the fiftieth and the seventy-fifth of an inch, we shall have precisely the same phenomena, with this difference only, that the restored light is brilliantly coloured. The colours vary in intensity, like the white light, disappearing in the positions 0°, 90°, 180°, and 270°, and reaching their maximum at 45°, 135°, 225°, and 315°. If we now turn the plate of tourmaline next the eye round 90°, so that the axis of the two plates are parallel, and the pencil polarised by the first freely transmitted by the second, and if we then introduce the same plate of sulphate of lime as before, we shall now find that in the four positions of it, viz. 0°, 90°, 180°, and 270°, where no light or colours were formerly nothing, but white light is visible; but that at the positions 45°, 135°, 225°, and 315°, where the maxima of light and colour took place, we have the maxima of a colour complementary to that formerly seen, the intensity of this colour gradually increasing from nothing at 0°, reaching its maximum at 45°, again diminishing to 90°, and so on with the other quadrants. These colours are called the colours of crystallized plates, or the colours of polarised light.
In the preceding experiment with plates of tourmaline, we see only one of the two complementary colours, while the position of the tourmaline remains the same. The disadvantage of using the tourmalines is, that, from their brown colour, the brilliancy of the polarised colours is greatly injured; and the tourmalines therefore cannot be employed, either when we wish to have the most brilliant representations of the phenomena, or when it is necessary to study the exact tints which are developed.
But if we use the plates of agate or of roughened Ice- land spar in the same manner, we shall not only have identically the same phenomena of colour in the bright and distinct image formed by the agate with a purer light; but we shall have the additional phenomenon of this bright coloured image placed in the middle of a nebula or haze of the complementary colour, so that we here see both the colours at the same time, and without any of the superadded brown colour imparted by the tourmaline. If the colour of the distinct image is green, it will be encircled by a haze of red light; if it is blue, with a haze of orange light; and so on.
2. The second method consists in placing the film of sulphate of lime GKHL between two bundles of glass plates A and B, which polarise the light by refraction, as shown in the annexed figure. A ray Rr emerges from A, polarised as at st, and will freely polarise the second bundle B, and emerge, as shown at rw, when the planes of refraction of A and B are parallel, as in the figure; but not a ray of st will emerge at v when the planes of refraction are perpendicular to each other. When we interpose, therefore, the sulphate of lime GKHL, it will exhibit identically the same phenomena as between the tourmaline plates, the bundle A corresponding to the fixed plate, and B to the moveable one, and the planes of refraction corresponding to the axes of the tourmaline. If we suppose the bundles A and B to be placed with their planes of refraction perpendicular to each other, then the position of the plate of sulphate of lime, in which no colour appears, is, when its axis of double refraction CD is its principal axis (viz. the line bisecting the resultant axis) if it is biaxial, parallel or perpendicular to the planes of refraction. When CD, therefore, or EF (perpendicular to it), is in the plane of refraction of A, or the plane of primitive polarisation, as it is called, not a ray of coloured light reaches the eye at v, and hence these have been called the neutral axes of the plate of sulphate of lime, because they produce no change upon the ray st. On the other hand, when the lines GH, KL, perpendicular to each other, and inclined forty-five degrees to the neutral axes, are in the plane of primitive polarisation, the coloured light depolarised is a maximum, and hence they have been called the depolarising axes of the plate GKHL, names which will be found very convenient in the description of phenomena.
It is an interesting fact, which was discovered by Sir David Brewster, that nature actually presents us, in the case of certain crystals of nitre, with the whole of the apparatus in the space of half an inch. One part of the crystal has its laminae inclined like the bundle A, while another part has them lying in a rectangular direction like B; so that such a crystal, by merely looking through it when the opposite faces are either polished by art or by a cement, exhibits its own coloured rings. Sir John Herschel subsequently observed the same fact in carbonate of potash, and proposes to call such crystals idiocyphalous, or those which show their own rings.
3. The third method is shown in the annexed figure, where A is a plate of black glass (or a bundle of 8 or 12 plates of thin transparent glass, for the purpose of increasing the light), which polarises in a horizontal plane a ray Rr, incident at an angle of 56°, reflecting it polarised in the direction rs, where it is received upon a second plate of black glass B at the same angle of 56°, so as to reflect it to the eye at O. If the plane of reflection from B is vertical, so as to be perpendicular to that from A, not a ray of the pencil Rr will be reflected, but the eye at O will perceive a large black spot on the part of the sky or other luminous surface from which the ray Rr proceeds. The plate or plates at A are called the polarising plates, and that at B the analysing plate.
The plate of sulphate of lime is then placed at GKHL anywhere between the plates, and it will exhibit the very same phenomena as between the tourmalines and the bundles of plates, though with more brilliancy and distinctness, as there is no brown colour to disturb the tints, and no haziness, as happens with the second bundle B of plates of glass. In this method the planes of reflection perform the same part as the axis of tourmaline and the planes of refraction in the other cases.
4. The last method which we shall mention, is to employ rhombs of Iceland spar, both for polarising and analysing the light. If the Iceland spar is converted into two of Nicol's prisms, then each prism performs exactly the same part as the tourmaline and the reflectors, exhibiting only one of the complementary colours. This ingenious contrivance, which derives its name from its inventor, William Nicol, Esq. of Edinburgh, consists of two pieces of calcareous spar cemented together so as to transmit only one of the polarised pencils; but the difficulty and expense of constructing it well, and the risk of a change in the state of the cement which unites the two parts, render it desirable to have a simple, a cheap, and a durable substitute for it. The polariser and analyser used by Sir David Brewster in many of his experiments in elliptical polarisation, and in the action of crystallized surfaces, was a single rhomb of calcareous spar, having thin plates of colourless glass cemented to its natural surfaces by Canada balsam, which, while it removes any imperfection of surface, protects the surfaces from any accidental injury, or from the deterioration of the polish arising from frequently cleaning them. This rhomb, shown at ABCD, may be of any thickness suited to the diameter of the pencil of light which we wish to have. By a rhomb one inch thick we can obtain a pencil of light 0·115 of an inch in diameter; by placing an aperture of that diameter at a on the lower side CD, we obtain two pencils b, c, polarised in opposite planes, and just touching each other.
If we wish to use only one pencil, we can conceal b or c with a wafer, and use the other; but for the purpose of our present experiment they may both be left clear. If we now construct an exactly similar, or a larger rhomb, without any aperture upon it, and place the two as we did the tourmaline plates, or in any of the positions shown in fig. 130, we shall see only two images, the other two being evanescent. Let the plate of sulphate of lime be now interposed as formerly, and we shall find, that when its principal axis CD forms angles of 0°, 90°, 180°, and 270° with the plane of the principal section of the rhombs, no light is depolarised; but that when the axis forms angles of 45°, 135°, 225°, and 315° with the principal section, the two evanescent pencils are restored and brilliantly coloured, let us suppose with green light, while the other two pencils formerly seen are brilliantly coloured with the complementary red colour. By turning round the rhomb next the eye, these two colours undergo all varieties of intensity, from their maximum tint to evanescence.
If we now enlarge the aperture \(a\), fig. 145, in the first rhomb, so that the two images \(b, c\) in place of being in contact, overlap each other, we shall have the parts that do not overlap exhibiting the two complementary colours as before, while the parts that do overlap form perfectly white light; thus proving that the two colours are exactly complementary to each other. Any of the images \(b, c\) may be enlarged, if necessary, by a convex lens placed before them, or by other means.
In order to obtain a large pencil of polarised light \(b\) or \(c\), we must make the rhomb very thick; but there is another way in which we may obtain the same effect in thin rhombs. There are particular specimens of the spar which are interrupted with veins, and which will be described in a separate section. If we obtain one of these in which the vein has a certain thickness, it will produce polarised images on each side of \(b\) and \(c\), and these images will be perfectly white. We may therefore use a much larger aperture \(a\), and obtain a very effective apparatus. This, however, will be better understood afterwards.
When the veins above mentioned are thin, these lateral images are beautifully coloured, the colours being portions of the coloured rings produced by the thin vein, the prism in front of the vein polarising the light, and the prism behind it analysing it.
These different methods of exhibiting the colours of crystalline plates may be varied, if we use a plate of glass as the polariser, and a tourmaline for the analyser; or, in short, we may use one half of each apparatus with one half of the other. When we wish to have a large surface of polarised light, the best and readiest is a large surface covered with black varnish.
In the preceding observations, with the four different kinds of apparatus, we have supposed the polariser and analyser to be fixed, either with their similar planes parallel or perpendicular to each other, and the plate of sulphate of lime to be moved round its axis. Let us suppose, however, that the sulphate of lime is fixed in the position where its colour, bright red, for example, is a maximum; that is, where any of the depolarising axes GH or KL, fig. 144, is parallel or perpendicular to the plane Rrs of primitive polarisation. In this position of GH, let the analysing plate B be made to revolve round the ray \(rs\), its motion commencing at 0°, and always keeping the same inclination to \(rs\), viz. 56°. The bright red visible at 0° will gradually diminish in intensity as B moves from 0° to 45°, when the red colour will wholly vanish, and the black spot be seen. Beyond 45° a faint green tint will appear, gradually increasing, and attaining its maximum of brightness at 90°. At an azimuth greater than 90°, the green becomes paler and paler, till it vanishes wholly at 135°. Here the red again begins, and reaches its maximum brightness at 180°. Similar changes take place while the plate B moves from 180° to its original position at 360° or 0°. Hence it appears, that when the sulphate of lime alone revolves, only one of the complementary colours is visible, whereas, when the plate B only revolves, both the complementary colours are visible during each half of its revolution.
When these experiments are repeated with plates of sulphate of lime, or any other mineral having different thicknesses, different colours will be produced, varying with the thickness; and in every case the two colours which are produced, either when we use the two polarising rhombs, or cause the reflector B to revolve, are always complementary to each other, or together make white light.
If we remove the plate B, and look through the sulphate of lime, we shall find that the light which it transmits is always white, whatever be the position of the sulphate of lime, whatever be the inclination which the ray \(Rr\) forms with the polariser \(A\), and whatever be the condition of the polariser itself. The decomposition of the white light, therefore, or its separation into two complementary colours, must be effected by reflection from the plate B. Now sulphate of lime is a doubly refracting crystal, giving two oppositely polarised images, lying above one another, and one of its neutral axes CD is the section of a plane passing through its principal axis of double refraction, while EF is the section of a plane perpendicular to that section. Let any of these planes, suppose EF, be placed, as in the figure, in the plane of primitive polarisation \(Rrs\), then the ray \(rs\) will not be doubled, but will pass into the ordinary ray of the sulphate of lime, and falling upon B, it will not suffer reflection. The very same will happen if CD is brought into the plane of primitive polarisation, so that in these two positions none of the light transmitted through the sulphate of lime will suffer reflection at B, and reach the eye at O. In all other positions, however, of the sulphate of lime, it forms two images or pencils of different intensities; and when either of the depolarising axes GH or KL is in the plane of primitive polarisation \(Rrs\), these two images or pencils will be of equal intensity, and polarised in opposite planes. Now, one of these images is red, and the other green, a fact which will be afterwards explained; and as the red is polarised in the plane of primitive polarisation, it will not suffer reflection from B; while the green, being polarised in the plane of reflection from B, will be reflected to the eye at O, and is therefore seen alone. From the same cause, when B is turned round 90°, the green will not suffer reflection from it; while the red will suffer reflection, and be seen by the eye at O. The plate B has therefore analysed the compound beam of red and green light by reflecting one and transmitting the other colour.
Now, if the sulphate of lime had been thicker than the fifthieth of an inch, the two pencils would have been both white; and when the plate was moved round, we should have had a white pencil reflected from B, and undergoing the very same changes that the coloured one did. Hence we see the cause of the depolarisation of light, both when the depolarising plate is thick and thin.
In the preceding experiments, the sulphate of lime has been supposed to be so thin as to give a red and a green tint; but if the plate is only 0·00046 of an English inch thick, it will depolarise no light at all, either coloured or uncoloured, and the black spot will be seen in every position of the sulphate of lime. If the thickness of the plate is 0·00124 of an inch, the light depolarised will be the white, the first order of Newton's scale, whose complementary colour is a deep violet; and if the plate is 0·01818 of an inch, or upwards, it will also polarise white light, composed of all the colours of the spectrum. When the plates of the mineral have an intermediate thickness between 0·00124 and 0·01818, they will give, at successive thicknesses, all the intermediate colours in Newton's table, between the white of the first order, and the white compounded of all the colours. The colours from the plate B will be those in col. 2 of Newton's table, while the colours seen in turning round the plate B will be the complementary ones in col. 3 of the same table, the one corresponding to the reflected and the other to the transmitted tints of thin plates.
By a variety of accurate experiments, M. Biot pointed out the connection between the colours of polarised light and of thin plates; and in the case of sulphate of lime, the films of which he measured with an ingenious instrument invented by M. Cauchoux, called the spherometer, he has proved that the thicknesses which produce the different colours in Newton's table are proportional to the value of the tints in the last column of Newton's table for glass; this substance having nearly the same refractive power as sulphate of lime. If we wish, for example, to know the thickness of sulphate of lime which will give the red of the first order of colours, the number in the last column opposite red is $\frac{3}{5}$ths; then, since the white of the first order is produced by a plate $0.00124$ of an inch thick, the number opposite which is $\frac{3}{5}$ths, we say, as $\frac{3}{5}$ths is to $\frac{3}{5}$ths, so is $0.00124$ to $0.00211$, which is the thickness at which sulphate of lime depolarises the red of the first order. By reversing this rule we can determine the colour which will be depolarised at any given thickness. This law of the colours is by no means general, and we shall presently see that the colours of polarised light bear frequently no relation whatever to those of thin plates.
The above phenomena may be beautifully exhibited to the eye by combining pieces of sulphate of lime into a painted Gothic window, so that when the window is exposed to common or to polarised light it appears perfectly colourless, but when seen by reflection, it will exhibit all the splendid colours of the separate films. Though this experiment is well suited for exhibition, yet the following one, which was also made by Sir David Brewster, is more instructive. Selecting an uniform plate of sulphate of lime, about the twenty-fifth of an inch thick, he ground down, with the powder of schistus, one of its faces, so as to make it a sort of wedge, in which the thickness varies from the twenty-fifth of an inch down to the thinnest edge that could be made. The plate was then placed in water, which acted upon it slowly, making its edge thinner, and giving a slight polish to its surface. When this film is placed at GH, in fig. 144, its figure will be seen covered with coloured fringes, as in fig. 146, parallel to its edges. At its thinnest edge the colours of the first order will commence from black, and gradually increasing, as in Newton's table, till, at the thick edge AD, we have the sixth or seventh, or higher orders of colours. Here we see at once how all the different tints and different orders are connected with the different thicknesses of the plate.
If we now cut this plate into two equal parts AB, CD, and cross them as in the figure, a new set of fringes will appear, parallel to a line joining the points where the two thinnest and the two thickest edges intersect, that is, parallel to NP, one of the diagonals of the intersectional square MNOP. The line NP will be black, and from NP to M and N the fringes will be exactly the same as those from the thin to the thick edge of the plate.
If the plates ab, cd have their axes at right angles to each other, there will be only one set of fringes, beginning at the angle where the two tints are minimum.
If we grind one side of the plate spherical, so that the thickness shall vary like the plate of air between a convex and a plane surface, then, by combining a plate of this kind with a prismatic plate, or by combining two similar plates, and by varying their maximum thicknesses, their breadths, &c., we shall produce, by their parallel or rectangular combination, intersectional fringes of figures of great variety and extreme beauty. These figures will be analogous, mutatis mutandis, to those observed by Mr Knox, and the far more splendid exhibitions produced by crossing plates of glass in the manner described in a subsequent chapter.
If we grind concave surfaces on one of the faces of the crystalline plate, we shall have circular rings, equidistant if the concavity is conical, but resembling those of thin plates if it is spherical. These plates may also be combined with those which give circular and rectilineal fringes, and very fine effects produced by the combination. Still more remarkable effects may be produced by turning beautiful patterns upon the sulphate of zinc, or etching them either by the action of pure water, or water slightly acidified. Lines of equal depths will be all equally coloured, and the slightest differences in depth, which can be easily regulated by a fine turning-lathe, will produce a great variety of different colours. Coloured figures and landscapes may be executed by scraping away the surface to the thickness necessary to produce the requisite tints. A cipher too, might be executed upon a mineral; and if we cover the surface upon which it is formed with a fluid of the same refractive power, it will be absolutely illegible by common light, but may be distinctly deciphered when placed between the polarising and analysing plates.
A sheet of ice irregularly frozen, and held in the position just mentioned, exhibits the colours of crystallized plates in a splendid manner; but if in a severe winter, when ice can be handled without melting, we take an uniform plate, and dissolve a pattern upon it by heat, which may be applied in many ways, so as to affect only the parts to be made thinner, we shall observe a phenomenon than which nothing can be more splendid.
**Sect. VIII.—On the System of Rings produced by Uniaxal Crystals.**
In the preceding experiments the crystallized plate is held at such a distance from the polarising plate that its uniaxal surface could be distinctly seen by the eye; and it was from observations made in this manner that M. Biot deduced his empirical formulae for expressing the variation of the tint, as depending on the thickness of the crystallized plate, and on the square of the sine of the inclination of the refracted ray to the axis of the crystal.
From this mode of observation M. Biot concluded, that arragonite, topaz, sulphate of lime, felspar, sulphate of barites, and sulphate of strontium, were all crystals with one axis of polarisation and double refraction. In 1813, however, Sir David Brewster was led to an entirely different mode of observation. He brought the crystal or plate as close to his eye as possible, and by using a large polarising plate, such as a black japanned tray, he was able to see, at the same instant, the colours produced at various angles of inclination, in place of determining the loci of these colours, as was done in the old way, by a great number of insulated observations.
When, for example, he looked through the ruby, the emerald, topaz, ice, nitre, and other bodies, he observed the most beautiful system of rings when the polarised light
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1 The plate may be cemented by one of its faces to a piece of plate-glass, which will protect it from breaking, and after it is finished and bisected, each half should be enclosed between two glass plates.
2 We would recommend to the practical optician the construction of such plates as a matter of trade. passed along the axis of double refraction of these bodies. The rings in crystals with one axis are essentially different from those with two axes.
The uniaxial system of rings is represented in Plate CCCLXXXIII., fig. 29, 30. I discovered them in all the doubly refracting crystals with one axis, already mentioned, excepting in Iceland spar, in which they were first observed by Dr Wollaston, and independently by M. Biot and Dr Seebeck. This system of rings is seen along the axis of double refraction; and it has been customary to cut faces upon the obtuse angles of the rhomb of calcareous spar perpendicular to the axis, in order to see them. Sir David Brewster, however, employed the simple method shown in the annexed figure, which does not require the aid of the lapislary. Let CDEF be the principal section of a rhomb of Iceland spar; cement upon CD and EF two small prisms DLK, FGH, having the angles LDK, GFH about 45° each; and let this rhomb be placed between the polarising and analysing plates, so that the polarised ray passes perpendicularly through the faces LD, FG. The rhomb must be held as close as possible to the reflector B, which need not be larger than the pupil of the eye. When the black spot is seen on the reflector B, in the apparatus as adjusted, then, when the rhomb is interposed, the observer, with his eye close to the plate B, and looking, as it were, through the reflected image of the rhomb, will see the beautiful system of rings shown in fig. 29, Plate CCCLXXXIII., intersected by a black rectangular cross AB, CD, the arms of which are parallel and perpendicular to the plane of primitive polarisation. The colours of the rings, of which seven or eight may be readily seen, are almost exactly the same as those of the reflected rings in thin plates, as seen in Newton's table or scale of colours.
If we turn the rhomb round its axis, the rings will suffer no change, the four arms of the black cross revolving round the circumference of the rings, or rather these four arms remain fixed, and the rings revolve with the rhomb.
If the rhomb is now fixed so that the rings are distinctly visible, and if we cause the plate to revolve from zero or 0°, then the rings will change into the form shown in fig. 30, Plate CCCLXXXIII., in which the black cross is broken up; and at 45° the rings will appear as in fig. 31, which is the complementary system, with a white centre exactly similar to the system seen by transmission in thin plates. These two systems of rings superposed or placed one above the other, would produce uniform white light, without any trace of rings. By continuing to turn B, the primary system, fig. 29, will re-appear at 90°, 180°, and 270°; and the secondary system, fig. 31, at 45°, 135°, 225°, and 315°; while the intermediate system, fig. 30, will be seen at intermediate azimuths.
It is very interesting to trace the passage of the primary into the secondary system. When B begins to turn, the arms of the black cross widen and become less black, and within them we can see segments of the complementary rings, whose dark intervals correspond to the bright ones of the primary rings, and vice versa. As B advances, the rings of the primary set grow less, and more dilute, while the others grow larger and brighter, till, at 45°, the secondary set is complete. When the light previous to polarisation has passed through ground glass, the diluted primary rings appear of a gray-white colour, and as if they were nearer the eye than the rest.
If, in place of the plate B, we use a polarising rhomb or an achromatised prism of Iceland spar, and look through it along the axis of the other rhomb, then, when the plane of its principal section is parallel or perpendicular to the plane of primitive polarisation, we shall see in one of its two images the primary system of rings, and in another the secondary system; and in intermediate positions we shall see the intermediate system, the one constantly passing into the other.
When all these experiments are repeated in homogeneous light, the system of rings will be smallest in violet, and largest in red light, and of intermediate sizes in intermediate colours.
If we divide the rhomb of spar, fig. 148, into two parts by the line MN, and examine the rings through each separately, we shall find that the rings produced by each part are larger than those produced by the whole, the thinnest piece producing the largest rings. Hence two rhombs united will give a system of rings corresponding to those produced by one rhomb of the same magnitude.
The systems, if formed by zircon, ice, and positive doubly-refracting crystals, are exactly the same as the preceding. But if we unite a positive system with an equal negative system, they will destroy each other; and if the two systems are unequal, we shall have a system equal to the difference of their effects. These experiments of combining systems of positive and negative rings, though rather troublesome, are extremely interesting. When such systems are combined, and the space between the crystals that form them left open, a series of splendid changes are induced upon the resulting system, by placing one or more crystallized films in one or more azimuths between them; but we shall have occasion to return to this subject in the section on the multiplication of images by Iceland spar.
In examining the phenomena of the primary rings, it is obvious that there is no polarisation, as there is no double refraction along the axis of the crystal. The tints polarised increase with the double refraction, that is, with the inclination of the polarised ray to the axis of double refraction; and their numerical value, as given in Newton's scale, increases with the square of the sine of that inclination. At any given inclination to the axis the tint increases with the thickness through which the polarised ray passes, so that when we have determined the tint at any given inclination and thickness, it is easy to find it for another inclination and thickness.
In crystals with one axis of double refraction, the lines of equal double refraction are circles when the thickness is equal, as in a sphere; in like manner, the lines of equal tint, or the isochromatic lines, are circles, the tints being a maximum in the equator, where the inclination to the axis is 90°. In crystals with great double refraction the same tint at the same inclination to the axis is produced at a much less thickness than in crystals with feeble double refraction. Quartz, for example, has a very feeble double refraction, and at the same inclination to the axis it would require a plate of quartz 115 times as thick as a plate of Iceland spar to produce the same tint in Newton's scale.
In some crystals with one axis, such as quartz, amethyst, beryl, the system of rings is disturbed by secondary causes, which we shall have occasion to refer to more fully in regard to the two first crystals. In other crystals imperfect crystallization is the general cause of these irregularities.
In Iceland spar, zircon, ice, tourmaline, and various other minerals, the tints of the rings are very nearly those of Newton's scale; but there are other crystals, such as apophyllite, in which Sir David Brewster and Sir John Herschel discovered remarkable deviations, which will be described in a subsequent section.
The intensity of the polarising force, or the value of the tint polarised at a given thickness, has been calculated by different persons for different crystals. The following have been given by Sir John Herschel for uniaxial crystals. Numerical Value Thicknesses that produce the same Tint.
Iceland spar..............35801...........0-000028 Hydrate of strontia.......1246.............0-000802 Tourmaline...............851..............0-001175 Hyposulphate of lime......470.............0-002129 Quartz..................312..............0-003024 Apophyllite, first variety..109............0-009150 Camphor................101..............0-009856 Vesuvian................41..............0-024170 Apophyllite, second variety..33...........0-030374 third variety.............3.............0-366520
These measures are calculated for the yellow rays.
Sect. IX.—On the System of Rings produced by Biaxal Crystals.
The biaxal system of rings was discovered by Sir David Brewster, while he was looking along one of the axes of topaz when the crystal happened to reflect the light of a part of the sky which was partially polarised, so that they were seen without the aid either of a polarising or an analysing plate.
Upon examining other minerals, he discovered that the possession of two systems of rings was the characteristic of far the greater number of crystallized bodies. In some of the crystals, such as topaz, the lines along which each system of rings is seen are so much inclined to each other, that we cannot see the two systems at once; whereas in others, where the inclination of the lines is small, both the systems may be distinctly seen at the same time. This will be understood, in the case of topaz, from fig. 149, where MN is a plate of topaz with parallel faces of cleavage perpendicular to PQ, the principal axis of double refraction. If we expose this plate to polarised light, so that the polarised ray passes along the line ABeE (the plane of incidence being in one of the two neutral axes of the plate); and if the eye at A receives this ray without using the analysing plate, it will see in the direction of that ray a system of oval rings of extreme beauty, like that shown in Plate CCCLXXXIII. fig. 32. When the polarised light is transmitted along the line CBdD, equally inclined to the perpendicular PQ, it will see a similar system. The lines BD, Be are therefore the resultant axes of topaz, along which the double refraction vanishes. The angle ABC is about 121° 16', but the inclination of the refracted rays or of the resultant axes is only 6°. Similar rings are seen by transmission in the direction Dd, Ee, but only when the analysing plate is used.
If we now receive the reflected ray upon the analysing plate at 0°, the system of rings will appear as in fig. 33, Plate CCCLXXXIII., which differs from fig. 32 only in the parts near the major axis. The colours are the same, but the central spots are much smaller, and the mass of darkness with which they are surrounded encroaches considerably upon the blue part of the first ring. The same system will be seen at 90°, 180°, 227°; but upon turning round the analysing plate, we shall see, at 45°, 135°, 225°, and 315°, a third set, shown in fig. 34, which is comparatively faint in its colour, but distinguished by its peculiarities. In its general structure it resembles the set in fig. 32, but in the middle of each central spot there is a darker spot, composed of blue and red chiefly, with a little green above the blue, and every ring is divided into two rings, each of which has the same colours as the original ring. This division of the rings occupies only a part of the semicircumference of each, and is not seen beyond the third ring. When the analysing plate begins to move from 0°, 70°, &c., where fig. 33 is seen, towards 45°, 135°, &c. two blue spots, and the division of the rings, begin to appear at A and A in all the rings, and in the two central spots, and move along each till they reach B at 45°, 135°, &c. Continuing to turn the analyser, the spots and divisions move onward from B to C in all the rings, &c., and disappear at C at 90°, 180°, &c. This curious system of rings is obviously the first set in fig. 32, seen at the same time with their complementary rings, and is a very rare phenomenon.
The biaxal system of rings is best seen in nitre or Rings in saltpetre, in which the inclination of the resultant axes is nitre, only about 5°, or forming an angle of 2½ with the axis of the six-sided prism. When a plate of nitre, about the sixth or eighth of an inch thick, is placed before a small analysing plate, and very close to it, and the eye also held as close to the analysing plate as possible, we shall see the beautiful biaxal system of rings discovered by Sir David Brewster, and shown in Plate CCCLXXXIII. fig. 35, where the plane passing through the two axes of nitre is parallel or perpendicular to the plane of primitive polarization. At angles inclined 45° to these planes, the rings assume the form shown in fig. 36. In passing from the one of these states to the other, the rings assume the forms shown in figs. 37 and 38. The colours begin at the centres A and B of each system; but at a certain distance, varying with the thickness of the plate, the rings, in place of returning and encircling each pole, encircle the two poles, as an ellipse does its foci. When the thickness of the plate is very small, the rings enlarge, and the fifth ring will surround both poles. At a less thickness the fourth, and so on, till at a very small thickness the first ring will surround both poles, and the system then resembles much the uniaxal system of rings. If the plate of nitre is very thick, the rings diminish in size. These colours deviate more and more from those of Newton's scale, and the tints do not begin at the poles A and B, but at virtual poles in their vicinity. The colours of the rings within the two poles are red, and beyond them blue, and the great body of the rings is pink and green. The rings have been called isochromatic lines, or lines of equal tint; and the axes passing through the poles A, B, optical axes, or axes of compensation, or resultant axes.
We have already given a long list of the various minerals and crystals which exhibit the biaxal system of rings, and also the position of the line which bisects the angular distance between the resultant axes, which is the principal axis of the crystal.
The following table, showing the inclination of the resultant axes in different crystals, was drawn up from the observations of Sir David Brewster. Several observations by other observers have been added.
| Names of Minerals | Character | Inclination of Principal Resultant Axis | |-------------------|-----------|----------------------------------------| | Glauberite | Negative | 2° or 3° | | Sulphate of nickel, certain specimens | Positive | 3° | | Carbonate of strontites | Negative | 6° | | Barytes | Ditto | | | Nitrate of potash | Ditto | 5° | | Mica, certain specimens | Ditto | 6° | | Talc | Ditto | 7° | | Carbonate of lead | Ditto | 10° | | Sulphate-carbonate of lead | Ditto | 10° | | Mother-of-pearl | Ditto | 11° | | Hydrate of barytes | Ditto | 13° | | Mica, certain specimens, about | Ditto | 14° | | Arragonite | Ditto | 18° |
Vol. XVI. | Names of Minerals | Character of Principal Axis | Inclination of Resultant Axis | |-------------------|----------------------------|-----------------------------| | Pyroxene (Miller) | Positive | 19° 30' | | Prussiate of potash, certain species | Ditto | 19° 34' | | Cymophane | Ditto | 27° 51' | | Borax | Ditto | 28° 42' | | Anhydrite | Ditto | 28° 7' | | Mica (Biot) | Negative | 30° to 37° | | Apophyllite, biaxal | Negative | 35° 8' | | Sulphate of magnesia | Ditto | 37° 24' | | Spermaceti, about | Positive | 37° 42' | | Tincal, or native borax | Negative | 38° 48' | | Nitrate of zinc, estimated at about | Positive | 40° 0' | | Stilbite | Positive | 41° 42' | | Sulphate of nickel | Ditto | 42° 4' | | Tartrate of ammonia (Miller) | Negative | 42° 20' | | Carbonate of ammonia | Negative | 43° 24' | | Anhydrite (Biot) | Negative | 44° 41' | | Lepidolite | Negative | 45° 0' | | Benzoate of ammonia | Positive | 45° 8' | | Sulphate of zinc | Negative | 44° 28' | | Hopeite | Negative | 48° 0' | | Brazilian topaz | Positive | 49° 50' | | Sulphate of ammonia | Ditto | 49° 42' | | Sugar | Negative | 50° 0' | | Sulphate of strontites | Positive | 50° 0' | | Murio-sulphate of magnesia and iron | Negative | 51° 16' | | Sulphate of ammonia and magnesia | Positive | 51° 22' | | Heulandite (Herschel) | Negative | 54° 17' | | Phosphate of soda | Negative | 55° 20' | | Comptonite | Positive | 56° 6' | | Felspar | Negative | 58° 30' | | Sulphate of lime | Positive | 60° 0' | | Oxynitrate of silver | Ditto | 62° 16' | | Dichroite or iolite | Negative | 60° 50' | | Topaz (Aberdeenshire) | Positive | 65° 0' | | Sulphate of potash | Ditto | 67° 0' | | Carbonate of soda | Negative | 70° 1' | | Acetate of lead | Ditto | 70° 25' | | Citric acid | Positive | 70° 29' | | Tartrate of potash | Negative | 71° 20' | | Benzoic acid (Miller) | Ditto | 75° 0' | | Tartaric acid | Ditto | 79° 0' | | Sulphate of oxide of iron and ammonia (Miller) | Negative | 79° 0' | | Tartrate of potash and soda | Positive | 80° 0' | | Carbonate of potash | Positive | 80° 30' | | Kyanite | Positive | 81° 48' | | Hyper-oxyumariate of potash | Negative | 82° 0' | | Muriate of copper | Positive | 84° 30' | | Epidote, about | Positive | 84° 19' | | Peridot | Positive | 87° 56' | | Crystallized Cheltenham salts | Negative | 88° 14' | | Hyposulphate of soda (Marx) | Positive | 89° 20' | | Succinic acid, estimated at about | Negative | 90° 0' | | Sulphate of iron, about | Positive | 90° 0' |
Many of the measures in the preceding table were taken with much care; but some of them were only estimated, and others will admit of correction by the use of better specimens than the author was able to procure.
In order to explain the biaxal system of rings, and to discover the law of the tints by which every point of the complex system of rings can be calculated, Sir David Brewster considered the optic or resultant axes P, P', produced by two or more rectangular axes, the principal one passing through O, and the other two at AB and CD. We shall suppose, however, the most simple case, where the axes are only two, viz. that at O, and another either coinciding with AB or CD, perpendicular to O. Now, if O is a negative axis, AB must also be negative; but if we take CD for the other axis, it must be positive; for as the axis CD compensates the negative axis O at P, acting in the same plane, it must be a positive axis, for a negative axis would have united its effect with that of O. For a similar reason, the axis at AB must be negative, in order to compensate the negative axis O in a plane at right angles to it.
Supposing, then, O and AB to be two negative axes, as in mica, let it be required to determine the system of rings which they will produce, or the tint at any point P. Now we may either assume the relative intensities of these axes, or, what is better, the angle formed by the two poles P, P', where the actions of the two axes are compensated, as this angle can be readily measured in any crystal. At the action of the axis AB (or the tint which it produces) at P is destroyed or compensated by the action of the axis O, or the tint which it produces, and as the tint is proportional to the square of the sine of the angle which the ray makes with the axis, it is evident that the intensity of the axis at O must be to the intensity of AB, as 1 to $\frac{1}{\sin^2 OP}$.
For as the tint produced at P by AB at an inclination of 90°, which is its maximum tint, is equal to the tint of O produced at P at an inclination equal to OP, the maximum tint produced at O will be found thus: $\frac{\sin^2 OP}{\sin^2 OP} = \frac{1}{\sin^2 OP}$. We have therefore obtained an accurate expression of the relative intensities of the two negative axes O and AB.
If we had supposed the rings to be produced by a negative axis O and a positive one CD (which would equally account for the phenomena), then the intensity of O will be to that of CD as $\cos^2 OP$ is to $\sin^2 OP$.
From a great number of observations made at all inclinations to the resultant axes, and from accurate measurement of the projected rings, Sir David Brewster found that all the phenomena of the rings, with all their varieties of form and curvature, were represented by the following law.
The tint produced at any point of the sphere, by the joint action of two axes, is equal to the diagonal of a parallelogram whose sides represent the tints produced by each axis separately, and whose angle is double of the angle formed by the two planes passing through that point of the sphere and the respective axes.
In showing the application of this law, let it be required to find the tint produced at E, fig. 150, by the joint action of the two axes O and AB, whose relative intensities are as 1 to $\frac{1}{\sin^2 OP}$. Through E draw three great circles AEF, CE, and OE; then let
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1 This must have been a different mineral from that examined by Biot. T = tint required at the point E; θ = the arch between the point E and the axis O; φ = the arch between the points E and C; a = the tint produced separately at E by the greater axis; b = the tint produced separately at E by the lesser axis; ψ = the angle of the forces; α = the angle CEF; ω = the angle OEF; A = the arch FO, or the angle OAF, or the azimuth on the great circle CPO passing through the poles P, P', or centres of the two systems of rings; D = the arch FE, or the declination or distance of the point E from the same great circle; ζ = half the difference of the angles at the base or at the diagonal of the parallelogram of forces.
Then we have
\[ \cos \phi = \cos A \times \cos D, \] \[ \phi = 90^\circ - D, \] \[ \cos \omega = \tan D, \] \[ \cos \alpha = \tan D, \] \[ \cos \psi = \tan \phi. \]
\[ 2 \text{ AEO} = \psi = 2 (180^\circ - \omega) = 2 \omega. \]
Since, then, the tints \( a, b \) produced at E by the axes O and AB are as the squares of the sines of the distance of E from these axes, or as \( \sin^2 \omega \) OE and \( \sin^2 \alpha \) AE, and as the relative intensities of the axes are as 1 to \( \frac{1}{\sin^2 OP} \), we shall have
\[ a = \sin^2 \omega \text{ OE}, \quad b = \sin^2 \alpha \text{ AE} \times \sin^2 OP. \]
Having thus found the sides of the parallelogram of forces whose angle is \( \psi \), the diagonal T of this parallelogram, or the compound tint at E, will be obtained in the following manner:
\[ \tan \zeta = \frac{a - b \tan \frac{1}{2} \psi}{a + b}, \] \[ \zeta + \frac{1}{2} \psi = \text{greater angle at the base}. \]
Hence \( T = \frac{\sin \zeta + \frac{1}{2} \psi}{\sin \zeta + \frac{1}{2} \psi}. \)
When \( a = b \), \( T = 2a (\cos \alpha + \omega) \).
When \( a = b \) and the two axes O and AB equal \( \pi = \omega \) and \( T + 2a (\cos 2\pi) \).
When twice the angle formed by the planes OE, AE or \( \psi = 90^\circ \), then \( T = \sqrt{a^2 + b^2} \).
When \( \psi = 180^\circ \), \( T = a - b \).
When \( \phi = 0^\circ \) or \( 360^\circ \), \( T = a + b \).
Such is the method of determining the tints, and consequently the form of the isochromatic curves in relation to the real axes to which the forces refer; but in relation to the poles P, P', the law of the tints may be more simply expressed by the formula \( T = \epsilon \sin PE \times \sin PE \), where \( \epsilon \) is the maximum tint at A, which must be previously determined. This rule was deduced by M. Biot mathematically from Sir David Brewster's law, and it was afterwards established experimentally by Sir John Herschel, without being aware of its having been deduced from the general law.
The preceding general law is equally applicable when the rings are formed in homogeneous light.
Sir John Herschel, in investigating the systems of biaxial rings, found that they resembled lemniscates, as represented in the annexed figure. In these curves the rectangle PE × PE is invariable throughout each curve, and the value of this constant rectangle in any one curve is \( a \times b \), \( b \) being the parameter of the curve, and \( a \) half the distance between the poles P, P'. The quantity \( a \) is of course the same for each curve, but \( b \) in different curves increases in the arithmetical progression 0, 1, 2, 3, &c., for the several dark intervals of the rings, beginning at the poles, and in the progression \( \frac{1}{2}, \frac{3}{4}, \frac{5}{6}, \ldots \) &c., for the brightest intermediate spaces.
Many interesting conclusions have been deduced from the preceding general law by Sir David Brewster. When the two negative axes O and AB, fig. 150, are of equal intensity, then their action will be compensated at C and D, and CD will be a single positive axis of double refraction, and also of an uniaxial system of rings like those in zircon; and the isochromatic curves given by the preceding formulae will be circles surrounding the axis. Hence it follows, that two negative rectangular axes of double refraction and polarisation compose a single positive axis. If we now suppose a negative axis at CD, equal in intensity to any of the other two, it will evidently destroy the positive axis of compensation at CD, so that three equal and similar negative axes in any crystal destroy each other; and hence our author was led to the conclusion, that all the tessular crystals in which there was neither double refraction nor polarisation had three such axes. But if the third axis at CD is not equal to O or AB, and if O and AB are unequal, then we shall have all the phenomena of a crystal with two axes. In order to compute the tints thus produced by three unequal axes, let O, AB, and CD be three axes, and P, P' the centres of the double systems of rings which they produce. We have already shown that the resulting tint of two axes O and AB, at any part E, is
\[ T = \frac{a \sin \psi}{\sin \zeta + \frac{1}{2} \psi}. \]
In order, however, to combine this tint with another, we must know the direction of it. Since \( \psi \) is the double of the real angle of the planes in which the forces from O and AB act, the direction of the new plane in which these forces act must form an angle with the real direction of O, whose complement is
\[ \frac{1}{2} \psi + \frac{\zeta}{2} \text{ or } \frac{\psi}{2} + \frac{\zeta}{2}, \]
or it forms with the real direction of A an angle whose complement is
\[ \frac{1}{2} \psi - \frac{\zeta}{2} \text{ or } \frac{\psi}{2} - \frac{\zeta}{2}. \]
Hence the direction of the resultant in relation to BE, the direction of the third force with which it is to be combined, is known.
In order to illustrate this in a case where the truth of the result will be immediately seen, we shall take the case of three equal axes, the general resultant of which is nothing. In fig. 150, let O, AB, and CD be the three equal axes, and E the point where we require to know the effect of their combined action. Take AE = 70°, CE = 60°, then EG = 30°, EF = 20°, AG = 66° 44', OG = 23° 16', OE = 37° 17'; then
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1 Treatise on Light, sect. 903. Polarisation.
\[ \begin{align*} a &= \sin^2 AE = 0.883104 \\ b &= \sin^2 CE = 0.7500 \\ c &= \sin^2 OE = 0.36694 \\ a + c &= 1.25004 \\ a - c &= 0.51616. \end{align*} \]
Combining then O and AB, we shall have
\[ T = 0.7500. \]
which will be \(+ \) or positive, as \( \psi \) is greater than 180°.
Then \( \frac{\psi}{4} + \frac{\zeta}{2} = 49° 19' \),
which gives 40° 21' for the direction of the new plane in which the forces O and A produce the combined tint of 0.7500. But the angle \( \omega \) or OEG = 40° 41', so that the resultant lies in the plane CEG; and hence, if we combine with this resultant, or \(+ \) 0.7500, the force — 0.7500, produced by CD, the result will be nothing. This method is also applicable to the combination of axes of double refraction, the numbers corresponding to \(a, b, c\) being in that case the difference between the squares of the velocities of the ordinary and extraordinary rays, as produced by each axis separately at the point E.
The following table of the intensities of the polarising force in biaxial crystals has been given by Sir John Herschel:
| Value of the Highest Tint | Thicknesses that produce the same Tint | |---------------------------|----------------------------------------| | Nitre | 7400 | | Anhydrite, angle of axes 43° 48', 1900 | 0.000135 | | Mica, angle of axes 45°, 1307 | 0.000526 | | Sulphate of barytes | 521 | | Heulandite (white), angle of axes 54° 17' | 0.001920 | | | 249 |
The numbers belong to the yellow ray.
Sect. X.—On Conical Refraction in Biaxal Crystals.
The phenomenon of conical refraction seen along the axes of biaxial crystals was deduced by Sir W. Hamilton from the undulatory theory, and was discovered experimentally and examined by Professor Lloyd in arragonite. It followed from the theory, that a single ray, proceeding from a point within the crystal, and emerging at each of the four poles, must be divided into an infinite number of emergent rays, constituting a conical surface; and that a single ray incident externally would be similarly divided.
In order to examine the emergent cone formed in air, Professor Lloyd placed a lens of short focus at its focal distance from the first surface of a plate of arragonite 0.49 of an inch thick, having its parallel faces perpendicular to the principal axis of the crystal, and so that the central part of the pencil might have an incidence nearly parallel to the optical axis. He then looked through the crystal at the light of a lamp placed at a considerable distance, and observed a point more luminous than the space around it, having a sort of stellar radiation. In order to examine this phenomenon, he placed a plate of thin metal, having a minute aperture, on the surface of the crystal next the eye, and adjusted the aperture so that the line connected with the luminous point on the first surface might be in the direction of the optical axis. When the adjustment was complete, there appeared at first a luminous circle with a small dark space in the centre, and in this dark central space were two bright points, separated by a narrow and well-defined dark line, as shown in Plate CCCLXXXIV., figs. 39 and 40. When the aperture in the plate was slightly shifted, the phenomena rapidly changed, assuming successively the forms shown in figs. 41, 42, 43. In the first stage of the change the central dark space became greatly enlarged, and a double sector appeared in the centre. The circle was reduced to about a quadrant, and was separated by a dark interval from the sector just mentioned. This is shown in fig. 41. The remote sector then disappeared, and the circular arch diminished, as in fig. 42; and as the inclination of the internal ray to the optical axis was farther increased, these two luminous portions merged gradually into two doubly-refracted pencils. This change is shown in fig. 43. In these experiments the emergent rays were received directly by the eye placed close to the aperture on the second surface.
Professor Lloyd succeeded in showing the phenomena on a screen with the sun's light, and he found the light sufficiently distinct when the diameter of the section was one and a half inch. Upon examining the cone with a tourmaline, Professor Lloyd was surprised to observe that one radius only of the circular section vanished in a given position of the tourmaline, and that the ray which disappeared ranged through 360°, as the tourmaline was turned through 180°, the rays of the cone being all polarised in different planes. Upon a more attentive examination he discovered the remarkable law, "that the angle between the planes of polarisation of any two rays of the cone is half the angle between the planes containing the rays themselves and the axis." The angle of the cone was found to be 6° 24', 5° 56', and 6° 22'; the mean of which is 6° 14'.
When the aperture was considerable, such as that formed by a large-sized pin, two concentric circles were seen to surround the axis, the inner one being nearly twice as bright as the outer one, and consisting of unpolarised light, while the outer one was polarised according to the preceding law. By using smaller apertures the inner circle grew less, until it became a point in the centre of the fainter exterior circle, which remained fixed. With a still less aperture a dark space sprung up in the centre, increasing as the aperture diminished, until, with a very minute aperture, the breadth of this central space increased to about three fourths of the entire diameter. In these cases the appearances are as shown in figs. 44 and 45. When the line joining the luminous point on the first surface was slightly inclined to the axis, the appearance was that shown in fig. 46.
Professor Lloyd observed an interesting variation in the phenomena, by substituting a narrow linear aperture for the circular one on the first surface of the crystal, this aperture and the one in the plate next the eye being in the plane passing through the optic axes. The line had the appearance shown in fig. 47, swelling out into the form of an oval curve round the optical axis. By using a very minute aperture next the eye, the phenomenon was as shown in fig. 48. When the plate next the eye was slightly shifted, so that the plane passing through the aperture did not coincide with the plane of the optic axes, the curves rapidly changed, preserving, however, the form of the conchoid, whose pole was the projection of the axis of the emergent cone, and asymptote the line on the first surface. These effects are represented in figs. 49 and 50.
The second kind of conical refraction deduced theoretically by Sir W. Hamilton takes place when a single external ray is incident upon a biaxial crystal, so that one refracted ray coincides with an optic axis. In this case there should be a cone of rays within the crystal, the angle of the cone in arragonite being 1° 55'. As this cone will have its rays refracted at emergence, in a direction parallel to the incident ray, they will form a small cylinder of rays in air, the character of whose section by the surface of emergence being only 1° 55' at a distance equal to the thickness of the crystal.
In order to detect the existence and measure the size of this cylinder, Professor Lloyd used the light of a lamp placed at some distance, and he made its light pass through two small apertures placed in a straight line, the one in a screen near the flame, and the other in a plate of metal close to the first circle of the crystal. Under ordinary circumstances, the incident ray will be doubly refracted within the crystal, and the two pencils will emerge parallel to the second surface. Professor Lloyd was able to distinguish these two pencils by means of a lens; and turning the crystal slowly, so as to vary the incidence, he observed a position in which the two rays changed their relative places rapidly on any slight change of incidence, and appeared at times to revolve round one another as the incidence was changed. Being convinced that the ray was now at the critical incidence, Professor Lloyd changed the position of the crystal relative to the incident ray very slowly; and after much care in the adjustment, he at last saw the two rays spread into a continuous circle, and exhibit the phenomena which we have already described in his own words in our history of Optics.
Professor Lloyd measured the angle of the cone by an indirect method, and found it $1^\circ 50'$, differing only $3'$ from the angle deduced from theory.
**Sect. XI.—On the Effect of Pressure and Heat on the Double Refraction of Crystals with One, Two, and Three Axes.**
The influence of pressure and heat in modifying the doubly-refracting structure of bodies that previously possessed that property; and of creating a new doubly-refractive structure in uncrystallized bodies, was first studied by Sir David Brewster.
By applying compressing and dilating forces to minerals, he succeeded in altering their doubly-refracting structure in every direction; but the effect was always most easily seen when it was produced along the real axes of uniaxial crystals, or the resultant axes of biaxial ones, where the effect of the natural forces was either nothing, or compensated. The following were some of the results to which he was led by applying the forces to parallel surfaces.
**Axis of Compression and Dilatation parallel to the Axis of the Crystal.**
| Positive crystals | Compressed... | Tints rise in Newton's scale. | |------------------|--------------|-------------------------------| | | Dilated......| Tints descend in Newton's scale. |
| Negative crystals | Compressed... | Tints descend in Newton's scale. | |------------------|--------------|---------------------------------| | | Dilated......| Tints rise in Newton's scale. |
**Axis of Compression and Dilatation perpendicular to the Axis of the Crystal.**
| Positive crystals | Compressed... | Tints descend in Newton's scale. | |------------------|--------------|---------------------------------| | | Dilated......| Tints rise in Newton's scale. |
| Negative crystals | Compressed... | Tints rise in Newton's scale. | |------------------|--------------|---------------------------------| | | Dilated......| Tints descend in Newton's scale. |
The axis of compression and dilatation is the line perpendicular to the two surfaces pressed together or drawn asunder.
The above results were obtained by experiments both on uniaxial and biaxial crystals.
When the axis of compression was perpendicular to the axis of double refraction of an uniaxial crystal, it was partially converted into a biaxial one with two axes, the poles of the two resultant axes being distinctly visible.
M. Fresnel was, we believe, the first person who observed the influence of heat in altering the tints of sulphate of lime perpendicular to the laminae, but we are not able to refer to the details of his experiments. Professor Mitscherlich, however, has investigated the action of heat upon this mineral so completely as to include all previous experiments in his results. Having found that heat acts upon calcareous spar differently in different directions, expanding it in the direction of its axis, and slightly contracting it in directions perpendicular to the axis, he sought to determine if any variation in the double refraction was produced by heat. By the method of interferences, and observing the compensation produced by crossing plates of crystals at different temperatures, he observed that a change in the double refraction was produced.
In extending these experiments, Professor Mitscherlich found that the two resultant axes of sulphate of lime inclined $60^\circ$ to each other at common temperatures, approached each other when heated, till they met, and constituted one axis of double refraction. By increasing the heat they again separated in a plane perpendicular to the laminae. In this experiment the principal axis of double refraction which bisected the optic axes gradually increased, while the second real axis perpendicular to the lamina diminished and disappeared when the crystal assumed the uniaxial state. A new axis then sprung up in the plane of the laminae perpendicular to the principal axis.
Sir John Herschel, in mentioning this remarkable experiment, states that he observed the tints of a plate of Herschel's sulphate of lime rise rapidly in the scale when the plate was moderately warmed by the heat of a candle held at some distance below it, and sink again when the heat was withdrawn. He found, on the contrary, "that mica similarly heated undergoes no apparent change in the position of its axes, or in the size of its rings, though heated nearly to ignition."
The extraordinary experiment of Professor Mitscherlich was repeated by Sir David Brewster with one of the specimens of sulphate of lime, in which he discovered one of the resultant axes of this mineral. The following is the account which he has given of this experiment, and of the discovery of a still more curious property in glauconite:
"The specimen of sulphate of lime was about one and a half inch thick in the plane of the laminae, and the system of rings which surrounded this axis was exceedingly minute, with the usual black brush at each end of them. The other system of rings could not be seen in this specimen, owing to the manner in which it was cut. Having brought the crystal to a considerable heat, and exposed it to polarised light, it was a singular sight to see the system of rings travelling along towards the line which bisects the optic axes, like a celestial body passing through the field of a telescope, and changing their form and size as they advanced. The specimen did not permit me to see the two systems unite, and still less to see them open out."
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1 See Irish Transactions, 1833, vol. xvii.; and London and Edinburgh Philosophical Magazine, 1833, No. 8, p. 112, No. 9, p. 207. 2 Phil. Trans. 1815, p. 160, 1816, p. 167; and Edinburgh Transactions, vol. viii. p. 281. 3 Edinburgh Transactions, vol. viii. p. 285. 4 Treatise on Light, sect. 1113. 5 Lond. and Edin. Phil. Mag. December 1832, vol. i. p. 417. again in a plane at right angles to the laminae; but from the degree of heat which I used, and which drove off the water of crystallization from part of the specimen, I pre- sume that the complete phenomenon cannot be developed without destroying the constitution of the crystal; that is, that after the two systems of rings have opened out in a new plane, they will not return, by cooling, through their state of union, into their primitive inclination of 60° in the plane of the lamina.
"A property of a similar kind, but perhaps a still more extraordinary one, I discovered some years ago, subsequent to Professor Mitscherlich's discovery; and I have slightly noticed it in a paper on glauberite, published in the Edin- burgh Transactions. This interesting mineral has at or- dinary temperatures the curious property of two axes of double refraction for red light, and only one axis for violet light. If we apply heat to it, the two optic axes for red light gradually close, and, at a temperature which the hand can endure, the two systems of rings for red light have united into one system, so that the crystal has now only one axis of double refraction for red light. By continuing to increase the heat, the two axes separated, and the single system of rings opened out into two systems, lying in a plane at right angles to that in which they were placed at first. The heat was now less than that of boiling water. By increasing it, the inclination of the optic axes gradu- ally increased.
"I now applied artificial cold to a crystal of glauberite at the ordinary temperature of the atmosphere. The in- clination of the optic axes for red light increased, as might have been predicted; but, what was very unexpected, a new axis was created for violet light, the plane of the two violet axes being coincident with the plane of the two red optic axes at and below the ordinary temperature. An in- crease of cold increased the inclination of the optic axes for all the colours of the spectrum; the inclination of the axes being least for the most refrangible, and greatest for the least refrangible rays.
"These results appear very complicated when we begin with the effects at an ordinary temperature, and view them in the manner in which they were observed; but if we commence the experiments at a low temperature, such as the freezing point, the order and connection of the pheno- mena will be more easily understood.
"At 32° glauberite has two axes of double refraction for rays of all colours, the inclination of the axes for the violet rays being least, and that for the red the greatest. As the temperature rises, the optic axes for all colours gra- dually approach, and the axes for violet first unite into one. At this time the crystal has two axes for all the other colours; but as the heat increases, all the other pairs of axes unite in succession, and form a single system of rings. But before this has taken place, the axes for violet rays have opened up again in a plane at right angles to that in which they originally lay, and they are followed by all the other pairs of axes; so that at a temperature much below that of boiling water, each pair of axes ap- pears with different inclinations arranged in a new direc- tion.
"During all the changes which have been described above, the crystal has preserved its constitution; and by abstracting the heat, the phenomena are all repeated in an inverse order.
"If the crystal should happen to be observed at that temperature, which very often occurs when the greenish- yellow or most luminous rays have the optic axes corre-
1 Edin. Trans. vol. xi. part ii. page 273. 2 This interesting paper was communicated by the author to Sir David Brewster, and published in the London and Edinburgh Philosophical Magazine, December 1832, vol. i. p. 499. author concluded, that the refractive power of calcareous spar for the ordinary ray either does not change at all with the temperature, or decreases with it by a quantity extremely small. The index of refraction of F' at the ordinary temperature was 1·66802.
b. Extraordinary Ray. By the difference of temperature of 64°, the deviation was increased 2' 25", or 2' 34" when corrected, which gives for the index of refraction,
\[ \frac{1}{1-49118} \text{at } 64° \] \[ \frac{1}{1-49075} \text{at ordinary temperature} \]
0·00043 increase.
Variation of the deviation
Variation of refracting angle
When corrected for the deviation of the plate of mica, they became
Variation of deviation
Variation of angle
Hence he obtained the following indices of refraction:
At ordinary temperature
At increased temperature
Changes on index
Hence, says our author, the double refraction of arragonite appears to decrease a little with the temperature, because the refracting power in the direction of the axis A has diminished in a smaller ratio than that along the axes B and C. In other respects arragonite comports itself quite differently from calcareous spar. The axis A of arragonite obviously corresponds with the axis of crystallization of the spar; but notwithstanding this, the refractive power in this direction diminishes in the former, and, on the contrary, increases in the latter, besides, that in the direction perpendicular to the axis A, the refractive power diminishes considerably in arragonite, whilst, on the contrary, it undergoes almost no change in calcareous spar.
While heat and pressure thus modify the doubly-refracting structure in minerals, they are capable of creating it with regular axes in several soft substances. This effect is quite different, as we shall soon see, from that which is produced upon bodies by pressure, where the result is modified by the external form of the body, and where the double refraction disappears when the heat or pressure is removed, or when the body is subdivided. A permanent change is induced upon the soft solids in question, and, when subdivided, each part of the mass or plate preserves the property communicated to it. Sir David Brewster described, in the Philosophical Transactions for 1815, the original experiment which he made on this subject with a mixture of rosin and white wax; but in the same work for 1830, he has given a detailed account of his experiments, and of the conclusions to which they lead respecting the origin of the doubly-refracting structure. The following is the fundamental experiment described by our author.
"I took a few drops of the melted compound (rosin and bees' wax), and placed them in succession on a plate of thick glass, so as to form a large drop. Before it was cold I laid above the drop a circular piece of glass about two thirds of an inch in diameter, and, by a strong vertical pressure on the centre of the piece of glass, I squeezed out the drop into a thin plate. This plate was now almost perfectly transparent, as if the pressure had brought the particles of the substance into optical contact.
"If we expose this plate to polarised light, we shall find that it possesses one positive axis of double refraction, and exhibits the polarised tints as perfectly as many crystals of the mineral kingdom. The structure thus communicated to the soft film by pressure does not belong to it as a whole, nor has it only one axis passing through its centre, like a circular piece of unannealed glass. In every point of it there is an axis of double refraction perpendicular to the plates, and the doubly-refracting force varies with the inclination of the incident ray to this axis, as in all regular uniaxial crystals.
"When the two plates of glass are drawn asunder, we can remove one or more portions of the compressed plate, and these portions act upon light exactly like plates of uniaxial mica or hydrate of magnesia, and develop a doubly-refracting force of nearly equal intensity."
By reasoning from this experiment, our author is led to cause of the opinion that double refraction is acquired by the particles of bodies at the instant of their aggregation, and arises from the pressures produced in the direction of three rectangular axes, by the forces of aggregation. When these forces are very weak, double refraction will not be produced; when they are sufficiently strong and of equal intensity, they will produce tessular crystals; when they are equal in two rectangular directions, they will produce uniaxial crystals; and when they are unequal in all the three directions, they will form biaxial ones. In this way all the phenomena of cleavage may be readily explained.
Upon some substances heat performs the same part as pressure; but our limits will not permit us to detail our author's experiments on this subject.
Sect. XII.—On the Deviation of the Polarised Tints from those of Newton's Scale.
In all his investigations respecting the colours of thin plates, M. Biot happened to use only such crystals as Polarisation.
gave polarised tints similar to those of Newton's scale, and he therefore considered this to be their character. In 1813, however, when Sir David Brewster described the rings in topaz, he gave a list of all the colours in different azimuths round the optic axes, and he not only found these colours to vary in these different azimuths in the same ring, but observed some colours at the extremity of the optic axes. In his paper on the laws of polarisation, published in the Philosophical Transactions for 1818, he remarks, that "in almost all crystals with two axes, the tints in the neighbourhood of the resultant axes, when the plate has a considerable thickness, lose their resemblance to those in Newton's scale, as will be more minutely described in another paper." In April 1817 he communicated to the Royal Society of Edinburgh an account of the extraordinary system of rings in the apophyllite from the Tyrol, which consisted of purple and yellow light, like the residual colours arising from the combination of irrational spectra.
In examining the colours of the polarised rings in biaxal crystals, he was led to divide them into two classes, viz. 1. Those that had the red ends of the rings inwards, or between the resultant axes, and the blue ends outwards. 2. Those that had the red ends of the rings outwards, and the blue ends of the rings inwards.
The crystals in which the deviation is very striking are given in the following table.
**Class 1.—Red ends inwards.**
| Nitre. | Carbonate of lead. | |--------|-------------------| | Sulphate of barytes. | Sulphate-bicarbonate of lead. | | —— of strontites. | Hyposulphate of strontia. | | Tartrate of potash and soda. | (Herschel). | | Phosphate of soda. | Tartrate of potash. | | Arragonite. |
**Class 2.—Red ends outwards.**
| Topaz. | Native borax. | | Mica. | Sulphate of magnesia. | | Anhydrite. | Arseniate of soda. |
**Unclassed.**
| Chromate of lead. | Superoxalate of potash. | | Muriate of mercury. | Oxalic acid. | | —— of copper. | Sulphate of iron. | | Oxynitrate of silver. | Cymophane. | | Sugar. | Felspar. | | Crystallized Cheltenham salts. | Benzoic acid. | | Nitrate of mercury. | Chromic acid. | | —— of zinc. | Nadelstein. | | —— of lime. | Hyposulphate of soda (Herschel). |
Sir John Herschel found anomalous tints in hyposulphate of lime, and Vesuvian, both crystals with one axis, a circumstance which we ascribe to that axis being the resultant of three axes, two of which are equal.
In examining the rings formed by biaxial crystals, Sir David Brewster found that the black spot at the point of compensation was not in the centre of the rings, and the position of this spot for topaz is given in his table of these colours.¹
It is to Sir John Herschel, however, that we owe the complete investigation of this subject in reference to biaxal crystals. By using homogeneous light, he found that the angle of the resultant axes POP, fig. 129, was different for the different colours of the spectrum, varying, in the case of tartrate of potash and soda, from 75° 42' in red light, to 55° 14' in violet light, so that with white light we have a system of rings, consisting of five rings of all colours, overlapping each other, and these five constituting an irregular system, unlike those produced by ordinary crystals.
In crystals where the displacement of the rings is great, the oval central spots seen in Plate CCCLXXXIII. figs. 32, 33, and 34, are drawn out, as Sir John Herschel remarked, into long spectra or tails of red, green, and violet light, and the extremities of the rings are distorted and highly coloured, as in fig. 51, Plate CCCLXXXIV. When we view these spectra with coloured media, they are found to consist of well-defined spots of the several simple colours, arranged on each side of the principal section, as shown in fig. 52.
These results are capable of being rigorously calculated by the law of resultant axes given by Sir David Brewster, and may be considered as a proof of that law. If this were not the case, tartrate of potash and soda would have two axes for every different ray of the spectrum, and four series of poles extending each over a space of ten degrees.
In order to show how these phenomena may be calculated by two axes, let O and A, fig. 153, be two negative axes, which in red light compensate each other at F and F; then, if O and A had the same proportional action on the violet and other rays as on the red rays, F would also be the point of compensation for the violet and other rays. In this case F would be the centre of all the systems of rings, as in uniaxal crystals, and the tints those of Newton's scale. But if the axis O has a greater proportional action upon the violet and other rays than A, the point of compensation will be at f, which will be the centre of the violet system of rings, the centres of all the other systems being between F and f if the action of O upon them is of an intermediate nature. This is the case with all the crystals in Class 1 of the foregoing table. On the other hand, if O has a less proportional action on the violet than on the red, c, c will be the points of compensation for the violet rays, and the centres of the two systems of violet rings.
The most remarkable instances of deviated tints are those in apophyllite, a crystal with one axis, discovered by Sir David Brewster. Sir John Herschel, in examining a number of apophyllites, found that some specimens exercise negative action upon the rays at one end of the spectrum, a positive action upon rays at the other end, and no action at all upon the mean refrangible rays, the doubly refracting action ceasing, in the one case, in the yellow rays, and in another in the indigo.² In other specimens, the diameter of the rings was nearly the same for all the colours of the spectrum, and hence the rings were approaching to a series of black and white ones. All these phenomena may be separately calculated by the law of resultant axes already mentioned, on the supposition that apophyllite has three rectangular axes of double refraction. Sir David Brewster had discovered, in the tesselated apophyllite, portions which had two axes co-existing with portions that had one axis; and in his coloured drawings of the phenomena exhibited by this mineral, he has pointed out a most extraordinary law of symmetry which regulates its varying double refraction; and as he had shown that a double dispersive power existed in the same crystal, the following explanation of the remarkable phenomena of apophyllite approaches to the character of demonstration.
Let O be the positive axis of uniaxial apophyllite, and let A and B be two positive axes which, if equal, would produce a negative axis at O. But as the real axis at O is a positive one, the apparent or finally resultant axis at
¹ Phil. Trans. 1814, p. 204, 205. ² Cambridge Trans. vol. i. p. 21-24. O will be a single axis, negative if the negative is the strongest, and positive if the positive is the strongest. Let us now suppose that the two axes at O have equal intensity, viz. \(+O = -O\) for yellow light \((-O\) being the resultant of \(+A\) and \(+B\)), and that \(-O\) acts more powerfully upon the red rays than \(+O\), while \(+O\) acts more powerfully upon the violet rays. In this case the two axes \(+O, -O\) will exactly compensate each other. In yellow light, a yellow ray will experience neither double refraction nor polarisation; whereas in red light, the predominance of \(-O\) will leave a single negative axis for red rays, and produce a negative system of rings; and in violet light the predominance of \(+O\) will leave a single positive axis of double refraction for violet rays, and consequently a positive system of rings. This compensation resembles that of a compound lens, consisting of a convex and concave lens of equal curvature, of such a glass that their refractive index for yellow light is equal, while the index of refraction for the violet rays is greater in the convex lens, and the index for the red rays greater in the concave lens. Such a lens will converge the violet rays, diverge the red rays, and produce no deviation at all in the yellow ones; that is, the same compound lens will be a plane lens in yellow light, a convex one in blue light, and a concave one in red light. Hence each order of colours in apophyllite is as it were a secondary or residual spectrum arising from the opposite action of unequal negative and positive axes.
From the fact of some apophyllites exercising a negative action, Sir David Brewster stated his expectation that apophyllites might be found in which the double refraction is negative for all the rays of the spectrum; and several years afterwards he discovered the remarkable mineral of oxalurate, which is an apophyllite with this property.\(^1\)
The phenomena of glauberite already described afford an additional illustration of these views.
**Sect. XIII.—On Crystals with Planes of Double Refraction exemplified in Analcime.**
Analcime or Cubizite, a mineral which has been ranked among the cubical crystals, might have been expected to have had no double refraction if this had been its form. It was found, however, by Sir David Brewster, to be a singular body in its action upon light, and to exhibit the extraordinary property of many planes of double refraction, or planes to which the doubly-refracting structure was related in the same manner as it is to one or two axes in other minerals.
Analcime crystallizes most commonly in the form of the icositetraedron, as in fig. 53, Plate CCCLXXXIV. If we suppose a complete crystal of it to be exposed to polarised light, it will give the remarkable figure shown in fig. 53, where the dark shaded lines are planes in which there is neither double refraction nor polarisation, the double refraction and the tints commencing at these planes, and reaching their maximum in the centre of the space enclosed by three of the dark lines. The tints are those of Newton's scale, and are negative in relation to each of the four axes of the icositetraedron. When light is transmitted through any pair of the four planes which are adjacent to any of the three axes of the solid, it is doubly refracted, the least refracted image being the extraordinary one, and consequently the double refraction negative in relation to the axes to which the doubly-refracted ray is perpendicular.
If we suppose the crystal to have the form of a cube, the planes of double refraction will be, as in fig. 54, a plane passing through the two diagonals of each face of the cube.
The tints vary as the square of the distance from the nearest plane of double refraction.
The tints shown in figs. 53 and 54 cannot of course be seen at the same time, but are deduced from observations made by transmitting polarised light in every direction through the crystal.
**Sect. XIV.—On the Double Refraction and Polarisation of Composite Crystals.**
In all the crystallized bodies whose action upon light we have been considering, excepting analcime, the phenomena are identical in all parallel directions, the smallest fragment having the same property as the largest, from whatever part of the crystal it is taken.
In the mineral world, however, and among the products of artificial crystallization, there occur crystals which are composed of several individual crystals whose axes are not parallel. These crystals sometimes occur in such regular symmetrical forms, that mineralogists have long regarded them as simple forms; and it is probable that they would have still been viewed in this light, if they had not been exposed to the scrutiny of polarised light.
One of the most remarkable of these composite crystals is Iceland spar, some specimens of which were observed, even by Bartholinus and Huygens, to exhibit phenomena quite different from those which we have already described. The property which they possessed of multiplying the images of a luminous body seen through them had been carefully studied by Benjamin Martin, Professor Robison, Lord Brougham, Malus, and others; but until the publication of Malus' work on Double Refraction, no sound explanation was given of these remarkable phenomena.
Malus describes the phenomena as produced by fissures parallel to the surface of the variety of this mineral described by Haüy under the name of chaux carbonatée équatoriale. He explains the duplication of the images on the supposition that there is a fissure or real opening between the conjoined faces of the spar, and he ascribes the varying tints to a cause not adequate to the production of such splendid phenomena,—to the colouring of the thin plate of air included in the fissure. He states correctly, that the secondary images are doubled by two fissures, tripled by three, &c.\(^2\)
This class of phenomena was particularly investigated by Sir David Brewster, who found that the fissures described by Malus were thin crystallized laminae of Iceland spar, having their axes of double refraction inclined to that of the portions of the crystal which it separated; that these laminae varied in thickness, the thinnest producing a large system of rings, and the thicker plates smaller systems, the plates being sometimes so thick that no colours whatever appeared. Hence it was obvious that each crystal of this kind was a polarising and an analysing apparatus, the thin laminae being the plate which exhibited its polarised tints in this singular position.
In order to understand this remarkable structure, we have represented the laminae in fig. 155 by the planes ABCD, efgh, afhd parallel to the edges EGFH, and also to the long diagonals of the rhomboidal faces, or perpendicular to the short diagonal EF. When we look through a crystal with only one of these laminae, we observe the two principal images of the candle A, B, or luminous
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\(^1\) Edinb. Journal of Science, No. xiii. p. 115.
\(^2\) Théorie de la Double Refraction, p. 194–199. Polarisation, while at a vertical incidence, and separated just as they would have appeared in a common crystal of the same thickness. But on each side of this double image is a single-polarised image C and D, C being polarised in the same plane as B, and D in the same plane as A. Let us suppose that these phenomena are seen through a rhomb with only one plane ABCD (fig. 155), and through the faces EACD, BEHC. Then, if we incline the rhomb in different directions slightly, we shall see the images C, D disappear when they have a certain distance from A, B. If we incline the rhomb, bringing EA nearer the eye than CD, the images C, D will approach to AB, and will be found to disappear nearer and nearer to AB as the inclination is increased, it being necessary to bring the edge EG nearer the eye than AD, to make C and D disappear. While C and D are thus approaching to A, B, they become less and less coloured till they are all white. If we incline the rhomb in an opposite direction, so that GD is brought nearer the eye than EA, the images C, D recede from A, B, and become more highly coloured, the two images A, B becoming also coloured. The images C, D sometimes appear doubled when this inclination is going on, but it is only a duplicity of colour, so to speak, in consequence of the spectrum being divided by portions of it passing into the reflected pencil. When we bring EC nearer the eye than AD, the colours increase, A and B become also coloured, and an apparent colorific duplication of the images C and D takes place. If the rhomb is inclined in an opposite direction, so that AD is brought nearer the eye than EC, the images C, D become also at first more coloured; but by increasing the inclination, the image C recedes rapidly on the right side from A, B, contracts in breadth, and becomes prismatically coloured, the spectrum which it exhibits being subdivided by several black lines or bands, the parts of the spectrum corresponding to these black lines or dark bands having passed into the reflected ray. The spectrum D recedes as rapidly to the left, expanding in breadth, and even disappearing, as well as the images A, B.
All these phenomena are more finely seen, and the law of their changes more easily detected, if, instead of a candle, we look at a long line of light, such as the narrow opening between the edges of the window-shutters.
If we look through the faces ADFH, ADEG of the rhomb, placing a prism of glass with an angle of 12° or 13° upon one of the faces, to permit the refracted rays to emerge at a moderate angle of deviation, the prismatic images formerly described will be large spectra, subdivided by black spaces into 4, 5, 6, &c. coloured images of the candle, or of the long luminous line, exhibiting one of the most magnificent phenomena that can be witnessed.
These phenomena vary, of course, with the thickness of the enclosed laminae, and as the laminae increase in thickness the subdivisions of the spectral images become more numerous.
When we reflect light from these laminae, ABCD, for example, by allowing the light to enter by the face BFHC, and emerge through the face AFHD, the boundary of total reflection is marked by a series of brilliant rectilineal fringes, polarised in the same manner as the image C, which is now the lowermost. When the light is transmitted through the laminae at the boundary of total reflection, by entering through the face BECG, and emerging through AFHD, a series of rectilineal fringes complementary to the former reflected series is seen. They also are polarised in the same plane as C, or the lowermost secondary image, and become much more distinct, by causing the oppositely polarised pencil to disappear.
The structure which produces the preceding phenomena, and the duplication of the images, will be understood from fig. 157, where ABCD is the principal section of a crystal of this kind of Iceland spar, having AD for its axis. One of the laminae oppositely crystallized is shown at MM, NN, but much thicker than they are generally, the angles AMM, DAN being 14° 44'. A ray of ordinary light Rb will be refracted in the lines bc, cd. These rays entering the laminae MN, will be again refracted doubly; but as the vein is so thin as to produce the system of uniaxial rings, the colours will vary with the thickness of the film and the inclination of the ray to the axis of the lamina. The four pencils will emerge from the lamina at e, f, and will be refracted again, as in the figure, into the pencils cm, cn, fo, fp, the colours of cm, fn being complementary to those of cm, fp.
That the multiplication and colour of the images are produced by the causes above explained, has been proved by Sir David Brewster, by actually placing laminae of different crystals between the prisms AN, BN. In this way, by introducing different films in different azimuths, most beautiful combinations may be produced.
If we grind down the angles A and B, so as to have two faces perpendicular to the axis AB, the uniaxial system of rings is beautifully modified by the action of the lamina MN; and that this was the cause of the singular transformations which the rings experienced in different crystals, was proved by the author above quoted, by inserting laminae between two plates of the chlorite carbonatite basis of Haiyi, whose natural faces are perpendicular to the axis. These transformations are exceedingly beautiful. Some of them are shown in figs. 55, 56, and 57; one of the rings consisting of eight dark radii, while the complementary system has its inner circle marked with eight dark-coloured spots. These rings suffer beautiful changes, both by the motion of the plane round its axes when the analysing plate is stationary, and by the motion of the analysing plate when the rhomb is stationary. In studying these phenomena in a great variety of crystals intersected with one or more laminae, Sir David Brewster noticed a very remarkable fact. A rhomb of spar which produced in one part of it the transformed systems already mentioned, exhibited a singular effect in another part, where the crystallization appeared perfect and simple, and where there were decidedly no veins or laminae. In one position or azimuth of this crystal, this portion gave, as might have been expected, the regular system of rings with the black cross shown in fig. 29. But upon turning it round 45°, all the rings became elliptical, as shown at ABCD in fig. 58, the first order of colours in one quadrant having joined the second order of colours in the adjacent quadrant. The arms of the black cross took the contorted position c, d, f, the continuations of it afterwards, viz. am, cm, dm, fm, being so very faint as to show the continuity of the elliptical rings. In this figure the rings of the same order are marked with the same figure. The very same phenomenon, which points to important theoretical consequences, was observed by the same author in another rhomb of spar wholly without veins, but the rings were not so elliptical as in fig. 58.
This composite structure was discovered by Sir David Brewster in various minerals, and he has described it very minutely in the case of Brazilian topaz, sulphate of potash, and apophyllite.
Sir John Herschel, we believe, first noticed this structure in Brazilian topaz, and observed that the central portion of the crystal had a different colour from the external portion, and that the plane of the principal section of the different parts made angles of $20^\circ$. Sir David Brewster found very remarkable arrangements of the coloured portions, which he has represented in coloured drawings in the Cambridge Transactions, vol. ii. In some of these crystals the structure was tesselated, as in fig. 158, where ABED, CBEF, are the two external tessellae, at one of the obtuse angles of the rhomboidal section. If we suppose
that these tessellae are divided into four laminae, 1, 2, 3, 4, and that MN is the principal section, or one of the neutral axes of the central portion of the crystal contiguous to DEF; then the laminae 1, I have their principal section in the direction ac', forming a very small angle with MN; the laminae 2, 2 have their principal section in the line bb', and so on to the superficial laminae 4, 4, which have their principal section in the direction dd', inclined from $10^\circ$ to $22^\circ$ to MN, the inclination varying in different crystals. The lines ac', bb', &c., are also the principal sections of the corresponding laminae on the side NC. In like manner, the principal sections ac', bb', &c., of the laminae in BCFE, are the principal sections of the corresponding laminae on the other side AN. As the laminae, however, are infinite in number, the principal sections have every possible division between dd' and bb'.
The bipyramidal sulphate of potash, which Count Bourdon supposed to be a simple crystal, was found by Sir David Brewster to be a tesselated crystal, composed of three pair of crystals of the prismatic sulphate of potash combined so that each pair had their principal axis parallel. When exposed to polarised light, each pair gave the system of biaxial rings, and when held at a distance from the eye, had the tesselated appearance shown in fig. 59, Plate CCCLXXXIV., each opposite pair of the triangles having the same tint.
The most remarkable of this class of minerals, and indeed the most remarkable body in the whole mineral kingdom, is the tesselated apophyllite. It crystallises most commonly in four-sided rectangular prisms, like CD, fig. 60. If we remove the uppermost slice A and the undermost B to the thickness of between the 50th and the 100th of an inch, and examine it either by the microscope or by polarised light, we shall find that it is like other uniaxal plates, giving a single system of rays having the very peculiar colours which have been already described. A number of veins appear at the edges, as shown in the figure. All the other slices lying below this exhibit the beautiful tessellated figure shown in fig. 61. The outer case MNOP, which, as it were, binds together the internal portions, consists of a great number of parallel veins or plates, which give the colours of grooved surfaces. This frame encloses nine different crystals, namely, the central lozenge abed, the four prisms A, B, C, D with trapezoidal bases, and the four triangular prisms ehl, inn, nkg, qfe, all of which are separated by distinct lines or veins, which are nearly all visible by the microscope by a proper method of illumination. In polarised light they are all seen with great facility.
The most extraordinary fact connected with this structure is, that the central lozenge has only one axis of double refraction, like the terminal plates A, B, fig. 60, while the four prisms A, B, C, D, and the four triangular spaces, have two axes. In A and D the planes of the resultant axes are coincident, as in the opposite triangles of sulphate of potash, and lie in the direction MN, while the planes of the resultant axes in B and C lie in the direction OP.
When the plate MONP is exposed to polarised light and turned round its axis before the analysing plate, the lozenge abed will be dark in every position of the plate, while the portions A, B, C, D will depolarise the light, or be luminous, when MO or ON are parallel or perpendicular to the plane of primitive polarisation.
Remarkable as is the structure which we have now described, it is greatly excelled in beauty by another variety of Faroe apophyllite, in which Sir David Brewster discovered the most extraordinary organization. He has given an enlarged coloured drawing of the fine symmetrical tints which it exhibits in polarised light, in the Edinburgh Transactions; but we hope its structure may be understood by the following description which he has given of it. The crystals have a greenish-white tinge, and are aggregated together in masses. The quadrangular prisms are in general below one twelfth of an inch in width; they are always unpolished on their terminal planes; they have the angles at the summit more deeply truncated than the other quadrangular prisms from Faroe; they are always perfectly transparent; and may sometimes be detached in a complete state, with both their terminal summits.
"In examining this variety of apophyllite, I was enabled, by the perfection of the crystals, to study their structure through the natural planes, and at right angles to their axes. The phenomena which this investigation presented to me were of a very singular and unexpected nature. In symmetry of form and splendour of colouring they far surpassed any of the optical arrangements that I had seen, while they developed a singular complexity of structure, and indicated the existence of new laws of mineral organization.
"When a complete crystal of this variety of apophyllite is exposed to polarised light, with its axes inclined $45^\circ$ to the plane of primitive polarisation, and is subsequently examined with an analysing prism, it exhibits, through both its pair of parallel planes, the appearance shown in fig. 62. In turning the crystal round the polarised ray, all the tints vanish, re-appear, and reach their maximum at the same time, so that they are not the result of any hemitropism, but arise wholly from a symmetrical combination of elementary crystals possessing different primitive forms and different refractive and polarising powers. The difference in the polarising powers is well shown by the variation of tint; and the difference of refractive power may be observed with equal distinctness by examining the..." crystal with the microscope under favourable circumstances of illumination, when the outlines of the symmetrical forms shown in fig. 62 will be clearly visible.
"In examining the splendid arrangement of tints exhibited in the figure, the perfect symmetry which appears in all its parts is particularly remarkable. The existence of the curvilinear solid in the centre; the gradual diminution in the length of the circumscribing plates, in consequence of which they taper, as it were, from the angles of the central rectangle to the truncated angles at the summits; but, above all, the reproduction of similar tints on each side of the central figure, and at equal distances from it, cannot fail to strike the observer with surprise and admiration.
"The tints exhibited by each crystal vary, of course, according to its thickness; but the range of tint in the same plate, and at the same thickness, generally amounts in the largest crystals to three of the orders of colours in Newton's scale. The central portion, and the two squares above and below it, have in general the same intensity, while the four segments round the central portion, and some of the parts beyond each of the squares, are also isochromatic. In the central part the colours have a decided termination; but towards the summit of the prism their outline is less regular, and less distinctly marked; though this irregularity has also its counterpart at the other termination. A part of these irregularities is sometimes owing to the longitudinal striae on the natural faces of the crystal, so that by carefully grinding these off, the beauty and regularity of the figure is greatly improved.
"In order to ascertain the order of the colours polarised by the crystal, and observe in what manner they passed into one another, I transmitted the polarised light in a direction parallel to one of the diagonals of the quadrangular prism, and thus obtained, as it were, a section of the different orders of colours from the zero of their scale. The result of this experiment, which is shown in fig. 63, was highly interesting, as it displayed to the eye not only the law according to which the intensity of the polarising forces varied in different parts of the crystal, but also the variation in the nature of the tints, and the connection between these two classes of phenomena. At the points in the diagonal mn, opposite to ba of the crystal, the tints rose to the seventh order of colours; and in other two places opposite to cd, they were only to the sixth; while near the summits at mn, they descended so low as the fourth order. Hence it follows that the four curvilinear segments, fig. 62, are next to these in intensity; that the central portions of the squares are again inferior to these; and that the weakest polarising force is near the summit of the prisms. At ab, the fourth, fifth, and sixth fringes have a singularly serrated outline, exhibiting, in a very interesting manner, the sudden variations which take place in the polarising forces of the successive laminae.
"Having thus described the structure and properties of the tessellated apophyllite, it becomes interesting to inquire how far such a combination of structures is compatible with the admitted laws of crystallography. The growth of a crystal, in virtue of the aggregation of minute particles endowed with polarity, and possessing certain primitive forms, is easily comprehended, whether we suppose the particles to exist in a state of igneous fluidity or aqueous solution. But it is a necessary consequence of this process, that the same law presides at the formation of every part of it, and that the crystal is homogeneous throughout, possessing the same mechanical and physical properties in all parallel directions.
"The tessellated apophyllite, however, could not have been formed by this process. It resembles more a work of art, in which the artist has varied, not only the materials, but the laws of their combination.
"A foundation appears to be first laid by means of an uniform homogeneous plate, the primitive form of which is pyramidal. A central pillar, whose section is a rectangular lozenge, then rises perpendicularly from the base, and consists of similar particles. Round this pillar are placed new materials, in the form of four trapezoidal solids, the primitive form of whose particles is prismatic; and in these solids the lines of similar properties are at right angles to each other. The crystal is then made quadrangular by the application of four triangular prisms of unusual acuteness. The nine solids arranged in this symmetrical manner, and joined by transparent veins performing the functions of cement, are then surrounded by a wall composed of numerous films, deposited in succession, and the whole of this singular assemblage is finally roofed in by a plate exactly similar to that which formed its foundation.
"The second variety of the tessellated apophyllite is still more complicated. Possessing the different combinations of the one which has just been described, it displays, in the direction of the length of the prism, an organization of the most singular kind. Forms unknown in crystallography occupy its central portion; and on each side of it particles of similar properties take their place at similar distances, now forming a zone of uniform polarising force, now another increasing to a maximum, and now a third descending in the scale by regular gradations. The boundaries of these corresponding though distant zones are marked with the greatest precision, and all their parts as nicely adjusted as if some skilful workman had selected the materials, measured the spaces they were to occupy, and finally combined them into the finest specimen of natural mosaic.
"The irregularities of crystallization which are known by the name of macle or hemitrope forms, and those compound groups which arise from the mutual penetration of crystals, are merely accidental deviations from particular laws which govern the crystallizations in which they occur. The aberrations themselves testify the predominance of the laws to which they form exceptions, and they are susceptible of explanation by assuming certain polarities in the integrant molecules. The compound structure of the apophyllite, however, cannot be referred to these capricious formations. It is itself the result of a general law, to which there are no exceptions, and when more deeply studied and better understood, it must ultimately lead to the introduction of some new principle of organization, of which crystallographers have at present no conception.
"The difficulty of accounting for the formation of apophyllite is in no way diminished by giving the utmost license to speculation. We cannot even avail ourselves of the extravagant supposition of a crystalline embryo, which, like that of animal and vegetable life, gradually expands to maturity. The germ of plants and animals is nourished by a series of organs, of which, however recondite be the operation, we yet see the action and witness the effects; but in the architecture of apophyllite no subsidiary organs are seen. The crystal appears only in its state of perfection; and we are left to admire the skill which presided at its formation, and to profit by the instruction which is so impressively conveyed by such mysterious organization."
We have represented in figs. 66 and 67 the figure produced by polarised light by an internal slice of the barrel or cylindrical apophyllite from Kudlisæt, in Disco Island,
Edinburgh Transactions, vol. ix. p. 323. brought home by Sir Charles Giessècke. The figures are from different specimens. The shaded part of them has only one axis of double refraction, while the four sectors have two axes, the luminous sectors being analogous to the prisms A, B, C, D, and the dark figure to the central lozenge abed, in fig. 61. The mechanical structure of the cleavage planes resembles the optical figure even after the planes are ground.
Sect. XV.—On the Absorption of Light by Uncrystallized Bodies.
When a beam of light passes through the most transparent media, such as air and water, a certain portion of it is lost. This loss of light is particularly apparent in such bodies when a beam of light has traversed a great thickness of the gas or the fluid. This loss of light has been called absorption, and the light lost is said to be absorbed; a term which we use at present as merely expressing a fact.
There are two kinds of absorption which may be noticed. 1. That in which all the rays of the spectrum are proportionally absorbed or lost; and 2. that in which different quantities of the differently coloured rays are lost. Those bodies in which the first kind of absorption takes place are colourless, and those in which the second kind takes place are coloured. In black ink, for example, the transmitted light of the sun is white. In red ink it is red, more of the most refrangible rays of the spectrum being lost than of the least refrangible ones.
When a beam of the sun's light falls upon a piece of charcoal, the light is almost wholly lost or absorbed. Sir Isaac Newton thought that the light was reflected or refracted "to and fro" within such bodies till it was lost; but still the question meets us, why is it lost? If it is scattered in all directions it must emerge again from the charcoal, and be visible in some way or other. In order to meet this and other difficulties, the light is supposed to be detained within the body, and somehow or other united to its substance.
In the case of red ink and similar bodies Sir Isaac Newton conceived that the blue rays which were lost were reflected by the particles of the ink, while the red rays were transmitted, as in the colours of thin plates; but as we cannot by any process see these blue rays, we can only say that they are lost, and the cause of their loss is as difficult to be found as in the phenomena of imperfect colourless transparency.
The following are the general phenomena of coloured absorptions in transparent bodies.
1. Red transparent solids or fluids absorb, generally speaking, the blue end of the spectrum. 2. Blue substances absorb, generally speaking, the red end of the spectrum. 3. Green bodies absorb both the blue and the red ends of the spectrum. 4. Yellow bodies absorb the blue and part of the green of the spectrum.
But when we examine more narrowly the action of coloured bodies on the spectrum, we find that a body may derive its peculiar tint from absorbing two, three, four, up to many hundred separate parts of the spectrum, so that the colour of such a body is the combination of all the parts of the spectrum which are not absorbed. We may infer, however, from the general tint of the body, what parts of the spectrum it has chiefly absorbed. Red nitrous gas, for example, must have acted most powerfully upon the blue end of the spectrum, as we have already seen that it does.
Three theories of absorption have been recently published, namely, by Sir John Herschel, Baron Wrede, and M. Herschel Lamé, all of which are founded upon the undulatory theory. Sir John Herschel conceives that light may be lost within bodies by the interference of different parts of a ray, which, after taking two routes of different lengths, meet again in a condition to interfere. This hypothesis has been modified by Mr Whewell in the following manner: "If we conceive," says he, "with Sir John Herschel, a medium which will not transmit vibrations except through certain canals, these canals must have a determinate direction, and therefore such a constitution would give different proportions in different directions. But let a medium consist of certain particles regularly distributed, the intervening space being filled by a medium capable of vibration. Let it be supposed, also, that each vibration, on reaching a medium so disposed, proceeds in part directly and in part by the indirect routes which go round some of the particles and rejoin the direct course. We have thus combinations of ramifying and uniting paths, which, though very complex in each direction, are the same in different directions, in consequence of the regular distribution of the particles. If the distribution, though regular, have a reference to certain axes, as in many crystals, the phenomena of absorption may be different in different directions with regard to these axes.
"In this way the theory of ramifying canals comes to coincide with the theory of vibrations, of which parts are differently retarded, and thus interfere with each other; a theory which has been suggested by other authors."
Baron Wrede supposes the particles of a transparent body placed regularly at equal distances, with the ether diffused between them. When a ray of light is propagated directly through this medium, a portion of it encounters some of the particles, and is reflected backwards, then forwards again, and emerges along with the direct ray, so that the reflected and direct portions will be in a state to interfere and destroy each other. This theory, which is analogous to Newton's, or rather the very same as Newton's, is liable to the objection we have already urged to every theory in which the light is supposed to be decomposed by interior reflections. Now, Baron Wrede's hypothesis may explain all the phenomena, such as dark bands and dark lines, which are known to be produced by thin plates of various thicknesses, as Dr Young has stated, and it may even explain those bands and absorptions which have been recently discovered in decomposed glass by Sir David Brewster, where the effects are clearly produced by a combination of a great number of thin plates, but where the reflected light is as copious as the transmitted light. But we cannot conceive it at all applicable to the cases of nitrous gas (to which he has attempted to apply it), and to solids and fluids, in which all attempts have failed to discover any of the reflected rays.
The hypothesis of M. Lamé, which we have already stated in our history of Optics, is incompatible with the actual phenomena of absorption. In place of the lines and bands depending on the surface of emergence, the system of lines is determined by the action of the first surface of the body, a fact independent of all theory.
Of all the theoretical views which we have mentioned, we consider those given by Sir John Herschel and Mr Whewell as the most consistent with observation. They re-
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1 Edinburgh Transactions, vol. ix. p. 326. 2 Taylor's Scientific Memoirs, part iii. 3 Phil. Trans. 1837, part ii. 4 Report of the Fourth Meeting of the British Association, 1834, p. 551. 5 Elements of Natural Philosophy, vol. i. p. 469. lieve us from the difficulty of accounting for the reflected light; but though they have this in their favour, they are not founded on any direct data, and indeed were brought forward, as Mr Whewell remarks, "to show that there is no incongruity between the undulatory theory and the phenomena of absorption."
An attempt has been recently made by Sir David Brewster to deduce an explanation of absorption, both in uncrystallized and doubly refracting bodies, from direct experiment; but as the details have not been published, we can only at present give a very general notice of it. Having been led to suppose, along with Dr Young,1 that a transparent medium transmits light in two separate portions, one passing through its ultimate particles, and the other through its pores, and that these portions reunite continually after each successive separation, Sir David Brewster endeavoured so to combine a number of thin plates, that the light transmitted through the whole should move with different velocities, the differences being so small as to produce interference. This task he found to be a very difficult one, but he succeeded, after many failures, in finding a doubly-refracting mineral upon which he could impress this mechanical condition.2 When he placed a small aperture upon this crystal, in order to exclude as much as possible all the light which had not experienced any change in its velocity, and looked through it at a well-formed prismatic spectrum, or, what was the same thing, analysed the light transmitted by the plate with a good prism, he was astonished at the sight which was presented to him, not only because it confirmed his views on a subject of such high scientific interest, but from the splendour of the phenomena which it displayed. The spectrum was covered with the sharpest bright and dark lines and bands grouped with all the irregularity of the solar and nitrous gas lines. The bright lines of maximum intensity were variously placed; and in almost every spectrum thus formed there was a narrow band of pure homogeneous white light, incapable of being decomposed by prismatic analysis. But the interest of this result did not stop here. The spectrum of the ordinary differed from that of the extraordinary ray, and the brightest lines had a different locality in each of them. Here then we have the phenomenon of dichroism, of the absorption of light in crystallized and uncrystallized bodies, and of bright and dark lines and bands in the spectrum, and of the unequal absorption of rays of the same refrangibility, but differing in colour, all displayed by an artificially produced combination, in which we know that the phenomena are owing to the interference of rays differently retarded. It can therefore scarcely be doubted that the varied phenomena of absorption arise from the differences of velocity in the rays which pass through the ultimate particles and through the pores of bodies. The constitution of the body, or the arrangement of the ultimate particles in the first film on which the light is incident, determines the nature of the absorptive action which the body exercises at different thicknesses.
Sect. XVI.—On Dichroism, and the Absorption of Common and Polarized Light by Doubly Refracting Crystals.
The name of dichroism or double colours, was given very appropriately, by M. Cordier, we believe, to a mineral called iolite, which in common light exhibited two different colours in different directions. Dr Wollaston and several mineralogists had observed this double colour in certain crystals of palladium, tourmaline, and other crystals. The origin of this singular property was not known till Sir David Brewster investigated its origin in iolite, and showed that it was connected with the doubly-refracting structure; that it never occurred in the tessular crystals which did not possess double refraction.
The connection of dichroism with double refraction, and its general laws, will be understood from the following observations. In a specimen of yellow Iceland spar the extraordinary image is of an orange-yellow colour, while the ordinary image is yellowish-white. Along the axis of double refraction the colour of the two pencils is exactly the same, and the difference of colour increases with the inclination of the refracted ray to the axis. Hence the difference of colour increases in proportion to the difference of the velocities of the two rays, and is consequently a maximum in the equator of double refraction, and is the same in all parallels; the colour along the axis being the natural colour of the mineral. This is the invariable law of the phenomena in uniaxial crystals. The following are the observations made by the author already referred to.
### Colours of the two Images in Crystals with one Axis
| Names of Crystals | Colour when its Axis is in the Plane of Primitive Polarisation. | Colour when its Axis is Perpendicular to that Plane. | |-------------------|-------------------------------------------------|--------------------------------------------------| | Zircon | Brownish white | A deeper brown | | Sapphire | Yellowish green | Blue | | Ruby | Pale yellow | Bright pink | | Emerald | Yellowish green | Bluish green | | Emerald | Bluish green | Yellowish green | | Beryl blue | Bluish white | Blue | | Beryl green | Whitish | Bluish green | | Beryl yellow green| Pale yellow | Pale green | | Rock crystal, almost transparent | Whitish | Faint brown | | Rock crystal, yellow | Yellowish white | Yellow | | Amethyst | Blue | Pink | | Amethyst | Grayish white | Ruby red | | Amethyst | Reddish yellow | Ruby red | | Tourmaline | Greenish white | Bluish green | | Rubellite | Reddish white | Paint red | | Idocrase | Yellow | Green | | Melilite | Yellow | Bluish white | | Phos. of lime (lilac) | Bluish green | Reddish | | (olive) | Bluish green | Yellowish green | | Phos. of lead | Bright green | Orange yellow | | Calcareous spar | Orange yellow | Yellowish white | | Octohedrite | Whitish brown | Yellowish brown |
Sir John Herschel has found this property beautifully displayed in the sub-oxy sulphate of iron, which crystallizes in six-sided prisms. Along the axis the colour is a deep blood-red, while through the sides of the prism it is of a light green colour. Several tourmalines have also been observed by Sir John to have these same colours along the axis, and at right angles to it. There can be little doubt that this property will be found in every crystal of sufficient thickness that has the property of double refraction. Even if the crystal is colourless, a slight inequality in the intensity of the two images may be observed; and when it is distinctly coloured, the difference of intensity is very easily seen, even when the two colours are not of a different kind.
The phenomena of dichroism are best seen in crystals with two axes of double refraction, and are well exemplified.
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1 Dr Young made this supposition, not to explain absorption, but to account for the dispersion of colours by refraction. Elem. Nat. Phil. vol. ii. p. 937. 2 The same property was afterwards found in several crystallized minerals. sied in iolite, a mineral which crystallizes in six or twelve sided prisms. These prisms are of a deep blue colour when seen along the axis, and of a yellowish brown colour when viewed in a direction perpendicular to the axis.
If abed is a section of the prism of iolite in a plane parallel to the axis of the prism, the transmitted light will be blue through the faces ab and cd, and yellowish-brown through ad, bc, and in every direction perpendicular to the axis of the prism. If we grind down the angles a, c, b, d, so as to replace them with faces mn, m'n', and op, o'p', inclined 31° 41' to cd, or to the axis of the prism; then, if the plane abed passes through the resultant axes of double refraction, we shall observe, by transmitting polarised light through the crystal in the directions ac, bd, and subsequently analysing it, a system of rings round each of these axes. The system will exhibit the individual rings very plainly if the crystal is thin; but if it is thick, we shall observe, when the plane abed is perpendicular to the plane of primitive polarisation, some branches of blue and white light, diverging in the form of a cross from the centre of the system of rings, or the poles of no polarisation, as shown at p and p', fig. 67, where the shaded branches represent the blue ones. The summits of the blue masses at p and p' are tipped with purple, and are separated by whitish light in some specimens, and yellowish light in others. The white light becomes more blue from p and p' to o, where it is quite blue, and more yellow from p and p' to c and d, where it is completely yellow. When the plane abcd is in the plane of primitive polarisation, the poles p, p' are marked by spots of white light, but everywhere else the light is a deep blue.
In the plane eadb, fig. 159, the mineral, when we look through it at common light, exhibits no other colour but yellow, mixed with a small quantity of blue polarised in an opposite plane. The ordinary image at c and d is yellowish brown, and the extraordinary image faint blue, the former receiving some blue rays, and the latter some yellow ones from c, and at a and b, where the difference of colour is still well marked. The yellow image becomes fainter from a and b to p and p', till it changes into blue, and the faint blue image is strengthened by other blue rays, till the intensity of the two blue images is nearly equal. As the incident ray advances from c and d to p and p', the faint blue image becomes more intense, and the yellow one, receiving an accession of blue rays, becomes of a bluish white colour. The ordinary image is whitish from p and p' to o, and the extraordinary a deep blue; but the whiteness gradually diminishes towards o, where they are both almost equally blue.
The principal axis of double refraction in iolite is negative. The most refracted image is purplish blue, and the least refracted one yellowish brown.
The following table shows the colours exhibited by crystals with two axes:
| Names of Crystals | Axis of Prism in the Plane of Primitive Polarisation | Axis of Prism perpendicular to the Plane of Primitive Polarisation | |-------------------|-----------------------------------------------------|---------------------------------------------------------------| | Mica | Blood red | Pale greenish yellow | | Acetate of copper | Blue | Greenish yellow | | Muriate of copper | Greenish white | Blue | | Olivine | Bluish green | Greenish yellow | | Spence | Yellow | Bluish |
In the last eight crystals of the preceding table the tints are not given in relation to any fixed line.
In Wiclhamite, a crystal whose principal axis is negative in relation to the axis of the prism, the dichroism is beautiful, and is exhibited both in common and polarised light. When common light is transmitted through the two parallel faces of the prism, the tint is of a crimson or amethyst colour, with a mixture of straw colour. Upon turning the crystal round, the yellow tint disappears, and the colour becomes a deep crimson-red. On continuing to turn the prism, the colour changes to a straw-yellow, and at the end of half a revolution the crystal resumes its compound tint. In the groups of crystals which have penetrated the quartz, some of them occupy accidentally the position which gives the yellow colour, others that which gives the red colour, and some that which gives the compound tint, so that, without a knowledge of their dichroitic property, the group might have been considered as composed of three different sets of crystals.
The following table contains the characters of the two pencils in crystals, the number of whose axes has not yet been determined.
| Names of Crystals | Axis of Prism in the Plane of Primitive Polarisation | Axis of Prism perpendicular to the Plane of Primitive Polarisation | |-------------------|-----------------------------------------------------|---------------------------------------------------------------| | Phosphate of iron | Fine blue | Bluish white | | Actynolite | Green | Greenish white | | Precious opal | Yellow | Lighter yellow | | Serpentinite | Dark green | Lighter green | | Asbestos | Greenish | Yellowish | | Blue carb. of copper | Violet blue | Greenish blue | | Orpiment | Sulphur yellow | Lighter yellow |
Sir David Brewster found that the dichroism of several crystals is changed by heat, and that in some cases this property may be communicated to them. In several coloured glasses, too, he found an analogous property, when they had received the doubly-refracting structure either temporarily or permanently.
Sect. XVII.—On the Action of the Surfaces of Crystalized Bodies upon Common and Polarised Light
It was remarked by Malus, that the action of the surface of Iceland spar upon light is independent of the position of its principal section, and that its surface acts like that of any common transparent body. In examining,
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1 Edinburgh Journal of Science, April 1825, vol. ii. p. 219. 2 See Phil. Trans. 1819, p. 11; or Edin. Philos. Jour. vol. ii. p. 346. 3 Théorie de la Double Refraction, p. 240, 241; and Biot's Traité de Physique, tom. iv. p. 333. however, the superficial action of this mineral, Sir David Brewster discovered, that all its surfaces, without exception, exercise a remarkable action upon light, and that its polarising angle varied in different azimuths, excepting when the surface was perpendicular to the axis.
If \( A \) and \( A' \) are the minimum and maximum polarising angles, viz. in azimuth 0°, or in the plane of the principal section, and in azimuth 90°, or perpendicular to that plane, he found that the variation of the polarising angle was represented by the following expression, where \( A' \) is the polarising angle required at the azimuth \( a \):
\[ A' = A + \sin^2 a (A'' - A) \]
in a plane perpendicular to the axis, \( A'' - A = 0 \), and consequently no change takes place in the polarising angle; in planes inclined 45° 23' to the axis on the actual faces of the rhomboid, \( A - A = 22° 18' \); and in planes coincident with the axis, \( A'' - A = 4° \) nearly.
The following were the measures which our author obtained on the natural faces of the rhomb:
| Polarising Angle | |------------------| | Azimuth | | 0° | 57° 7' | | 50° | 58° 32' | | 90° | 59° 32' |
On faces nearly parallel to the axes:
| Azimuth | |------------------| | 0° | 54° 18' | | 90° | 58° 14' |
Sir David Brewster also observed that the polarisation was more complete in azimuth 0° than in azimuth 90° on the faces of the rhomb; but more complete in azimuth 90° than in azimuth 0° in faces parallel to the axis.
As these experiments clearly proved that the forces which produced double refraction extended beyond the surface of Iceland spar, our author became desirous of ascertaining if the light polarised by reflection from the spar suffered any change from the same cause. He therefore thought of weakening the force which produces reflection, in order to allow the interior force to show its weaker influence; and he accomplished this by placing oil of cassia on its surface, and examining the light reflected at the separating surface of the spar and the oil. The experiments which were thus made, and which are detailed in the Philosophical Transactions for 1819, completely proved that the interior force polarised common light out of the plane of reflection, and modified the law of intensity, according to which light is reflected at different angles of incidence.
These experiments excited no attention till 1835, when Professor Macculagh, of Trinity College, Dublin, began to investigate the laws which regulate the reflection and refraction of light at the separating surface of two media. He had anticipated from theory effects the reverse of those deduced from the preceding experiments; and in order to account for the latter, he was obliged to modify his theoretical views, and was thus led to the result, that when a ray is polarised by reflection from a doubly-reflecting surface, the plane of polarisation deviates from the plane of incidence, except when the axis lies in the latter plane. The formula which expresses this deviation represents very accurately the measures of the polarising angles in different azimuths in the natural faces of the rhomb, the only surface in which the exception is true; but at all other inclinations of the reflecting planes to the axis, the theory and the formula are in fault, for there is a large deviation when the axis or principal section of the crystal is in the plane of reflection.
Professor Macculagh's success in deducing theoretically the general fact of a deviation increasing as the refractive power of the medium approached to that of the spar, induced Sir David Brewster to resume his inquiries, the general result of which he communicated to the British Association at Bristol in 1836, in the following very brief abstract.
"When light is reflected at the separating surface of two media, the lowermost of which is a doubly-refracting one, the reflected ray is exposed to the action of two forces, one of which is the ordinary reflecting force, and the other a force which emanates from the interior of the doubly-refracting crystal. When the first medium is air, or even water, the first of these forces overpowers the second; and, in general, the effects of the one are so masked by the effects of the other, that I was obliged to use oil of cassia, a fluid of high refractive power, in order that the interior force of the calcareous spar which I wished to examine might exhibit its effects independently of those which arise from ordinary reflection. The separating surface, therefore, which I used, had a small refractive power; and the reflecting pencil is so attenuated, especially in using polarised light, that it is almost impossible to use any other light than that of the sun.
"When a pencil of common light is reflected from the separating surface of oil of cassia and calcareous spar, the general action of the spar is to polarise a part of the ray in a plane perpendicular to that of reflection, and thus to produce by reflection the very same effect that other surfaces do by refraction.
"On the face of calcareous spar, perpendicular to the axis of the crystal, the effect is exactly the same in all azimuths; but in every other face the effect varies in different azimuths, and depends upon the inclination of the face to the axis of double refraction. On the natural face of the rhomb common light is polarised in the plane of reflection, in 0° of azimuth, or in the plane of the principal section; but at 88° of azimuth, the whole pencil is polarised at right angles to the plane of reflection; and in other azimuths the effect is nearly the same as I have stated in my printed paper.
"In order, however, to observe the change which is actually produced upon light, it is necessary to use two pencils, one polarised + 45°, and the other — 45°, to the plane of incidence. The planes of polarisation of these pencils are inclined 90° to each other, and the invariable effect of the new force is to augment that angle in the same manner as is done by a refracting surface, while the tendency of the ordinary reflective force is to diminish the same angle. Hence I was led to make an experiment in which these opposite forces might compensate one another. I mixed oil of olives and oil of cassia, till I obtained a compound of such a refractive power that its action in bringing together the planes of polarisation should be equal to the action of the new force in separating them. Upon reflecting the compound pencil from this surface, I was delighted to find that the inclination of the planes was still 90°, and I thus obtained the extraordinary result of a reflecting surface which possessed no action whatever upon common or upon polarised light.
"The action of the new force when the plane of reflection coincides with the principal section of the crystal is obviously inexplicable by any theory of light, though I have no doubt that the undulatory theory may ultimately accommodate itself to this as well as to other classes of phenomena which it does not at present embrace. The difficulty, however, is increased by another result of my experiments, which it is important to notice. On the faces of the spar which are inclined 0°, 45°, 90°, to the axis of double refraction, the action of the new force is symmetrical upon the two pencils of polarised light whose planes are inclined + 45° and — 45° to the plane of incidence; whereas in intermediate faces, whose inclination to the axes is 22° and 67°, the plane of one of the polarised rays remains stationary, while that of the other is turned round 15°.
"This effect is undoubtedly a very extraordinary one, and indicates some singular structure in calcareous spar, the nature of which it is not easy to conjecture." "I have examined these phenomena by using, in place of oil of cassia, various fluids whose refractive powers descend gradually to that of water; but it would be a waste of time to give any detailed account of them at present. I shall only state, that the action of the new force becomes weaker and weaker as the force of ordinary reflection is increased, by diminishing the refractive power of the oil which is placed in contact with the spar. With an oil of the highest refractive index, the action of the new force predominates over the feeble power of the ordinary force of reflection. With an oil of a lower index the two forces exactly balance each other; while with oils of still lower indices of refraction, the ordinary force overcomes and conceals the action of the new one.
"Although I have obtained pretty accurate measures of the amount of the deviations produced by the new force, on eight surfaces, differently inclined to the axis, and in various azimuths on these surfaces, yet many experiments are still necessary before we can hope to discover the physical law of the phenomena; and if this should be done, I have no doubt that Mr Macculagh will be equally successful in the higher attempt of accounting for them by some modification of the undulatory theory."
Sect. XVIII.—On the Mutual Action of Polarised Rays.
This curious subject has, we believe, not been studied by any other philosophers but M. Arago and M. Fresnel, and we shall therefore make no apology for giving an account of the results which they obtained, in the words of M. Fresnel himself:
"In studying the interferences of polarised rays, M. Arago and I found that they no longer exercise any influence upon one another when their planes of polarisation are perpendicular to each other; that is, that they cannot then produce fringes, although all the necessary conditions for their appearance in ordinary cases are scrupulously fulfilled. I shall describe the three principal experiments which illustrate this fact, beginning with that which belongs to M. Arago. It consists in making two pencils, emanating from the same luminous point, and introduced by two parallel slits, traverse two piles of very thin transparent plates, such as those of mica or blown-glass, which are sufficiently inclined to each other to polarise almost completely each of the two pencils, taking care that the two planes are perpendicularly inclined to each other. In this case no fringes can be perceived, whatever care may have been taken thus to compensate the differences of both in varying very gently the inclination of one of the piles; but when the planes of incidence of the piles are no longer perpendicular to each other, they always cause the fringes to appear. In proportion as the planes cease to be parallel, the fringes become weaker, and they disappear altogether when they are rectangular, provided the polarisation of the two pencils has been sufficiently complete. It results from these experiments, that rays polarised in the same plane influence one another, like the rays of light not modified; but this influence diminishes in proportion as the planes of polarisation deviate from one another, and becomes nothing when they are rectangular.
The following is another experiment which leads to the same results. Take a plate of sulphate of lime or of rock-crystal parallel to its axis, and of a uniform thickness; cut it in two, and place each of the halves upon one of the slits of the screen. I suppose that we have turned the two halves in such a way that the edges which were in contact before the division of the plate are parallel, and the axes will also be parallel. But, in this case, we only perceive a single group of fringes in the middle of the bright space as it was before the division of the plates. But, if we turn one of these halves in its plane, thus disturbing the parallelism of their axes, we make two other groups of fainter fringes spring up, situated one on the right, and the other on the left of the group in the middle, and which are completely separated from it, in the white light when the plates of rock-crystal or of sulphate of lime which are used, are only a millimetre thick. It is to be remarked, that the number and breadth of the fringes lying between the middle of one of these groups and the central group, is proportional to the thickness of the plates for crystals of the same kind, or whose double refraction is of the same strength, like rock-crystal and sulphate of lime. In proportion as the angle of the two axes increases, these new groups of fringes become more and more distinct, and attain at last their maximum intensity when the axes of the two plates are perpendicular to each other. In this position, the central group, which had been gradually weakened, altogether disappeared, and is replaced by a uniform light. From this we may conclude, that the rays which produce them, by their interference, are no longer capable of influencing one another.
From the position of these fringes, it is easy to see that they result from the interference of the rays, which have undergone the same kind of refraction in the two plates, since, having travelled with equal velocities, they ought to arrive simultaneously at the middle of the bright space which corresponds to the equal routes, if as we suppose the two plates have the same thickness, and always continue perpendicular to the rays. Hence, the fringes of the central groups were formed by the superposition of those which arise—1. From the interference of the ordinary rays of the left plate with the ordinary ray of the right plate; 2. From the interference of the extraordinary rays of the first plate with the extraordinary rays of the second. The two eccentric groups, on the contrary, arise from the interference of the rays which have undergone different refractions in the two plates; and as it is the ordinary rays which move with the greatest velocity in sulphate of lime or rock-crystal, we see that, if we employ one of these two species of crystals, the left group ought to be formed by the union of the extraordinary rays of the left plate with the ordinary rays of the right plate, and the right group by the union of the extraordinary rays of the right plate with the ordinary rays of the left plate. This being established, it remains now to determine the direction of polarization in each of the pencils which interfere, in order that we may deduce from it what are the relative directions of the planes of polarization which favour or hinder their mutual influence. Analogy shews that the mode of the polarization of light ought to be the same as in the small plates in crystals thick enough to divide it into two distinct pencils. But as this hypothesis might perhaps be an object of discussion, and contradict an ingenious theory of one of our most celebrated philosophers, we shall not present it at first as a certain principle, and we shall have recourse to a direct experiment to determine the planes of polarisation, both of the ordinary and extraordinary rays which emerge from these plates, which we suppose to be one or two millimetres thick. This thickness is sufficient to allow us to cut one of their edges obliquely, and obtain by this prismatic form the separation of the ordinary and extraordinary rays. We then find that they are effectually polarised, the first in the principal section, and the others in a perpendicular direction. If we are not sufficiently convinced that this is their manner of polarisation in emerg-
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1 We have not seen the original Memoir, but have translated this portion of it from M. Pouillet's very valuable work, entitled "Elements de Physique Experimental et de Meteorologie," liv. viii. chap. iii. ing from each plate when these two surfaces are parallel, we shall find a new demonstration of this in the facts that we are about to describe, setting out from the principles established by the experiments of M. Arago, and which are besides confirmed by others, of which we are going to speak. If, on the contrary, we no longer question the direction of the polarisation of the ordinary and extraordinary rays, actual experiment will afford us a second demonstration of these principles. When the axes, indeed, of the two plates were parallel, the rays which had experienced the same refractions in the two crystals were found polarised in the same direction, and those of contrary colours in rectangular directions. It is thus that the group of fringes in the middle which proceeded from the interference of rays of the same name, had a maximum intensity, and the two others, which resulted from the interference of the rays of contrary names, did not appear again. But when the axes of the two plates formed an oblique angle, of 45° for instance, the rays of a contrary name, and those of the same name, could act at the same time one upon the other, since their polarising planes were no longer rectangular, and the three groups of fringes were produced. When, in short, the axes became perpendicular to one another, the rays of the same colour were polarised in rectangular directions, and the central group, to which it had given birth, vanished, while the ordinary rays of the left plate were then polarised parallel to the extraordinary rays of the right plate, and this is the cause why the right group which they produce attained its maximum intensity. It is the same with the left group, arising from the interference of the ordinary rays of the right plate with the extraordinary rays of the left plate.
The following is a third experiment, which entirely confirms the results which we have drawn from the first. Having polished a rhomb of calcareous spar upon two opposite faces, wrought with care so as to be quite parallel, I sawed it perpendicularly to these faces, and I obtained in this manner two rhombs of equal thickness, and in which the routes of the ordinary and extraordinary rays were exactly parallel at the same incidence. I placed them one before the other, so that the rays proceeding from the luminous point which had traversed the first rhomb passed through the second, taking care that their faces were perpendicular to the direction of the incident rays; moreover, the principal section of the second rhomb was perpendicular to that of the first, in such a manner that the four pencils which they produced were in general reduced to two; the ordinary pencil of the first rhomb was refracted extraordinarily in the second, and the extraordinary pencil of the first was refracted ordinarily in the second. From this arrangement it followed, that the difference of the route, proceeding from the difference of velocity of the ordinary and extraordinary rays, was found compensated for by the two emerging pencils. They crossed each other, too, at a very small angle, and such that the fringes ought to have had a magnitude much more than sufficient to be seen; and, notwithstanding, though the necessary conditions for the production of the fringes, in ordinary circumstances, were carefully observed, I never could succeed in making them appear.
While I was searching for them with care, holding a magnifying-glass to my eye, I gently varied the direction of one of the rhombs, sometimes to the right, sometimes to the left, in order to compensate the effect resulting from any difference of thickness that might have existed; but in spite of this trial, repeated a great many times, I never could perceive any fringes; and this is not surprising, after what the other experiments have taught us, since the two pencils emerged polarised at right angles to each other. But what proves that the absence of fringes does not depend upon the difficulty of arriving by trials at an exact compensation is, that I easily succeeded in making them appear by employing the light which had been polarised before it entered the rhombs, and in causing it to receive a new polarisation after its emergence. It is, then, completely demonstrable from the experiments which I have described, that rays polarised at right angles cannot exert any sensible influence upon one another; or in other words, that their union always produces the same intensity of light, whatever may be the difference of route of the two pencils which interfere.
Another remarkable fact is, that when they have been once polarised in rectangular directions, it is not sufficient that they are brought back to a common plane of polarization, in order that they may give apparent signs of their mutual influence. If in the experiment of M. Arago, indeed, or that which I have described, we cause the rays which have emerged from the two slits, and which are polarized at right angles, to pass through a pile of inclined plates of glass, no fringes were perceived, in whatever direction its ray of incidence was turned. Instead of a pile, we may employ a rhomb of calcareous spar, if we incline its principal section at 45° to the plane of polarisation of the incident pencils, so that it divides into two equal parts the angle which they make with each other, each image will contain the half of each pencil, and these two halves having the same plane of polarisation in the same image, ought to produce fringes there, if it is sufficient to bring back the rays to a plane of common polarisation, to re-establish the apparent effects of their mutual influence. But the fringes can never be obtained by this method, as long as the rays have not been polarised in the same plane, before they were divided into two pencils polarised at right angles.
When the light has experienced this previous polarisation, on the contrary, the interposition of the rhomb makes the fringes reappear. The most advantageous direction to give the primitive plane of polarisation, is that which divides into two equal parts the angle of the rectangular planes, in which the two pencils are polarised in the second instance, because then the incident light is equally divided between them. Suppose, in order to fix our ideas, that the primitive plane of polarisation is horizontal, it will be necessary that the planes of polarisation impressed upon each of the two pencils, is inclined 45° to the horizontal plane, the one above it and the other below it, in such a manner that they remain perpendicular. We can obtain this rectangular polarisation either with the help of the two little piles employed in the experiment of M. Arago, or with two plates whose axes are rectangular axes, or, finally, with a single crystallized plate: We shall only consider this last case, the two others presenting phenomena precisely similar.
To divide the light into two pencils, which cross under a small angle, and which may thus produce fringes, the apparatus of two mirrors is generally better than the screen pierced by two slits, because it produces more brilliant fringes. It has, besides, the advantage of giving immediately to the two pencils the previous polarisation necessary to our experiment; it is sufficient for this purpose that the two mirrors should be of glass not tinned, and inclined 35° to the incident rays; care must be taken to blacken them behind, to destroy the second reflection. We place near them, in the line of the reflected rays, and perpendicularly to their direction, a plate of sulphate of lime or of rock-crystal parallel to the axis, and one or two millimetres thick, inclining its principal section 45° to the plane of primitive polarisation, which we have supposed horizontal. The apparatus being thus placed, we only see a single group of fringes across the plate as before its interposition, and it occupies the same place. But if we put before the magnifying-glass, a pile of glass plates, inclined in a horizontal or vertical direction, we dis- cover on each side of the central group another group of fringes, which is the more distant as the crystallized plate is thicker. If we replace the pile of plates by a rhomb of calcareous spar, whose principal section is divided horizontally or vertically, we shall see in each of the two images which it produces, the two systems of additional fringes which the interposition of the pile of plates has caused to appear; and it deserves to be remarked, that these two images are complementary to one another; that is to say, that the obscure bands of the one correspond to the brilliant bands of the other.
We see in this experiment a new confirmation of the principles demonstrated by the preceding ones. The rays which have experienced the refractions of opposite names, cannot influence each other, because, in emerging from the same plate in the case we are now considering, they are found polarised in rectangular directions; consequently the groups to the right and the left cannot exist, at least while we have not re-established the mutual influence of those rays by bringing them to a common plane of polarisation: this is what is effected by the interposition of the pile of plates or of the rhomb. The fringes thus produced are the more distinct, as the two pencils of contrary names which concur in their formation are more equal in intensity; and this is the reason why the direction of the principal section of the rhomb, which makes an angle of $45^\circ$ with the axis of the plate, is the most favourable to the appearance of the fringes. When the principal section of the rhomb is parallel or perpendicular to that of the plate, the rays refracted ordinarily by the plate pass entirely into one image, instead of being divided between the two, and all the extraordinary rays pass into the other image, so that there can be no more interference between them; and the additional groups of fringes disappear, each image presenting only the fringes which resulted from the interference of the rays of the same name, that is to say, those which compose the central group.
These two groups of additional fringes which polarized light exhibited in the first position of the rhomb, afford one of the most exact methods of measuring double refraction and of studying its law. Their eccentric position, indeed, depends on the difference of route of the ordinary and extraordinary rays which emerge from the plate; and we can judge of the number of undulations by which the extraordinary rays of the right pencil remain behind the ordinary rays of the left one, by the number and width of the fringes comprised between the middle of the right group, and that of the central group. We may determine this difference of route still better, by measuring the interval between the middle of the two extreme groups, which is the double of their distance from the middle of the central group. White light is best suited for this kind of observations, because it is brightest, and renders the central band of each group easier to be observed. By comparing the thickness of the plate with the difference of the observed route, we may deduce from it the ratio of the velocities of the ordinary and extraordinary rays."
Sect. XIX.—On the Production of Double Refraction by Heat, Cold, Pressure, and slow and rapid Induration.
The various phenomena of double refraction and the system of polarised rings, may be produced either transiently or permanently in glass and other substances, by heat and cold, compression and dilatation, and by slow and gradual aduration. The phenomena thus produced in polarised light are exceedingly beautiful, and throw much light on the subject of double refraction.
Art. I. On the Transient Influence of Heat and Cold.
The influence of heat and cold may be exhibited in cylinders, tubes, spheres, cubes, and rectangular plates of Palarisa glass, all the phenomena of which were discovered by Sir David Brewster.
1. Cylinders of Glass with one axis of double refraction.
1. Negative Axis.—If we take a cylinder of glass, ACBD, fig. 69, from half an inch to an inch in diameter or upwards, about one-half an inch or more in thickness, and transmit heat uniformly from its circumference to its centre, it will exhibit, when placed between the polarising and the analysing plate, and held about eight or ten inches from the eye, the system of uniaxial rings shewn in fig. 69, exactly similar to those in fig. 29; and by turning round the analysing plate, we shall see the complementary set, as in fig. 31, these rings will be seen as if they were in the substance of the glass, and hence, if we cover up any part of the circular surface, we shall cover up a corresponding part of the system of rings. The axis of the system or of double refraction is here fixed in the axis of the cylinder, and does not lie in every direction parallel to that axis, as in regularly crystallized bodies.
The system of rings thus produced is negative, like those in calcareous spar.
As soon as the heat reaches the axis of the cylinder, the rings become less bright, and they disappear entirely when the heat is uniformly distributed through the glass.
2. Positive Axis.—If we heat a similar cylinder of glass uniformly in boiling oil or otherwise, and cool it rapidly at its edges by encircling its margin with a cold and good conducting material, it will exhibit a similar system, which will vanish when the glass is uniformly cold. This system of rings, however, is positive, like those of Zircon, and if it is placed above the equal negative system, produced as already described, they will destroy each other. If the two systems are not equal, we shall have a system equal to their difference, as in positive and negative uniaxial crystals. In both these systems the tint at any point varies as the square of the distance of that point from the axis, so that if $T$ is the tint at any distance $D$, the tint $t$ corresponding to any distance $d$, will be $t = \frac{T D^2}{d^2}$. If $V$ is the velocity of the ordinary ray, we shall have the velocity of the extraordinary ray $e = V^2 + a d^2$.
If we transmit polarized light through the cylindrical surface of these cylinders, we shall observe the phenomena of biaxial systems exactly the same, as in rectangular plates, fig. 81, the tints being produced by the action of the positive or negative axis of the cylinder acting in opposition to an axis passing through each diameter of the cylinder, drawn perpendicular to any point in the middle line of the cylindrical surface.
2. Oval Cylinders, with two axes of double refraction.
If we perform the two experiments above described with oval cylinders, as in fig. 70, we shall have a system of rings with two axes. A new axis is developed perpendicular to the axis of the cylinder, and in the case of the heated cylinder the new axis at O is a positive one, while in the cooled cylinder it is a negative one, the neutral black lines AD, CB separating the two classes of tints, and corresponding to the dark hyperbolic branches in biaxial systems of rings. The figure referred to is that shewn in azimuths inclined $45^\circ$ to the plane of primitive polarization; but in the azimuths of $0^\circ$ and $90^\circ$, the branches AD, CB resume the form of the rectangular cross.
3. Cubes and Parallelopipeds of glass with double refraction.
Cubes of Glass.—When the shape of the glass is that of a cube as in fig. 71, the figure which it produces in azimuth $0^\circ$ by the two processes of heating and cooling is that shewn in the figure, the tints being negative or positive according as we apply heat or cold.
The complementary system is shewn in fig. 72.
**Parallelopipeds of Glass.**—In a parallelopiped 0.38 of an inch square, and 1.11 long, the direct and complementary systems at 0° of azimuth are shewn in fig. 73 and 74. The first of these consist of a black cross, surrounded with beautiful fringes of contrary flexure, and four bright green spots of the third order. The coloured spots at the angles of fig. 74 are of a brilliant pink colour, with a spot of blue in the middle of each. When the azimuth is 45°, the direct and complementary systems change into fig. 75 and 76.
4. **Cylindrical Tubes of Glass with two axes of double refraction.**
When the cylinder has the form of a tube, as in fig. 77, the double refraction is singularly distributed by the application of heat or cold either to the outside ABCD of the cylinder, or to its inside abcd, or to both. A black circular fringe mno p, is the central line which separates the outside or positive structure from the internal negative structure, and vice versa. The breadth of the internal annulus ao is always less than Ao, that of the external one. They approach to equality as the bore of the cylinder widens, and the negative structure grows very small, as shewn in fig. 78, when the bore diminishes; so that when the bore becomes infinitely small, the system becomes either wholly negative or positive, according as heat or cold has been used. If when fig. 77 is fully developed, we cut a notch EF in the cylinder, we shall have a biaxial system of fringes in which there is a positive structure between two negative ones, or vice versa, as shewn in fig. 79.
The diameter op, fig. 77 and 78, is a geometrical mean between the interior and exterior diameters of the tube, that is \( op = \sqrt{(AB \times ab)} \).
5. **Rectangular Plates of Glass, with planes of double refraction.**
If a well annealed parallelopiped of glass EFCD, fig. 80, is submitted to the processes already described, or even if we lay one edge of it CD on a piece of iron almost red hot, and place the whole between the polarizing and analyzing plates, so that if the heated edge CD is inclined 45° to the planes of primitive polarization, and the eye can see the whole surface of the glass, a series of remarkable phenomena will be produced. The moment the heat enters the lower surface at CD, fringes of brilliant colours are seen above CD, and almost at the same instant, before the heat has reached the upper surface EF, or even the central line ab, similar fringes will appear at EF. Colours, at first faint blue, then white, yellow, orange, red, &c. of the first order spring up at ab, and these central colours are separated from those at the edges by two dark lines or planes MN, OP, corresponding to the hyperbolic branches in biaxial crystals, the double refraction between MN and OP being negative, in reference to a line perpendicular to the fringes, while it is positive without MN and OP. The tints thus developed are those of Newton's scale.
If T is the central tint in the line ab, D the distance of either of the lines MN, OP, from ab, the tint t at any other distance d, will be \( t = T - \frac{Td^2}{D^2} \), a formula deduced from the opposition of two axes. The term \( \frac{Td^2}{D^2} \) represents the influence of the principal axis or axes perpendicular to the line ab at every point of it; but as the axis in the plane of the plate produces an uniform tint T, whose maximum is in the line ab, where the action of the other axis disappears, and as these axes oppose each other by acting rectangularly, they will compensate each other in the lines MN, OP, and the tint at any point must always be equal to the difference of the tints, or to \( T - \frac{Td^2}{D^2} \).
The magnitude 2D, or the distance between MN and OP, is a function of the breadth of the plate or b, and 2D: V = 10 : 16.02 and D = .312 B².
The fringes seen through the thickness of the plates is shewn in fig. 81, and the one seen through the ends in fig. 82.
If we wish to find the tints in reference to the lines MN, OP, let \( \delta, \gamma \) be the distance of any point from these lines, whose distance from ab was d, then we have \( \delta = 1 - d, \gamma = 1 - d \), and \( \delta + \gamma = 1 - d^2 \), that is the tint at any point varies as the product of the distances of that point from the planes of the resultant axes MN, OP. If we make \( v = V^2 + \delta \gamma \), an expression which gives v the velocity of the extraordinary ray, we shall have the extraordinary refraction in such plates.
6. **Spheres and Spheroids of Glass with double Refraction.**
**Spheres.**—If we place a sphere of glass in a glass-trough of hot oil or otherwise heat it regularly, we shall find that when the heat is passing to the centre of the sphere, it will exhibit a positive uniaxial system of rings like that in fig. 69, in every direction in which we transmit the polarized light; that is, it will have an infinite number of positive axes of double refraction.
If a hot sphere of glass is immersed in a glass-trough of cold oil, a similar system of rings will be produced in every possible direction; but it will be negative.
**Spheroids.** If we substitute oblate and prolate spheroids in place of the sphere in these experiments, we shall find that they will have each a positive system of rings round their axis of revolution. If the polarised light is transmitted through an equatorial diameter, we shall find that there are two axes of double refraction, the black cross opening out when the axis of revolution is inclined 45° to the plane of primitive polarisation.
In the prolate spheroid the black cross opens out in a different plane.
7. **On the Effects produced by combining Plates of Glass under the transient influence of heat and cold.**
If we combine any two plates of the same shape, the resulting system of fringes will be equal to the sum of their systems or effects, if the plates are of the same name, that is, both positive or both negative, or to the difference of their effects, if they are of different names. When the plates are solids or symmetrical forms, such as cylinders, cubes, or quadrangular plates, no essential variation of figure is produced by the combination; but when the plates are rectangular, very interesting phenomena are exhibited when plates of the same or of different names are crossed rectangularly. Sir David Brewster has given formulæ for calculating the forms of the compound or isochromatic curves, as he calls them, but our space will only permit us to exhibit the effects to the eye.
When equal rectangular plates of similar names, that is, both negative or both positive, are crossed, the phenomena of the intersectional fringes, as they may be called, are shewn in fig. 83, where the isochromatic curves are hyperbolas.
When the plates are of different names, the one positive and the other negative, and of the same breadth, and the same number of fringes, the isochromatic curves are circles, as in fig. 84. When the plates are of different names, and of different breadths, but containing the same number of fringes, the isochromatic curves will be ellipses, as in fig. 85.
8. On the effects produced by altering the form of, or subdividing plates of glass under the influence of heat or cold.
If we alter the shape of any of the plates above described, the form of the isochromatic curves is immediately changed. If we cut any rectangular plate into two by a line passing through its middle, each of the two plates thus produced has the property of the whole plate, though the fringes are less numerous. If a plate ABCD gives the tints shewn in fig. 86, OP and MN being the dark neutral lines; then if we cut it with a diamond at ab, so as to subdivide it into two plates, as in fig. 87, each of the plates EF r s, GH r s, will have the same structure as ABCD, viz. two neutral lines o p, m n, and assume positive and negative tints.
Art. II. On the Permanent Influence of Rapid Cooling.
In March 1814, Sir David Brewster found that glass melted and suddenly cooled, as in the case of Prince Rupert's drops, possessed a permanently doubly refracting power, and he communicated this fact in a letter to Sir Joseph Banks, dated April 8, 1814; and without knowing that Dr Seebeck had published in December 1814 similar results with cubes of glass, our author had discovered that cubes, cylinders, plates, spheres, and spheroids of glass, exhibiting permanently the phenomena described in the preceding pages, may be formed by bringing them to a red heat, and cooling them rapidly and equally on their edges. A great variety of beautiful optical figures, developed in polarised light, may thus be obtained by cooling the glass on metallic patterns. A very simple effect of this is shewn in fig. 88, where the plate of glass was cooled by resting it at its centre on a cylinder of iron.
Art. III. On the production of Double Refraction by Compression and Dilatation.
The effects of compression and dilatation in producing double refraction were discovered by Sir David Brewster, and communicated to the Royal Society in 1815. Our limits will permit us to only give a slight notice of them.
The phenomena both of compression and dilatation or extension may be well seen by bending, merely with the force of the hands, a square rod, or a long and narrow plate of glass, as in fig. 89. When it is held between the polarising and analysing plate, eight or ten inches from the latter, with its edge AB inclined 45° to the plane of primitive polarisation, the whole thickness of the glass will be covered with two series of coloured fringes, like those in the figure separated by a dark neutral line MN, where there is neither compression nor dilatation. The fringes on the convex side are negative, being produced by the extension of the glass in the direction mA, mB, while those on the concave side are produced by the compression of the glass in the directions Cn, Dn. The isochromatic curves marked by similar figures, indicating the orders of the colours, are bent as in the figure.
When a plate of bent glass producing fringes crosses another at right angles, the effect at the intersectional space is shewn in fig. 90, where the tint is supposed only to be the white of the first order.
When a plate of bent glass is crossed with a plate crystallized by heat, the fringes in the intersectional square will be parabolas, as shewn in fig. 91, whose vertex will be towards the convex side of the bent plate, if the principal axis of the other plate is positive, but towards the concave side, if that axis is negative.
Art. IV. On the Double Refraction produced by gradual Induration and difference of Density in soft Solids.
The phenomena of luminous sectors, separated by a black cross at the central part of the uniaxial system of rings, which Sir David Brewster discovered round cavities in the diamond, in glass, and in various gums, arise from the gradual induration of the mass, combined with the elastic pressure of the air included in the cavities: They are therefore not properly cases of induration alone.
When isinglass is dried in a circular trough, it exhibits, by polarised light, the uniaxial system of rings like glass, in fig. 69.
When it is indurated in the form of a rectangular mass, by the exposure of two sides, fringes are produced parallel to these sides, and biaxial like those in rectangular plates of heated glass.
A sphere of transparent jelly or isinglass, when allowed to indurate gradually, will have an axis of double refraction in every direction, like a sphere of glass heated. In like manner an indurated spheroid will exhibit the biaxial structure of a spheroid of glass.
The most splendid examples, however, of this class of facts are exhibited in the lenses of fishes and animals, as shewn in figs. 92 and 93. The first of these shews the doubly refracting structure of the crystalline lens of a cod, along the axis of vision. The central and the external luminous sectors have a negative doubly refracting structure, while the intermediate ones have a positive structure.
The figure given by the crystalline lens of a cow is shewn in fig. 93, where there are four series of luminous sectors, the central ones being positive, the next negative, the next positive, and the last negative.
CHAP. II.—ON CIRCULAR POLARISATION.
1. Circular Polarisation in Rock-crystal and Amethyst.
The general phenomena of circular polarisation were discovered by M. Arago in 1811. He found that in plates of rock-crystal, the colours polarised along its axis were different from those which he had studied in plates of other crystallised bodies. When they were analysed by a prism of Iceland spar, he found that the two images had complementary colours in the ordinary tints, but, what was remarkable, they descended in Newton's scale as the prism revolved, so that if the tint of the extraordinary image was red, it became in succession orange, yellow, green, and blue. Hence he concluded that the differently coloured rays had been polarised in different planes in passing along the axis of the rock-crystal.
M. Biot took up the subject at this point, and investigated it with his usual ingenuity and success. He found that while in some crystals of quartz the tints descended in the scale of colours, by turning the analysing prism from right to left, in others they descended in the scale by turning the prism from left to right. The one he called left-handed quartz, and the other right-handed quartz. He took a plate of quartz, for example, \( \frac{1}{8} \)th of an inch thick, and having polarised the homogeneous colours of the spectrum, he transmitted them in succession along the axis of this plate, and obtained the following
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1 See Phil. Trans., 1816, p. 311. 2 See Phil. Trans., 1837, vol. ii. Circular results. When the analysing plate was in 0° of azimuth, polarisation the red light polarised by the plate was a maximum. When the analysing plate was turned from right to left, the red tint gradually diminished, and after a rotation of 17½°, it wholly vanished. With a plate 2-25ths of an inch thick, the red tint did not vanish till after a rotation of 35°; and so on, every additional 25th of an inch of rock crystal requiring an additional rotation of 17½° to make the tint vanish. A whole inch of quartz, for example, would require $25 \times 17\frac{1}{2}° = 437\frac{1}{2}°$, or one whole turn, and 77½° more to cause the red tint to vanish. It is obvious that twenty-five plates of quartz, 1-25th of an inch thick, would produce the same effect as one inch of it.
When right-handed plates, however, are combined with left-handed ones, the rotation produced is equal to the difference of their actions; thus a plate of left-handed quartz 1-25th of an inch, combined with a plate 2-25ths of an inch, would produce a rotation of only 17½°. The following table contains the rotations produced upon the other coloured rays of the spectrum, as given by M. Biot:
| Names of the ray | Are of rotation for 1-25th of an inch in quartz | |------------------|------------------------------------------------| | Extreme red......| 17° 49' 6" | | Limit of red and orange | 20° 47' 8" | | Limit of orange and yellow | 22° 31' 8" | | Limit of yellow and green | 25° 6' 52" | | Limit of green and blue | 30° 4' 60" | | Limit of blue and indigo | 34° 57' 17" | | Limit of indigo and violet | 37° 6' 29" | | Extreme violet...| 44° 8' 27" |
M. Biot conceived that this property of quartz belonged to its ultimate molecules, but Sir David Brewster proved that this was not the case, by shewing that heat deprived quartz of the property of circular polarisation; and Sir John Herschel's beautiful discovery, that it was connected with the crystallisation of the mineral, put this result beyond a doubt. He found that those crystals in which the plagia- ral faces described by Haury, went round the crystal from right to left, exhibited the optical properties of left-handed crystals, and those crystals in which the plagial faces leant round the crystals from left to right, had the properties of right-handed crystals. Hence he concluded that whatever be the cause which determined the direction of rotation, the same law acted in determining the direction of the plagieral faces.
When Sir David Brewster discovered the system of rings in quartz, he found the tints of circular polarisation occupying, as might have been expected, the inner circle of the rings as shewn in fig. 94, only small portions of the black cross being visible; but these black portions were larger as the plate became thinner.
In examining the structure and properties of the amethyst, Sir David Brewster found that this singular mineral was actually composed of the two different kinds of quartz, viz., the right-handed and the left-handed. These two kinds of quartz are arranged in veins, as represented in figs. 95, 96. In fig. 95, the shaded veins which correspond to each alternate face of the pyramid turns the planes of polarisation from right to left, while all the rest of the crystal turns the same planes from left to right; and what is very interesting, the black lines where these two structures unite have no action whatever on the planes of polarisation. In some specimens these opposite veins are so very minute, that they destroy each other's action upon the polarised ray, and when this happens, the single system of rings appears with its black cross, and entirely free of any of the tints of circular polarisation. The colouring matter of the amethyst is arranged in a very singular manner in relation to these veins; and the fracture across the veins exhibits a beautiful, and sometimes a regular rippled structure, resembling the engine-turning of a watch, and affords an infallible mi- neralogical character of the amethyst, whether its colour is yellow, orange, olive, green, blue, or perfectly colourless.
The general structure of well crystallised amethyst is shewn in fig. 96, which is of the natural size, and is taken from one of the finest specimens that Sir David Brewster met with. "On the three alternate sides of the prism," says he, "viz., MN, OP, and QR, are placed sectors MeN, OdP, QaR, which are divided into two parts by dark lines ce', dd', aa', which separate the direct structures of A, C, and E from the retrograde structures of B, D, and F. On the other three alternate faces of the prisms are placed the three veined sectors MeBaR, NcbdO, and PdbaQ, which meet at b in angles of 120°, and consist of veins of opposite structure, alternating with each other, and so minute, that in many places the circular tints are almost wholly extinguished by their mutual action. The direct sectors A, C, and E, are all connected together by the three radial veins ba, bc, bd, and are therefore to be considered as the expanded terminations of these veins. The retrograde sectors B, D, and F, are expansions of the first retrograde veins next to bde, dba, and abc, and the lines ce', dd', and aa', are continuations of the dark or neutral lines which separate the first retrograde vein from the direct radial veins.
"All the sectors A, B, C, D, E, and F, are of a yellowish brown colour, and all the rest of the crystal is of a pale lilac colour, the lilac tints being arranged in the manner previously described. The phenomena which I have now mentioned as existing in this specimen are very common in the amethyst; and I have never yet found a specimen in which the yellow tints were not confined to those portions which formed the expanded terminations of veins; a fact which indicates that this would have been the colour of the crystal, whether its action were direct or retrograde, and that the lilac colour affects in general those portions which are composed of opposite veins."
The subject of circular polarisation received great accep- tions from the genius of M. Fresnel. He conceived that a ray passing along the axis of quartz should be refracted in two pencils, and he ascertained this to be the case by the following experiment. He took a prism ACB of right-handed quartz, having its faces AC, BC, equally inclined to its axis AB, so that a ray PV should be incident at angles of 75° on either face. As a ray, however, refracted at R, would not emerge at all from the other side CB, he took another similar prism, but from a crystal of left-handed quartz, and having cut it into two halves, he placed these two halves ACD, BCE, as in the figure, so that he had an achromatic combination of three prisms.
Now a ray PQ incident perpendicularly at Q, should pass straight on without deviation, or double refraction, if quartz were like other uniaxial crystals. But if the pencil PQ suffer any double refraction at Q, this double refraction will be doubled at R, because ABC has an opposite kind of double refraction. The same effect takes place at C, so that the ray PQ at its emergence at T, ought to have a very sensible double refraction, even if that at Q was very small. Now M. Fresnel found that this double refraction actually existed, but upon examining the image he found that they had suffered a new kind of double refraction, and acquired new properties. In place of their being polarised in opposite planes, like other doubly refracted pencils, which, when examined with a doubly refracting prism, give two unequal images alternating in brightness during the revolution of the spar prism, they exhibit the following properties:
1. Either of the quartz rays, when examined with a spar prism, gave two images of equal intensity in every position. of the prism. Hence they resemble unpolarised light, as if they consisted of two rectangularly polarised rays.
2. They differ from unpolarised light in having the following remarkable and characteristic property. If either of them are incident at right angles, as shewn at RP; fig. 169, upon the face AB of a parallelopiped of crown glass, having its refracting index 1.51, and its angles ABC and ADC 45°; it will suffer two total reflexions at Q and S, emerging perpendicularly from the surface DC in the direction ST. Now this ray ST is found to be completely polarised in a plane inclined 45° to the plane of its reflexions, whatever may be the position of that plane. If the other ray is incident at rp, and is reflected at q and s, so as to emerge in the direction st, the one ray ST will be polarised in a plane 45° to the right, and the other st 45° to the left of the plane of reflexion. Hence they emerge when superimposed in a state of common light. The two rays RP, rp, are said to be circularly polarised.
3. If a ray thus circularly polarised is transmitted through a thin crystallized plate, and parallel to its axis, it is divided by double refraction into two rays of complementary tints, thus shewing a decided difference from a ray of common light; and these complementary colours always differ from those that are produced from light, polarised and analysed in the usual way, by an exact quarter of a tint either in defect or in excess.
4. A circularly polarised ray transmitted again along the axis of rock crystal, and subsequently analysed, produces, like common light, no colours, and differs in this respect from polarised light.
As two circularly polarised rays RP, rp, emerge from Fresnel's rhomb, (as the parallelopiped of glass ABCD, fig. 169, has been called), in rays ST, st, polarised ± 45° to the plane of reflexion, it occurred to Fresnel, and he found it to be so, that a ray TS polarised 45° to the plane of reflexion in the rhomb, would emerge in the direction PR, as a circularly polarised ray, possessing all the properties of one of the rays formed along the axis of quartz.
In an extensive series of experiments, of which we shall give some account in the following chapter, Sir David Brewster had occasion to examine some of the kindred phenomena of circular polarisation. His first experiments on this subject preceded those of Fresnel. He found that total reflexion produced polarised tints analogous to those of crystallised laminae, and he supposed that these colours were produced by the interference of two portions of light, the one partially reflected in the first instance, and the other beginning to be refracted, and caused to return, by the continued operation of the same power. In continuing his experiments, he found that the colours produced by total reflexion did not rise in the scale by successive reflexions; and at the end of 1816, he announced in the Journal of the Royal Institution, that he had discovered "a new species of moveable polarisation, in which the complementary tints never rise above the white (the bluish white) of the first order, by the successive application of the polarising influence," &c. He determined experimentally the angles at which this tint was successively produced and destroyed, and thus discovered some of the leading properties of total reflexion. It was Fresnel, however, that discovered this new species of polarisation to be circular, and made those other splendid discoveries which we have just detailed. We owe to Sir David Brewster, however, the discovery of the inversion of the spectrum in the phenomena of total reflexion, of which we shall give some account in the next chapter. Circular
In giving the name of circular polarisation to that which polarisation is impressed on the two rays along the axis of quartz, Mr. Fresnel was guided by theoretical considerations. Mr. Airy has, however, taken a different view of the condition of the light forming these two rays in quartz, and has been led to results of very high interest. The following are the experiments, which we shall give in his own words, on which he founded his deductions. They were made with a Fresnel's rhomb, fitted up as in the annexed figure, where the rhomb is shewn at rr.
"1. If Fresnel's rhomb, mounted as in the annexed figure, be placed to receive the polarised light, so that the plane of reflexion passes through the divisions 45° and 225°, the calc spar will present another appearance 97. The rings are abruptly and absolutely dislocated: those in the upper right hand quadrant and the quadrant opposite to it, are pushed from the centre by one-fourth of an interval, and those in the other quadrants are drawn nearer to the centre by the same quality. The line separating the quadrants is nowhere black; the intensity of its light is uniform, and about equal to the mean intensity. If the plane of incidence pass through 135° and 315°, the phenomena of adjacent quadrants are exactly interchanged. No alteration is made by turning the analysing plate round the incident ray; the lines dividing the quadrants are always parallel and perpendicular to the plane of reflexion at the analysing plate.
"2. If the plane of reflexion in the rhomb pass through 0° and 180°, or through 90° and 360°, the phenomena are precisely the same, and undergo the same changes as those in ordinary rings. If while the plates are crossed, the rhomb be turned gradually from the position 0° towards 45°, the rings are gradually changed, at first becoming (as far as the eye can judge) elliptical, and then assuming the form represented in fig. 98.
"3. If a plate of quartz, whether right or left handed, be interposed between the crossed plates, a set of rings is seen like those in fig. 94. As far as the eye can judge, the rings Fig. 94. are exactly circular, but there is no black cross, and the central tint is not black, but removed from it by a number of tints in Newton's scale proportional to the thickness of the quartz. Thus with a thickness of 0·48 inch, the central tint is pale pink; with a thickness 0·38 inch, the central tint is bright yellowish green; with thickness 0·26 inch it is a rich red plum colour; with thickness 0·17 inch, it is a rich yellow.
"The colours then appear to be nearly the same, beginning from the centre, as in Newton's scale, beginning with the tint representing this central tint. At a considerable distance from the centre, four dark brushes begin to be visible, in the same directions as the arms of the black cross in calc spar.
"4. Now, (supposing the crystal right handed), if the plate of quartz be thin, and the analysing plate be turned, the upper part towards the observer's left hand, a bluish short-armed cross appears in the centre, which, on turning further, becomes yellow, and the rings are enlarged. On turning still further, the cross breaks into four dots. The rings are no longer circular, but of a form intermediate between a circle and a square, their diagonals (as well as the cross) being inclined to the left of the parallel, and perpendicular to the plane of reflexion. See fig. 99. If the analysing plate be turned the other way, there is no cross; the form of the rings is changed from circular nearly as in the former case.
"5. If the plate of quartz be thick, the dilatation, and the rings, and the change of form are all the perceptible phe-
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1 See Chromatics, vol. vi. p. 646, and Phil. Trans. 1830, p. 310. 2 Vol. iii. p. 213. Circular nomena. And on turning the analysing plate continually polarisation to the left, the rings continually dilate, and new spots start up continually in the centre, and become rings. If the crystal be left handed, the remarks in this and the last article apply equally well, supposing the analysing plate turned in the opposite direction.
"6. If Fresnel's rhomb be placed in the position 45°, and the light thus circularly polarised pass through the quartz, on applying the analysing plate instead of rings, there are seen two spirals naturally inwrapping each other, as in fig. 100. If the rhomb be placed in position 135°, the figure is turned through a quadrant. If the quartz be left handed, the spirals are turned in the opposite direction. The central tint appears to be white. With the rhomb which I have commonly used, (which is of plate glass, but with the angles given by Fresnel for crown glass), there is at the centre an extremely dilute tint of pink. I think it likely that this arises from the cross in the angles, as the intensity of the colours have no proportion to that in other parts of the spirals. The figure was drawn from the appearances given by a plate of quartz 0·26 inch thick.
"7. If two plates of quartz of equal thickness, but cut one from a right handed, and the other from a left handed crystal, be attached together, and put between the polarising and analysing plates, the left handed slice nearest to the polarising plate, the appearance presented is that of fig. 101. Four spirals (proceeding from a black cross in the centre, which is inclined to the plane of reflexion), cut a series of circles at every quadrant. The points of intersection are in the plane of reflexion, and perpendicular to it. This is the simplest way of describing the form; but if we followed the colours which graduate most gently, we should say that the form of each is alternately a spiral and circular arc, quadrant after quadrant.
"At a distance from the centre, the black brushes are seen. If the combination be turned, so that the right handed slice is nearest to the polarising plate, the spirals are turned in the opposite direction. This is one of the most beautiful phenomena of optics. The slices from whose appearance the figure was drawn, are each 0·16 inch thick."
The preceding phenomena are described as they appear when examined with an analysing plate of unsilvered glass. The following are the theoretical views which Mr. Airy considers as consonant with these experiments. They had been originally suggested to him by the desire of finding some connecting link between the peculiar double refraction in quartz, and the common double refraction.
"1. I suppose the ordinary rays to consist of light elliptically polarised, the greater axis of the ellipse being perpendicular to the principal plane, and the extraordinary rays to consist of light elliptically polarised, the greater axis of the ellipse being in the principal plane.
"2. I suppose that when the ordinary ray is right-elliptically polarised, the extraordinary ray is left-elliptically polarised, and vice versa.
"3. I suppose that the proportions of the axes of the two ellipses are the same, each proportion being one of equality when the direction of the ray coincides with the axis, and becoming more unequal, according to some unknown law, as the direction is more inclined to the axis; the minor axes of the ellipses having sensible magnitudes when the rays are inclined 10° to the axis.
"4. I suppose that the course of the rays after refraction can be determined by the construction given by Huygens for calc spar, with this difference only, that the prolate spheroid for determining the course of the extraordinary ray must not be supposed to touch the sphere for determining the course of the ordinary ray, but must be entirely contained within it."
In a supplement to his investigations, Mr. Airy remarks, that he has not yet ascertained the law which connects the polar ellipticity of the rays with the angle that they make with the axis. He considers, however, the following points as made out:
"One of the rays is certainly right-handed elliptical, and the other certainly left-handed elliptical. The major axis of one is certainly perpendicular to the principal plane of the crystal, and the major axis of the other is certainly in that plane. Mr. Airy remarks, that in some trials for measuring the ellipticities of the rays, he seems to have arrived at the conclusion, that the proportion of the axes of the ordinary ray is more nearly one of equality than the proportion of the axes of the extraordinary ray."
This subject has recently been investigated by Professor Macculagh, whose object was to pave the way for a mechanical theory, by shewing that all the phenomena may be grouped together by means of a simple geometrical hypothesis. Setting out from this hypothesis, he arrives immediately at all the known laws, and obtains at the same time a law that was previously unknown, and which is technically called the law of ellipticity. By this law Professor Macculagh has been able to compute the ellipticities observed by Mr. Airy in rays inclined to the axis of quartz, from the angles of rotation observed by M. Biot in rays parallel to that axis.
The phenomena of circular polarisation were discovered by Sir David Brewster, in plates of glass possessing the doubly refracting structure. M. Dove of Berlin has found it also in compressed glass and has published an interesting memoir on the subject, in which he takes little notice of the previous labours of others, on the various subjects of which he treats. As our limits will not permit us to give any account of its contents, we must refer our readers to the original memoir.
Sect. I.—On the Circular Polarisation of Fluids.
Although this remarkable property was discovered in some fluids by M. Seebeck by independent observation, yet M. Biot had anticipated him in it, and has made this subject so completely his own, by a series of the most elaborate and beautiful researches, that if he had done nothing else for science, they would have ensured him a high reputation. We regret extremely that our limits will not permit us to give anything like a full and satisfactory account of his discoveries, particularly those contained in his valuable paper of 1832. We must therefore refer the reader to his original memoirs, and present to them in an abridged form as possible his leading results.
M. Biot discovered that some fluids turn the planes of a polarised ray from right to left, and others from left to right. He found also that the tints rose in the scale as in quartz by an increase in the thickness of the fluid. The following table contains some of his results:
I. Fluids that turn the Plane of Polarisation from Left to Right.
| Fluids | Colours | |--------|---------| | Oil of fennel seeds | Palish green | | Caraway seeds | Colourless | | Lavender | Greenish | | Rosemary | Greenish | | Sassafras | Orange yellow | | Savine | Yellow | | Bitter oranges | Greenish yellow | | Bergamot | Colourless | | Lemons | Colourless | | Neroli, common | Yellow | | Fine | Orange yellow | | Neroli, fine | Reddish orange |
Report of the British Association for 1836, p. 18.
This memoir has been translated and published in Taylor's Scientific Memoirs, vol. i. part i. p. 73. II. Fluids that turn the Plane of Polarisation from Right to Left.
Essential oil of turpentine, Greenish..............59°21 Naphtha.............................................15°21 Oil of anise seeds.................................1°51 mint..................................................32°28 rhus..................................................Yellowish, conjectured.
Oil of mustard and oil of bitter almonds exercise no action upon polarised light.
M. Biot found that in a solution of natural camphor in alcohol, in which there was 0·37117 of camphor in weight to 1 of the solution, and its density 0·87221, the rotation for red light, and a thickness of 152 millimetres, was 17°56' from left to right. A solution of artificial camphor in alcohol, on the other hand, with 0·0917 of camphor to 1 of the solution, and having its density 0·8455, and in a thickness of 1357 millimetres, produced only a rotation of 2°4', but in the opposite direction, from right to left. In gum from Senegal, of which 47·4 parts was dissolved in 99·1 of distilled water there was a rotation from right to left, of 12°13'20", with a thickness of 152 millimetres.
The following table contains the results which our author obtained from different kinds of sugar:
| From Left to Right. | |---------------------| | Proportion of sugar in 1 of the solution | Arc of rotation of red light in a thickness of 152 millimetres | Molecular power of rotation | | Sugar of canes, syrup of...0·25 | 1·0592 | 23°23'45" | | | 0·50 | 1·2311 | 32°7'30" | | | 0·65 | 1·3114 | 37°11'15" | | Sugar of milk...0·14 | 1·0597 | 10°21'40" | | Starch...0·65 | 1·2460 | 48°30'0" | | Crystallisable principle of honey...0·34 | 1·1329 | 16°47'30" | | Sugar of grapes... | | |
| From right to left. | |---------------------| | Juice of grapes white... | 160 millimetres | 3°2 | | Do. red and white mixed... | | 6°25 | | Chasselas... | | 5°50 | | Muscat... | | 3°33 | | Verjus... | | 1°81 | | Chasselas of Fountainbleau... | | 8°00 | | Common black grape... | | 10°00 | | Apples for cider... | | 3°33 | | Red gooseberries very ripe... | | 1°81 | | Berries of the service tree... | | 2°5 |
M. Biot also found, that claret, white champagne, alcohol, sulphuric ether, citric acid dissolved in the proportion of 30 to 37 of water, sulphuric acid pure and colourless, and olive oil, produced no effect upon polarised light.
He found, however, that tartaric acid dissolved in the proportion of 53 of acid to 52 of water produced a rotation of 8° from right to left.
One of the most curious discoveries contained in M. Biot's memoir, is that of the powerful rotatory property of dextrine, an uncrystallisable syrup which is found in the farina of rice, wheat, and even of ligneous tissue. It differs from gum in producing an opposite rotation, and from sugar in its superior power of rotation, which is almost triple of that which is exerted by sugar of canes. It surpasses all animal and vegetable substances at present known, at equal densities and thicknesses, in turning round the plane of a polarised ray, polarisation rock crystal only being superior to it. The name of dextrine has been given to it in order to mark the direction as well as the powerful energy of its force of rotation.
M. Biot and M. Persoz have more recently found a sugar of starch, whose power of rotation is almost equal to that of crystallised sugar of canes. They have also discovered that, when sugar of canes is dissolved in water mixed with dilute sulphuric acid, and heated below the boiling point, it loses its power of turning the planes of polarisation from left to right, like sugar of grapes not solidified. Sugar of starch submitted to the same process did not experience this inversion.
M. Biot likewise found, that the colouration of liquids did not exercise any influence on their rotatory property.
We regret that want of room prevents us from giving any account of M. Biot's more recent, and extremely interesting researches on the growth and nutrition of plants. These researches have a high practical value, and shew in a striking manner that the most abstract researches in physical science will sooner or later find some useful application.
CHAP. III.—ON ELLIPTICAL POLARISATION.
The phenomena and laws of elliptical polarisation, as they are at present known, have been investigated only by Sir David Brewster. This species of polarisation forms the connecting link between plane polarisation and circular polarisation, passing nearly into the former when exhibited by galena and into the latter when exhibited by silver.
Sir David Brewster found at an early period, what Malus had previously observed, that the light reflected from metals was polarised in different planes. The former, however, found, that the pencil polarised in the plane of reflection was always more intense than that polarised in a perpendicular plane, and which he conceived had entered the metal and been partially absorbed. He found the difference between the intensities of these pencils to be least in silver and greatest in galena, metals which had an intermediate effect, being arranged as in the following table.
Order in which the metals polarise most light in the plane of reflection.
| Galena. | Steel. | Copper. | Fine gold. | |---------|-------|--------|-----------| | Lead. | Zinc. | Tin plate. | Common silver. | | Grey cobalt. | Speculum metal. | Brass. | Pure silver. | | Arsenical cobalt. | Platinum. | Grain tin. | Total reflection | | Iron pyrites. | Bismuth. | Jewellers' gold. | from glass. | | Antimony. | Mercury. |
He found also that, by increasing the number of reflexions, the whole of the light could be polarised in one plane. The white light of a wax candle 10 feet distant, is polarised by eight reflexions from steel between 60° and 80°, whereas it requires more than thirty-six from silver, and in total reflexion where the elliptical polarisation becomes circular; and when the two pencils are equal, the total polarisation of the pencil cannot be effected by any number of reflexions.
The action of metals upon polarised light presents us with a series of very extraordinary phenomena. One of the first of these which was discovered by our author, was the action of successive reflections from silver and gold upon polarised light. These reflexions are made from two metallic plates placed between the polarising and analysing plate or rhomb, and when the plane of metallic reflexions is parallel or perpendicular to the plane of primitive polarisation, its Elliptical azimuth will be $0^\circ$, $90^\circ$, $180^\circ$, &c., and when it is inclined $45^\circ$ polarisation to that plane, its azimuth will be $45^\circ$, $135^\circ$, &c.
In azimuth $0^\circ$ no colours are observed by reflexions from two plates of silver placed parallel to one another, just as in the case of crystallised laminae whose axes are in $0^\circ$ of azimuth.
At $45^\circ$, $135^\circ$ of azimuth, the most brilliant complementary colours are seen, either by turning round the analysing plate, or by using a rhomb of spar that gives two images. These colours become fainter and fainter while the azimuth changes from $45^\circ$ to $90^\circ$, or from $45^\circ$ back to $0^\circ$, exactly like those of crystallised laminae.
When a small number of reflexions are used, the tints are fainter and less brilliant, and they increase with the number of reflexions. There is an angle of reflexion about $75^\circ$, at which the tints are brightest, and they become fainter as the reflexions are performed at greater or at less angles.
All the other metals in the preceding table, as well as total reflexion from glass, give analogous colours, but they are most brilliant in silver, and diminish towards galena.
Now, it is obvious, that at about $75^\circ$ of incidence (if we suppose the metal to be steel) the polarised light which it reflects has acquired some new physical property.
1. It is neither common light nor partially polarised light, because if we reflect it a second time at $75^\circ$ it is restored to light polarised in one plane.
2. It is not polarised light, because it does not vanish during the revolution of the analysing rhomb.
In order then to discover its nature, let it be transmitted along the axis of Iceland spar. The common uniaxial system of rings is changed into that shewn in fig. 102, which is similar to the effect produced by crossing the uniaxial system of rings with a thin film polarising the pale blue of the first order. If we substitute for the Iceland spar, films of sulphate of lime, we shall find that their tints are increased or diminished according as the metallic action coincides with or opposes that of the crystal. This experiment led our author into the erroneous opinion that metals acted like crystallised plates; but when he found that a second reflexion at $75^\circ$ destroyed the effect of the first reflexion, he saw that this opinion was untenable; and was led to consider the phenomena as having some resemblance to those of circular polarisation.
We have already seen, that a circularly polarised ray RP, fig. 161, emerges after two total reflexions in the direction ST, polarised $45^\circ$ to the plane of the two reflexions in every azimuth. Now, if we reflect a ray of light R polarised $+45^\circ$ at Q from one plate of silver CD, the rays QS will have acquired a property analogous to that of circular polarisation; for if it is reflected a second time at S, the reflected ray ST will emerge polarised $39^\circ$ $48'$ to the plane of reflexion. Now the difference between this result and that from total reflexion, is that one reflexion from silver impresses the same character upon light, whereas in total reflexion two reflexions are necessary. Another point of difference is, that when the ray is restored by the same number of reflexions, it is not wholly restored to a plane $-45^\circ$, but only to a plane $-39^\circ$ $48'$. But there is another difference of a very interesting kind. In circular polarisation, the ray has the same properties on all its sides, and the angles of reflexion at which it is restored to polarised light in different azimuths, are all equal to the radius of a circle described round the rays. "Hence," says our author, "without any theoretical reference the term circular polarisation is from this and other facts experimentally appropriate." In like manner, without referring to the theoretical existence of elliptical vibrations produced by the interference of two rectilineal vibrations of unequal amplitudes, we may give to the new phenomenon the name of elliptic polarisation, because the angles of reflexion at which this kind of light is restored to polarised light, may be represented by the variable radius of an ellipse.
Now, it is a curious fact, that while silver restores the ray to angles of $39^\circ$ $48'$, other metals restore it to angles deviating more and more from $45^\circ$, as is shewn in the following table:
| Metal | Angle of reflexion | |----------------------|--------------------| | Bismuth | $21^\circ$ $0'$ | | Speculum metal | $21^\circ$ $0'$ | | Zinc | $19^\circ$ $10'$ | | Steel | $17^\circ$ $0'$ | | Iron pyrites | $14^\circ$ $0'$ | | Antimony | $15^\circ$ $15'$ | | Arsenical cobalt | $13^\circ$ $30'$ | | Cobalt | $12^\circ$ $30'$ | | Lead | $11^\circ$ $0'$ | | Galena | $10^\circ$ $0'$ | | Specular iron | $2^\circ$ $0'$ |
Hence it appears that the elliptic polarisation passes into circular nearly in the case of silver, and into plane polarisation in the case of galena; the ellipse becoming nearly a circle in the former case, and a straight line in the latter.
As light polarised $+45^\circ$ suffers different degrees of elliptical polarisation by one reflexion from metals, and is restored again to polarised light, though in different planes, by a second reflexion, so it exhibits the same phenomena at $3$, $5$, $7$, $9$ in reflexions, and is restored to polarised light by $4$, $6$, $8$, $10$ reflexions at the same angle. The following table shews the inclination of the plane of polarisation of the restored ray to the plane of reflexion, in various numbers of reflexions from silver and steel.
| Number of Reflexions | Inclination of the plane of the polarised ray | |----------------------|-----------------------------------------------| | | Steel | | 2 | $17^\circ$ $0'$ | | 4 | $5^\circ$ $22'$ | | 8 | $1^\circ$ $38'$ | | 16 | $0^\circ$ $90'$ | | 32 | $0^\circ$ $3^\circ$ | | 64 | $0^\circ$ | | 128 | $0^\circ$ | | 256 | $0^\circ$ |
These results shew in the clearest manner the reason why common light is polarised by $8$ reflexions from steel, and not till after $36$ reflexions from silver, the planes of inclination of the two rectangularly polarised rays requiring in each case that number of reflexions to bring them into a state of parallelism.
The angles at which elliptical polarisation is produced by one reflexion, may be regarded in the present state of our knowledge of the subject as the angle of maximum polarisation, and its tangent as the index of refraction of the metal, as given in the following table:
| Name of metal | Angles of maximum polarisation | Index of refraction | |------------------------|--------------------------------|---------------------| | Grain tin | $78^\circ$ $39'$ | $4.913$ | | Mercury | $78^\circ$ $27'$ | $4.698$ | | Galena | $78^\circ$ $10'$ | $4.733$ | | Iron pyrites | $77^\circ$ $30'$ | $4.311$ | | Grey cobalt | $76^\circ$ $56'$ | $4.309$ | | Speculum metal | $76^\circ$ $0'$ | $4.011$ | | Antimony melted | $75^\circ$ $25'$ | $3.844$ | | Steel | $75^\circ$ $0'$ | $3.782$ | | Bismuth | $74^\circ$ $50'$ | $3.689$ | | Pure silver | $73^\circ$ $0'$ | $3.271$ | | Zinc | $72^\circ$ $30'$ | $3.172$ | | Tin plate hammered | $70^\circ$ $50'$ | $2.879$ | | Jewellers' gold | $70^\circ$ $45'$ | $2.864$ |
1 See Sir John Herschel's Treatise on Light, § 1050. We may produce elliptical polarisation by a sufficient number of reflexions at any given angle, in the same manner as in plane polarisation. The following table contains the results of observations made with steel:
| Number of reflexions at which elliptical polarisation is produced | Number of reflexions at which the pencil is restored to a single plane | Observed angle of incidence | |---------------------------------------------------------------|---------------------------------------------------------------|-----------------------------| | 3 9 15, &c. | 6 12 18, &c. | 86° 0' | | 24 74 124, &c. | 5 10 15, &c. | 84° 0' | | 2 6 10, &c. | 4 8 12, &c. | 82° 20' | | 14 44 74, &c. | 3 6 9, &c. | 79° 0' | | 1 3 5, &c. | 2 4 6, &c. | 75° 0' | | 14 44 74, &c. | 3 6 9, &c. | 67° 40' | | 2 6 10, &c. | 4 8 12, &c. | 60° 20' | | 2; 74 124, &c. | 5 10 15, &c. | 56° 25' | | 3 9 12, &c. | 6 12 18, &c. | 52° 29' |
At an incidence of 67° 40' elliptical polarisation is produced by 14, 44, 74 reflexions. Hence we draw the interesting conclusion, that the ray must have completed its elliptical polarisation in the middle of the second and fifth reflection; that is, when it had reached its greatest depth within the metallic surface. It then begins to resume its state of polarisation in a single plane, and recovers it at the end of the 3rd, 5th, and 7th reflection. Another very interesting effect is produced when one reflection is made on one side of the polarising angle, and the other reflection on the other side. A ray that has been partially elliptically polarised by one reflection at 85° does not as in plane polarisation acquire more by a reflection at 54°, but it retraces its course, and recovers its state of single polarisation.
We have seen that by two reflexions, there is only one angle, viz. 73° for silver, at which the elliptically polarised ray can be restored to plane polarisation. At three reflexions there are two angles, viz. 63° 43', and 79° 40', at which the restoration can take place, at four reflexions three angles, and so on. This phenomena is exhibited to the eye in fig. 103, where the concentric arches I, I; II, II, &c., represent the quadrant of incidence from one, two, &c., reflexions, B being the point of 90° and C that of 0°. The point D on the line A is the point or line of maximum elliptic polarisation, viz. 73° for silver, and the figures 1, 2, 3, 4, 5, indicate the points or nodes of restoration, and their distances from C, the corresponding angles of incidence at which the restoration takes place. The loops or double curves lying between the points 1, 2, 3, &c., are drawn to give an idea of the intensity of the elliptic polarisation which has its minimum at 1, 2, 3, &c., and its maximum at the white intermediate parts. These points of maximum intensity do not bisect the loops, or are not equidistant from the minima; but such is their relation that the maximum for n reflexions is the minimum for 2n reflexions. These phenomena lead us to the explanation and analysis of the complementary colours which accompany elliptical and circular polarisation.
On the Colours of Elliptical and Circular Polarisation.
When the preceding experiments are made with homogeneous light, we find that the points and angles of restoration vary for the differently coloured rays. Thus in silver we have the maximum polarising angle as follows:
| Corresponding index of refraction | |----------------------------------| | For red light.................. 76° 3' 3-866 | | For yellow light................. 73° 3-271 | | For blue light.................. 70° 2-824 |
Hence it is obvious that at the point of restoration where the blue rays are restored, and vanish, the red rays are not restored, and consequently will appear when the principal section of the analysing rhomb is in the plane of reflection.
Here, then, we have the cause of the phenomena of the elliptical complementary colours seen in reflection from metals. They polarisation are analogous to the colours in oil of cassia, and chromate of lead at the maximum polarising angle.
But the remarkable result of the preceding measures is, that in metallic as well as in total reflection, the index of refraction is less for blue than for red light, or in the language of the undulatory theory, the refractive index increases with the length of the wave. In a recent communication to the Royal Irish Academy, on the propagation of light in uncrystallized media, Professor Lloyd has obtained an expression for the velocity of the propagation of light, each of its terms consisting of two parts with opposite signs, one of which is due to the action of the ether and the other to that of the body. Conceiving, therefore, that there may be bodies in which the principal term is nearly nothing, the principal part of the expression will be that derived from the second term; and if that term be taken as an approximate value, it will follow that the refractive index of the substance must be in the subduplicate ratio of the length of the wave nearly. "Now," says Professor Lloyd, "it is remarkable that this law of dispersion, so unlike anything observed in transparent media, agrees pretty nearly with the results obtained by Sir David Brewster in some of the metals. In all these bodies, the refractive index (inferred from the angle of maximum polarisation) increases with the length of the wave. Its values for the red, mean, and blue, ray in silver, are 3-866, 3-271, 2-824, the ratios of the second and third to the first being .86 and .73. According to the law above given, these ratios should be .88 and .79."
Professor Maccullagh has very recently endeavoured to represent the phenomena described in the preceding pages by empirical formulae, in the same manner as Fresnel represented those of total reflection. The following is a brief abstract of Professor Maccullagh's researches, which we shall give in his own words.
"The author observes, that the theory of the action of metals upon light is among the desiderata of physical optics, whatever information we possess upon this subject being derived from the experiments of Sir David Brewster. But, in the absence of a real theory, it is important that we should be able to represent the phenomena by means of empirical formulae; and accordingly, the author has endeavoured to obtain such formulae by a method analogous to that which Fresnel employed in the case of total reflection at the surface of a rarer medium, and which, as is well known, depends on a peculiar interpretation of the sign $\sqrt{-1}$. For the case of metallic reflection, the author assumes that the velocity of propagation in the metal, or the reciprocal of the refractive index, is of the form
$$m (\cos \chi + \sqrt{-1} \sin \chi);$$
without attaching to this form any physical signification, but using it rather as a means of introducing two constants (for there must be two constants, m and $\chi$, for each metal) into Fresnel's formulae for ordinary reflection, which contain only one constant, namely, the refractive index.
"Then if $i$ be the angle of incidence on the metal, and $i'$ the angle of refraction, we have
$$\sin i' = m (\cos \chi + \sqrt{-1} \sin \chi) \sin i,$$
and therefore we may put
$$\cos i' = m' (\cos \chi - \sqrt{-1} \sin \chi) \cos i,$$
if
$$m'^2 \cos^2 i = 1 - 2m' \cos 2\chi \sin^2 i + m'^4 \sin^4 i,$$
and
$$\tan 2\chi = \frac{m'^2 \sin 2\chi \sin^2 i}{1 - m'^2 \cos 2\chi \sin^2 i}.$$
"Now, first, if the incident light be polarised in the plane..." Elliptical of reflexion, and if the preceding values of $\sin \psi$, $\cos \psi$, be substituted in Fresnel's expression
$$\frac{\sin (i-\psi)}{\sin (i+\psi)}$$
for the amplitude of the reflected vibration, the result may be reduced to the form
$$a (\cos \delta - \sqrt{1 - \sin^2 \delta}),$$
if we put
$$\tan \psi = \frac{m}{m'},$$
$$\tan \delta = \tan 2\psi \sin (\chi + \chi'),$$
$$a^2 = \frac{1 - \sin 2\psi \cos (\chi + \chi')}{1 + \sin 2\psi \cos (\chi + \chi')}.$$
Then according to the interpretation before alluded to, of $\sqrt{-1}$, the angle $\delta$ will denote the change of phase, or the retardation of the reflected light; and $a$ will be the amplitude of the reflected vibration, that of the incident vibration being unity. The values of $m'$, $\chi'$, for any angle of incidence, are found by formulae (3), (4), the quantities $m$, $\chi$ being given for each metal. The angle $\chi'$ is very small, and may in general be neglected.
"Secondly, when the incident light is polarised perpendicularly to the plane of reflexion, the expression
$$\tan (i-\psi)$$
$$\tan (i+\psi)$$
treated in the same manner, will become
$$a'(\cos \delta' - \sqrt{1 - \sin^2 \delta'})$$
if we make
$$\tan \psi' = mm',$$
$$\tan \delta' = \tan 2\psi' \sin (\chi - \chi'),$$
$$a'^2 = \frac{1 - \sin 2\psi' \cos (\chi - \chi')}{1 + \sin 2\psi' \cos (\chi - \chi')}.$$
and here, as before, $\delta'$ will be the retardation of the reflected light, and $a'$ the amplitude of its vibration.
"The number $m = \frac{1}{m}$ may be called the modulus, and the angle $\chi$ the characteristic of the metal. The modulus is something less than the tangent of the angle which Sir David Brewster has called the maximum polarising angle. After two reflections at this angle, a ray originally polarised in a plane inclined $45^\circ$ to that of reflexion, will again be plane polarised in a plane inclined at a certain angle $\phi$ (which is $17^\circ$ for steel) to the plane of reflexion; and we must have
$$\tan \phi = \frac{a'^2}{a^2}.$$
Also, at the maximum polarising angle we must have
$$\delta' = 90^\circ.$$
And these two conditions will enable us to determine the constants $m$ and $\chi$ for any metal, when we know its maximum polarising angle and the value of $\phi$; both of which have been found for a great number of metals by Sir David Brewster. The following table is computed for steel, taking $m = 3$, $\chi = 54^\circ$.
| $\delta$ | $\delta'$ | $a^2$ | $a'^2$ | $\frac{1}{2}(a^2 + a'^2)$ | |---------|----------|-------|--------|------------------| | 0° | 27° | .526 | .526 | .526 | | 30° | 31° | .575 | .475 | .525 | | 45° | 38° | .638 | .407 | .522 | | 60° | 54° | .729 | .308 | .518 | | 75° | 98° | .850 | .240 | .545 | | 85° | 152° | .947 | .491 | .719 | | 90° | 180° | 1 | 1 | 1 |
"The most remarkable thing in this table is the last column, which gives the intensity of the light reflected when polarised common light is incident. The intensity decreases very slowly up to a large angle of incidence, (less than $75^\circ$,) and then increases up to $90^\circ$, where there is total reflexion. This singular fact, that the intensity decreases with the obliquity of incidence, was discovered by Mr. Potter, whose experiments extend as far as an incidence of $70^\circ$. Whether the subsequent increase which appears from the table indicates a real phenomenon, or arises from an error in the empirical formulae, cannot be determined without more experiments. It should be observed, however, that in these very oblique incidences Fresnel's formulae for transparent media do not represent the actual phenomena for such media, a great quantity of the light being stopped, when the formulae give a reflexion, very nearly total.
"The value of $\delta' - \delta$, or the difference of phase, increases from $0^\circ$ to $180^\circ$. When a plane-polarised ray is twice reflected from a metal, it will still be plane-polarized if the sum of the values of $\delta' - \delta$ for the two angles of incidence be equal to $180^\circ$.
"It appears from the formulæ, that when the characteristic $\chi$ is very small, the value of $\delta'$ will continue very small up to the neighbourhood of the polarising angle. It will pass through $90^\circ$, when $mm' = 1$; after which the change will be very rapid, and the value of $\delta'$ will soon rise to nearly $180^\circ$. This is exactly the phenomenon which Mr. Airy observed in the diamond.
"Another set of phenomena to which the author has applied his formulæ are those of the coloured rings formed between a glass lens and a metallic reflector; and he has thus been enabled to account for the singular appearances described by M. Arago in the Memoires d'Arcueil, tom. 3, particularly the succession of changes which are observed when common light is incident, the intrusion of a new ring, &c. But there is one curious appearance which he does not find described by any former author. It is this. Through the last twenty or thirty degrees of incidence the first dark ring, surrounding the central spot, which is comparatively bright, remains constantly of the same magnitude; although the other rings, like Newton's rings formed between two glass lenses, dilate greatly with the obliquity of incidence. This appearance was observed at the same time by Professor Lloyd. The explanation is easy. It depends simply on this circumstance, (which is evident from the table,) that the angle $180^\circ - \delta'$, at these oblique incidences, is nearly proportional to $\cos i$.
"As to the index of refraction in metals, the author conjectures that it is equal to $\frac{m}{\cos \chi}$.
In concluding our account of the phenomena of Physical Undulations of Optics, we could have wished to have given a popular account of the undulatory theory of light, and of the explanation which it affords of a great variety of the most interesting phenomena in optics. This however has been done to such an extent by Dr. Thomas Young, in the article Chromatics in this work, and in the article Polarisation by M. Arago, that we would not be justified in entering again upon the subject. Notwithstanding, however, its acknowledged power in accounting for such a variety of facts, there are many of the principal phenomena in optics of which it is not capable of giving the least explanation; and there are others to which it seems in its present form to be totally inapplicable. We look forward, however, with much hope, to the labours of Professor Lloyd, and Professor Macculagh, and Professor Powell, in this country, who, in place of checking the progress of enquiry, as has been lately done, by declaring the theory of undulations to be
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1 See Edinburgh Journal of Science, N. S. Oct. 1830, vol. iii. p. 278. PART VIII. ON THE APPLICATION OF OPTICS TO THE EXPLANATION OF NATURAL PHENOMENA.
As several of the subjects which belong to this branch of Optics have been treated pretty fully in other parts of this work, we must confine our attention to topics which have not been previously discussed.
Sect. I.—On the Rainbow.
A general description of the rainbow has already been given among the optical phenomena in Meteorology. In order to explain the progress of the rays of light which form the two bows, let R, R, R, fig. 104, be parallel rays proceeding from the sun, situated at the back of the observer, placed at O, and let them fall on drops of rain E, F, G, H, in front of the observer. Some of these rays of light will enter the spherical drops of rain, and those which fall perpendicularly and nearly so, will be transmitted through the drop, and of course never reach the observer at O. Other rays, however, especially those which fall obliquely, will be separated into the prismatic colours at the first refraction, and will subsequently be reflected once, twice, and more times, within the drop, and emerge after one, two, or more reflections in different directions. Now, it is obvious that there will be some position of the drops, such as E, F, at which rays that have suffered one reflection will reach the eye of the observer at O. Drops above these will throw the rays which they reflect after one reflection above O, and drops below these will throw the same rays below O. In like manner, there must be some position, as at H and G, at which other drops in which the light that has suffered two reflections will fall upon the eye at O, the drops above these throwing the rays above, and the drops below these throwing the rays below O. Now, each drop forms by refraction a prismatic spectrum, or coloured and elongated image of the sun. The rays RE, RF, which reach the eye at O, must fall upon the lower drops E, F, on their upper side, as shown in the figure, and consequently (as may be found by tracing the rays through the drops) the spectrum which they form will have the red rays uppermost, as at r, near F, and the violet rays downwards, as at v, near E; and as the same effect will be produced from all the other drops which reflect the sun's rays to the point O, there will appear to an eye at O, a coloured bow, such as we see it in the heavens, with all the colours of the spectrum, as if they had been formed from the sun's image, by a prism of water that produced the same degree of refraction. In like manner, the rays that enter the lower side of the drops will form an inverted spectrum, after two reflexions, in which the red rays are below, and the violet ones above; and as this spectrum is much fainter, it will give a second coloured bow, fainter than the first, and having its red side below, and its violet side above. The following are the dimensions of the two bows:
| Radius of the red edge of the inner bow | ...42° 2' | |----------------------------------------|-----------| | Violet edge | ...40° 17' | | Breadth of the inner bow | ...1° 45' | | Radius of the violet edge of the outer bow | ...54° 7' | | Red | ...50° 57' | | Breadth of the outer bow | ...3° 10' | | Distance between the bows | ...8° 55' |
Dr. Halley has shewn that the solar rays which suffer three reflexions, will form a bow round the sun at the distance of 40° 28', and that those which suffer four reflexions will form a bow at the distance of 45° 33' from the sun; but the light which reaches the eye after so many reflexions is too faint to be seen, and these bows have consequently never been discovered.
Supernumerary bows of red and green light, to the number of three, have been seen in contact with the violet arch of the inner bow, and we have seen them also on the outside of the outer or secondary bow. The cause of these is not known, but a very ingenious explanation of them has been given in Chromatics, vol. xi. p. 634, sect. iii.
Sir David Brewster, upon examining the two rainbows with a rhomb of Iceland spar, found that they consisted wholly of light polarised in the plane of reflexion within the drop, or in planes coincident with the radii of the bow. The two bows present a case of conical polarisation, the part of the bow vanishing as the principal section of the rhomb becomes parallel to its radius. It is strange that the polarisation of the bow, and consequently of light, had not been discovered when it happened to be seen by reflexion on panes of glass, or other reflecting substances, lying with their planes of reflexion perpendicular to the planes of reflexion within the drop, and near the angle of maximum polarisation. See Meteorology, vol. xiv. p. 749.
Sect. II.—On Halos and Parhelia.
The name of halos and parhelia have been given to circles round the sun and moon, some of which are extremely complicated and beautiful. The general theory of this class of phenomena has been given by Dr. Young, in the article Chromatics, vol. vi. p. 634, sect. ii., and a description of the phenomena themselves, in the article Meteorology, vol. xiv. p. 749.
The production of halos, which have their origin in the artificial refraction and reflexion of crystals of ice floating in the atmosphere, may be illustrated by the following method given by Sir David Brewster. A few drops of a saturated solution of alum spread over a plate of glass, so as to crystallise rapidly, will cover the glass with an imperfect crust, which is composed, when examined by the microscope, of flat octahedral crystals, scarcely visible to the eye. If the observer places his eye behind this plate, and close to its smooth side, he will see the sun or the candle encircled with three fine halos, placed at different distances. The interior one, which is the whitest, is formed by the refraction of the rays through two of the faces that have the least inclination to each other, and consequently give a spectrum in which the colours are not greatly dispersed; and as a similar pair of refracting planes lie in every direction, there will be a spectrum in every direction, and consequently a rainbow of a circular form. The second halo, which is blue without and red within, with all the intermediate prismatic colours more highly dispersed, is formed by a pair of faces more inclined. The third halo, which is larger and more brilliantly coloured, is produced by a third pair of refracting planes, having a greater refracting power, and consequently giving a higher dispersion. When the granular crystals have double refraction, and when they crystallise with their axes perpendicular to the plates, combinations of greater variety and beauty will be produced.
We have endeavoured, by looking through hoar frost upon glass, to produce halos actually resembling those seen in nature; but we have not succeeded, though we have no doubt that it may be effected by causing vapour deposited under a variety of circumstances, to be frozen in different ways.
Sect. III.—On the Unusual Refraction of the Atmosphere.
One of the most interesting applications of optical science The explanation which it affords of the extraordinary phenomena which arise from difference of density in different parts of the atmosphere. As this subject has been treated very fully in our article Meteorology, vol. xiv., and illustrated with some interesting figures, we shall confine ourselves at present to an account of the most extraordinary of all the phenomena of this kind which have been observed and correctly described. It was observed by Dr. Vince, on the 6th of August 1806, about seven o'clock in the evening. Between Ramsgate and Dover there is a hill, over which the tops of the four turrets of Dover castle are usually seen to a person at Ramsgate. At the time above mentioned, however, Dr. Vince, when at Ramsgate, not only saw the four turrets e, x, w, y, but the whole of the castle, m, n, s, r, appearing as if it were situated on the side of the hill next to Ramsgate, and rising as much above the hill AB as usual, as if it had been brought over and placed on the Ramsgate side of the hill, fig. 105. This appearance continued about twenty minutes. Between Ramsgate and the land from which the hill rises, there is about six miles of sea, and from thence to the top of the hill about the same distance, the height of the eye above the surface of the sea being about seventy feet. It is a very singular circumstance in this phenomenon, that the image of the castle was so very strong and well defined that the hill itself did not appear through the image.
In order to explain this phenomenon, Dr. Vince supposes AB, fig. 106, to represent the castle, FC the cliff of Ramsgate, BTD the hill, DC the sea, E the place of the spectator, T the top of the hill, AyeE a ray of light coming from the top of the castle to the observer, and Bxe another ray coming from the bottom of the castle, and TxeE a ray from the summit of the hill, reaching the eye at E, in a direction between those of the other two rays; then it is obvious that such a disposition of the rays will produce the observed appearance. In order to give such a refraction, the density of the air between yxe and xeE must have varied with great rapidity, so as to increase the curvature of the ray TxeE, after it cuts BxeE in x, in order to make the ray TxeE fall between the other two rays. See Edinburgh Transactions, vol. vi. p. 245.
Sect. IV.—On the Colours of the Atmosphere.
As the earth's atmosphere acts upon light like all other transparent bodies, and is continually changing its chemical, mechanical, and hygrometrical condition, it acts upon light in very different ways, under different circumstances. As the colour of the sky is absolutely black on the tops of the highest mountains, its blue colour in the regions which we inhabit is owing to the action of the atmosphere.
That the blue light of the sky is light that has suffered reflexion from the particles of our atmosphere, is proved by the fact observed by Sir David Brewster, that this blue light is polarized in a plane passing through the observer's eye and the sun. This fact is well illustrated by the discovery which we owe to the same author, of atmospheric lines in the spectrum, formed by the blue sky. These lines are principally in the more refrangible spaces, as already described, and hence the prevailing light is blue.
The splendid colours which mark the rising and the setting of the sun, varying from the deepest red to orange, yellow, and even green, arise from the same cause; for when we analyse these various lights with the prism, we find that they are owing to definite parts of the spectrum having been absorbed by the atmosphere.
The phenomenon of blue shadows is finely seen when the sky is particularly blue. It arises solely from the shadows being illuminated by the blue sky, while the part round the shadow is illuminated by the sun, or by the light of a candle. If the light of the sun passes at the time through vapours so as to make it yellow or orange, the contrast of the shadow is still more striking and beautiful. The phenomenon of blue shadows in great perfection.
The subject of the blue colour of the sky has been treated at considerable length in our article on Meteorology, vol. xiv. p. 760, and an instrument called a Cyanometer, for measuring its blue colour, has been well described, and an account of the experiments made with it fully detailed.
Sect. V.—On the Colours of Natural Bodies.
The splendid colours which appear in the natural world have long attracted the attention of philosophers; but no natural person ever had the courage to give a philosophical theory of them but Sir Isaac Newton. When he had completed his analysis of the colours of thin plates, he conceived that they furnished the true cause of the colours of natural bodies. If we take a thin film of mica, a few millionths of an inch in thickness, it appears to the eye of a bright blue colour. Sir Isaac Newton maintained, that if this film could be cut into a great number of minute parts of the same thickness as itself, these particles would "keep their colour, and a heap of them constitute a mass or powder of the same colour which the plate exhibited before it was broken." A plate of mica of a different thickness, would be green, another yellow, and another red, and all these, if broken down into particles "of the same thickness with the plates," would, of course, according to our author, give a green, a yellow, or a red mass.
We have already seen that different thicknesses of a transparent plate like mica, give various orders of colours; each different tint corresponding with a particular thickness. Considering then the particles of all bodies whatever as transparent, and as having different sizes, they will produce colours corresponding to these different sizes, and consequently we shall have as great a variety of tints in nature as there are varieties in the sizes of the particles of bodies. A difficulty, however, here presents itself. The colours arising from thin plates vary rapidly by inclining them to the incident light, whereas those of coloured media suffer no such change. Hence Sir Isaac Newton, was driven to the supposition that the particles of bodies have such an enormous refractive power, that the paths of the rays refracted by a parallel film will not differ much in length from, and consequently not be very oblique to, a perpendicular line. After explaining this theory, Sir Isaac ventures to affix to different natural colours the order to which they belong, the very tint of that order, and consequently the thickness of the particles which produce the colour. He says, for example, that the green colour of all vegetables, the most general tint in nature, is a green of the third order, and that the blue colour of the sky is a blue of the first order. Now we know the composition of a green of the third order, and of a blue of the first order, as given by Sir Isaac Newton himself. The green of the third order "is principally constituted of original green, but not without a mixture of some blue and yellow;" that is, it consists of all the rays of the green space, with the least refrangible rays of the blue space, and the most refrangible rays of the yellow space, and it does not contain a single ray of indigo, violet, orange, or red light.
Such being the case, it occurred to Sir David Brewster that the green colour of plants could be accurately analysed by the prism, and having extracted by means of alcohol the green juice of a great variety of vegetable bodies, he analysed their colours by the prism. In all bodies he found the composition of this green colour to be identically the same; but it had no relation whatever to the green of the first order. It contained portions of all the colours of the spectrum; and the prismatic spectrum seen through these green juices was divided unequally into six luminous bands of various breadths, separated by dark intervals. In the same manner he found that the blue colour of the sky was not a blue of the first order.
From a series of experiments in which the same author has been engaged, he has been led to the conclusion that absorption is the cause of this extensive class of colours; and that all the colours of natural bodies arise from the interference of light, by which certain rays are extinguished. When the interference takes place as in thin plates, between the light reflected from the two surfaces, and between the direct transmitted ray and other transmitted rays which suffer reflexion within the thin plates, we have two colours complementary to each other; but even in this case, when the number of films is great, as in decomposed glass, the transmitted colours lose all their resemblance to the colours of thin plates, while the reflected tints are exceedingly brilliant and metallic in their lustre.
In coloured fluids and coloured glasses, and coloured gaseous media, the interference arises from rays that acquire different velocities in passing through the coloured medium, one part of the introducted light passing through the particles, and the other through the intervening spaces. Hence there are no reflected tints in such coloured media.
Those who wish to study the Newtonian theory of the colours of natural bodies, are referred to Biot's Traité de Physique, where it is illustrated and defended with all the sagacity and talent of that distinguished philosopher. See also an account of various ingenious experiments hostile to the theory, in a paper by Mr. Delaval in the Manchester Memoirs, vol. ii. p. 131.
Sect. V.—On the Eye and on Vision.
In our article on Anatomy, we have already given a full description of the organ of vision, and Plate XXXIII, fig. 3, exhibits a fine section of the eye after Soemmerring. The following dimensions of the eye have been given by Dr. Thomas Young, the measures being taken with great care from his own eye:
| Dimensions of the eye | |-----------------------| | Length of optical axis | 94 | | Vertical chord of the cornea | 45 | | Versed sine of ditto | 11 | | Horizontal chord of the cornea | 49 | | Aperture of the pupil seen through the cornea | 27 to 13 | | Diminished in consequence of the magnifying power of the cornea | 25 to 12 | | Radius of the anterior surface of the crystalline lens | 30 | | Radius of the posterior surface | 22 | | Distance of the optical centre from the anterior surface of the lens | 10 | | Distance of the optical centre of the lens from the cornea | 22 | | Focal length of the cornea for objects 10 inches distant | 115 | | Joint focus of cornea and lens | 69 | | Principal focal distance of lens | 173 | | Distance of the centre of the optic nerve from the point opposite the pupil | 11 | | Range of the eye, or field of vision | 110 |
The following measures of the crystalline lens and cornea were taken by Sir David Brewster and Dr. Gordon, from the eye of a female, above fifty years of age, a few hours after death.
| Dimensions of the crystalline lens | |-----------------------------------| | Diameter of the crystalline lens | 0.378 | | Diameter of the cornea | 0.400 | | Thickness of the crystalline lens | 0.172 | | Thickness of the cornea | 0.042 |
The following measures of the refractive powers of the humours of the eye, were taken by the same authors from the same eye:
| Humours of the eye | |-------------------| | Refractive power of water | 1.3333 | | Ditto of aqueous humour | 1.3866 | | Ditto of vitreous humour | 1.3394 | | Ditto of outer coat of vitreous | 1.3767 | | Ditto of middle coat of ditto | 1.3786 | | Ditto of central part of ditto | 1.3900 | | Ditto of the whole crystalline | 1.3839 |
The following measures may be occasionally useful:
| Measures | |----------| | From aqueous humour into the crystalline | 1.0466 | | Do., taking the mean index of the crystalline | 1.0353 | | From the crystalline into the vitreous humour | 0.930 |
If we execute a large diagram of the eye, and by means of Vision, the above indices of refraction trace the progress of parallel rays from the cornea to the retina, we shall find that they converge to points in or near to that membrane. The increase of density in the crystalline lens towards its centre is calculated to correct the spherical aberration, by bringing the central rays to the same focus with the marginal rays; but there is no provision in the eye for correcting the aberration of colour, because the purposes of vision do not require it to be corrected. It may be readily proved by tracing the rays diverging from both extremities of any object to the retina, and it may be also shown by direct experiment, that an inverted image of the object is formed upon that membrane. Now, it is a law of vision, that when a ray of light falls upon any point of the retina, the mind infers that the ray proceeded from a point in some line perpendicular to that point of the retina. Hence, as rays from the upper part of an object fall upon the lower part of the retina, and vice versa, such rays will seem to proceed from the upper part of the object, and all points of an object will be seen in the direction of the rays which issue from them, and consequently the object itself must appear erect, though its image is inverted.
As it is a law of vision that an object seen with a single eye is seen in a fixed direction, arising from the form of the retina as a whole, or from the form of its individual parts, then if rays from the same object fall upon another eye, or upon a hundred other eyes which have the power of placing the retina of all the eyes, so as to see the same object in the same direction, then the object thus seen must appear single. The only difference will be, that the object will be seen twice as bright with two eyes, and a hundred times as bright with a hundred eyes. If we place a hundred shillings in the same straight line, an eye whose axis coincides with the axis of the cylinder which they compose, will only see one shilling, and the same effect would be produced if the shillings were transparent. If the hundred eyes were placed with their axes in a hundred different directions, an hundred objects will be seen. Small objects are seen double and even triple with one eye when the crystalline lens is not uniform in its refractive power.
The defect of squinting may arise from several causes. It may be an original defect, in which the axis of the eye, or the line in which objects are seen most distinctly does not pass through the centre of the pupil. In this case it is incurable; but it is generally speaking a disease arising from an imperfection in one eye, from its having a different focal length from the other, from its giving a less distinct vision of objects, or from its muscles not being able to direct it as quickly as the other to visible objects. The consequence of this is, that as the observer can do without it, and uses only his best eye, the imperfect one does not follow the movements of the other, and therefore squints.
When we wish to see any object very distinctly, we invariably direct to it the axis of the eye, and it is a curious fact that there is no retina at the point where the axis meets the back of the eye, the foramen centrale corresponding to the extremity of the axis. When the eye thus sees an object with perfect distinctness, every other point of the same object is seen distinctly, and there is no adjustment of the eye by which distinctness of vision can be obtained at any distance from the axis of the eye, the only way of seeing distinctly being to direct the axis to the point we wish to examine.
The preceding fact leads to the opinion, that parts of an object seen in the retina cannot be seen distinctly, and yet if this were the case, we should be able to perceive upon an uniform ground the foramen centrale.
The opinion that the retina, though sensible to light, does not give perfectly distinct vision, is favoured by the fact, that when the image of any object falls upon the round base of the optic nerve, shewn in Plate XXXIII. fig. 3 of Anatomy, the object is not distinctly visible. This may be easily proved by fixing on the wall of a room, at the height of the eye, three wafers each two feet distant. Stand in front of the middle wafer with one eye shut, and beginning near the wall, withdraw gradually from it, (continuing to view the left hand wafer if the right eye is open, and the right hand wafer if the left eye is open,) till the middle wafer vanishes. This will be found to take place at five times the distance from the wall at which the wafers are placed, that is, at the distance of ten feet in the present case. If we use three candles, the middle one will not vanish like the wafer, but will become a cloudy mass of light.
The occasional insensibility of the retina to objects seen obliquely, was discovered by Sir David Brewster, who has illustrated it in the following manner. If we fix one eye on a particular point, such as the head of a pin stuck into a green cloth, and lay down a quill or strip of paper upon the green cloth, some inches distant from the pin, and then keep looking steadily at the pin's head, part of the quill, or the whole of it, will occasionally disappear, as if it had been wholly removed from the cloth. In a short time it will reappear, and again vanish. The very same effect is produced, though less readily, when both eyes are used, and when a luminous body is used in place of the quill. In this case the luminous body does not disappear, but expands into a mass of nebulous light, which is of a bluish white colour, encircled with a bright ring of yellow light.
But though we cannot see objects distinctly by oblique vision, yet they appear much brighter, and minute objects, especially luminous ones, are more easily seen by turning the axis of the eye away from them. Various astronomers have found that very faint stars, and the satellites of Saturn, which disappear when the eye is turned fully upon them, may be distinctly seen by directing the eye to another part of the field. This effect seems to arise from the expansion and enlargement of luminous points seen obliquely.
It has long been disputed, but the question has not been much agitated in modern times, whether the retina or the choroid coat behind it is the seat of vision. The insensibility of the base of the optic nerve, and the fact that vision is most distinct where there is no retina, are arguments in favour of the choroid coat being the seat of vision, as Mariotte believed. The transparency of the retina, and the opacity of the choroid coat were considered as additional arguments in favour of that opinion. Dr. Knox has shewn, that in the eye of the cuttle fish, there is a membranous opaque pigment in front of the retina, so that in this case the retina must receive the influence of light from the vibrations of this membrane just as it may receive them in other cases, from the same membrane placed behind it. M. Lehot, a living French author, has endeavoured to shew that the vitreous humour is the seat of vision; that we see the length, breadth, and thickness of every object as we do an image formed in smoke; and that the retina takes cognizance of this image in its three dimensions, by nervous filaments extending so far into the vitreous humour as to render any adjustment of the eye at different distances unnecessary.
The insensibility of the retina to direct impressions of faint light, was discovered by Sir David Brewster, who found, that when the eye directed its axis to objects faintly illuminated, it could not keep up a sustained vision of them. They disappeared and reappeared, and the eye was thrown into a state of painful agitation.
When we shut the eye quickly after looking at an object, we see it for an instant (about the seventh part of a second) in its own colours; but this impression is instantly followed by an image of the object in its complementary colours. If we look at a window at the end of a long passage, we shall first see, after shutting our eyes, a picture of the window, with black bars and white panes; but after the seventh of a second the picture is one with white bars and black panes. When we whirl a burning stick, we see a complete circle of red light, although the burning end of the stick can only be in one part of the circle at the same instant. This phenomenon of the retina is the principle of Dr. Paris's pretty toy, called the Thaumatrope, or Wonder-turner. A circular card with two strings is made to whirl rapidly round one of its diameters. If we draw a cage on one side of a card, and a bird on the other, and whirl the card round, we shall see the bird within the cage, the retina retaining the impression of both even when none of them are seen, which is the case when the edge of the card is directed to the eye.
The same property of the retina to preserve impressions of light, is the principle on which that beautiful instrument, called the phenakistoscope, or magic disc, is founded. This instrument was, we believe, originally invented by Dr. Roget, and improved by M. Plateau, at Brussels, and Mr. Faraday. It consists of a circular disc from six to twelve inches in diameter, with rectilineal apertures on its margin in the direction of its radii. A series of figures, of a rider for example, leaping a fence, is drawn on the circumference of a circle, parallel to the rim of the disc. The first figure represents the rider and horse standing before the fence; and the last figure represents them standing over the fence when the leap is completed. Between these two figures there are several others, representing the rider and the horse in different parts of the leap. The observer then stands in front of a looking-glass, with the disc in his left hand, attached to a handle, and by a piece of simple mechanism he whirls it rapidly round, looking at its image on the glass through the notches in its margin. He is then surprised to see the horse and his rider actually leaping the fence, as if they were alive, and returning and leaping again as the disc revolves. If we look over the margin of the disc, at the reflected picture on the face of the disc, all the figures are effaced, and entirely invisible; but when we look through the notches, we only see the figure of the horse and rider at the instant the notch or aperture passes the eye, so that the picture instantaneously formed on the retina is not obliterated by preceding or subsequent impressions. Hence the eye receives in succession the pictures of the horse and rider in all the attitudes of the leap, which are blended, as it were, into one action. The apparent velocity with which the horse and rider advances (supposing the disc always to have the same velocity) depends on the proportion between the number of apertures in the margin of the disc, and the number of figures of the horse and rider.
If we use a disc with three concentric circles of apertures, each containing different numbers, 8, 10, and 12, for example, then considering that these apertures revolve in the opposite direction in the reflected image, it is obvious that when we look through the circle of 10, which moves from left to right at the image of 10, revolving in the mirror from right to left, these opposite motions will de- PART IX. DESCRIPTION OF OPTICAL INSTRUMENTS.
The great number of optical instruments which have been described in different parts of this work renders it scarcely necessary to treat this subject in the general article. Under the articles Burning Instruments, Camera Lucida, Kaleidoscope, Micrometer, Microscope, Photometers, and Telescope, the reader will find the information, some of which he might have expected here. There are instruments, however, so intimately connected with Optics, and not previously described, which we must shortly notice, namely, the Cylindrical Mirror, the Camera Obscura, the Magic Lantern, and the Phantasmagoric Machine.
1. Cylindrical Mirror.
We have already (see p. 382) described the principle of Cylindrical mirrors. If we suppose one of these mirrors, AB, fig. 107, to be placed on a table, with the portrait of Fig. 107, any person laid before it on the table, the reflected picture of the portrait in the cylindrical mirror will be distorted. If we take an accurate drawing of this distorted picture and lay it before a cylindrical mirror, as shown at MN, where the human form can scarcely be recognised, we shall see in the cylindrical mirror its image reduced to symmetry.
2. On the Camera Obscura.
We have already explained the principle of the camera obscura in treating of the images formed by convex lenses. The instrument is indeed nothing more than a convex lens placed in a suitable box, on the side or bottom of which an image of external objects is formed by the lens.
A convenient portable camera obscura for drawing objects is shown in fig. 108. The external object or landscape is reflected down into the lens AB by an inclined mirror CD. The rays thus falling vertically upon the lens, are refracted to their foci, and form a distinct image of the landscape on the paper placed at EF. On one side of the box there is an opening through which the observer introduces his head and hand, care being taken, by a curtain of black cloth behind him, to exclude all extraneous light. M. Cauchoux of Paris has found that the best form of the lens for a camera is a meniscus, having its convex surface towards the image, and its concave surface towards the object, and their radii of curvature as 5 to 8.
3. On the Magic Lantern.
The magic lantern, an invention of the celebrated Athanasius Kircher, is shewn in fig. 109. It consists merely of a lens AB, which forms on the wall of a dark room a picture of any object placed before it, and at a greater distance than its anterior principal focus. The light of an argand lamp is thrown in a condensed state by the illuminating lens D, upon transparent varnished pictures painted on long sliders. The lens AB forms a large circle of light upon the wall, which, if it is not smooth and white, should be covered with a white smooth cloth, and the images of the coloured figure appear within this circle.
A magic lantern is the same as a solar microscope, the Optical sun being used for the source of light in the latter case, and instruments natural objects in place of pictures. The solar camera microscope invented by Dr. Gorling, and fully described in our article Microscope, vol. xv. p. 46, and the oxyhydrogen microscope, described in the same article, may be considered as the most perfect magic lanterns that have been constructed, there being no difficulty in adapting them to give magnified representations of minute transparent paintings.
4. On the Phantasmagoric Apparatus.
The apparatus for the phantasmagoria, or the raising of spectres, is nothing more than a magic lantern mounted upon wheels, which, in place of throwing its pictures upon an opaque white ground, upon which the spectator looks, throws them upon one side of an imperfectly transparent screen, the spectator viewing them on the other side of the screen. The direct light of a lamp A, fig. 110, and the light reflected from the concave mirror B, is thrown upon the two illuminating lenses C, D, which condense it, and thus strongly illuminate the spectres and figures painted upon sliders at E. These sliders are placed a little before the anterior focus of the magnifying lens F, which forms a highly magnified picture of the figures on the transparent screen at G. When this apparatus is mounted upon a carriage with wheels, as at H, it may be made to approach to, or recede from, the screen G, in consequence of which the figures may be made to contract into dwarfs, disappearing in a point of light, or swell out into giants of enormous magnitude. In order, however, to have the pictures distinct at different distances of the apparatus from the screen, an adjustment is necessary, to make the distance EF increase as the apparatus approaches to G, and diminish as it recedes from it. With this view, the lens F is fixed to a slider, which may be drawn out by the general frame H. When this frame H is drawn away from the screen, the point K is brought lower by means of the rod IK, connected with another rod KN fixed to the frame of the screen at N, where there is a joint or centre of motion. The descent of the point K causes another lever KL to move the horizontal slider (which carries the lens F) in such a manner as to keep the screen always in the focus of F, and consequently the picture upon it always distinct. When the frame H, on the other hand, advances to the screen, the point K rises, and the lens F is again adjusted by the motion of the slider. When the images diminish and appear to vanish, the support of the lens F permits the screen M to fall and intercept part of the light. The screen M may have a triangular opening, so as to uncover the middle only of the lens F. In this adjusting apparatus the rods KN and KL must be equal, and the point I must be twice the focal length of the lens F before the object, L being immediately under the focus of the lens. The object of the screen M is to diminish the illumination of the objects as they get smaller and appear to retire from the spectator, because in the instrument they actually become brighter.
When M. Robertson exhibited this remarkable instrument, living persons were often strongly illuminated and introduced into the picture. The effect of life, however, was better given when the shadows of living objects only were introduced.
For farther information on the subject of this article, see Astronomy, vol. iv. p. 101, chap. iv. Achromatic Telescopes, Burning Instruments, Camera Lucida, Chromatics, Colours, Kaleidoscope, Meteorology, Micrometer, Microscope, Photometer, and Telescope.