THAT science which treats of the element of light, and the various phenomena of vision.

HISTORY.

§ 1. Discoveries concerning the Light.

The element of light has occupied much of the attention of thinking men ever since the phenomena of nature have been the objects of rational investigation. The discoveries that have from time to time been made concerning it, are so fully inserted under the article Light, that there is little room for any further addition here. The nature of that subtle element is indeed very little known as yet, notwithstanding all the endeavours of philosophers; and whatever side is taken with regard to it, whether we suppose it to consist of an infinity of small particles propagated by a repulsive power from the luminous body, or whether we suppose it to consist in the vibrations of a subtile fluid, there are prodigious difficulties, almost, if not totally insuperable, which will attend the explanation of its phenomena. In many parts of this work the identity of light and of the electric fluid is asserted; this, however, doth not in the least interfere with the phenomena of optics; all of which are guided by the same invariable laws, whether we suppose light to be a vibration of that fluid, or any thing else. We shall therefore proceed to,

§ 2. Discoveries concerning the Refraction of Light.

We find that the ancients, though they made very few optical experiments, nevertheless knew, that when light passed through mediums of different densities, it did not move forward in a straight line, but was bent, or refracted, out of its course. This was probably suggested to them by the appearance of a straight stick partly immersed in water; and we find many questions concerning this and other optical appearances in Aristotle; to which, however, his answers are insignificant. Archimedes is even said to have written a treatise concerning the appearance of a ring or circle under water, and therefore could not have been ignorant. rant of the common phenomena of refraction. But the ancients were not only acquainted with these more ordinary appearances of refraction, but knew also the production of colours by refracted light. Seneca says, that if the light of the sun shines through an angular piece of glass, it will show all the colours of the rainbow. These colours, however, he says, are false, such as are seen in a pigeon's neck when it changes its position; and of the same nature, he says, is a spectacle, which, without having any colour of its own, assumes that of any other body. It appears also, that the ancients were not unacquainted with the magnifying power of glass globes filled with water, though they do not seem to have known anything of the reason of this power; and the ancient engravers are supposed to have made use of a glass globe filled with water to magnify their figures, and thereby to work to more advantage. That the power of transparent bodies of a spherical form in magnifying or burning was not wholly unknown to the ancients, is further probable from certain gems preserved in the cabinets of the curious, which are supposed to have belonged to the Druids. They are made of rock-crystal of various forms, amongst which are found some that are lenticular and others that are spherical; and though they are not sufficiently wrought to perform their office as well as they might have done if they had been more judiciously executed, yet it is hardly possible that their effect, in magnifying at least, could have escaped the notice of those who had often occasion to handle them; if indeed, in the spherical or lenticular form, they were not solely intended for the purposes of burning. One of these, of the spherical kind, of about an inch and a half diameter, is preserved among the fossils given to the university of Cambridge by Dr Woodward.

The first treatise of any note written on the subject of optics, was by the celebrated astronomer Claudius Ptolemy, who lived about the middle of the second century. The treatise is lost; but from the accounts of others we find that he treated of astronomical refractions. Though refraction in general had been observed very early, it is possible that it might not have occurred to any philosopher much before his time, that the light of the sun, moon, and stars, must undergo a similar refraction in consequence of falling obliquely upon the gross atmosphere that surrounds the earth; and that they must, by that means, be turned out of their rectilinear course, so as to cause those luminaries to appear higher in the heavens than they would otherwise do. The first astronomers were not aware that the intervals between stars appear less near the horizon than near the meridian; and, on this account, they must have been much embarrassed in their observations. But it is evident that Ptolemy was aware of this circumstance, by the caution that he gives to allow something for it, upon every recourse to ancient observations.

This philosopher also advances a very sensible hypothesis to account for the remarkably greater apparent size of the sun and moon when seen near the horizon. The mind, he says, judges of the size of objects by means of a pre-conceived idea of their distance from us; and this distance is fancied to be greater when a number of objects are interposed between the eye and the body we are viewing; which is the case when we see the heavenly bodies near the horizon. In his Almagest, however, he ascribes this appearance to a refraction of the rays by vapours, which actually enlarge the angle under which the luminaries appear; just as the angle is enlarged by which an object is seen from under water.

In the 12th century, the nature of refraction was largely considered by Alhazen an Arabian writer; in so much that, having made experiments upon it at the common surface between air and water, air and glass, water and glass or crystal; and, being prepossessed with discoveries the ancient opinion of crystalline orbs in the regions of Alhazen, above the atmosphere, he even suspected a refraction there also, and fancied he could prove it by astronomical observations. This author deduces from hence several properties of atmospheric refraction, as that it increases the altitudes of all objects in the heavens; and he first advanced, that the stars are sometimes seen above the horizon by means of refraction, when they are really below it. This observation was confirmed by Vitello, B. Waltherus, and especially by the excellent observations of Tycho Brahe. Alhazen observed, that refraction contracts the vertical diameters and distances of the heavenly bodies, and that it is the cause of the twinkling of the stars. But we do not find that either he, or his follower Vitello, knew any of its just quantity. Indeed it is too small to be determined except by very accurate instruments, and therefore we hear little more of it till about the year 1500; at which time great attention was paid to it by Bernard Walther, Maestlin, and others, but chiefly by Tycho Brahe.

Alhazen supposed that the refraction of the atmosphere did not depend upon the vapours in it, as was probably the opinion of philosophers before his time, but on the different transparency; by which, as Montucla conjectures, he meant the density of the gross air contiguous to the earth, and the ether or subtle air that lies beyond it. In examining the effects of refraction, he endeavours to prove that it is so far from being the cause of the heavenly bodies appearing larger near the horizon, that it would make them appear less; two stars, he says, appearing nearer together in the horizon, than near the meridian. This phenomenon he ranks among optical deceptions. We judge of distance, he says, by comparing the angle under which objects appear, with their supposed distance; so that if these angles be nearly equal, and the distance of one object be conceived greater than that of the other, it will be imagined to be larger. And the sky near the horizon, he says, is always imagined to be further from us than any other part of the concave surface. Roger Bacon ascribes this account of the horizontal moon to Ptolemy; and as such it is examined, and objected to by B. Porta.

In the writings of this Bacon, whose genius perhaps equalled that of his great namesake Lord Verulam, we find the first distinct account of the magnifying power of glasses; and it is not improbable, that what he wrote upon this subject gave rise to that most useful invention of spectacles. For he says, that if an object be applied close to the base of the larger segment of a sphere of glass, it will appear magnified. He also treats of the appearance of an object through a globe, and says that he was the first who observed the refraction of rays into it. In 1270, Vitellio, a native of Poland, published a treatise of optics, containing all that was valuable in Alhazen, and digested in a much more intelligible and methodical manner. He observes, that light is always lost by refraction, in consequence of which the objects seen by refracted light always appear less luminous; but he does not pretend to estimate the quantity of this loss. He reduced into a table the result of his experiments on the refractive powers of air, water, and glass, corresponding to different angles of incidence. In his account of the horizontal moon he agrees exactly with Alhazen; observing, that in the horizon she seems to touch the earth, and appears much more distant from us than in the zenith, on account of the intermediate space containing a greater variety of objects upon the visible surface of the earth. He ascribes the twinkling of the stars to the motion of the air in which the light is refracted; and to illustrate this hypothesis, he observes, that they twinkle still more when viewed in water put in motion. He also shows, that refraction is necessary as well as reflection, to form the rainbow; because the body which the rays fall upon is a transparent substance, at the surface of which one part of the light is always reflected and another refracted. But he seems to consider refraction as serving only to condense the light, and thereby enabling it to make a stronger impression upon the eye. This writer also makes some ingenious attempts to explain refraction, or to ascertain the law of it. He also considers the foci of glass spheres, and the apparent size of objects seen through them; though upon these subjects he is not at all exact. It is sufficient indeed to show the state of knowledge, or rather of ignorance, at that time, to observe, that both Vitellio, and his master Alhazen, endeavour to account for objects appearing larger when they are seen under water by the circular figure of its surface; since, being fluid, it conforms to the figure of the earth.

Contemporary with Vitellio was Roger Bacon, a man of very extensive genius, and who wrote upon almost every branch of science; yet in this branch he does not seem to have made any considerable advances beyond what Alhazen had done before him. Even some of the wildest and most absurd of the opinions of the ancients have had the sanction of his authority. He does not hesitate to assert an opinion adopted by many of the ancients, and indeed by most philosophers till his time, that visual rays proceed from the eye; giving this reason for it, that everything in nature is qualified to discharge its proper functions by its own powers, in the same manner as the sun and other celestial bodies. In his Specula Mathematica, he added some observations on the refraction of the light of the stars; the apparent size of objects; the extraordinary size of the sun and moon in the horizon; but in all this he is not very exact, and advances but little. In his Opus Majus he demonstrates, that if a transparent body interposed between the eye and an object, be convex towards the eye, the object will appear magnified. This observation, however, he certainly had from Alhazen; the only difference between them is, that Bacon prefers the smaller segment of a sphere, and Alhazen the larger, in which the latter certainly was right.

From this time, to that of the revival of learning in Europe, we have no farther treatise on the subject of refraction, or indeed on any other part of optics. One of the first who distinguished himself in this way was Maurolycus, teacher of mathematics at Messina. In a treatise, De Lumine et Umbra, published in 1575, he demonstrates that the crystalline humour of the eye is a lens that collects the rays of light issuing from the objects, and throws them upon the retina where is the focus of each pencil. From this principle he discovered the reason why some people were short-sighted and others long-sighted; and why the former are relieved by concave, and the others by convex, glasses.

About the same time that Maurolycus made such discoveries advances towards the discovery of the nature of vision, Joannes Baptista Porta of Naples discovered the camera obscura, which throws still more light on the same subject. His house was constantly resorted to by all the ingenious persons at Naples, whom he formed into what he called an academy of secrets; each member being obliged to contribute something that was not generally known, and might be useful. By this means he was furnished with materials for his Magia Naturalis, which contains his account of the camera obscura, and the first edition of which was published, as he informs us, when he was not quite 15 years old. He also gave the first hint of the magic lantern; which Kircher afterwards followed and improved. His experiments with the camera obscura convinced him, that vision is performed by the introduction of something into the eye, and not by visual rays proceeding from the eye, as had been formerly imagined; and he was the first who fully satisfied himself and others upon this subject. Indeed the resemblance between experiments with the camera obscura and the manner in which vision is performed in the eye, was too striking to escape the observation of a less ingenious person. But when he says that the eye is a camera obscura, and the pupil the hole in the window-shutter, he was so far mistaken as to suppose that it was the crystalline humour that corresponds to the wall which receives the images; nor was it discovered till the year 1604, that this office is performed by the retina. He makes a variety of just observations concerning vision; and particularly explains several cases in which we imagine things to be without the eye, when the appearances are occasioned by some affection of the eye itself, or some motion within the eye. He observes also, that in certain circumstances, vision will be affected by convex or concave glasses; and he seems also to have made some small advances towards the discovery of telescopes. He takes notice, that a round and flat surface plunged into water, will appear hollow as well as magnified to an eye perpendicularly over it; and he very well explains by a figure the manner in which it is done.

All this time, however, the great problem concerning the measuring of refractions had remained unrefraction solved. Alhazen and Vitellio, indeed, had attempted to discover it; but failed, by attempting to measure the angle itself instead of its line. At last it was discovered by Snellius, professor of mathematics at Leyden. This philosopher, however, did not perfectly understand his own discovery, nor did he live to publish any account of it himself. It was afterwards explained by professor Hortensius both publicly and privately before fore it appeared in the writings of Descartes, who published it under a different form, without making any acknowledgment of his obligations to Snellius, whose papers Huygens affirms us, from his own knowledge, Descartes had seen. Before this time Kepler had published a New Table of refracted Angles, determined by his own experiments for every degree of incidence. Kircher had done the same, and attempted a rational or physical theory of refraction, on principle, and on a mode of investigation, which, if conducted with precision, would have led him to the law assumed or discovered by Snellius.

Descartes undertook to explain the cause of refraction by the reflection of forces, on the principles of mechanics. In consequence of this, he was obliged to suppose that light passes with more ease through a dense medium, than through a rare one. The truth of this explanation was first questioned by M. Fermat, counsellor to the parliament of Toulouse, and an able mathematician. He asserted, contrary to the opinion of Descartes, that light suffers more resistance in water than air, and more in glass than in water; and he maintained, that the resistance of different mediums with respect to light is in proportion to their densities. M. Leibnitz adopted the same general idea; and these gentlemen argued upon the subject in the following manner.

Nature, say they, accomplishes her ends by the shortest methods. Light therefore ought to pass from one point to another, either by the shortest road, or that in which the least time is required. But it is plain that the line in which light passes, when it falls obliquely upon a denser medium, is not the most direct or the shortest; so that it must be that in which the least time is spent. And whereas it is demonstrable, that light falling obliquely upon a denser medium (in order to take up the least time possible in passing from a point in one medium to a point in the other) must be refracted in such a manner, that the sines of the angles of incidence and refraction must be to one another, as the different facilities with which light is transmitted in those mediums; it follows, that since light approaches the perpendicular when it passes obliquely from air into water, so that the sine of the angle of refraction is less than that of the angle of incidence, the facility with which water suffers light to pass through it is less than that of the air; so that light meets with more resistance in water than air.

Arguments of this kind could not give satisfaction; and a little time showed the fallacy of the hypothesis. At a meeting of the Royal Society, Aug. 31, 1664, an experiment for measuring the refraction of common water was made with a new instrument which they had prepared for that purpose; and, the angle of incidence being 40 degrees, that of refraction was found to be 30. About this time also we find the first mention of mediums not refracting the light in an exact proportion to their densities. For Mr Boyle, in a letter to Mr Oldenburgh, dated Nov. 3, 1664, observes, that in spirit-of-wine, the proportion of the sines of the angles of incidence to the sines of the angles of refraction was nearly the same as 4 to 3; and that, as spirit-of-wine occasions a greater refraction than common water, so oil of turpentine, which is lighter than spirit-of-wine, produces not only a greater refraction than common water, but a much greater than salt water. And at a meeting held Nov. 9, the same year, Dr Hooke (who had been ordered to prosecute the experiment) brought in an account of one that he had made with pure and clear salad oil, which was found to have produced a much greater refraction than any liquor which he had then tried; the angle of refraction that answered to an angle of incidence of 35° being found no less than 40° 30', and the angle of refraction that answered to an angle of incidence of 20° being 29° 47'.—M. de la Hire also made several experiments to ascertain the refractive power of oil with respect to that of water and air, and found the sine of the angle of incidence to that of refraction to be as 60 to 42; which, he observes, is a little nearer to that of glass than to that of water, though oil is much lighter than water, and glass much heavier.

The members of the Royal Society finding that the refraction of salt water exceeded that of fresh, pursued the experiment farther with solutions of vitriol, saltpetre, and alum, in water; when they found the refraction of the solution of vitriol and saltpetre a little more, but that of alum a little less, than common water.

Dr Hooke made an experiment before the Royal Society, Feb. 11, 1663, which clearly proves that ice refracts the light less than water; which he took to be a good argument that the lightness of ice, which causes it to swim in water, is not caused only by the small bubbles which are visible in it, but that it arises from the uniform constitution or general texture of the whole mass. M. de la Hire also took a good deal of pains to determine whether, as was then the common opinion, the refractive power of ice and water were the same; and he found, as Dr Hooke had done before, that ice refracts less than water.

By a most accurate and elaborate experiment made in the year 1698, in which a ray of light was transmitted through a Torricellian vacuum, Mr Lowthorp found, that the refractive power of air is to that of water as 36 to 34,400. He concludes his account of the experiment with observing, that the refractive power of bodies is not proportioned to the density, at least not to the gravity, of the refracting medium. For the refractive power of glass to that of water is as 55 to 34, whereas its gravity is as 87 to 34; that is, the squares of their refractive powers are very nearly as their respective gravities. And there are some fluids, which, though they are lighter than water, yet have a greater power of refraction. Thus the refractive power of spirit-of-wine, according to Dr Hooke's experiment, is to that of water as 36 to 33, and its gravity reciprocally as 33 to 36, or 364. But the refractive powers of air and water seem to observe the simple proportion of their gravities directly. And if this should be confirmed by succeeding experiments, it is probable, he says, that the refractive powers of the atmosphere are everywhere, and at all heights above the earth, proportioned to its density and expansion; and then it would be no difficult matter to trace the light through it, so as to terminate the shadow of the earth; and, together with proper expedients for measuring the quantity of light illuminating an opaque body, to examine at what distances the moon must be from the earth to suffer eclipses of the observed durations. Cassini the younger happened to be present when Mr Lowthorp made the above-mentioned experiment before the Royal Society; and upon his return home, having made a report of it to the members of the Royal Academy of Sciences, those gentlemen endeavoured to repeat the experiment in 1700; but they did not succeed.—For, as they said, beams of light passed through the vacuum without suffering any refraction. The Royal Society being informed of this, were desirous that it might be put past dispute, by repeated and well-attested trials; and ordered Mr Haucksee to make an instrument for the purpose, by the direction of Dr Halley. It consisted of a strong brass prism, two sides of which had sockets to receive two plane glasses, whereby the air in the prism might either be exhausted or condensed. The prism had also a mercurial gage fixed to it, to discover the density of the contained air; and was contrived to turn upon its axis, in order to make the refractions equal on each side when it was fixed to the end of a telescope. The refracting angle was near 64°; and the length of the telescope was about 10 feet, having a fine hair in its focus. The event of this accurate experiment was as follows:

Having chosen a proper and very distinct erect object, whose distance was 2588 feet, June 15. O.S. 1708, in the morning, the barometer being then at 29.73, and the thermometer at 60, they first exhausted the prism, and then applying it to the telescope, the horizontal hair in the focus covered a mark on the object distinctly seen through the vacuum, the two glasses being equally inclined to the visual ray. Then admitting the air into the prism, the object was seen to rise above the hair gradually as the air entered, and in the end the hair was observed to hide a mark 10½ inches below the former mark. This they often repeated, and with the same success.

After this they applied the condensing engine to the prism; and having forced in another atmosphere, so that the density of the included air was double to that of the outward, they again placed it before the telescope, and, letting out the air, the object which before seemed to rise, appeared gradually to descend, and the hair at length rested on an object higher than before by the same interval of 10½ inches. This experiment they likewise frequently repeated without any variation in the event.

They then forced in another atmosphere; and upon discharging the condensed air, the object was seen near 21 inches lower than before.

Now the radius, in this case being 2588 feet, 10½ inches will subtend an angle of one minute and 8 seconds, and the angle of incidence of the visual ray being 32 degrees (because the angle of the glass planes was 64), it follows from the known laws of refraction, that as the sine of 39° is to that of 31°, 59', 26", differing from 32° by 34" the half of 1', 8"; so is the sine of any other incidence, to the sine of its angle of refraction; and so is radius, or 1000000, to 999716; which, therefore, is the proportion between the sine of incidence in vacuo and the sine of refraction from thence into common air.

It appears, by these experiments, that the refractive power of the air is proportional to its density. And since the density of the atmosphere is as its weight directly, and its heat inversely, the ratio of its density, at any given time, may be had by comparing the heights of the barometer and thermometer; and thence he concludes that this will also be the ratio of the refraction of the air. But Dr Smith observes, that, before we can depend upon the accuracy of this conclusion, we ought to examine whether heat and cold alone may not alter the refractive power of air, while its density continues the same. This, he says, may be tried, by heating the condensed or rarefied air, shut up in the prism, just before it is fixed to the telescope, and by observing whether the hair in its focus will continue to cover the same mark all the while that the air is cooling.

The French academicians, being informed of the result of the above-mentioned experiment, employed M. Delisle the younger to repeat their former experiment with more care; and he presently found, that their operators had never made any vacuum at all, there being chinks in their instrument, through which the air had insinuated itself. He therefore annexed a gage to his instrument, by which means he was sure of his vacuum; and then the result of the experiment was the same with that in England. The refraction was always in proportion to the density of the air, excepting when the mercury was very low, and consequently the air very rare; in which case the whole quantity being very small, he could not perceive much difference in them. Comparing, however, the refractive power of the atmosphere, observed at Paris, with the result of his experiment, he found, that the best vacuum he could make was far short of that of the etherial regions above the atmosphere.

Dr Hooke first suggested the thought of making allowance for the effect of the refraction of light, in passing from the higher and rarer, to the lower and denser regions of the atmosphere, in the computed height of mountains. To this he ascribes the different opinions of authors concerning the height of several very high hills. He could not account for the appearance of the Peak of Teneriff, and several very high mountains, at so great a distance as that at which they are actually seen, but upon the supposition of the curvature of the visual ray, that is made by its passing obliquely through a medium of such different density, from the top of them to the eye, very far distant in the horizon. All calculations of the height of mountains that are made upon the supposition that the rays of light come from the tops of them, to our eyes, in straight lines, must, he says, be very erroneous.

Dr Hooke gives a very good account of the twinkling of the stars; ascribing it to the irregular and unequal refraction of the rays of light, which is also the reason why the limbs of the sun, moon, and planets appear to wave or dance. And that there is such an unequal distribution of the parts of the atmosphere, he says, is manifest from the different degrees of heat and cold in the air. This, he says, will be evident by looking upon distant objects, over a piece of hot glass, which cannot be supposed to throw out any kind of exhalation from itself, as well as through ascending streams of water.

About this time Grimaldi first observed that the coloured image of the sun refracted through a prism is always... always oblong, and that colours proceed from refraction.—The way in which he first discovered this was by Vitello's experiment above mentioned, in which a piece of white paper placed at the bottom of a glass vessel filled with water, and exposed to the light of the sun, appears coloured. However, he observed, that in case the two surfaces of the refracting medium were exactly parallel to each other, no colours were produced. But of the true cause of those colours, viz. the different refrangibility of the rays of light, he had not the least suspicion. This discovery was referred to Sir Isaac Newton, and which occurred to him in the year 1666. At that time he was busy in grinding optic glasses, and procured a triangular glass prism to satisfy himself concerning the phenomena of colours. While he amused himself with this, the oblong figure of the coloured spectrum first struck him. He was surprised at the great disproportion between its length and breadth; the former being about five times the measure of the latter. He could hardly think that any difference in the thickness of the glass, or in the composition of it, could have such an influence on the light. However, without concluding anything a priori, he proceeded to examine the effects of these circumstances, and particularly tried what would be the consequence of transmitting the light through parts of the glass that were of different thicknesses, or through holes in the window-shutter of different sizes; or by setting the prism on the outside of the shutter, that the light might pass through it, and be refracted before it was terminated by the hole.

He then suspected that these colours might arise from the light being dilated by some unevenness in the glass, or some other accidental irregularity; and to try this, he took another prism, like the former, and placed it in such a manner, as that the light, passing through them both, might be refracted contrarywise, and so be returned by the latter into the same course from which it had been diverted by the former. In this manner he thought that the regular effects of the first prism would be destroyed by the second; but that the irregular ones would be augmented by the multiplicity of refractions. The event was, that the light, which by the first prism was diffused into an oblong form, was by the second reduced into a circular one, with as much regularity as if it had not passed through either of them.

At last, after various experiments and conjectures, he hit upon what he calls the experimentum crucis, and which completed this great discovery. He took two boards, and placed one of them close behind the prism at the windows, so that the light might pass through a small hole made in it for the purpose, and fall on the other board, which he placed at the distance of about 12 feet; having first made a small hole in it also, for some of that incident light to pass through. He then placed another prism behind the second board, so that the light which was transmitted through both the boards might pass through that also, and be again refracted before it arrived at the wall. This being done, he took the first prism in his hand, and turned it about its axis, so much as to make the several parts of the image, cast on the second board, successively to pass through the hole in it, that he might observe to what places on the wall the second prism would refract them; and he saw, by the change of those places, that the light tending to that end of the image towards which the refraction of the first prism was made, did, in the second prism, suffer a refraction considerably greater than the light which tended to the other end. The true cause, therefore, of the length of the image was discovered to be no other, than that light is not similar, or homogeneous; but that it consists of rays, some of which are more refrangible than others: so that, without any difference in their incidence on the same medium, some of them shall be more refracted than others; and therefore, that, according to their particular degrees of refrangibility, they will be transmitted through the prism to different parts of the opposite wall.

Since it appears from these experiments that different rays of light have different degrees of refrangibility, it necessarily follows, that the rules laid down by preceding philosophers concerning the refractive power of water, glass, &c. must be limited to the middle kind of rays. Sir Isaac, however, proves that the fine of the incidence of every kind of light, considered apart, is to its fine of refraction in a given ratio. This he deduces, both by experiment, and also geometrically, from the supposition that bodies refract the light by acting upon its rays in lines perpendicular to their surfaces.

The most important discovery with regard to refraction since the time of Sir Isaac Newton is that of Mr Dollond, who found out a method of curing the faults of refracting telescopes arising from the different refrangibility of the rays, and which had been generally thought impossible to be removed.—Notwithstanding the great discovery of Sir Isaac Newton concerning the different refrangibility of the rays of light, he had no idea but that they were all affected in the same proportion by every medium, so that the refrangibility of the extreme rays might be determined if that of the mean ones was given. From this it would follow, as Mr Dollond observes, that equal and contrary refractions must not only destroy each other, but that the divergency of the colours from one refraction would likewise be corrected by the other, and that there could be no possibility of producing any such thing as refraction which would not be affected by the different refrangibility of light; or, in other words, that however a ray of light might be refracted backwards and forwards by different mediums, as water, glass, &c. provided it was so done, that the emergent ray should be parallel to the incident one, it would ever after be white; and consequently, if it should come out inclined to the incident, it would diverge, and ever after be coloured; and from this it was natural to infer, that all spherical object-glasses of telescopes must be equally affected by the different refrangibility of light, in proportion to their apertures, of whatever materials they may be formed.

For this reason, Sir Isaac Newton, and all other philosophers and opticians, had despaired of bringing refracting telescopes to any great degree of perfection, without making them of an immoderate and very inconvenient length. They therefore applied themselves chiefly to the improvement of the reflecting telescope; and the business of refraction was dropped till about about the year 1747, when M. Euler, improving upon a hint of Sir Isaac Newton's, formed a scheme of making object-glasses of two materials, of different refractive powers; hoping, that by this difference, the refractions would balance one another, and thereby prevent the dispersion of the rays that is occasioned by the difference of refrangibility. These object glasses were composed of two lenses of glass with water between them. This memoir of M. Euler excited the attention of Mr Dollond. He carefully went over all M. Euler's calculations, substituting for his hypothetical laws of refraction those which had been actually affected by the experiments of Newton; and found, that, after this necessary substitution, it followed from M. Euler's own principles, that there could be no union of the foci of all kinds of colours, but in a lens infinitely large.

M. Euler did not mean to controvert the experiments of Newton; but he said, that they were not contrary to his hypothesis, but in so small a degree as might be neglected; and asserted, that, if they were admitted in all their extent, it would be impossible to correct the difference of refrangibility occasioned by the transmission of the rays from one medium into another of different density; a correction which he thought was very possible, since he supposed it to be actually effected in the structure of the eye, which in his opinion was made to consist of different mediums for that very purpose. To this kind of reasoning Mr Dollond made no reply, but by appealing to the experiments of Newton, and the great circumspection with which it was known that he conducted all his inquiries.

In this state of the controversy, the friends of M. Clairaut engaged him to attend to it; and it appeared to him, that, since the experiments of Newton cited by Mr Dollond could not be questioned, the speculations of M. Euler were more ingenious than useless.

The same paper of M. Euler was also particularly noticed by M. Klingenstierna of Sweden, who gave a considerable degree of attention to the subject, and discovered, that, from Newton's own principles, the result of the 8th experiment of the second book of his Optics could not answer his description of it.

He found, he says, that when light goes out of air through several contiguous refracting mediums, as through water and glass, and thence goes out again into air, whether the refracting surfaces be parallel or inclined to one another, that light, as often as by contrary refractions it is so corrected as to emerge in lines parallel to those in which it was incident, continues ever after to be white; but if the emergent rays be inclined to the incident, the whiteness of the emerging light will, by degrees, in passing on from the place of emergence, become tinged at its edges with colours. This he tried by refracting light with prisms of glass, placed within a prismatic vessel of water.

By theorems deduced from this experiment he infers, that the refractions of the rays of every sort, made out of any medium into air, are known by having the refraction of the rays of any one sort; and also that the refraction out of one medium into another is found as often as we have the refractions out of them both into any third medium.

On the contrary, the Swedish philosopher observes, that, in this experiment, the rays of light, after passing through the water and the glass, though they come out parallel to the incident rays, will be coloured; but that the smaller the glass prism is, the nearer will the result of it approach to Newton's description.

This paper of M. Klingenstierna being communicated to Mr Dollond by M. Mallet, made him entertain doubts concerning Newton's report, and determined him to have recourse to experiment.

He therefore cemented together two plates of parallel glass at their edges, so as to form a prismatic vessel, when stopped at the ends or bases; and the edge being turned downwards, he placed in it a glass prism, with one of its edges upwards, and filled up the vacancy with clear water; so that the refraction of the prism was contrived to be contrary to that of the water, in order that a ray of light, transmitted through both these refracting mediums, might be affected by the difference only between the two refractions. As he found the water to refract more or less than the glass prism, he diminished or increased the angle between the glass plates, till he found the two contrary refractions to be equal; which he discovered by viewing an object thro' this double prism. For when it appeared neither raised nor depressed, he was satisfied that the refractions were equal, and that the emergent rays were parallel to the incident.

Now, according to the prevailing opinion, he observes, the object should have appeared through this double prism in its natural colour; for if the difference of refrangibility had been in all respects equal in the two equal refractions, they would have rectified each other. But this experiment fully proved the fallacy of the received opinion, by showing the divergency of the light by the glass prism to be almost double of that by the water; for the image of the object, though not at all refracted, was yet as much infected with prismatic colours, as if it had been seen through a glass wedge only, whose refracting angle was near 30 degrees.

This experiment is the very same with that of Sir Isaac Newton's above-mentioned, notwithstanding the result was so remarkably different; but Mr Dollond assures us, that he used all possible precaution and care in his process; and he kept his apparatus by him, that he might evince the truth of what he wrote, whenever he should be properly required to do it.

He plainly saw, however, that if the refracting angle of the water vessel could have admitted of a sufficient increase, the divergency of the coloured rays would have been greatly diminished, or entirely rectified; and that there would have been a very great refraction without colour, as he had already produced a great discolouring without refraction; but the inconvenience of so large an angle as that of the prismatic vessel must have been, to bring the light to an equal divergency with that of the glass prism whose angle was about 60 degrees, made it necessary to try some experiments of the same kind with smaller angles.

Accordingly, he got a wedge of plate glass, the angle of which was only nine degrees; and using it in the same circumstances, he increased the angle of the water wedge, in which it was placed, till the divergency of the light by the water was equal to that by the glass; that is, till the image of the object, though considerably refracted by the excess of the refraction of the water, appeared nevertheless quite free from any colours proceeding from the different refrangibility of the light; and, as near as he could then measure, the refraction by the water was about $\frac{1}{4}$ of that by the glass. He acknowledges, indeed, that he was not very exact in taking the measures, because his business was not at that time to determine the exact proportions, so much as to show that the divergency of the colours, by different substances, was by no means in proportion to the refractions, and that there was a possibility of refraction without any divergency of the light at all.

As these experiments clearly proved, that different substances made the light to diverge very differently in proportion to their general refractive power, Mr Dollond began to suspect that such variety might possibly be found in different kinds of glasses, especially as experience had already shown that some of the kinds made much better object glasses in the usual way than others; and as no satisfactory cause had been assigned for such difference, he thought there was great reason to presume that it might be owing to the different divergency of the light in the same refractions.

His next business, therefore, was to grind wedges of different kinds of glasses, and apply them together; so that the refractions might be made in contrary directions, in order to discover, as in the above-mentioned experiments, whether the refraction and the divergency of the colours would vanish together. But a considerable time elapsed before he could set about that work: for though he was determined to try it at his leisure, for satisfying his own curiosity, he did not expect to meet with a difference sufficient to give room for any great improvement of telescopes, so that it was not till the latter end of the year 1757 that he undertook it; but his first trials convinced him that the business deserved his utmost attention and application.

He discovered a difference far beyond his hopes in the refractive qualities of different kinds of glasses, with respect to the divergency of colours. The yellow or straw-coloured foreign sort, commonly called Venice glasses; and the English crown glasses, proved to be very nearly alike in that respect; though, in general, the crown glasses seemed to make the light diverge less than the two. The common English plate-glasses made the light diverge more; and the white crystal, or English flint glasses, most of all.

It was now his business to examine the particular qualities of every kind of glasses that he could come at, not to amuse himself with conjectures about the cause of this difference, but to fix upon two sorts in which it should be the greatest; and he soon found these to be the crown glasses and the white flint glasses. He therefore ground one wedge of white flint, of about 25 degrees; and another of crown glasses, of about 29 degrees; which refracted very nearly alike, but their power of making the colours diverge was very different. He then ground several others of crown glasses to different angles, till he got one which was equal, with respect to the divergency of the light, to that in the white flint glasses: for when they were put together, so as to refract in contrary directions, the refracted light was entirely free from colours. Then measuring the refraction of each wedge with these different angles, he found that of the white glasses to be that of the crown glasses nearly as two to three: and this proportion held very nearly in all small angles; so that any two wedges made in this proportion, and applied together, so as to refract in a contrary direction, would refract the light without any dispersion of the rays.

In a letter to M. Klingenspierna, quoted by M. Clairaut, Mr Dollond says, that the sine of incidence in crown glasses is to that of its general refraction as 1 to 1.53, and in flint glasses as 1 to 1.583.

To apply this knowledge to practice, Mr Dollond went to work upon the object-glasses of telescopes; not doubting but that, upon the same principles on which a refracted colourless ray was produced by prisms, it might be done by lenses also, made of similar materials. And he succeeded, by considering, that, in order to make two spherical glasses that should refract the light in contrary directions, the one must be concave and the other convex; and as the rays are to converge to a real focus, the excess of refraction must evidently be in the convex lens. Also, as the convex glass is to refract the most, it appeared from his experiments, that it must be made of crown glasses, and the concave of white flint glasses. Farther, as the refractions of spherical glasses are in an inverse ratio of their focal distances, it follows, that the focal distances of the two glasses shall be inversely as the ratios of the refractions of the wedges; for being thus proportioned, every ray of light that passes through this combined glass, at whatever distance it may pass from its axis, will constantly be refracted, by the difference between two contrary refractions, in the proportion required; and therefore the different refrangibility of the light will be entirely removed.

Notwithstanding our author had these clear grounds in theory and experiment to go upon, he found that he had many difficulties to struggle with when he came to reduce them into actual practice; but with great patience and address, he at length got into a ready method of making telescopes upon these new principles.

His principal difficulties arose from the following circumstances. In the first place, the focal distances, as well as the particular surfaces, must be very nicely proportioned to the densities or refracting powers of the glasses, which are very apt to vary in the same sort of glasses made at different times. Secondly, The centres of the two glasses must be placed truly in the common axis of the telescope, otherwise the desired effect will be in a great measure destroyed. Add to these, that there are four surfaces to be wrought perfectly spherical; and any person, he says, but moderately practised in optical operations, will allow, that there must be the greatest accuracy throughout the whole work. At length, however, after numerous trials, and a resolute perseverance, he was able to construct refracting telescopes, with such apertures and magnifying powers, under limited lengths, as, in the opinion of the best judges, far exceeded anything that had been produced. duced before, representing objects with great distinctness, and in their true colours.

It was objected to Mr Dollond's discovery, that the small dispersion of the rays in crown glass is only apparent, owing to the opacity of that kind of glass which does not transmit the fainter coloured rays in a sufficient quantity; but this objection is particularly considered, and answered by M. Beguelin.

As Mr Dollond did not explain the methods which he took in the choice of different spheres proper to destroy the effect of the different refrangibility of the rays of light, and gave no hint that he himself had any rule to direct himself in it; and as the calculation of the dispersion of the rays, in so complicated an affair, is very delicate; M. Clairaut, who had given a good deal of attention to this subject, from the beginning of the controversy, endeavoured to make out a complete theory of it.

Without some assistance of this kind, it is impossible, says this author, to construct telescopes of equal goodness with those of Mr Dollond, except by a servile imitation of his; which, however, on many accounts, would be very unlikely to answer. Besides, Mr Dollond only gave his proportions in general, and pretty near the truth; whereas the greatest possible precision is necessary. Also the best of Mr Dollond's telescopes were far short of the Newtonian ones (A); whereas it might be expected that they should exceed them, if the foci of all the coloured rays could be as perfectly united after refraction through glass, as after reflexion from a mirror; since there is more light lost in the latter case than in the former.

With a view, therefore, to assist the artist, he endeavoured to ascertain the refractive power of different kinds of glass, and also their property of separating the rays of light, by the following exact methods. He made use of two prisms placed close to one another, as Mr Dollond had done: but, instead of looking through them, he placed them in a darkened room; and when the image of the sun, transmitted through them, was perfectly white, he concluded that the different refrangibility of the rays was corrected.

In order to ascertain with more ease the true angles that prisms ought to have to destroy the effect of the difference of refrangibility, he constructed one which had one of its surfaces cylindrical, with several degrees of amplitude. By this means, without changing his prisms, he had the choice of an infinity of angles; among which, by examining the point of the curve surface, which, receiving the solar ray, gave a white image, he could easily find the true one.

He also ascertained the proportion in which different kinds of glass separated the rays of light, by measuring, with proper precautions, the oblong image of the sun, made by transmitting a beam of light through them. In making these experiments, he hit upon an easy method of convincing any person of the greater refractive power of English flint-glass above the common French glass, both with respect to the mean refraction, and the different refrangibility of the colours; for having taken two prisms, of these two kinds of glass, but equal in all other respects, and placed them so that they received, at the same time, two rays of the sun, with the same degree of incidence, he saw, that, of the two images, that which was produced by the English flint-glass was a little higher up on the wall than the other, and longer by more than one half.

M. Clairaut was assisted in these experiments by M. De Tourmieres, and the results agreed with Mr Dollond's in general; but whereas Mr Dollond had made the dispersion of the rays in glass and in water to be as five to four (acknowledging, however, that he did not pretend to do it with exactness), these gentlemen, who took more pains, and used more precautions, found it to be as three to two. For the theorems and problems deduced by M. Clairaut from these new principles of optics, with a view to the perfection of telescopes, we must refer the reader to Mem. Acad. Par. 1756, 1757.

The labours of M. Clairaut were succeeded by those of M. D'Alembert, which seem to have given the makers of these achromatic telescopes all the aid that calculations can afford them. This excellent mathematician has likewise proposed a variety of new constructions of these telescopes, the advantages and disadvantages of which he distinctly notes; at the same time that he points out several methods of correcting the errors to which they are liable: as by placing the object glasses, in some cases, at a small distance from one another, and sometimes by using eye-glasses of different refractive powers; which is an expedient that seems not to have occurred to any person before him. He even shows, that telescopes may be made to advantage, consisting of only one object-glass, and an eye-glass of a different refractive power. Some of his constructions have two or more eye-glasses of different kinds of glass. This subject he considered at large in one of the volumes of his Opuscules Mathematiques. We have also three memoirs of M. D'Alembert upon this subject, among those of the French Academy; one in the year 1764, another in 1765, and a third in 1767.

At the conclusion of his second memoir he says, that he does not doubt, but, by the different methods he proposes, achromatic telescopes may be made to far greater degrees of perfection than any that have been seen hitherto, and even such as is hardly credible: And though the crown glass, by its greenish colour, may absorb some part of the red or violet rays, which, however, is not found to be the case in fact; that objection cannot be made to the common French glass, which is white, and which on this account he thinks must be preferable to the English crown glass.

Notwithstanding Messrs Clairaut and D'Alembert seemed to have exhausted the business of calculation on the subject of Mr Dollond's telescopes, no use could be made of their labours by foreign artists. For still the telescopes made in England, according to no exact rule,

(A) This assertion of M. Clairaut might be true at the time that it was made, but it is by no means so at present. rule, as foreigners supposed, were greatly superior to any that could be made elsewhere, though under the immediate direction of those able calculators. For this M. Beguelin assigned several reasons. Among others, he thought that their geometrical theorems were too general, and their calculations too complicated, for the use of workmen. He also thought, that in consequence of neglecting small quantities, which these calculators professedly did, in order to make their algebraical expressions more commodious, their conclusions were not sufficiently exact. But what he thought to be of the most consequence, was the want of an exact method of measuring the refractive and dispersing powers of the different kinds of glass; and for want of this, the greatest precision in calculation was altogether useless.

These considerations induced this gentleman to take another view of this subject; but still he could not reconcile the actual effect of Mr Dollond's telescopes with his own conclusions: so that he imagined, either that he had not the true refraction and dispersion of the two kinds of glass given him; or else, that the aberration which still remained after his calculations, must have been destroyed by some irregularity in the surfaces of the lenses. He finds several errors in the calculations both of M. D'Alembert and Clairaut, and concludes with expressing his design to pursue this subject much farther.

M. Euler, who first gave occasion to this inquiry, which terminated so happily for the advancement of science, being persuaded both by his reasoning and calculations, that Mr Dollond had discovered no new principle in optics, and yet not being able to controvert Mr Short's testimony in favour of the goodness of his telescopes, concluded that this extraordinary effect was owing, in part, to the crown glass not transmitting all the red light, which would otherwise have come to a different focus, and have distorted the image; but principally to his happening to hit on a just curvature of his glass, which he did not doubt would have produced the same effect if his lenses had all been made of the same kind of glass. In another place he imagines that the goodness of Mr Dollond's telescope might be owing to the eye-glass. If my theory, says he, be true, this disagreeable consequence follows, that Mr Dollond's object-glasses cannot be exempt from the dispersion of colours: yet a regard to so respectable a testimony embarrasses me extremely, it being as difficult to question such express authority, as to abandon a theory which appears to me perfectly well founded, and to embrace an opinion, which is as contrary to all the established laws of nature as it is strange and seemingly absurd. He even appeals to experiments made in a darkened room; in which, he says, he is confident that Mr Dollond's object-glasses would appear to have the same defects that others are subject to.

Not doubting, however, but that Mr Dollond, either by chance, or otherwise, had made some considerable improvement in the construction of telescopes, by the combination of glasses, he abandoned his former project, in which he had recourse to different mediums, and confined his attention to the correction of the errors which arise from the curvature of lenses. But while he was proceeding, as he imagined, upon the true principles of optics, of which, however, he made but little use, he could not help expressing his surprise that Mr Dollond should have been led to so important a discovery by reasoning in a manner quite contrary to the nature of things. At length, however, M. Euler was convinced of the reality and importance of Mr Dollond's discoveries; and very frankly acknowledges, that he should perhaps never have been brought to assent to it, had not his friend M. Clairaut assured him that the experiments of the English optician might be depended upon. However, the experiments of M. Zeither of Peterburgh gave him the most complete satisfaction with respect to this new law of refraction.

This gentleman demonstrated, that it is the lead in the composition of glass that gives it this remarkable property, that while the refraction of the mean rays is nearly the same, that of the extremes differs considerably. And, by increasing the quantity of lead in the mixture, he produced a kind of glass, which occasioned a much greater separation of the extreme rays than the flint-glass which Mr Dollond had made use of. By this evidence M. Euler owns that he was compelled to renounce the principle which, before this time, had been considered as incontrovertible, viz. that the dispersion of the extreme rays depends upon the refraction of the mean; and that the former varies with the quality of the glass, while the latter is not affected by it.

From these new principles M. Euler deduces theorems concerning the combination of the lenses, and, in a manner similar to M. Clairaut and D'Alembert, points out methods of constructing achromatic telescopes.

While he was employed upon this subject, he informs us, that he received a letter from M. Zeither, dated 30th of January 1764, in which he gives account of him a particular account of the success of his experiments on the composition of glass; and that, having corrected mixed minium and sand in different proportions, the result of the mean refraction and the dispersion of the refracting rays varied according to the following table:

| Proportion of minium to flint | Mean refraction from air into glass | Dispersion of the rays in comparison of crown-glass | |-----------------------------|-------------------------------------|--------------------------------------------------| | I. — 3 : 1 | 2028 : 1000 | 4800 : 1000 | | II. — 2 : 1 | 1830 : 1000 | 3550 : 1080 | | III. — 1 : 1 | 1787 : 1000 | 3259 : 1000 | | IV. — ½ : 1 | 1732 : 1000 | 2207 : 1000 | | V. — ¼ : 1 | 1724 : 1000 | 1800 : 1000 | | VI. — ⅛ : 1 | 1664 : 1000 | 2354 : 1000 |

By this table it is evident, that a greater quantity of lead not only occasions a greater dispersion of the rays, but also considerably increases the mean refraction. The first of these kinds of glass, which contains three times as much minium as flint, will appear very extraordinary; since, hitherto, no transparent substance has been known, whose refractive power exceeded the ratio of two to one, and that the dispersion occasioned by this glass is almost five times as great as that of crown glass, which could not be believed by those who enter- entertained any doubt concerning the same property in flint glass, the effect of which is three times as great as crown glass. One may observe, however, in these kinds of glasses, something of a proportion between the mean refraction and the dispersion of rays, which may enable us to reconcile these surprising effects with other principles already known.

Here, however, M. Euler announces to us another discovery of the same M. Zeiller, no less surprising than the former, and which disconcerted all his schemes for reconciling the above-mentioned phenomena. As the six kinds of glasses mentioned in the above table were composed of nothing but minium and flint, M. Zeiller happened to think of mixing alkaline salts with them, in order to give the glasses a consistence more proper for dioptric uses; when he was much surprised to find this mixture greatly diminished the mean refraction, almost without making any change in the dispersion. After many trials, he at length obtained a kind of glass greatly superior to the flint-glass of Mr Dollond, with respect to the construction of telescopes; since it occasioned three times as great a dispersion of the rays as the common glass, at the same time that the mean refraction was only as 1.61 to 1.

M. Euler also gives particular instructions how to find both the mean and extreme refractive power of different kinds of glasses; and particularly advises to make use of prisms with very large refracting angles, not less than 70°.

Notwithstanding it evidently appeared, we may say, to almost all philosophers, that Mr Dollond had made a real discovery of something not comprehended in the optical principles of Sir Isaac Newton, it did not appear so to Mr Murdoch. Upon this occasion, he interposed in the defence, as he imagined, of Sir Isaac Newton; maintaining, that Mr Dollond's positions, which, he says, he knows not by what mishap have been deemed paradoxes in Sir Isaac's theory of light, are really the necessary consequences of it. He also endeavours to show that Sir Isaac might not be mistaken in his account of the experiment above-mentioned. But, admitting all that he advances in this part of his defence, Newton must have made use of a prism with a much smaller refracting angle than, from his own account of his experiments, we have any reason to believe that he ever did make use of.

The fact probably was, that Sir Isaac deceived himself in this case, by attending to what he imagined to be the clear consequence of his other experiments; and though the light he saw was certainly tinged with colours, and he must have seen it to be so, yet he might imagine that this circumstance arose from some imperfection in his prisms, or in the disposition of them, which he did not think it worth his while to examine. It is also observable, that Sir Isaac is not so particular in his description of his prisms, and other parts of his apparatus, in his account of this experiment, as he generally is in other cases; and therefore, probably, wrote his account of it from his memory only. In reality, it is no reflection upon Sir Isaac Newton, who did so much, to say that he was mistaken in this particular case, and that he did not make the discovery that Mr Dollond did; though it be great praise to Mr Dollond, and all those persons who contributed to this discovery, that they ventured to call in question the authority of so great a man.

Mr Dollond, however, was not the only optician who had the merit of making this discovery; it had been made and applied to the same purpose by a private gentleman—Mr Cheft of Cheft-hall. He had observed that prisms of flint glass gave larger spectrums than prisms of water when the mean refraction was the same in both, i.e. when the deviation of the refracted ray from the direction of the incident was the same. He tried prisms of other glasses, and found similar differences; and he employed the discovery in the same manner, and made achromatic experiments some time before Dollond. These facts came out in a process raised at the instance of Watkins optician at Charing-cross, as also in a publication by Mr Ramsden optician. There is, however, no evidence that Dollond stole the idea from Mr Cheft, or that they had not both claims to the discovery.

Still the best refracting telescopes, constructed on the principles of Mr Dollond, are defective, on account of that colour which, by the aberration of the rays, they give to objects viewed through them, unless the object glasses be of small diameter. This defect men of genius and science have laboured to remove, some by one contrivance and some by another. Father Boxford, whom every branch of optics is much indebted, has, in his attempts for this purpose, displayed much ingenuity; but the philosopher whose exertions have been crowned with most success, and who has perhaps made the most important discovery in this branch of science since the era of Newton, is Dr Robert Blair regius professor of astronomy in the college of Edinburgh. By a judicious set of experiments ably conducted, he has proved, that the quality of dispersing the rays in a greater degree than crown glass, is not confined to a few mediums, but is possessed by a great variety of fluids, and by some of these in a most extraordinary degree. He has shown, that although the greater refrangibility of the violet rays than of the red rays, when light passes from any medium whatever into a vacuum, may be considered as a law of nature; yet in the passages of light from one medium into another, it depends entirely on the qualities of the mediums which of these rays shall be the most refrangible, or whether there shall be any difference in their refrangibility. In order to correct the aberration arising from difference of refrangibility among the rays of light, he instituted a set of experiments, in the conducting of which he detected a very singular and important quality in the muriatic acid. In all the dispersive mediums hitherto examined, the green rays, which are the mean refrangible in crown glasses, were found among the least refrangible; but in the muriatic acid, these same rays were by him found to make a part of the more refrangible. This discovery led to complete success in removing the great defect of optical instruments, viz. that dissipation or aberration of the rays which arise from their unequal refrangibility, and has hitherto rendered it impossible to converge all of them to one point either by single or opposite refractions. A fluid, in which the particles of marine acid and metallic particles hold a due proportion, at the same time that it separates the extreme rays of the spectrum. spectrum much more than crown glass, refracts all the orders of the rays in the same proportion that glass does; and hence rays of all colours made to diverge by the refraction of the glass may either be rendered parallel by a subsequent refraction made in the confine of the glass and this fluid; or, by weakening the refractive density of the fluid, the refraction which takes place in the confine of it and glass may be rendered as regular as reflection, without the least colour whatever.

The Doctor has a telescope, not exceeding 15 inches in length, with a compound object glass of this kind, which equals in all respects, if it does not surpass, the best of Dollond's 42 inches long. Of this object glass a figure will be found in the third volume of the Transactions of the Royal Society of Edinburgh; and to that volume we must refer our readers for a full and perspicuous account of the experiments which led to this discovery, as well as of the important purposes to which it may be applied.

We shall conclude the history of the discoveries concerning refraction, with some account of the refractions of the atmosphere.—Tales of this have been calculated by Mr Lambert, with a view to correct the inaccuracies of geometrical observations of the altitudes of mountains. The observations of Mr Lambert, however, go upon the supposition that the refractive power of the atmosphere is invariable: But this is by no means the case; and therefore his rules must be considered as true for the mean state of the air only.

A most remarkable variety in the refractive power of the atmosphere was observed by Dr Nettleton, near Halifax in Yorkshire, which demonstrates how little we can depend upon the calculated heights of mountains, when the observations are made with an instrument, and the refractive power of the air is to be allowed for. Being desirous to learn, by observation, how far the mercury would descend in the barometer at any given elevation (for which there is the best opportunity in that hilly country), he proposed to take the height of some of their highest hills; but when he attempted it, he found his observation to much disturbed by refraction, that he could come to no certainty. Having measured one hill of a considerable height, in a clear day, and observed the mercury at the bottom and at the top, he found, according to that estimation, that about 90 feet or more were required to make the mercury fall 1/3 of an inch; but afterwards, repeating the experiment on a cloudy day, when the air was rather grofs and hazy, he found the small angles so much increased by refraction as to make the hill much higher than before. He afterwards frequently made observations at his own house, by pointing a quadrant to the tops of some neighbouring hills, and observed that they would appear higher in the morning before sunrise, and also late in the evening, than at noon in a clear day, by several minutes. In one case the elevations of the same hill differed more than 30 minutes. From this he infers, that observations made on very high hills, especially when viewed at a distance, and under small angles, as they generally are, are probably uncertain, and not much to be depended upon.

M. Euler considered with great accuracy the refractive power of the atmosphere, as affected by different degrees of heat and elasticity; in which he shows, that its refractive power, to a considerable distance from the zenith, is sufficiently near the proportion of the tangent of that distance, and that the law of refraction follows the direct ratio of the height of the barometer, and the inverse ratio of the difference marked by the thermometer; but when stars are in the horizon, the changes are in a ratio somewhat greater than this, more especially on account of the variation in the heat.

The cause of the twinkling of the stars is now generally acknowledged to be the unequal refraction of light, in consequence of inequalities and undulations in the atmosphere.

Mr Mitchell supposes that the arrival of fewer or more rays at one time, especially from the smaller or the more remote fixed stars, may make such an unequal impression upon the eye, as may, at least, have some share in producing this effect; since it may be supposed, that even a single particle of light is sufficient to make a sensible impression upon the organs of sight; so that very few particles arriving at the eye in a second of time, perhaps no more than three or four, may be sufficient to make an object constantly visible. For though the impression may be considered as momentary, yet the perception occasioned by it is of some duration. Hence, he says, it is not improbable that the number of the particles of light which enter the eye in a second of time, even from Sirius himself (the light of which does not exceed that of the smallest visible fixed star, in a greater proportion than that of about 100 to 1), may not exceed 3000 or 4000, and from stars of the second magnitude they may, therefore, probably not exceed 1000. Now the apparent increase and diminution of the light which we observe in the twinkling of the stars, seems to be repeated at not very unequal intervals, perhaps about four or five times in a second. He therefore thought it reasonable to suppose, that the inequalities which will naturally arise from the chance of the rays coming sometimes a little denser, and sometimes a little rarer, in so small a number of them as must fall upon the eye in the fourth or fifth part of a second, may be sufficient to account for this appearance. An addition of two or three particles of light, or perhaps a single one, upon 20, especially if there should be an equal deficiency out of the next 20, would, he supposed, be very sensible, as he thought was probable from the very great difference in the appearance of stars, the light of which does not differ so much as is commonly imagined. The light of the middlemost star in the tail of the Great Bear does not, he thinks, exceed the light of the very small star that is next to it in a greater proportion than that of about 16 or 20 to 1; and M. Bouger found, that a difference in the light of objects of one part in 66 was sufficiently distinguishable.

It will perhaps, he says, be objected, that the rays coming from Sirius are too numerous to admit of a sufficient inequality arising from the common effect of chance to frequently as would be necessary to produce this effect, whatever might happen with respect to the smaller stars; but he observes, that till we know what inequality is necessary to produce this effect, we can only guess at it one way or the other.

Since these observations were published, Mr Michell has entertained some suspicion that the unequal History.

§ 3. Discoveries concerning the Reflection of Light.

However much the ancients might have been mistaken with regard to the nature of light, we find that they were acquainted with two very important observations concerning it; viz. that light is propagated in right lines, and that the angle of incidence is equal to the angle of reflection. Who it was that first made these important observations is not known. But indeed, important as they are, and the foundation of a great part of even the present system of optics, it is possible that, if he were known, he might not be allowed to have any share of merit, at least for the former of them; the fact is so very obvious, and so easily ascertained. As to the latter, that the angle of incidence is equal to the angle of reflection, it was probably first discovered by observing a ray of the sun reflected from the surface of water, or some other polished body; or from observing the images of objects reflected by such surfaces. If philosophers attended to this phenomenon at all, they could not but take notice, that, if the ray fell nearly perpendicular upon such a surface, it was reflected near the perpendicular; and if it fell obliquely, it was reflected obliquely: and if they thought of applying any kind of measures to these angles, however coarse and imperfect, they could not but see that there was sufficient reason to assert their equality. At the same time they could not but know that the incident and reflected rays were both in the same plane.

Aristotle was sensible that it is the reflection of light from the atmosphere which prevents total darkness after the sun sets, and in places where he does not shine in the day-time. He was also of opinion, that rainbows, halos, and mock suns, were all occasioned by the reflection of the sun-beams in different circumstances, by which an imperfect image of his body was produced, the colour only being exhibited, and not his proper figure. The image, he says, is not single, as in a mirror; for each drop of rain is too small to reflect a visible image, but the conjunction of all the images is visible.

Without inquiring any farther into the nature of Euclid's light or vision, the ancient geometricians contented themselves with deducing a system of optics from the optics, two observations mentioned above, viz. the rectilinear progress of light, and the equality of the angles of incidence and reflection. The treatise of optics which has been ascribed to Euclid is employed about determining the apparent size and figure of objects, from the angle under which they appear, or which the extremities of them subtend at the eye, and the apparent place of the image of an object reflected from a polished mirror; which he fixes at the place where the reflected ray meets a perpendicular to the mirror drawn through the object. But this work is so imperfect, and so inaccurately drawn up, that it is not generally thought to be the production of that great geometrician.

It appears from a circumstance in the history of Socrates, that the effects of burning-glasses had also been observed by the ancients; and it is probable that the Romans had a method of lighting their sacred fire by means of a concave speculum. It seems indeed to have been known pretty early, that there is an in-

Mr Mutchtenbrock suspects, that the twinkling of the stars arises from some affection of the eye, as well as the state of the atmosphere. For he says, that in Holland, when the weather is frothy, and the sky very clear, the stars twinkle most manifestly to the naked eye, though not in telescopes; and since he does not suppose that there is any great exhalation, or dancing of the vapour at that time, he questions whether the vivacity of the light affecting the eye may not be concerned in the phenomenon.

But this philosopher might very easily have satisfied himself with respect to this hypothesis, by looking at the stars near the zenith, when the light traverses but a small part of the atmosphere, and therefore might be expected to affect the eye the most sensibly. For he would not have perceived them to twinkle near so much, as they do near the horizon, when much more of their light is intercepted by the atmosphere.

Some astronomers have lately endeavoured to explain the twinkling of the fixed stars by the extreme minuteness of their apparent diameter; so that they suppose the sight of them is intercepted by every mote that floats in the air. But Mr Michell observes, that no object can hide a star from us that is not large enough to exceed the apparent diameter of the star, by the diameter of the pupil of the eye; so that if a star was a mathematical point, the intercepting object must still be equal in size to the pupil of the eye; nay, it must be large enough to hide the star from both eyes at the same time.

Besides a variation in the quantity of light, a momentary change of colour has likewise been observed in some of the fixed stars. Mr Melville says, that when one looks steadily at Sirius, or any bright star not much elevated above the horizon, its colour seems not to be constantly white, but appears tinged, at every twinkling, with red and blue. This observation Mr Melville puts among his queries, with respect to which he could not entirely satisfy himself; and he observes, that the separation of the colours by the refractive power of the atmosphere is, probably, too small to be perceived. But the supposition of Mr Michell above-mentioned will pretty well account for this circumstance, though it may be thought inadequate to the former case. For the red and blue rays being much fewer than those of the intermediate colours, and therefore much more liable to inequalities, from the common effect of chance, a small excess or defect in either of them will make a very sensible difference in the colour of the stars. crease of heat in the place where the rays of light meet, when they are reflected from a concave mirror. The burning power of concave mirrors is taken notice of by Euclid in the second book of the treatise above-mentioned. If we give but a small degree of credit to what some ancient historians are said to have written concerning the exploits of Archimedes, we shall be induced to think that he made great use of this principle, in constructing some very powerful burning-mirrors; but nothing being said of other persons making use of his inventions, the whole account is very doubtful. It is allowed, however, that this eminent geometrician did write a treatise on the subject of burning-mirrors, though it be not now extant.

B. Porta supposes that the burning-mirrors of the ancients were of metal, in the form of a section of a parabola. It follows from the properties of this curve, that all the rays which fall upon it, parallel to its axis, will meet in the same point at the focus. Consequently, if the vertex of the parabola be cut off, as in fig. 1, it will make a convenient burning-mirror. In some drawings of this instrument the frustrum is so small, as to look like a ring. With an instrument of this kind, it is thought, that the Romans lighted their sacred fire. Some have also thought that this was the form of the mirror with which Archimedes burnt the Roman fleet; using either a lens, to throw the rays parallel, when they had been brought to a focus; or applying a smaller parabolic mirror for this purpose, as is represented fig. 2. But Dechales shows, that it is impossible to convey any rays in a direction parallel to one another, except those that come from the same point in the sun's disk.

All this time, however, the nature of reflection was very far from being understood. Even lord Bacon, who made much greater advances in natural philosophy than his predecessors, and who pointed out the true method of improving it, was so far deceived with regard to the nature of reflection and refraction, that he supposed it possible to see the image reflected from a looking-glass, without seeing the glass itself; and to this purpose he quotes a story of friar Bacon, who is reported to have apparently walked in the air between two steeples, and which was thought to have been effected by reflection from glasses while he walked upon the ground.

The whole business of seeing images in the air may be traced up to Vitellio; and what he said upon the subject seems to have passed from writer to writer, with considerable additions, to the time of lord Bacon. What Vitellio endeavours to show is, that it is possible, by means of a cylindrical convex speculum, to see the images of objects in the air, out of the speculum, when the objects themselves cannot be seen. But, if his description of the apparatus requisite for this experiment be attended to, it will be seen that the eye was to be directed towards the speculum, which was placed within a room, while both the object and the spectator were without it. But though he recommends this observation to the diligent study of his readers, he has not described it in such a manner as is very intelligible; and, indeed, it is certain, that no such effect can be produced by a convex mirror. If he himself did make any trial with the apparatus that he describes for this purpose, he must have been under some deception with respect to it.

B. Porta says, that this effect may be produced by a plane mirror only; and that an ingenious person may succeed in it; but his more particular description of a method to produce this extraordinary appearance is by a plane mirror and a concave one combined.

Kircher also speaks of the possibility of exhibiting these pendulous images, and supposes that they are reflected from the dense air; and the most perfect and pleasing deception depending upon the images in the air is one of which this writer gives a particular account in his Ars Magna Lucis et Umbrae, p. 783. In this case the image is placed at the bottom of a hollow polished cylinder, by which means it appears like a real solid substance, suspended within the mouth of the vessel. In this manner, he says, he once exhibited a representation of the ascension of Christ; when the images were so perfect, that the spectators could not be persuaded, but by attempting to handle them, that they were not real substances.

Among other amusing things that were either invented or improved by Kircher, was the method of throwing the appearance of letters, and other forms of things, into a darkened room from without, by means of a lens and a plane mirror. The figures or letters were written upon the face of the mirror, and inverted; and the focus of the lens was contrived to fall upon the screen or wall that received their images. In this manner, he says, that with the light of the sun he could throw a plain and distinct image 500 feet.

It was Kepler who first discovered the true reason of the apparent places of objects seen by reflecting mirrors, as it depends upon the angle which the rays of light, issuing from the extreme part of an object, make with one another after such reflections. In plane mirrors these rays are reflected with the same degree of inclination to one another that they had before their incidence; but he shows that this inclination is changed in convex and concave mirrors.

Mr Boyle made some curious observations concerning the reflecting powers of differently coloured substances. Many learned men, he says, imagined that snow affects the eyes, not by a borrowed, but by a native light; but having placed a quantity of snow in a room from which all foreign light was excluded, neither he nor any body else was able to perceive it. To try whether white bodies reflect more light than others, he held a sheet of white paper in a sun-beam admitted into a darkened room; and observed that it reflected much more light than a paper of any other colour, a considerable part of the room being enlightened by it. Farther, to show that white bodies reflect the rays outwards, he adds, that common burning-glasses will not of a long time burn or discolour white paper. When he was a boy, he says, and took great pleasure in making experiments with these glasses, he was much surprised at this remarkable circumstance; and it set him very early upon guessing at the nature of whiteness, especially as he observed that the image of the sun was not so well defined upon white paper as upon black; and as, when he put ink upon the paper, the moisture would be quickly dried up, and the paper, which he could not burn before, would presently take fire. He also found, that, by exposing his hand to the sun, with a thin black glove upon it, it would be suddenly and more considerably heated, than if he held his naked hand to the rays, or put on a glove of thin white leather.

To prove that black is the reverse of white, with respect to its property of reflecting the rays of the sun, he procured a large piece of black marble; and having got it ground into the form of a large spherical concave speculum, he found that the image of the sun reflected from it was far from offending or dazzling his eyes, as it would have done from another speculum; and though this was large, he could not in a long time set a piece of wood on fire with it; though a far less speculum, of the same form, and of a more reflecting substance, would presently have made it flame.

To satisfy himself still farther with respect to this subject, he took a broad and large tile; and having made one half of its surface white and the other black, he exposed it to the summer sun. And having let it lie there some time, he found, that while the whitened part remained cool, the part that was black was grown very hot. For his farther satisfaction, he sometimes left part of the tile of its native red; and, after exposing the whole to the sun, observed that this part grew hotter than the white, but was not so hot as the black part. He also observes, that rooms hung with black are not only darker than they would otherwise be, but warmer too; and he knew several persons, who found great inconvenience from rooms hung with black. As another proof of his hypothesis, he informs us, that a virtuoso, of unsuspected credit, acquainted him, that, in a hot climate, he had seen eggs well roasted in a short time, by first blacking the shells, and then exposing them to the sun.

We have already taken notice of the remarkable property of lignum nephriticum first observed by Kircher. (See Guilandina.) However, all his observations with regard to it fell very short of Mr Boyle. He describes this lignum nephriticum to be a whitish kind of wood, that was brought from Mexico, which the natives call coal or taparaztli, and which had been thought to tinge water of a green colour only; but he says that he found it to communicate all kinds of colours. If, says he, an infusion of this wood be put into a glass globe, and exposed to a strong light, it will be as colourless as pure water; but if it be carried into a place a little shaded, it will be a most beautiful green. In a place still more shaded, it will incline to red; and in a very shady place, or in an opaque vessel, it will be green again.

A cup of this remarkable wood was sent to Kircher by the procurator of his society at Mexico, and was presented by him to the emperor as a great curiosity. It is called lignum nephriticum, because the infusion of it was imagined to be of service in diseases of the kidneys and bladder, and the natives of the country where it grows do make use of it for that purpose.

Mr Boyle corrected several of the hasty observations of Kircher concerning the colours that appear in the infusion of lignum nephriticum, and he diversified the experiments with it in a very pleasing manner. He first distinctly noted the two very different colours which this remarkable tincture exhibits by transmitted and reflected light. If, says he, it be held directly between the light and the eye, it will appear tinged (excepting the very top of it, where a sky-coloured circle sometimes appears) almost of a golden colour, except the infusion be too strong; in which case it will be dark or reddish, and requires to be diluted with water. But if it be held from the light, so that the eye be between the light and the phial, it will appear of a deep lovely blue colour; as will also the drops, if any lie on the outside of the glass.

When a little of this tincture was poured upon a sheet of white paper, and placed in a window where the sun could shine upon it, he observed, that if he turned his back upon the sun, the shadow of his pen, or any such slender substance, projected upon the liquor, would not be all dark, like other shadows; but that part of it would be curiously coloured, the edge of it next the body being almost of a lively golden colour, and the more remote part blue. These, and other experiments of a similar nature, many of his friends, he says, beheld with wonder; and he remembered an excellent occultist, who accidentally meeting with a phial full of this liquor, and being unacquainted with this remarkable property of it, imagined, after he had viewed it a long time, that some new and strange distemper had seized his eyes; and Mr Boyle himself acknowledges, that the oddness of the phenomenon made him very desirous to find out the cause of it; and his inquiries were not altogether unsuccessful.

Observing that this tincture, if it were too deep, was not tinged in so beautiful a manner, and that the impregnating virtue of the wood did, by being frequently infused in fresh water, gradually decay, he conjectured that the tincture contained much of the essential salt of the wood; and to try whether the subtle parts, on which the colour depended, were volatile enough to be distilled, without dissolving their texture, he applied some of it to the gentle heat of a lamp-furnace; but he found all that came over was as limpid and colourless as rock water, while that which remained behind was of so deep a blue, that it was only in a very strong light that it appeared of any colour.

Suspecting that the tingling particles abounded with salts, whose texture, and the colour thence arising, would probably be altered by acids, he poured into a small quantity of it a very little spirit of vinegar, and found that the blue colour immediately vanished, while the golden one remained, on which ever side it was viewed with respect to the light.

Upon this he imagined, that as the acid salts of the vinegar had been able to deprive the liquor of its blue colour, a fulphurous salt, which is of a contrary nature, would destroy their effects; and having placed himself betwixt the window and the phial, and let fall into the same liquor a few drops of oil of tartar per deliquium, he found that it was immediately restored to its former blue colour, and exhibited all the same phenomena which it had done at the first.

Having sometimes brought a round long-necked phial, filled with this tincture, into a darkened room, into which a beam of the sun was admitted by a small aperture; and holding the phial sometimes near the sun-beams, and sometimes partly in them and partly out. out of them, changing also the position of the glass, and viewing it from several parts of the room, it exhibited a much greater variety of colours than it did in an enlightened room. Besides the usual colours, it was red in some places and green in others, and within were intermediate colours produced by the different degrees and odd mixtures of light and shade.

It was not only in this tincture of lignum nephriticum that Mr Boyle observed the difference between reflected and transmitted light. He observed it even in gold, though no person explained the cause of these effects before Sir Isaac Newton. He took a piece of leaf-gold, and holding it betwixt his eye and the light, observed that it did not appear of a golden colour, but of a greenish blue. He also observed the same change of colour by candle-light; but the experiment did not succeed with a leaf of silver.

The constitution of the atmosphere and of the sea, we shall find, by observations made in later periods, to be similar to that of this infusion; for the blue rays, and others of a faint colour, do not penetrate so far into them as the red, and others of a stronger colour; but what this constitution is, which is common to them all, deserves to be inquired into. For almost all other tinctures, and this of lignum nephriticum too, after some change made in it by Mr Boyle, as well as all other semi-transparent coloured substances, as glass, appear of the same hue in all positions of the eye. To increase or diminish the quantity makes no difference, but to make the colour deeper or more dilute.

Mr Boyle's account of the colours exhibited by thin plates of various substances, are met with among the colours of Mr Boyle. To show the chemists that colours may be made to appear or vanish, where there is no accession or change either of the sulphurous, the saline, or the mercurial principle of bodies, he observes, that all chemical essential oils, as also good spirit of wine, being shaken 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. He then mentions the colours that appear in bubbles of soap and water, and also in turpentine. He sometimes got glass blown so thin as to exhibit similar colours; and observes, that a feather, of a proper shape and size, and also a black ribbon, held at a proper distance, between his eye and the sun, showed a variety of little rainbows, as he calls them, with very vivid colours, none of which were constantly to be seen in the same objects.

Much more pains were taken with this subject, and a much greater number of observations respecting it were made, by Dr Hooke. As he loved to give preference by his discoveries, he promised, at a meeting of the society on the 7th of March 1672, to exhibit, at their next meeting, something which had neither reflection nor refraction, and yet was diaphanous. Accordingly, at the time appointed, he produced the famous coloured bubble of soap and water, of which such admirable use was afterwards made by Sir Isaac Newton, but which Dr Hooke and his contemporaries seem to have overlooked in Mr Boyle's treatise on colours, though it was published nine years before. It is no wonder that so curious an appearance excited the attention of that inquisitive body, and that they should desire him to bring an account of it in writing at their next meeting.

By the help of a small glass pipe, there were blown several small bubbles, out of a mixture of soap and water; where it was observable, that, at first, they appeared white and clear; but that, after some time, the film of water growing thinner, there appeared upon it all the colours of the rainbow: first a pale yellow; then orange, red, purple, blue, green, &c. with the same series of colours repeated; in which it was farther observable, that the first and last series were very faint, and that the middlemost order or series was very bright. After these colours had passed over the changes above-mentioned, the film of the bubble began to appear white again; and presently, in several parts of this second white film, there appeared several holes, which by degrees grew very big, several of them running into one another. After reciting other observations, which are not of much consequence, he says it is strange, that though both the encompassing and encompassed air have surfaces, yet he could not observe that they afforded either reflection or refraction, which all the other parts of the encompassed air did. This experiment, he says, at first sight, may appear very trivial, yet, as to the finding out the nature and cause of reflection, refraction, colours, congruity and incongruity, and several other properties of bodies, he looked upon it as one of the most instructive. And he promised to consider it more afterwards; but we do not find that ever he did: nor indeed is it to be much regretted, as we shall soon find this business in better hands. He adds, that that which gives one colour by reflection, gives another by refraction; not much unlike the tincture of lignum nephriticum.

Dr Hooke was the first to observe, if not to describe, the beautiful colours that appear in thin plates of mucov glass. These, he says, are very beautiful to the naked eye, but much more when they are viewed with a microscope. With this instrument he could perceive that these colours were ranged in rings surrounding the white specks or flaws in this thin substance, that the order of the colours was the very same as in the rainbow, and that they were often repeated ten times. But the colours, he says, were disposed as in the outer bow, and not the inner. Some of them also were much brighter than others, and some of them very much broader. He also observed, that if there was a place where the colours were very broad, and conspicuous to the naked eye, they might be made, by pressing the place with the finger, to change places, and move from one part to another. Lastly, he observed, that if great care be used, this substance may be split into plates of $\frac{1}{8}$ or $\frac{1}{4}$ of an inch in diameter, each of which will appear through a microscope to be uniformly adorned with some one vivid colour, and that these plates will be found upon examination to be of the same thickness throughout.

As a fact similar to this, but observed previous to it, we shall here mention that Lord Brereton, at a meeting of the Royal Society in 1666, produced some pieces pieces of glass taken out of a window of a church, both on the north and on the south side of it; observing, that they were all eaten in by the air, but that the piece taken from the south side had some colours like those of the rainbow upon it, which the others on the north side had not. This phenomenon has been frequently observed since, and in other circumstances. It is not to be doubted, but that in all these cases, the glass is divided into thin plates, which exhibit colours, upon the same principle with those which Dr Hooke observed in the bubble of soap and water, and in the thin plate of air, which we shall find more fully explained by Sir Isaac Newton. With care the thin plates of the glass may be separated, and the theory verified.

An observation made by Otto Guericke, well explains the reason why stars are visible at the bottom of a deep well. It is, says he, because the light that proceeds from them is not overpowered by the rays of the sun, which are lost in the number of reflections which they must undergo in the pit, so that they can never reach the eye of a spectator at the bottom of it.

But of all those who have given their attention to this subject of the reflection of light, none seems to have given such satisfaction as M. Bouguer; and next to those of Sir Isaac Newton, his labours seem to have been the most successful. The object of his curious and elaborate experiments was to measure the degrees of light, whether emitted, reflected, or refracted, by different bodies. They were originally occasioned by an article of M. Maury's in the memoirs of the French academy for 1721, in which the proportion of the light of the sun at the two solstices were supposed to be known; and his laudable attempt to verify what had been before taken for granted, suggested a variety of new experiments, and opened to him and to the world a new field of optical knowledge. His first production upon this subject was a treatise entitled Étude d'Optique, which was received with general approbation. Afterwards, giving more attention to this subject, he formed an idea of a much larger work, to which many more experiments were necessary: but he was prevented, by a variety of interruptions, from executing his design so soon as he had proposed; and he had hardly completed it at the time of his death, in 1788; so that we are obliged to his friend M. de la Caille for the care of the publication. At length, however, it was printed at Paris in 1760, under the title of Traité d'Optique.

At the entrance upon this treatise, we are induced to form the most pleasing expectations from our author's experiments, by his account of the variety, the singular accuracy, and circumference, with which he made them; whereby he must, to all appearance, have guarded against every avenue to error and particularly against those objections to which the few attempts that had been made, of a similar nature, before him had been liable. In order to compare different degrees of light, he always contrived to place the bodies from which it proceeded, or other bodies illuminated by them, in such a manner as that he could view them distinctly at the same time; and he either varied the distances of these bodies, or modified their light in some other way, till he could perceive no difference between them. Then, considering their different distances, or the other circumstances by which their light was affected, he calculated the proportion on which they would have borne to each other at the same distance, or in the same circumstances.

To ascertain the quantity of light lost by reflection, he placed the mirror, or reflecting surface, B, on which the experiment was to be made, truly upright; and having taken two tablets, of precisely the same colour, or of an equal degree of whiteness, he placed them exactly parallel to one another at E and D, and threw light upon them by means of a lamp or candle, P, placed in a right line between them. He then placed himself so, that with his eye at A he could see the tablet E, and the image of the tablet D, reflected from the mirror B, at the same time; making them, as it were, to touch one another. He then moved the candle along the line ED, so as to throw more or less light upon either of them, till he could perceive no difference in the strength of the light that came to his eye from them. After this, he had nothing more to do than to measure the distances EP and DP; for the squares of those distances expressed the degree in which the reflection of the mirror diminished the quantity of light. It is evident, that if the mirror reflected all the rays it received, the candle P must have been placed at C, at an equal distance from each of the tablets, in order to make them appear equally illuminated; but because much of the light is lost in reflection, they can only be made to appear equally bright by placing the candle nearer the tablet D, which is seen by reflection only.

To find how much light is lost by oblique reflection, he took two equally polished plates, D and E, and caused them to be enlightened by the candle P; and while one of them, D, was seen at A, by reflection from B, placed in a position oblique to the eye, the other, E, was so placed, as to appear contiguous to it; and removing the plate E, till the light which it reflected was no stronger than that which came from the image D, seen by reflection at B, he estimated the quantity of light that was lost by this oblique reflection, by the squares of the distances of the two objects from the candle.

It need scarcely be added, that in these experiments all foreign light was excluded, that his eye was shaded, and that every other precaution was observed in order to make his conclusions unquestionable.

In order to ascertain the quantity of light lost by reflection with the greatest exactness, M. Bouguer introduced two beams of light into a darkened room, as by the apertures P and Q; which he had so contrived, that he could place them higher or lower, and enlarge or contract them at pleasure; and the reflecting surface (as that of a fluid contained in a vessel) was placed horizontally at O, from whence the light coming through the hole P, was reflected to R, upon the screen GH, where it was compared with another beam of light that fell upon S, through the hole Q; which he made so much less than P, as that the spaces S and R were equally illuminated; and by the proportion that the apertures P and Q bore to each other, he calculated what quantity of light was lost by the reflection at O.

It was necessary, he observes, that the two beams of light light PO and QS (which he usually made 7 or 8 feet long) should be exactly parallel, that they might come from two points of the sky equally elevated above the horizon, and having precisely the same intensity of light. It was also necessary that the hole Q should be a little higher than P, in order that the two images should be at the same height, and near one another. It is no less necessary, he says, that the screen GH be exactly vertical, in order that the direct and reflected beams may fall upon it with the same inclination; since, otherwise, though the two lights were perfectly equal, they would not illuminate the screen equally. This disposition, he says, serves to answer another important condition in these experiments; for the direct ray QS must be of the same length with the sum of the incident and reflected rays, PO and OR, in order that the quantity of light introduced into the room may be sensibly proportional to the sizes of the apertures.

We shall now proceed to recite the result of the experiments which he made to measure the quantity of light that is lost by reflection in a great variety of circumstances; but we shall introduce them by the recital of some which were made previous to them on the diminution of light by reflection, and the transmission of it to considerable distances through the air, by M. Basset, at the time that he was constructing his machine to burn at great distances, mentioned under the article Burning-Glass.

Receiving the light of the sun in a dark place, and comparing it with the same light of the sun reflected by a mirror, he found, that at small distances, as four or five feet, about one half was lost by reflection; as he judged by throwing two reflected beams upon the same place, and comparing them with a beam of direct light; for then the intensity of them both seemed to be the same.

Having received the light at greater distances, as at 100, 200, and 300 feet, he could hardly perceive that it lost any of its intensity by being transmitted through such a space of air.

He afterwards made the same experiments with candles, in the following manner: He placed himself opposite to a looking-glass, with a book in his hand, in a room perfectly dark; and having one candle lighted in the next room, at the distance of about 40 feet, he had it brought nearer to him by degrees, till he was just able to distinguish the letters of the book, which was then 24 feet from the candle. He then received the light of the candle, reflected by the looking-glass, upon his book, carefully excluding all the light that was reflected from anything else; and he found that the distance of the book from the candle, including the distance from the book to the looking-glass (which was only half a foot) was in all 15 feet. He repeated the experiment several times, and always with nearly the same result; and therefore concluded, that the quantity of direct light is to that of reflected as 576 to 225; so that the light of five candles reflected from a plane mirror is about equal to that of two candles.

From these experiments it appeared, that more light was lost by reflection of the candles than of the sun, which M. Buffon thought was owing to this circumstance, that the light issuing from the candle diverges, and therefore falls more obliquely upon the mirror than the light of the sun, the rays of which are nearly parallel.

These experiments and observations of M. Buffon are curious; though it will be seen that they fall far short of those of M. Bouguer, both in extent and accuracy. We shall begin with those which he made to ascertain the difference in the quantity of light reflected by glass and polished metal.

Using a smooth piece of glass one line in thickness, Mr Bouguer found, that when it was placed at an angle of 15 degrees with the incident rays, it reflected 628 parts of 1000 which fell upon it; at the same time that a metallic mirror, which he tried in the same circumstances, reflected only 561 of them. At a less angle of incidence much more light was reflected; so that at an angle of three degrees the glass reflected 700 parts, and the metal something less, as in the former case.

Trying the reflection of bodies that were not polished, he found that a piece of white plaster, placed at an angle of 75°, with the incident rays, reflected $\frac{1}{19}$ part of the light that is received from a candle nine inches from it. White paper, in the same circumstances, reflected in the same proportion; but at the distance of three inches, they both reflected 150 parts of 1000 that were incident.

Proceeding to make farther observations on the subject of reflected light, he premises the two following theorems, which he demonstrates geometrically.

1. When the luminous body is at an infinite distance, and its light is received by a globe, the surface of which has a perfect polish, and absorbs no light, it reflects the light equally in all directions, provided it be received at a considerable distance. He only excepts the place where the shadow of the globe falls; but this, he says, is no more than a single point, with respect to the immensity of the spherical surface which receives its light.

2. The quantity of light reflected in one certain direction will always be exactly the same, whether it be reflected by a very great number of small polished hemispheres, by a less number of larger hemispheres, or by a single hemisphere, provided they occupy the same base, or cover the same ground-plan.

The use he proposes to make of these theorems is to assist him in distinguishing whether the light reflected from bodies be owing to the extinction of it within them, or whether the roughnesses or eminences which cover them have not the same effect with the small polished hemispheres above-mentioned.

He begins with observing, that, of the light reflected from Mercury, $\frac{1}{4}$ at least is lost, and that probably no substances reflect more than this. The rays were received at an angle of 115 degrees of incidence, that is measured from the surface of the reflecting body, and not from the perpendicular, which, he says, is what we are from this place to understand whenever he mentions the angle of incidence.

The most striking observations which he made with great respect to this subject, are those which relate to the very great difference in the quantity of light reflected at different angles of incidence. In general, he says, that reflection is stronger at small angles of incidence, and weaker at large ones. The difference is excessive when the rays strike the surface of transparent substances. stances, with different degrees of obliquity; but it is almost as great in some opaque substances, and it was always more or less so in every thing that he tried. He found the greatest inequality in black marble; in which he was astonished to find, that, with an angle of $3^\circ 35'$ of incidence, though not perfectly polished, yet it reflected almost as well as quicksilver. Of 1000 rays which it received, it returned 600; but when the angle of incidence was 14 degrees, it reflected only 150; when it was 30, it reflected 51; and when it was 80, it reflected only 23.

Similar experiments made with metallic mirrors always gave the differences much less considerable. The greatest was hardly ever an eighth or a ninth part of it, but they were always in the same way.

The great difference between the quantity of light reflected from the surface of water, at different angles of incidence, is truly surprising; but our author observes, that this difference was greater when the smallest inclinations were compared with those which were near to a right angle. He sometimes suspected, that, at very small angles of incidence, the reflection from water was even greater than from quicksilver. All things considered, he thought it was not quite so great, though it was very difficult to determine the precise difference between them. In very small angles, he says, that water reflects nearly $\frac{1}{2}$ of the direct light.

There is no person, he says, but has sometimes felt the force of this strong reflection from water, when he has been walking in still weather on the brink of a lake opposite to the sun. In this case, the reflected light is $\frac{1}{2}$, or sometimes a greater proportion of the light that comes directly from the sun, which is an addition to the direct rays of the sun that cannot fail to be very sensible. The direct light of the sun diminishes gradually as it approaches the horizon, while the reflected light at the same time grows stronger; so that there is a certain elevation of the sun, in which the united force of the direct and reflected light will be the greatest possible, and this he says is 12 or 13 degrees.

On the other hand, the light reflected from water at great angles of incidence is extremely small. Our author was assured, that, when the light was perpendicular, it reflected no more than the 37th part that quicksilver does in the same circumstances; for it did not appear, from all his observations, that water reflects more than the 60th, or rather the 55th, part of perpendicular light. When the angle of incidence was 50 degrees, the light reflected from the surface of water was about the 32d part of that which mercury reflected; and as the reflection from water increases with the diminution of the angle of incidence, it was twice as strong in proportion at 39 degrees; for it was then the 16th part of the quantity that mercury reflected.

In order to procure a common standard by which to measure the proportion of light reflected from various fluid substances, he pitched upon water as the most commodious; and partly by observation, and partly by calculation, which he always found to agree with his observations, he drew up the following table of the quantity of light reflected from the surface of water, at different angles with the surface.

| Angles of incidence | Rays reflected of 1000 | |---------------------|-----------------------| | $\frac{1}{2}$ | 721 | | 1 | 692 | | 1 $\frac{1}{2}$ | 669 | | 2 | 639 | | 2 $\frac{1}{2}$ | 614 | | 3 | 501 | | 7 $\frac{1}{2}$ | 409 | | 10 | 333 | | 12 $\frac{1}{2}$ | 271 | | 15 | 211 |

| Angles of incidence | Rays reflected of 1000 | |---------------------|-----------------------| | 17 $\frac{1}{2}$ | 178 | | 20 | 145 | | 25 | 97 | | 30 | 65 | | 40 | 34 | | 50 | 22 | | 60 | 19 | | 70 | 18 | | 80 | 18 | | 90 | 18 |

In the same manner, he drew up the following table of the quantity of light reflected from the looking-glass not quicksilvered.

| Angles of incidence | Rays reflected of 1000 | |---------------------|-----------------------| | 2 $\frac{1}{2}$ | 584 | | 5 | 543 | | 7 $\frac{1}{2}$ | 474 | | 10 | 412 | | 12 $\frac{1}{2}$ | 356 | | 15 | 299 | | 20 | 222 | | 25 | 157 |

| Angles of incidence | Rays reflected of 1000 | |---------------------|-----------------------| | 30 | 112 | | 40 | 57 | | 50 | 34 | | 60 | 27 | | 70 | 25 | | 80 | 25 | | 90 | 25 |

Pouring a quantity of water into a vessel containing quicksilver, it is evident that there will be two images of any objects seen by reflection from them, one at the surface of the water, and the other at that of the quicksilver. In the largest angles of incidence, the image at the surface of the water will disappear, which will happen when it is about a 60th or an 80th part less luminous than the image at the surface of the quicksilver. Depressing the eye, the image on the water will grow stronger, and that on the quicksilver weaker in proportion; till at last, the latter will be incomparably weaker than the former, and at an angle of about 10 degrees they will be equally luminous. According to the table, $\frac{1}{111}$ of the incident rays are reflected from the water at this angle of 10 degrees. At the surface of the mercury they were reduced to 500; and of these, part being reflected back upon it from the under surface of the water, only 333 remained to make the image from the mercury.

It has been observed by several persons, particularly by Mr Edwards, (see Phil. Trans. vol. 53. p. 229.) that there is a remarkably strong reflection into water, by the air, with respect to rays issuing from the water; and persons under water have seen images of things in the air in a manner peculiarly distinct and beautiful; but this fact had not been observed with a sufficient degree of attention, till it came into M. Bouguer's way to do it, and he acknowledges it to be very remarkable. In this case, he says, that from the smallest angles of incidence, to a certain number of degrees, the greatest part of the rays are reflected, perhaps in as great a proportion as at the surface of metallic mirrors, or of quicksilver; while the other part, which does not escape into the air, is extinguished or absorbed; so that the surface of the transparent body appears opaque on the inside. If the angle of incidence be increased only a few degrees, the strong reflection ceases altogether, a great number of rays escape into the air, and very few are absorbed or extinguished. In proportion as the angle of incidence is farther increased, the quantity of the light reflected becomes less and less; and when it is near 90 degrees, almost all the rays escape out of the transparent body, its surface losing almost all its power of reflection, and becoming almost as transparent as it is in other cases, or when the light falls upon it from without.

This property belonging to the surfaces of transparent bodies, of absorbing or extinguishing the rays of light, is truly remarkable, and, as there is reason to believe, had not been noticed by any person before Mr. Bouguer. It had been conjectured by Sir Isaac Newton, that rays of light become extinct only by impinging upon the solid parts of bodies; but these observations of M. Bouguer show that the fact is quite otherwise; and that this effect is to be attributed, not to the solid parts of bodies, which are certainly more numerous in a long tract of water than just in the passage out of water into air, but to some power lodged at the surfaces of bodies only, and therefore probably the same with that which reflects, refracts, and infuses the light.

One of the above-mentioned observations, viz. all the light being reflected at certain angles of incidence from air into denser substances, had frequently been made, especially in glass prisms; so that Newton made use of one of them, instead of a reflecting mirror, in the construction of his telescope. If a beam of light fall upon the air from within these prisms, at an angle of 10°, 20°, or 30° degrees, the effect will be nearly the same as at the surface of quicksilver, a fourth or a third of the rays being extinguished, and \( \frac{1}{4} \) or \( \frac{1}{3} \) this reflected. This property retains its full force as far as an incidence of 49° 49' (supposing the proportion of the fines of refraction to be 31 and 20 for the mean refrangible rays); but if the angle of incidence be increased but one degree, the quantity of light reflected inwards decreases suddenly, and a great part of the rays escape out of the glass, so that the surface becomes suddenly transparent.

All transparent bodies have the same property, with this difference, that the angle of incidence at which the strong reflection ceases, and at which the light which is not reflected is extinguished, is greater in some than in others. In water this angle is about 41° 32'; and in every medium it depends so much on the invariable proportion of the sine of the angle of refraction to the fine of the angle of incidence, that this law alone is sufficient to determine all the phenomena of this new circumstance, at least as to this accidental opacity of the surface.

When our author proceeded to measure the quantity of light reflected by these internal surfaces at great angles of incidence, he found many difficulties, especially on account of the many alterations which the light underwent before it came to his eye; but at length, using a plate of crystal, he found, that, at an angle of 75 degrees, this internal reflection diminished the light 27 or 28 times; and as the external reflection at the same angle diminished the light only 26 times, it follows that the internal reflection is a little stronger than the other.

Repeating these experiments with the same and different pieces of crystal, he sometimes found the two reflections to be equally strong; but, in general, the internal was the stronger. Also, the image reflected internally was always a little redder than an object which was seen directly through the plate of crystal.

Referring his observations on the diminution of light, occasioned by the reflection of opaque bodies obliquely situated, he compared it with the appearances of similar substances which reflected the light perpendicularly. Using pieces of silver made very white, he found, that, when one of them was placed at an angle of 75 degrees with respect to the light, it reflected only 640 parts out of 1000. He then varied the angle, and also used white plaster and fine Dutch paper, and drew up the following table of the proportion of the light reflected from each of those substances at certain angles.

| Angles of incidence | Silver | Plaster | Dutch Paper | |--------------------|--------|---------|-------------| | 90 | 1000 | 1000 | 1000 | | 75 | 802 | 762 | 971 | | 60 | 640 | 640 | 743 | | 45 | 455 | 529 | 507 | | 30 | 319 | 352 | 332 | | 15 | 209 | 194 | 203 |

Supposing the asperities of opaque bodies to consist of very small planes, it appears from these observations, that there are fewer of them in those bodies which reflect the light at small angles of incidence than at greater; and our author says, that the case was nearly the same with respect to all the opaque bodies that he tried. None of them had their roughness equivalent to small hemispheres, which would have dispersed the light equally in all directions; and, from the data in the preceding table, he deduces mathematically the number of the little planes that compose those surfaces, and that are inclined to the general surface at the angles above-mentioned, supposing that the whole surface contains 1000 of them that are parallel to itself, so as to reflect the light perpendicularly, when the luminous body is situated at right angles with respect to it. His conclusions reduced to a table, corresponding to the preceding, are as follow:

| Inclinations of the small surfaces with respect to the large one. | The distribution of the small planes that constitute the asperities of the opaque surface in the | |---------------------------------------------------------------|----------------------------------| | | Silver | Plaster | Paper | | 0 | 1000 | 1000 | 1000 | | 15 | 777 | 736 | 937 | | 30 | 554 | 554 | 545 | | 45 | 333 | 374 | 358 | | 60 | 161 | 176 | 166 | | 75 | 53 | 50 | 52 |

These These variations in the number of little planes, or surfaces, he expresses in the form of a curve; and afterwards he shows, geometrically, what would be the effect if the bodies were enlightened in one direction, and viewed in another; upon which subject he has several curious theorems and problems: as, the position of the eye being given, to find the angle at which the luminous body must be placed, in order to its reflecting the most light; or, the situation of the luminous body being given, to find a proper situation for the eye, in order to see it the most enlightened, &c. But it would carry us too far into geometry to follow him through all these disquisitions.

Since the planets, as this accurate observer takes notice, are more luminous at their edges than at their centres, he concludes, from the above-mentioned principles, &c., that the bodies which form them are constituted in a manner different from ours; particularly that their opaque surfaces consist of small planes, more of which are inclined to the general surface than they are in terrestrial substances; and that there are in them an infinity of points, which have exactly the same splendour.

Our philosopher and geometrician next proceeds to ascertain the quantity of surface occupied by the small planes of each particular inclination, from considering the quantity of light reflected by each, allowing those that have a greater inclination to the common surface to take up proportionably less space than those which are parallel to it. And comparing the quantity of light that would be reflected by small planes thus disposed, with the quantity of light that was actually reflected by the three substances above-mentioned, he found, that plaster, notwithstanding its extreme whiteness, absorbs much light; for that, of 1000 rays that fall upon it, of which 166 or 167 ought to be reflected at an angle of 77 degrees, only 67 are in fact returned; so that 100 out of 167 were extinguished, that is, about three-fifths.

With respect to the planets, our author concludes, that of 300,000 rays which the moon receives, 172,000 are absorbed, or perhaps 204,100.

Having considered the surfaces of bodies as consisting of planes only, he thus explains himself.—Each small surface, separately taken, is extremely irregular, and some of them are really concave, and others convex; but, in reducing them to a middle state, they are to be considered as planes. Nevertheless he considers them as planes only with respect to the reception of the rays; for as they are almost all curves, and as, besides this, many of those whose situation is different from others contribute to the same effect, the rays always issue from an actual or imaginary focus, and after reflection always diverge from another.

If it be asked, what becomes of those rays that are reflected from one aperture to another? he shows that very few of the rays can be in those circumstances; since they must fall upon planes which have more than 45 degrees of obliquity to the surface, of which there are very few in natural bodies. These rays must also fall at the bottom of those planes, and must meet with other planes similarly situated to receive them; and considering the great irregularity of the surfaces of opaque bodies, it may be concluded that very few of the rays are thus reflected upon the body itself; and that the little that is so reflected is probably lost to the spectators, being extinguished in the body.

We are obliged to Mr Melville for some ingenious observations on the manner in which bodies are heated by light. He observes, that, as each colorific particle of an opaque body must be somewhat moved by the reaction of the particles of light, when it is reflected backwards and forwards between the same particles, it is manifest that they must likewise be agitated with a vibratory motion, and the time of a vibration will be equal to that which light takes up in moving from one particle of a body to another adjoining. This distance, in the most solid opaque bodies, cannot be supposed greater than \( \frac{1}{1000000} \)th of an inch, which space a particle of light describes in the \( \frac{1}{1000000000000} \)th of a second. With so rapid a motion, therefore, may the internal parts of bodies be agitated by the influence of light, as to perform 125,000,000,000 vibrations, or more, in a second of time.

The arrival of different particles of light at the surface of the same colorific particle, in the same or different rays, may disturb the regularity of its vibrations, but will evidently increase their frequency, or raise still smaller vibrations among the parts which compose those particles; by which means the intestine motion will become more minute, and be thoroughly diffused. If the quantity of light admitted into the body be increased, the vibrations of the particles must likewise increase in magnitude and velocity, till at last they may be so violent, as to make all the component particles dash one another to pieces by their mutual collision; in which case, the colour and texture of the body must be destroyed.

Since there is no reflection of light but at the surface of a medium, the same person observes, that the greatest quantity of rays, though crowded into the smallest space, will not of themselves produce any heat. From hence it follows, that the portion of air which lies in the focus of the most potent speculum, is not at all affected by the passage of light through it, but continues of the same temperature with the ambient air; though any opaque body, or even any transparent body denser than air, when put in the same place, would be intensely heated in an instant.

This consequence, evidently flowing from the plainest and most certain principles, not seeming to have been rightly understood by many philosophers, and even the silence of most physical writers concerning this paradoxical truth making it probable that they were unacquainted with it, he thought it worth his while to say something in explication of it. He observes, that the easiest way to be satisfied of the matter experimentally is, to hold a hair, or a piece of down, immediately above the focus of a lens or speculum, or to blow a stream of smoke from a pipe horizontally over it; for if the air in the focus were hotter than the surrounding fluid, it would continually ascend upon account of its rarefaction, and thereby sensibly agitate those slender bodies. Or a lens may be so placed as to form its focus within a body of water, or some other transparent substance, the heat of which may be examined from time to time with a thermometer; but care must be taken, in this experiment, to hold the lens as near as possible to the transparent body, lest the rays, by falling closer than ordinary on its surface, surface, should warm it more than the common sun- beams (b).

To apply these observations to the explication of natural phenomena, he observes, that the atmosphere is not much warmed by the passage of the sun's light through it, but chiefly by its contact with the heated surface of the globe. This, he thought, furnished one very simple and plausible reason why it is coldest in all climates on the tops of very high mountains; namely, because they are removed to the greatest dis- tance from the general surface of the earth. For it is well known, that a fluid heated by its contact with a solid body, decreases in heat in some inverse propor- tion to the distance from the body. He himself found, by repeated trials, that the heat of water in deep lakes decreases regularly from the surface downwards. But to have this question fully determined, the temperature of the air in the valley and on the mountain-top must be observed every hour, both night and day, and care- fully compared together.

From this doctrine he thinks it reasonable to sup- pose, that the heat produced by a given number of rays, in an opaque body of a given magnitude, must be greater when the rays are more inclined to one another, than when they are less so; for the direction of the vibrations raised by the action of the light, whether in the colorific particles, or those of an in- ferior order, will more interfere with one another; from whence the intestine shocks and collisions must increase. Besides this, the colorific particles of opaque bodies being disposed in various situations, perhaps, upon the whole, the rays will fall more directly on each, the more they are inclined to one another. Is not this, says he, the reason of what has been remarked by philosophers, that the heat of the sun's light, col- lected into a cone, increases in approaching the focus in a much higher proportion than according to its density? That the difference of the angle in which the rays fall on any particle of a given magnitude, placed at different distances from the focus, is but small, is no proof that the phenomenon cannot be attributed to it; since we know not in what high pro- portion one or both the circumstances now mentioned may operate. However, that it proceeds not from any unknown action of the rays upon one another, as has been insinuated, is evident from this, that each particular ray, after passing through the focus, pre- serves its own colour and its own direction, in the same manner as if it were alone.

The attempts of the Abbé Nollet to fire inflam- mable substances by the power of the solar rays col- lected in the foci of burning mirrors, have a near rela- tion to the present subject. Considering the great power of burning mirrors and lenses, especially those of late construction, it will appear surprising that this celebrated experimental philosopher should not be able to fire any liquid substance. But though he made the trial with all the care imaginable on the 19th of Fe- bruary 1757, he was not able to do it either with spirit of wine, olive-oil, oil-of-turpentine, or æther;

and though he could fire sulphur, yet he could not succeed with Spanish-wax, rosin, black pitch, or fuel. He both threw the focus of these mirrors upon the substances themselves, and also upon the fumes that rose from them; but all the effect was, that the li- quor boiled, and was dispersed in vapour or very small drops, but would not take fire. When linen-rags, and other solid substances, were moistened with any of these inflammable liquids, they would not take fire till the liquid was dispersed in a copious fume; so that rags thus prepared were longer in burning than those that were dry.

M. Beaume, who assisted M. Nollet in some of M. Beu- these experiments, observed farther, that the same effects expe- riences which were easily fired by the flame of flint- burning bodies, could not be set on fire by the contact of the hottest bodies that did not actually flame. Nei- ther æther nor spirit-of-wine could be fired with a hot coal, or even red-hot iron, unless they were of a white heat. From these experiments our author concludes, that supposing the electric matter to be the same thing with fire or light, it must fire spirit-of-wine by means of some other principle. The members of the academy Del Cimento had attempted to fire several of these substances, though without success; but this was so early in the history of philosophy, that nobody seems to have concluded, that, because they failed in this attempt, the thing could not be done. However, the Abbé informs us, that he read an account of his experiments to the Royal Academy at Paris several years before he attended to what had been done by the Italian philosophers.

By the help of optical principles, and especially by Bodies observations on the reflection of light, Mr Melville dis- covered that bodies which seem to touch one another to touch are not always in actual contact. "It is common one ano- (fays he) to admire the volatility and lustre of drops in actual of rain that lie on the leaves of colewort, and some contact other vegetables;" but no philosopher, as far as he knew, had put himself to the trouble of explaining this curious phenomenon. Upon inspecting them narrowly, he found that the lustre of the drop is pro- duced by a copious reflection of light from the flattened part of its surface contiguous to the plant. He ob- served farther, that, when the drop rolls along a part which has been wetted, it immediately loses all its lustre, the green plant being then seen clearly through it; whereas, in the other case, it is hardly to be dis- cerned.

From these two observations put together, he con- cluded, that the drop does not really touch the plant, when it has the mercurial appearance, but is suspended in the air at some distance from it by the force of a repulsive power. For there could not be any copious reflection of white light from its under-surface, unless there were a real interval between it and the surface of the plant.

If that surface were perfectly smooth, the under- surface of the drop would be so likewise, and would therefore show an image of the illuminating body by

(b) To these observations objections might be made which it would not perhaps be easy to answer; but we are at present giving only the history of optics. reflection, like a piece of polished silver; but as it is considerably rough and unequal, the under-surface becomes rough likewise, and so, by reflecting the light copiously in different directions, assumes the resplendent white colour of unpolished silver.

It being thus proved by an optical argument, that the drop is not really in contact with the plant which supports it, it may easily be conceived whence its volatility arises, and why it leaves no moisture where it rolls.

Before we conclude the history of the observations concerning the reflection of light, we must not omit to take notice of two curious miscellaneous ones. Baron Alexander Funk, visiting some silver mines in Sweden, observed, that, in a clear day, it was as dark as pitch underground in the eye of a pit, at 60 or 70 fathoms deep; whereas, in a cloudy or rainy day, he could even see to read at 106 fathoms deep. Inquiring of the miners, he was informed that this is always the case; and, reflecting upon it, he imagined that it arose from this circumstance, that when the atmosphere is full of clouds, light is reflected from them into the pit in all directions, and that thereby a considerable proportion of the rays are reflected perpendicularly upon the earth; whereas, when the atmosphere is clear, there are no opaque bodies to reflect the light in this manner, at least in a sufficient quantity; and rays from the sun itself can never fall perpendicularly in that country. The other was that of the ingenious Mr Grey, who makes such a figure in the history of electricity. This gentleman took a piece of stiff brown paper, and pricking a small hole in it, he held it at a little distance before him; when, applying a needle to his eye, he was surprised to see the point of it inverted. The nearer the needle was to the hole, the more it was magnified, but the less distinct; and if it was held, as that its image was near the edge of the hole, its point seemed crooked. From these appearances he concluded, that these small holes, or something in them, produce the effects of concave speculums; and from this circumstance he took the liberty to call them aerial speculums.

§ 4. Discoveries concerning the Inflection of Light.

This property of light was not discovered till about the middle of the last century. The person who first made the discovery was Father Grimaldi; at least he first published an account of it in his treatise De lumine, coloribus, et iride, printed in 1666. Dr Hooke, however, laid claim to the same discovery, though he did not publish his observations till six years after Grimaldi; having probably never seen his performance.

Dr Hooke having made his room completely dark, admitted into it a beam of the sun's light by a very small hole in a brass plate fixed in the window-shutter. This beam spreading itself, formed a cone, the apex of which was in the hole, and the base was on a paper, so placed as to receive it at some distance. In this image of the sun, thus painted on the paper, he observed that the middle was much brighter than the edges, and that there was a kind of dark penumbra about it, of about a 16th part of the diameter of the circle; which penumbra, he says, must be ascribed to a property of light, which he promised to explain.

Having observed this, at the distance of about two inches from the former he let in another cone of light; and receiving the bases of them, at such a distance from the holes as that the circles intersected each other, he observed that there was not only a penumbra, or darker ring, encompassing the lighter circle, but a manifest dark line, or circle, which appeared even where the limb of the one interfered with that of the other. This appearance is distinctly represented, fig. 6.

Comparing the diameter of this base with its distance from the hole, he found it to be by no means the same as it would have been if it had been formed by straight lines drawn from the extremities of the sun's disk, but varied with the size of the holes, and the distance of the paper.

Struck with this appearance, he proceeded to make farther experiments concerning the nature of light thus transmitted. To give a just idea of which, he held an opaque body BB, fig. 7, so as to intercept the light that entered at a hole in the window-shutter O, and was received on the screen AP. In these circumstances, he observed, that the shadow of the opaque body (which was a round piece of wood, not bright or polished) was all over somewhat enlightened, but more especially towards the edge. Some persons who were present, imagining that this light within the shadow might be produced by some kind of reflection from the side of this opaque body, on account of its roundness; and others supposing it might proceed from some reflection from the sides of the hole in the piece of brass through which the light was admitted into the room; to obviate both these objections, he admitted the light through a hole burnt in a piece of pasteboard, and intercepted it with a razor which had a very sharp edge; but still the appearances were the very same as before: so that, upon the whole, he concluded that they were occasioned by a new property of light, different from any that had been observed by preceding writers.

He farther diversified this experiment, by placing the razor so as to divide the cone of light into two parts, the hole in the shutter remaining as before, and placing the paper so as that none of the enlightened part of the circle fell upon it, but only the shadow of the razor; and, to his great surprise, he observed what he calls a very brisk and visible radiation striking down upon the paper, of the same breadth with the diameter of the lucid circle; and this radiation always struck perpendicularly from the line of shadow, and, like the tail of a comet, extended more than 10 times, and probably more than 100 times the breadth of the remaining part of the circle: nay, as far as he could find, by many trials, the light from the edge struck downwards into the shadow very near to a quadrant, though the greater were the deflections of this new light from the direct radiations of the cone, the more faint they were.

Observing this appearance with more attention, he found, wherever there was a part of the interposed body higher than the rest, that, opposite to it, the radiation of light into the shadow was brighter, as in the figure; and wherever there was a notch or gap in it, there would be a dark stroke in the half-enlightened shadow. From all these appearances, he concluded, that they were... were to be ascribed to a new property of light, whereby it is deflected from straight lines, contrary to what had been before asserted by optical writers.

It does not appear, however, that our philosopher ever prosecuted this experiment to any purpose; as all that we find of his on the subject of light, after this time, are some crude thoughts which he read at a meeting of the Royal Society, on the 18th of March 1675; which, however, as they are only short hints, we shall copy.

They consist of eight articles: and, as he thought, contained an account of several properties of light that had not been noticed before. There is a deflection of light, differing both from reflection and refraction, and seeming to depend on the unequal density of the constituent parts of the ray, whereby the light is dispersed from the place of condensation, and rarified, or gradually diverged into a quadrant. 2. This deflection is made towards the superficies of the opaque body perpendicularly. 3. Those parts of the diverged radiations which are deflected by the greatest angle from the straight or direct radiations are the faintest, and those that are deflected by the least angles are the strongest. 4. Rays cutting each other in one common foramen do not make the angles at the vertex equal. 5. Colours may be made without refraction. 6. The diameter of the sun cannot be truly taken with common sights. 7. The same rays of light, falling upon the same point of an object, will turn into all sorts of colours, by the various inclinations of the object. 8. Colours begin to appear when two pulses of light are blended so well, and so near together, that the sense takes them for one.

We shall now proceed to the discoveries of Father Grimaldi. Having introduced a ray of light, through a very small hole, AB, fig. 8, into a darkened room, he observed that the light was diffused in the form of a cone, the base of which was CD; and that if any opaque body, FE, was placed in this cone of light, at a considerable distance from the hole, and the shadow was received upon a piece of white paper, the boundaries of it were not confined within GH, or the penumbra LL, occasioned by the light proceeding from different parts of the aperture, and of the disk of the sun, but extended to MN; at which he was very much surprized, suspecting, and finding by calculation, that it was considerably broader than it could have been made by rays passing in right lines by the edges of the object.

But the most remarkable circumstance in this appearance was, that upon the lucid part of the base, CM and ND, streaks of coloured light were plainly distinguished, each being terminated by blue on the side next to the shadow, and by red on the other; and though these coloured streaks depended, in some measure, on the size of the aperture AB, because they could not be made to appear if it was large, yet he found that they were not limited either by it, or by the diameter of the sun's disk.

He farther observed, that these coloured streaks were not all of the same breadth, but grew narrower as they receded from the shadow, and were each of them broader the farther the shadow was received from the opaque body, and also the more obliquely the paper on which they were received was held with respect to it. He never observed more than three of these streaks.

To give a clearer idea of these coloured streaks, he drew the representation of them, exhibited in fig. 9, in which NMO represents the broadest and most luminous streak, next to the dark shadow X. In the space in which M is placed there was no distinction of colour, but the space NN was blue, and the space OO, on the other side of it, was red. The second streak, QPR, was narrower than the former; and of the three parts of which it consisted, the space P had no particular colour, but QQ was a faint blue, and RR a faint red. The third streak, TSV, was exactly similar to the two others, but narrower than either of them, and the colours still fainter.

These coloured streaks he observed to lie parallel to the shadow of the opaque body; but when it was of an angular form, they did not make the same acute angles, but were bent into a curve, the outermost being rounder than those that were next the shadow, as is represented in fig. 10. If it was an inward angle, as DCH, the coloured streaks, parallel to each other of the two sides, crossed without obliterating one another; only the colours were thereby rendered either more intense or mixed.

The light that formed these coloured streaks, the reader will perceive, must have been bent from the body; but this attentive observer has likewise given an account of other appearances, which must have been produced by the light bending towards the body. For within the shadow itself he sometimes perceived coloured streaks, similar to those above-mentioned on the outside of the shadow. Sometimes he saw more of them, and sometimes fewer; but for this purpose a very strong light was requisite, and the opaque body was obliged to be long, and of a moderate breadth; which, he says, is easily found by experience. A hair, for instance, or a fine needle, did not answer so well as a thin and narrow plate; and the streaks were most distinguishable when the shadow was taken at the greatest distance; but then the light grew fainter in the same proportion.

The number of these streaks within the shadow was greater in proportion to the breadth of the plate. They were at least two, and sometimes four, if a thicker rod were made use of. But, with the same plate or rod, more or fewer streaks appeared, in proportion to the distance at which the shadow was received; but they were broader when they were few, and narrower when there were more of them; and they were all much more distinct when the paper was held obliquely.

These coloured streaks within the shadow, like those on the outside of it, were bent in an arch, round the acute angles of the shadow, as they are represented in fig. 11. At this angle also, as at D, other shorter lucid streaks were visible, bent in the form of a plume, as they are drawn between D and C, each bending round and meeting again in D. These angular streaks appeared, though the plate or rod was not wholly immersed in the beam of light, but the angle of it only; and there were more or fewer in number in proportion to the breadth of the rod or plate. If the plate or rod was very thin, the coloured streaks within the shadow might be seen to bend round from the opposite sides, and meet one another, as at B. A only represents a section of the figure, and not a proper termination of the shadow, and the streaks within each side of it. The coloured streaks without the shadow, he also observes, bend round it in the same manner.

Our author acknowledges, that he omits several observations of less consequence, which cannot but occur to any person who shall make the experiment; and he says, that he was not able to give a perfectly clear idea of what he has attempted to describe, nor does he think it in the power of words to do it.

In order to obtain the more satisfactory proof that rays of light do not always proceed in straight lines, but really bend, in passing by the edges of bodies, he diversified the first of the above mentioned experiments in the following manner. He admitted a beam of light, by a very small aperture, into a darkened room, as before; and, at a great distance from it, he fixed a plate EF, fig. 12, with a small aperture, GH, which admitted only a part of the beam of light, and found, that when the light transmitted through this plate was received at some distance upon a white paper, the base IK was considerably larger than it could possibly have been made by rays issuing in right lines through the two apertures, as the other straight lines drawn close to their edges plainly demonstrate.

That those who choose to repeat these experiments may not be disappointed in their expectations from them, our author gives the following more particular instructions. The sun's light must be very intense, and the apertures through which it is transmitted very narrow, particularly the first, CD, and the white paper, IK, on which it is received, must be at a considerable distance from the hole GH; otherwise it will not much exceed NO, which would be the breadth of the beam of light proceeding in straight lines. He generally made the aperture CD \( \frac{3}{4} \) or \( \frac{1}{4} \) part of an ancient Roman foot, and the second aperture, GH, \( \frac{3}{8} \) or \( \frac{1}{8} \); and the distances DG and GN were, at least, 12 such feet. The observation was made in the summer time, when the atmosphere was free from all vapours, and about mid-day.

F. Grimaldi also made the same experiment that has been recited from Dr Hooke, in which two beams of light, entering a darkened room by two small apertures near to one another, projected cones of light, which, at a certain distance, in part coincided; and he particularly observed that the dark boundaries of each of them were visible within the lucid ground of the other.

To these discoveries of Grimaldi, we shall subjoin an additional observation of Dechales; who took notice, that if small scratches be made in any piece of polished metal, and it be exposed to the beams of the sun in a darkened room, it will reflect the rays streaked with colours in the direction of the scratches; as will appear if the reflected light be received upon a piece of white paper. That these colours are not produced by refraction, he says, is manifest; for that, if the scratches be made upon glass, the effect will be the same; and in this case, if the light had been refracted at the surface of the glass, it would have been transmitted through it. From these, and many other observations, he concludes that colour does not depend upon the refraction of light only, nor upon a variety of other circumstances, which he particularly enumerates, and the effects of which he discusses, but upon the intensity of the light only.

We shall here give an account of a phenomenon of vision observed by M. De la Hire, because the Hire subject of this section, viz. the inflection of light, seems to supply the true solution of it, though the author himself thought otherwise. It is observable, he says, that when we look at a candle, or any luminous body, with our eyes nearly shut, rays of light are extended from it, in several directions, to a considerable distance, like the tails of comets. This appearance excited the sagacity of Descartes and Rohault, as well as of our author; but all three seem to have been mistaken with respect to it. Descartes ascribed this effect to certain wrinkles in the surface of the humours of the eye. Rohault says, that when the eyelids are nearly closed, the edges of them act like convex lenses. But our author says, that the moisture on the surface of the eye, adhering partly to the eye itself, and partly to the edge of the eyelid, makes a concave mirror, and so disperses the rays at their entrance into the eye. But the true reason seems to be, that the light passing among the eyelashes, in this situation of the eye, is reflected by its near approach to them, and therefore enters the eye in a great variety of directions. The two former of these opinions are particularly stated and objected to by our author.

The experiments of Father Grimaldi and Dr Hooke Sir Isaac Newton were not only repeated with the greatest care by Sir Isaac Newton, but carried much farther than they had thought of. So little use had been made of Grimaldi's observations, that all philosophers before Newton had ascribed the broad shadows, and even the fringes of light which he described, to the ordinary refraction of the air; but we shall see them placed in a very different point of view by our author.

He made in a piece of lead a small hole with a pin, the breadth of which was the 42d part of an inch. Through this hole he let into his darkened chamber a beam of the sun's light; and found, that the shadows of hairs, and other slender substances placed in it, were considerably broader than they would have been if the rays of light had passed by those bodies in right lines. He therefore concluded, that they must have passed as they are represented in fig. 1, in which X represents a section of the hair, and AD, BE, &c., rays of light passing by at different distances, and then falling upon the wall GQ. Since, when the paper which receives the rays is at a great distance from the hair, the shadow is broad, it must follow, as he observes, that the hair acts upon the rays of light at some considerable distance from it, the action being strongest on those rays which are at the least distance, and growing weaker and weaker on those which are farther off, as is represented in this figure; and from hence it comes to pass that the shadow of the hair is much broader in proportion to the distance of the paper from the hair when it is nearer than when it is at a great distance.

He found, that it was not material whether the hair was surrounded with air, or with any other pellucid... substance; for he wetted a polished plate of glass, and laid the hair in the water upon the glass, and then laying another polished plate of glass upon it, so that the water might fill up the space between the glasses, and holding them in the beam of light, he found the shadow at the same distances was as big as before. Also the shadows of scratches made in polished plates of glass, and the veins in the glass, cast the like broad shadows: so that this breadth of shadow must proceed from some other cause than the refraction of the air.

The shadows of all bodies, metals, stones, glass, wood, horn, ice, &c. in this light were bordered with three parallel fringes, or bands of coloured light, of which that which was contiguous to the shadow was the broadest and most luminous, while that which was the most remote was the narrowest, and so faint as not easily to be visible. It was difficult to distinguish these colours, unless when the light fell very obliquely upon a smooth paper, or some other smooth white body, so as to make them appear much broader than they would otherwise have done; but in these circumstances the colours were plainly visible, and in the following order. The first or innermost fringe was violet, and deep blue next the shadow, light blue, green, and yellow in the middle, and red without. The second fringe was almost contiguous to the first, and the third to the second; and both were blue within, and yellow and red without; but their colours were very faint, especially those of the third. The colours, therefore, proceeded in the following order from the shadow; violet, indigo, pale blue, green, yellow, red; blue, yellow, red; pale blue, pale yellow, and red. The shadows made by scratches and bubbles in polished plates of glass were bordered with the like fringes of coloured light.

He also observes, that by looking on the sun through a feather, or black ribbon, held close to the eye, several rainbows will appear, the shadows which the fibres or threads cast on the retina being bordered with the like fringes of colours.

Measuring these fringes and their intervals with the greatest accuracy, he found the former to be in the progression of the numbers $1, \sqrt{\frac{1}{2}}, \sqrt{\frac{1}{3}}, \sqrt{\frac{1}{4}}$, and their intervals to be in the same progression with them; that is, the fringes and their intervals together to be in continual progression of the numbers $1, \sqrt{\frac{1}{2}}, \sqrt{\frac{1}{3}}, \sqrt{\frac{1}{4}}, \sqrt{\frac{1}{5}}, \sqrt{\frac{1}{6}}$, or thereabouts. And these proportions held the same very nearly at all distances from the hair, the dark intervals of the fringes being as broad in proportion to the breadth of the fringes at their first appearance as afterwards, at great distances from the hair, though not so dark and distinct.

In the next observation of our author, we find a very remarkable and curious appearance, which we should hardly have expected from the circumstances, though it is pretty similar to one that was noticed by Dr Hooke. The sun shining into his darkened chamber, through a hole $\frac{1}{2}$ of an inch broad, he placed, at the distance of two or three feet from the hole, a sheet of pasteboard, black on both sides; and in the middle of it he had made a hole about $\frac{1}{2}$ of an inch square, for the light to pass through; and behind the hole he fastened to the pasteboard the blade of a sharp knife, to intercept some part of the light which passed through the hole. The planes of the pasteboard and blade of the knife were parallel to one another, and perpendicular to the rays; and when they were so placed that none of the light fell on the pasteboard, but all of it passed through the hole to the knife, and there part of it fell upon the blade of the knife, and part of it passed by its edge, he let that part of the light which passed by fall on a white paper, 2 or 3 feet beyond the knife, and there saw two streams of faint light shoot out both ways from the beam of light into the shadow, like the tails of comets. But because the sun's direct light, by its brightness upon the paper, obscured these faint streams, so that he could scarce see them, he made a little hole in the midst of the paper for that light to pass through and fall on a black cloth behind it; and then he saw the two streams plainly. They were like one another, and pretty nearly equal in length, breadth, and quantity of light. Their light, at that end which was next to the sun's direct light, was pretty strong for the space of about $\frac{1}{2}$ of an inch, or $\frac{1}{2}$ of an inch, and decreased gradually till it became insensible.

The whole length of either of these streams, measured upon the paper, at the distance of 3 feet from the knife, was about 6 or 8 inches; so that it subtended an angle, at the edge of the knife, of about 10 or 12, or at most 14, degrees. Yet sometimes he thought he saw it shoot 3 or 4 degrees farther; but with a light so very faint, that he could hardly perceive it. This light he suspected might, in part at least, arise from some other cause than the two streams. For, placing his eye in that light, beyond the end of that stream which was behind the knife, and looking towards the knife, he could see a line of light upon its edge; and that not only when his eye was in the line of the streams, but also when it was out of that line, either towards the point of the knife, or towards the handle. This line of light appeared contiguous to the edge of the knife, and was narrower than the light of the innermost fringe, and narrower when his eye was farthest from the direct light; and therefore seemed to pass between the light of that fringe and the edge of the knife; and that which passed nearest the edge seemed to be most bent, though not all of it.

He then placed another knife by the former, so that their edges might be parallel, and look towards one another, and that the beam of light might fall upon both the knives, and some part of it pass between their edges. In this situation he observed, that when the distance of their edges was about the 400th part of an inch, the stream divided in the middle, and left a shadow between the two parts. This shadow was so black and dark, that all the light which passed between the knives seemed to be bent and turned aside to the one hand or the other; and as the knives still approached one another, the shadow grew broader and the streams shorter next to it, till, upon the contact of the knives, all the light vanished.

From this experiment our author concludes, that the light which is least bent, and which goes to the inward ends of the streams, passes by the edges of the knives at the greatest distance; and this distance, when the shadow began to appear between the streams, was about the 800th part of an inch; and the light which passed by the edges of the knives at distances still less... and less, was more and more faint, and went to those parts of the streams which were farther from the direct light; because, when the knives approached one another till they touched, those parts of the streams vanished last which were farthest from the direct light.

In the experiment of one knife only, the coloured fringes did not appear; but, on account of the breadth of the hole in the window, became so broad as to run into one another, and, by joining, to make one continual light in the beginning of the streams; but in the last experiment, as the knives approached one another, a little before the shadow appeared between the two streams, the fringes began to appear on the inner ends of the streams, on either side of the direct light; three on one side, made by the edge of one knife, and three on the other side, made by the edge of the other knife. They were the most distinct when the knives were placed at the greatest distance from the hole in the window, and became still more distinct by making the hole less; so that he could sometimes see a faint trace of a fourth fringe beyond the three above-mentioned: and as the knives approached one another, the fringes grew more distinct and larger, till they vanished; the outermost vanishing first, and the innermost last. After they were all vanished, and the line of light which was in the middle between them was grown very broad, extending itself on both sides into the streams of light described before, the above-mentioned shadow began to appear in the middle of this line, and to divide it along the middle into two lines of light, and increased till all the light vanished. This enlargement of the fringes was so great, that the rays which went to the innermost fringe seemed to be bent about 20 times more when the fringe was ready to vanish, than when one of the knives was taken away.

From both these experiments compared together, our author concluded, that the light of the first fringe passed by the edge of the knife at a distance greater than the 800th part of an inch; that the light of the second fringe passed by the edge of the knife at a greater distance than the light of the first fringe, and that of the third at a greater distance than that of the second; and that the light of which the streams above-mentioned consisted, passed by the edges of the knives at less distances than that of any of the fringes.

He then got the edges of two knives ground truly straight, and pricking their points into a board, so that their edges might look towards one another, and meeting near their points, contain a rectilinear angle, he fastened their handles together, to make the angle invariable. The distance of the edges of the knives from one another, at the distance of 4 inches from the angular point, where the edges of the knives met, was the 8th part of an inch; so that the angle contained by their edges was about 1° 54′. The knives being thus fixed together, he placed them in a beam of the sun's light let into his darkened chamber, through a hole the 42d part of an inch wide, at the distance of 10 or 13 feet from the hole; and he let the light which passed between their edges fall very obliquely on a smooth white ruler, at the distance of ¼ inch, or an inch, from the knives; and there he saw the fringes made by the two edges of the knives run along the edges of the shadows of the knives, in lines parallel to those edges, without growing sensibly broader, till they met in angles equal to the angle contained by the edges of the knives; and where they met and joined, they ended, without crossing one another. But if the ruler was held at a much greater distance from the knives, the fringes, where they were farther from the place of their meeting, were a little narrower, and they became something broader as they approached nearer to one another, and after they met they crossed one another, and then became much broader than before.

From these observations he concluded, that the distances at which the light composing the fringes passed by the knives were not increased or altered by the approach of the knives, but that the angles in which the rays were there bent were much increased by that approach; and that the knife which was nearest to any ray determined which way the ray should be bent, but that the other knife increased the bending.

When the rays fell very obliquely upon the ruler, at the distance of a third part of an inch from the knives, the dark line between the first and second fringe of the shadow of one knife, and the dark line between the first and second fringe of the shadow of the other knife, met one another, at the distance of the fifth part of an inch from the end of the light which passed between the knives, where their edges met one another; so that the distance of the edges of the knives, at the meeting of the dark lines, was the 160th part of an inch; and one half of that light passed by the edge of one knife, at a distance not greater than the 320th part of an inch, and, falling upon the paper, made the fringes of the shadow of that knife; while the other half passed by the edge of the other knife, at a distance not greater than the 320th part of an inch, and, falling upon the paper, made the fringes of the shadow of the other knife. But if the paper was held at a distance from the knives greater than the third part of an inch, the dark lines above mentioned met at a greater distance than the fifth part of an inch from the end of the light which passed between the knives, at the meeting of their edges; so that the light which fell upon the paper where those dark lines met passed between the knives, where their edges were farther distant than the 160th part of an inch. For at another time, when the two knives were 8 feet and 5 inches from the little hole in the window, the light which fell upon the paper where the above-mentioned dark lines met passed between the knives, where the distance between their edges was, as in the following table, at the distances from the paper there noted.

| Distances of the paper from the knives in inches | Distances between the edges of the knives in millimetal parts of an inch | |--------------------------------------------------|--------------------------------------------------| | 1 ½ | 0.012 | | 3 ¼ | 0.020 | | 8 ½ | 0.034 | | 32 | 0.057 | | 96 | 0.081 | | 131 | 0.087 |

From these observations he concluded, that the light... light which makes the fringes upon the paper is not the same light at all distances of the paper from the knives; but that, when the paper is held near the knives, the fringes are made by light which passes by the edges of the knives at a less distance, and is more bent than when the paper is held at a greater distance from the knives.

When the fringes of the shadows of the knives fell perpendicularly upon the paper, at a great distance from the knives, they were in the form of hyperbolas, their dimensions being as follows. Let CA, CB, represent lines drawn upon the paper, parallel to the edges of the knives; and between which all the light would fall if it suffered no inflection. DE is a right line drawn through C, making the angles ACD, BCE, equal to one another, and terminating all the light which falls upon the paper, from the point where the edges of the knives meet. Then ei, ef, et, and glv, will be three hyperbolical lines, representing the boundaries of the shadow of one of the knives, the dark line between the first and second fringes of that shadow, and the dark line between the second and third fringes of the same shadow. Also xi, ykq, and zlr, will be three other hyperbolical lines, representing the boundaries of the shadow of the other knife, the dark line between the first and second fringes of that shadow, and the dark line between the second and third fringes of the same shadow. These three hyperbolas are similar, and equal to the former three, and cross them in the points i, k, and l; so that the shadows of the knives are terminated, and distinguished from the first luminous fringes, by the lines ei, ef, et, till the meeting and crossing of the fringes; and then those lines cross the fringes in the form of dark lines terminating the first luminous fringes on the inside, and distinguishing them from another light, which begins to appear at i, and illuminates all the triangular space i p DEs, comprehended by these dark lines and the right line DE. Of these hyperbolas one asymptote is the line DE, and the other asymptotes are parallel to the lines CA and CB.

The sun shining into his darkened room through the small hole mentioned above, he placed at the hole a prism to refract the light, and to form on the opposite wall the coloured image of the sun; and he found, that the shadows of all bodies held in the coloured light between the prism and the wall, were bordered with fringes of the colour of that light in which they were held; and comparing the fringes made in the several coloured lights, he found, that those made in the red light were the largest, those made in the violet were the least, and those made in the green were of a middle bigness. For the fringes with which the shadow of a man's hair were bordered, being measured across the shadow, at the distance of six inches from the hair, the distance between the middle and most luminous part of the first or innermost fringe on one side of the shadow, and that of the like fringe on the other side of the shadow, was, in the full red light \( \frac{1}{2} \) of an inch, and in the full violet \( \frac{1}{3} \). The like distance between the middle and most luminous parts of the second fringes, on either side of the shadow, was in the full red light \( \frac{1}{2} \), and the violet \( \frac{1}{7} \) of an inch; and these distances of the fringes held the same proportion at all distances from the hair, without any sensible variation.

From these observations it was evident, that the rays which made the fringes in the red light, passed by the hair at a greater distance than those which made the like fringes in the violet; so that the hair, in causing these fringes, acted alike upon the red light or least refrangible rays at a greater distance, and upon the violet or most refrangible rays at a less distance; and thereby occasioned fringes of different sizes, without any change in the colour of any sort of light.

It may therefore be concluded, that when the hair in the first observation was held in the white beam of the sun's light, and cast a shadow which was bordered with three fringes of coloured light, those colours arose not from any new modifications impressed upon the rays of light by the hair, but only from the various inflections whereby the several sorts of rays were separated from one another, which before separation, by the mixture of all their colours, composed the white beam of the sun's light; but, when separated, composed lights of the several colours which they are originally disposed to exhibit.

The person whose name we find first upon the list of those who pursued any experiments similar to those Maraldi's of Newton on inflected light is M. Miraldi; whose observations chiefly respect the inflection of light towards other bodies, whereby their shadows are partially illuminated; and many of the circumstances which he noticed relating to it are well worthy of our attention, as the reader will be convinced from the following account of them.

He exposed in the light of the sun a cylinder of wood three feet long, and 6½ inches in diameter; when its shadow, being received upon a paper held close to concerning it, was everywhere equally black and well defined, shadows of continued to be so to the distance of 23 inches cylinders from it. At a greater distance the shadow appeared to be of two different densities; for the two extremities of the shadow, in the direction of the length of the cylinder, were terminated by two dark strokes, a little more than a line in breadth. Within these dark lines there was a faint light, equally dispersed through the shadow, which formed an uniform penumbra, much lighter than the dark strokes at the extremity, or than the shadow received near the cylinder. This appearance is represented in Plate CCCLIII. fig. 3.

As the cylinder was removed to a greater distance from the paper, the two black lines continued to be nearly of the same breadth, and the same degree of obscurity; but the penumbra in the middle grew lighter, and its breadth diminished, so that the two dark lines at the extremity of the shadow approached one another, till, at the distance of 60 inches, they coincided, and the penumbra in the middle entirely vanished. At a still greater distance a faint penumbra was visible; but it was ill defined, and grew broader as the cylinder was removed farther off, but was sensible at a very great distance.

Besides the black and dark shadow, which the cylinder formed near the opaque body, a narrow and faint penumbra was seen on the outside of the dark shadow. And on the outside of this there was a tract more strongly illuminated than the rest of the paper. The breadth of the external penumbra increased with the distance of the shadow from the cylinder, and the breadth of the tract of light on the outside of it was also enlarged; but its splendor diminished with the distance.

He repeated these experiments with three other cylinders of different dimensions; and from them all he inferred, that every opaque cylindrical body, exposed to the light of the sun, makes a shadow which is black and dark to the distance of 38 to 45 diameters of the cylinder which forms it; and that, at a greater distance, the middle part begins to be illuminated in the manner described above.

In explaining these appearances, our author supposes that the light which diluted the middle part of the shadow was occasioned by the inflection of the rays, which, bending inwards on their near approach to the body, did at a certain distance enlighten all the shadow, except the edges, which was left undisturbed. At the same time other rays were deflected from the body, and formed a strong light on the outside of the shadow, and which might at the same time contribute to dilute the outer shadow, though he supposed that penumbra to be occasioned principally by that part of the paper not being enlightened, except by a part of the sun's disk only, according to the known principles of optics.

The same experiments he made with globes of several diameters; but he found, that, whereas the shadows of the cylinders did not disappear but at the distance of 41 of their diameters, those of the globes were not visible beyond 15 of their diameters; which he thought was owing to the light being inflected on every side of a globe, and consequently in such a quantity as to disperse the shadows sooner than in the case of the cylinders.

In all these cases, the penumbra occasioned by the inflected light began to be visible at a less distance from the body in the stronger light of the sun than in a weaker, on account of the greater quantity of rays inflected in those circumstances.

Considering the analogy between these experiments and the phenomena of an eclipse of the moon, immersed in the shadow of the earth, he imagined, that part of the light by which she is then visible is inflected light, and not that which is refracted by the atmosphere; though this may be so copious as to efface several of the above-mentioned appearances, occasioned by inflected light only. But this gentleman should have considered, that as no light is inflected but what passes exceedingly near to any body, perhaps so near as the distance of \( \frac{1}{2} \) part of an inch, this cause must be altogether inadequate to the effect.

Being sensible that the above-mentioned phenomena of the shadows were caused by inflected light, he was induced to give more particular attention to this remarkable property; and, in order to it, to repeat the experiments of Grimaldi and Sir Isaac Newton in a darkened room. In doing this, he presently observed, that, besides the enlarged shadow of a hair, a fine needle, &c. the bright gleam of light that bordered it, and the three coloured rings next to this enlightened part, when the shadow was at a considerable distance from the hair, the dark central shadow was divided in the middle by a mixture of light; and that it was not of the same density, except when it was very near the hair.

This new appearance will be seen to be exactly similar to what our philosopher had observed with respect to the shadows in the open day-light above mentioned; but the following observations, which he made with some variation of his apparatus, are much more curious and striking, though they arise from the same cause.

Having placed a bristle, which is thicker than a common hair, in the rays of the sun, admitted into a dark chamber by a small hole, at the distance of nine feet from the hole, it made a shadow, which, being received at five or six feet from the object, he observed to consist of several streaks of light and shade. The middle part was a faint shadow, or rather a kind of penumbra, bordered by a darker shadow, and after that by a narrower penumbra; next to which was a light streak broader than the dark part, and next to the streak of light, the red, violet, and blue colours were seen as in the shadow of the hair.

In the same manner he placed, in the same rays of the sun, several needles of different sizes; but the appearances were so exceedingly various, tho' sufficientlyingular, that he does not recite them particularly, but chooses rather to give, at some length, the observations he made on the shadows of two plates, as by that means he could better explain the phenomena of the round bodies.

He exposed in the rays of the sun, admitted by a small hole into a dark chamber, a plate that was two inches long, and a little more than half a line broad. This plate being fixed perpendicularly to the rays, at the distance of nine feet from the hole, a faint light was seen uniformly dispersed over the shadow, when it was received perpendicularly to it, and very near. The shadow of the same plate being received at the distance of two feet and a half, was divided into four very narrow black streaks, separated by small lighter intervals equal to them. The boundaries of this shadow on each side had a penumbra, which was terminated by a very strong light, next to which were the coloured streaks of red, violet, and blue, as before. This is represented in Plate CCCCLI. fig. 4.

The shadow of the same plate, at 4½ feet distance from it, was divided into two black streaks only, the two outermost having disappeared, as in fig. 5.; but these two black streaks which remained were broader than before, and separated by a lighter shade, twice as broad as one of the former black streaks, when the shadow was taken at 2½ feet. This penumbra in the middle had a tinge of red. After the two black streaks there appeared a pretty strong penumbra, terminated by the two streaks of light, which were now broad and splendid, after which followed the coloured streaks.

A second plate, two inches long and a line broad, being placed like the former, 14 feet from the hole by which the rays of the sun were admitted, its shadow being received perpendicularly very near the plate, was illuminated by a faint light, equally dispersed, as in the case of the preceding plate. But being received at the distance of 13 feet from the plate, six small black streaks streaks began to be visible, as in fig. 6. At 17 feet from the plate, the black streaks were broader, more distinct, and more separated from the streaks that were less dark. At 42 feet from the plate, only two black streaks were seen in the middle of the penumbra, as in fig. 7. This middle penumbra between the two black streaks was tinged with red. Next to the black streaks there always appeared the streaks of light, which were broad, and the coloured streaks next to them.

Receiving the shadow of the same plate at the distance of 72 feet, the appearances were the same as in the former situation, except that the two black streaks were broader, and the interval between them, occupied by the penumbra, was broader also, and tinged with a deeper red.

In the same rays of the sun he placed different plates, and larger than the former, one of them a line and a half, another two lines, another three lines broad, &c. but receiving their shadows upon paper, he could not perceive in them those streaks of faint light which he had observed in the shadows of the small plates, though he received these shadows at the distance of 56 feet. Nothing was seen but a weak light, equally diffused, as in the shadows of the two smallest plates, received very near them. But had his dark chamber been large enough, he did not doubt, but that, at a proper distance, there would have been the same appearances in the shadows of the larger plates as in those of the smallest. For the same reason, he supposed, that, if the shadows of the small needles could have been distinctly viewed very near those bodies, the different streaks of light and shade would have been as visible in them as in those of the small plates; and indeed he did observe the same appearances in the shadows of needles of a middling size.

The streaks of light in these shadows our author ascribed to the rays of light which are inflected at different distances from the bodies; and he imagined that their crossing one another was sufficient to account for the variations observable in them at different distances.

The extraordinary size of the shadows of these small substances M. Maraldi thought to be occasioned by the shadow from the enlightened part of the sky, added to that which was made by the light of the sun, and also to a vortex occasioned by the circulation of the inflected light behind the object; but our readers will probably not think it necessary for us either to produce all his reasons for this hypothesis, or to enter into a refutation of them.

Our author having made the preceding experiments upon single long substances, had the curiosity to place two of them so as to cross one another in a beam of the sun's light. The shadows of two hairs placed in this manner, and received at some distance from them, appeared to be painted reciprocally one upon the other, so that the obscure part of one of them was visible upon the obscure part of the other. The streaks of light also crossed one another, and the coloured streaks did the same.

Having placed a needle and a hair crossing one another, their shadows, at the same distance, exhibited the same appearances as the shadows of the two hairs, though the shadow of the needle was the stronger.

He also placed in the rays of the sun a brittle and a plate of iron a line thick, so that they crossed one another obliquely; and when their shadows were received at the same distance, the light and dark streaks of the shadow of the brittle were visible so far as the middle of the shadow of the plate on the side of the acute angle, but not on the side of the obtuse angle, whether the brittle or the plate were placed next to the rays. The plate made a shadow sufficiently dark, divided into six black streaks; and these were again divided by as many light ones equal to them; and yet all the streaks belonging to the shadow of the brittle were visible upon it, as in fig. 8. To explain this appearance, he supposed that the rays of the sun glided a little along the brittle, so as to enlighten part of that which was behind the plate. But this seems to be an arbitrary and improbable supposition.

Our philosopher did not fail to expose several small globes in the light of the sun in his dark chamber, and to compare their shadows with those of the long substances, as he had done in the day-light, and the appearances were still similar. It was particularly evident, that there was much more light in the shadows of the globes than in those of the cylinders, not only when they were both of an equal diameter, but when that of the globe was larger than that of the cylinder, and the shadows of both the bodies were received at the same distance. He also observed, that he could perceive no difference of light in the shadows of the plates which were a little more than one line broad, though they were received at the distance of 72 feet; but he could easily see a difference of shades in those of the globes, taken at the same distance, tho' they were 2½ lines in diameter.

In order to explain the colours at the edges of these shadows, he contrived to throw some of the shadows upon others; and the following observations, though they did not enable him to accomplish what he intended, are curious and worth reciting.

Having thrown several of the similar colours upon one another, and thereby produced a tinge more lively than before, he threw the gleam of light, which always intervened between the colours and the darker part of the shadow, upon different parts of other shadows; and observed, that, when it fell upon the exterior penumbra made by another needle, it produced a beautiful sky-blue colour, almost like that which was produced by two blue colours thrown together. When the same gleam of light fell upon the deeper shadow in the middle, it produced a red colour; which seemed to prove, that the reddish colour in the middle of several of the shadows might come from the little light inflected into that place. But here our author seems to have been misled by some false hypotheses concerning colours.

He placed two plates of iron, each three or four lines broad, very near one another, but with a very small interval between them; and having placed them in the rays of the sun, and received their shadows at the distance of 15 or 20 feet from them, he saw no light between them but a continued shadow, in the middle middle of which were some streaks of a lively purple, parallel to one another, and separated by other black streaks; but between them there were other streaks, both of a very faint green, and also of a pale yellow. He also informs us, that M. Delisle had observed colours in the streaks of light and shade, which are observable in shadows taken near the bodies.

Among those who followed Sir Isaac Newton in his observations on the inflection of light, we also find the ingenious M. Mairan: but, without attempting the discovery of new facts, he only endeavoured to explain the old ones, by the hypothesis of an atmosphere surrounding all bodies; and consequently making two reflections and refractions of the light that impinges upon them, one at the surface of the atmosphere, and the other at that of the body itself. This atmosphere he supposed to be of a variable density and refractive power, like the air.

M. Mairan was succeeded by M. Du Tour, who thought the variable atmosphere superfluous, and imagined that he could account for all the phenomena by the help of an atmosphere of uniform density, and of a less refractive power than the air surrounding all bodies. But what we are most obliged to this gentleman for, is not his ingenious hypothesis, but the beautiful variety with which he has exhibited the experiments, which will render it much easier for any person to investigate the true causes of them.

Before M. Du Tour gave his attention to this subject, only three fringes had been observed in the colours produced by the inflection of light; but he was accidentally led to observe a greater number of them, and adopted from Grimaldi the following ingenious method of making them all appear very distinct.

He took a circular board A.B.E.D (fig. 9.), 13 inches in diameter, the surface of which was black, except at the edge, where there was a ring of white paper about three lines broad, in order to trace the circumference of a circle, divided into 360 degrees, beginning at the point A, and reckoning 180 degrees on each hand to the point E; B and D being each of them placed at 90 degrees. A slip of parchment three inches broad, and disposed in the form of a hoop, was fastened round the board, and pierced at the point E with a square hole, each side being four or five lines, in order to introduce a ray of the sun's light. Lastly, in the centre of the board C, and perpendicular to it, he fixed a pin about ¼ of a line in diameter.

This hoop being so disposed, that a ray of light entering the dark chamber, through a vertical cleft of two lines and a half in length, and about as wide as the diameter of the pin, went through the hole at E, and passing parallel to the plane of the board, projected the image of the sun and the shadow of the pin at A. In these circumstances he observed,

1. That quite round the concave surface of this hoop, there were a multitude of coloured streaks; but that the space mAn, of about 18 degrees, the middle of which was occupied by the image of the sun, was covered with a faint light only.

2. The order of the colours in these streaks was generally such that the most refrangible rays were the nearest to the incident ray ECA; so that, beginning from the point A, the violet was the first and the red the last colour in each of the streaks. In some of them, however, the colours were disposed in a contrary order.

3. The image of the sun, projected on each side of the point A, was divided by the shadow of the pin, which was bordered by two luminous streaks.

4. The coloured streaks were narrower in some parts of the hoop than others, and generally decreased in breadth in receding from the point A.

5. Among these coloured streaks, there were sometimes others which were white, a line or a line and an half in breadth, which were always bordered on both sides by a streak of orange colour, at least when the light of the sun was intense, and the chamber sufficiently dark.

From this experiment he thought it was evident, that the rays which passed beyond the pin were not the only ones that were decomposed, for that those which were reflected back from the pin were decomposed also; from which he concluded, that they must have undergone some refraction. He also thought that those which went beyond the pin suffered a reflection, so that they were all affected in a similar manner.

In order to account for these facts, our author describes the progress of a ray of light through a uniform atmosphere, which he supposes to surround the pin; and shows, that the differently refrangible rays will be separated at their emergence from it: but he refers to some experiments and observations in a future memoir, to demonstrate that all the coloured streaks are produced by rays that are both reflected and refracted.

To give some idea of his hypothesis, he shows that the ray a b, fig. 10. after being refracted at b, reflected at r and u, and again refracted at s and t, will be divided into its proper colours; the least refrangible or the red rays issuing at x, and the most refrangible or violet at y; which agrees with his observations. Those streaks in which the colours appear in a contrary order he thinks are to be ascribed to inequalities in the surface of the pin. This might easily have been ascertained by turning the pin round; in which case these differently-coloured streaks would have changed their places.

If any person should choose to repeat these experiments, he observes that it requires that the sky be very clear and free from vapours, in order to exhibit the colours with the greatest distinctness; since even the vapours that are imperceptible will diminish the lustre of the colours on every part of the hoop, and even efface some of them, especially those which are on that part in which the beam of light enters, as at E, fig. 9., where the colours are always fainter than in any other place, and indeed can never be distinguished except when the hole E is confined by black substances, so as to intercept a part of the light that might reach the pin; and unless also those rays which go beyond the pin to form the image of the sun at A be stopped, so that no rays are visible except those that are reflected towards the hole, and which make the faint streaks.

The coloured streaks that are next the shadow of the pin, he shows, are formed by those rays which, entering the atmosphere, do not fall upon the pin; and, without any reflection, are only refracted at their entering. entering and leaving the atmosphere, as at b and r u, fig. 11. In this case, the red or least refrangible rays will issue at r, and the violet at u.

To distinguish the rays which fell upon the hoop in any particular direction, from those that came in any other, he made an opening in the hoop, as at P, fig. 9, by which means he could, with advantage, and at any distance from the centre, observe those rays unmixed with any other.

To account for the coloured streaks being larger next the shadow of the pin, and growing narrower to the place where the light was admitted, he shows, by fig. 12, that the rays a b are farther separated by both the refractions than the rays c d.

Sometimes our author observed, that the broader streaks were not disposed in this regular order; but then he found, that by turning the pin they changed their places, so that this circumstance must have been an irregularity depending upon the accidental surface of the pin.

The white streaks intermixed with the coloured ones he ascribes to small cavities in the surface of the pin, or some other foreign circumstance; for they also changed their places when the pin was made to turn upon its axis.

Other observations of our author seem to prove that the refracting atmospheres surrounding all kinds of bodies are of the same size; for when he placed a great variety of substances, and of different sizes also, he always found the coloured streaks of the same dimensions.

M. Du Tour observes that his hypothesis contradicts an observation of Sir Isaac Newton, that those rays which pass the nearest to any body are the most inflected; but he thinks that Newton's observations were not sufficiently accurate. Besides, he observes, that Newton only said that he thought it to be so, without asserting it positively.

Since the rays which formed these coloured streaks are but little diverted out of their way, our author infers that this atmosphere is of small extent, and that its refractive power is not much less than that of air.

Exposing two pieces of paper in the beam of light, so that part of it passed between two planes formed by them, M. Du Tour observed, that the edges of this light, received upon paper, were bordered with two orange coloured streaks, which Newton had not taken notice of in any of his experiments. To account for them, he supposes, that, in fig. 13, the more refrangible of the rays which enter at b are so refracted, that they do not reach the surface of the body itself at R; so that the red and orange-coloured light may be reflected from thence in the direction dM, where the orange-coloured streaks will be formed; and, for the same reason, another streak of orange will be formed at m, by the rays which enter the atmosphere on the other side of the chink. In a similar manner he accounts for the orange-coloured fringes at the borders of the white streaks, in the experiment of the hoop.

The blue rays, which are not reflected at R, he supposes, pass on to l; and that of these rays the blue tinge observable in the shadows of some bodies are formed.

We may here make a general observation, applicable to all the attempts of philosophers to explain these phenomena by atmospheres. These attempts give no explanation whatever of what is attempted, founded, i.e., the physical cause of the phenomena. A phenomenon is some individual fact or event in nature. We are said to explain it when we point out the general fact in which it is comprehended, and show the manner in which it is so comprehended, or the particular modification of the general fact. Philosophy resembles natural history, having for its subject the events of nature; and its investigations are nothing but the classification of these events, or the arrangement of them under the general facts of which they are individual instances. In the present instance there is no general fact referred to. The atmosphere is a mere gratuitous supposition; and all that is done is to show a resemblance between the phenomena of inflection of light to what would be the phenomena were bodies surrounded with such atmospheres; and even in this point of view, the discussions of Mairan and Du Tour are extremely deficient. They have been satisfied with very vague resemblances to a fact observed in one single instance, and not sufficiently examined or described in that instance, namely, the refraction of light through the atmosphere of this globe.

The attempt is to explain how light is turned out of its direction by passing near the surface of bodies. This indicates the action of forces in a direction transverse to that of the light. Newton took the right road of investigation, by taking the phenomenon in its original simplicity, and attending merely to this, that the rays are deflected from their former course; and the sole aim of his investigation was to discover the laws, i.e., the more general facts in this deflection. He deduced from the phenomena, that some rays are more deflected than others, and endeavoured to determine in what rays the deflections are most remarkable; and no experiment of M. Du Tour has shown that he was mistaken in his modified assertion, that those rays are most inflected which pass nearest to the body. We say modified assertion; for Newton points out with great sagacity many instances of alternate fits of inflection and deflection; and takes it for granted, that the law of continuity is observed in these phenomena, and that the change of inflection into deflection is gradual.

But these analogical discussions are eminently deficient in another respect: They are (prima facie) held out as mechanical explanations of the changes of motion observed in rays of light. When it shall be shown, that these are precisely such as are observed in refracting atmospheres, nothing is done towards deciding the original question; for the action of refracting atmospheres presents it in all its difficulties, and we must still ask how do these atmospheres produce this effect? No advance whatever is gained in science by thrusting in this hypothetical atmosphere; and Newton did wisely in attaching himself to the simple fact; and he thus gives us another step in science, by showing us a fact unknown before, viz., that the action of bodies' inflection on light is not confined to transparent bodies. He probably added another general fact to our former stock, that light as well as other matter is acted on at a distance; he made a very important deduction, that forces... When therefore the glass ball mentioned above puts the other in motion by striking it, we are entitled to say, that unless the pressure during the stroke has been equal to 800 pounds for every square inch of contact, the motion has been produced without contact or real impulse, by the action of repulsive forces exerted between the balls, in the same manner as would happen between two magnets floating on cork with their north poles facing each other; in which case (if the motion has been sufficiently slow) the striking magnet will be brought to rest, and the other move off, with its original velocity, in the same manner as happens to the glass balls. Many such communications of motion happen, where we cannot say that the impulsive pressure is greater than that now mentioned; without and in such cases we are well entitled to say, that the impulse, motion has been produced without real impulse, by repulsive forces acting at a distance. This evidently diminishes to a great degree the familiarity of the fact of impulse.

But we conclude too hastily, from the phenomena of the object glasses, that a pressure exceeding 800 pounds on the square inch will produce contact.

Blow a soap bubble, and let it fall on a piece of cloth, and cover it with a glass bell: after some time you will observe rings of colours on its upper part, which will increase in number and breadth, and be in every respect similar to those between the object-glasses. These arise from the gradual thinning of the upper part of the soap bubble; a certain thickness of this, as well as of the interval between the glasses, invariably reflecting a certain colour. At last a black spot appears atop, which is sharply defined, and increases in diameter. Soon after this the bubble bursts. Thus then there is a certain thickness necessary for enabling the plate of soap fuds to reflect light so as to be very sensible. Analogy obliges us to extend this to the object-glasses, and to say, not that the glasses touch each other through the extent of the black spot, but that their distance is there too small for the sensible reflection of light; and it remains undecided whether any pressure, however great, can annihilate all distance between them. So far, therefore, from impulse being a double familiar fact, and its supposed laws being proper and logical principles of reasoning and explanation, it appears extremely doubtful whether the fact has ever been observed; and it must therefore be against the observed rules of logic to adduce the laws of impulse for the explanation of any arbitrary phenomenon.

Ether and other fluid atmospheres have often been referred to by philosophers puzzled for an explanation; and all this trouble has been taken to avoid the supposed difficulty of bodies acting at a distance. We now see that this is only putting the difficulty a step farther off. We may here add, that in all these attempts the very thing is supposed which the philosophers wish to avoid. These ethers have been fitted for their tasks by supposing them of variable densities. It is quite easy to show, that such a variation in density cannot be conceived without supposing the particles to act on particles not in contact with them, and to a distance as great as that to which the change of density extends. The very simplest form of an elastic fluid supposes this, either with respect to its own particles, or with respect to the particles of a still more subtle... subtle fluid, from the interspersion of which it derives its elasticity. To get rid of one action at a distance, therefore, we introduce millions. Instead, therefore, of naturalists pluming themselves on such explanations, and having recourse, in all their difficulties, to the ether of Sir Isaac Newton, which they make a drudge, a Mungo here, Mungo there, Mungo everywhere; let us rather wonder how that great man, not more eminent for penetration and invention than for accuracy of conception and justness of reasoning, should so far forget himself, and deviate from that path of logical investigation in which he had most successfully advanced, and should, in his fabrication of ether, and application of it to explain the more abstruse phenomena of nature, at once transgress all the rules of philosophizing which he had prescribed to himself and others. Let this slip, this mark of frail mortality, put us on our guard, lest we also be seduced by the specious offers of explanation which are held out to us by means of invisible atmospheres of every kind.

M. Le Cat has well explained a phenomenon of vision depending upon the inflection of light, which shows, that, in some cases objects appear magnified by this means. Looking at a distant steeple, when a wire, of a less diameter than the pupil of his eye, was held pretty near to it, and drawing it several times betwixt his eye and that object, he was surprized to find, that, every time the wire passed before his pupil, the steeple seemed to change its place, and some hills beyond the steeple seemed to have the same motion, just as if a lens had been drawn betwixt his eye and them.

Examining this appearance more attentively, he found that there was a position of the wire, but very difficult to keep, in which the steeple seemed not to have any motion, when the wire was passed before his eye; and in this case the steeple appeared less distinctly, and seemed to be magnified. These effects being similar to those of a lens, he attended to them more particularly; and placed his eye in such a manner with respect to the steeple, that the rays of light by which he saw it must come very close to the edge of a window, where he had placed himself to make his observations. Then passing the wire once more before his eye, he observed, that, when it was in the visual axis, the steeple appeared nearer to the window, on whichever side the wire was made to approach. He repeated this experiment, and constantly with the same result, the object being always magnified, and nearly doubled, by this means.

This phenomenon is easily explained by fig. 14, in which B represents the eye, A the steeple, and C the diameter of the wire. The black lines express the cone of light by which the natural image of the steeple A is formed, and which is much narrower than the diameter of the wire C; but the dotted lines include not only that cone of light, stopped and turned out of its course by the wire, but also more distant rays inflected by the wire, and thereby thrown more converging into the pupil; just as would have been the effect of the interposition of a lens between the eye and the object. The result of this experiment was the same, whatever substances he made use of in the place of the wire, provided they were of the same diameter.

Plate CCCLII.

§ 5. Discoveries concerning Vision.

Maurolycus was the first who showed the true theory of vision, by demonstrating that the crystalline humour of the eye is a lens which collects the light of Maurolycus, Kepli, &c., the retina, where is the focus of each pencil. He did not however find out, that, by means of this refraction, an image of every visible object was formed upon the retina, though this seems hardly to have been a step beyond the discovery he had already made. Montucla indeed conjectures, that he was prevented from mentioning this part of the discovery by the difficulty of accounting for the upright appearance of objects, as the image on the retina is always inverted. This discovery was made by Kepler; but he, too, was much perplexed with the inverted position of the image. The rectification of these images, he says, is the business of the mind; which, when it perceives an impression on the lower part of the retina, considers it as made by rays proceeding from the higher parts of objects; tracing the rays back to the pupil, where they cross one another. But this hypothesis can scarcely be deemed satisfactory.—Kepler did not pretend to account for the manner in which the mind perceives the images upon the retina, and very much blames Vitello for attempting prematurely to determine a question of this nature, and which indeed, he says, does not belong to optics. He accounts, however, though not in a satisfactory manner, for the power we have of seeing distinctly at different distances.

The discovery concerning vision was completed by Scheiner. For, in cutting away the coats of the back Discoveries part of the eyes of sheep and oxen, and presenting several objects before them, within the usual distance of vision, he saw their images distinctly and beautifully painted upon the retina. He did the same thing with the human eye, and exhibited this curious experiment at Rome in 1625. He takes particular notice of the resemblance between the eye and the camera obscura, and explains a variety of methods to make the images of objects erect. As to the images of objects being inverted in the eye, he acquiesces in the reason given for it by Kepler. He knew that the pupil of the eye is enlarged in order to view remote objects, and that it is contracted while we are viewing those that are near; and this he proved by experiment, and illustrated by figures.

Scheiner also took a good deal of pains to ascertain the density and refractive power of all the humours of the eye, by comparing their magnifying power with that of water or glass in the same form and circumstances. The result of his inquiries was, that the aqueous humour doth not differ much from water in this respect, nor the crystalline from glass; and that the vitreous humour is a medium between both. He also very accurately and minutely traces the progress of the rays of light through all the humours of the eye; and after discarding every possible hypothesis concerning the proper seat of vision, he demonstrates that it is in the retina, and shows that this was the opinion of Alhazen, Vitello, Kepler, and all the most eminent philosophers. He produces many reasons of his own for this hypothesis; answers a great number of objections. objections to it; and, by a variety of arguments, refutes the opinion of former times, that the seat of vision is in the crystalline.

Descartes makes a good number of observations on the phenomena of vision. He explains satisfactorily the natural methods of judging of the magnitudes, situations, and distances, of objects by the direction of the optic axes; comparing it to a blind man's judging of the size and distance of an object, by feeling at it with two sticks of a known length, when the hands in which he holds them are at a known distance from each other. He also observes, that having been accustomed to judge of the situation of objects by their images falling on a particular part of the eye; if by any distortion of the eye they fall on a different place, we are apt to mistake their situation, or imagine one object to be two; as, till we become accustomed to it, we imagine one stick to be two, when it is placed between two contiguous fingers laid across one another. But he observes, that all the methods we have of judging of the distances of objects are very uncertain, and extend but to narrow limits. The direction of the optic axes, he says, will not serve us beyond 15 or 20 feet, and the change of form of the crystalline not more than three or four feet. For he imagined that the eye conforms itself to the view of near or distant objects by a change in the curvature of the crystalline, which he supposed to be a muscle, the tendons of it being the processus ciliares. In another place, he says, that the change in the conformation of the eye is of no use to us for the purpose of judging of distances beyond four or five feet, and the angle of the optic axes not more than 100 or 200 feet: for this reason, he says, that the sun and moon are conceived to be much more nearly of the same size than they are in reality. White and luminous objects, he says, appear larger than others, and also the parts contiguous to those on which the rays actually impinge; and for the same reason, if the objects be small, and placed at a great distance, they will always appear round, the figure of the angles disappearing.

The celebrated Berkeley bishop of Cloyne published, in 1709, An Essay towards a New Theory of Vision, which contains the solution of many difficulties. He does not admit that it is by means of those lines and angles, which are extremely useful in explaining the theory of optics, that different distances are judged of by the sense of sight; neither does he think that the mere direction of the optic axes, or the greater or less divergency of the rays of light, are sufficient for this purpose. "I appeal (says he) to any one's experience, whether, upon sight of an object, he computes its distance by the bigness of the angle made by the meeting of the two optic axes? or whether he ever thinks of the greater or less divergency of the rays which arrive from any point to his pupil? Nay, whether it be not perfectly impossible for him to perceive, by sense, the various angles wherewith the rays according to their greater or lesser divergency fall upon his eye?" That there is a necessary connection between these various angles, &c. and different degrees of distance, and that this connection is known to every person skilled in optics, he readily acknowledges; but "in vain (says he) shall all the mathematicians in the world tell me, that I perceive certain lines and angles, which introduce into my mind the various notions of distance, so long as I am myself conscious of no such thing."

Distance, magnitude, and even figure, he maintains to be the objects of immediate perception only by the sense of touch; and that when we judge of them by sight, it is from different sensations felt in the eye which experience has taught us to be the consequence of viewing objects of greater or less magnitude, of different figures, and at different distances. These various sensations, with the respective distances, figures, and magnitudes by which they are occasioned, become so closely associated in the mind long before the period of distinct recollection, that the presence of the one instantly suggests the other; and we attribute to the sense of sight those notions which are acquired by the sense of touch, and of which certain visual sensations are merely the signs or symbols, just as words are the symbols of ideas. Upon these principles he accounts, in a manner worthy of the reader's attention, for single vision by both eyes, and for our perceiving objects erect by inverted images of them on the retina tunic. Subsequent writers have made great discoveries in the theory of vision; and among them there is hardly any one to whom this branch of science is so much indebted as to Dr Reid. Their reasonings, however, our limits will not permit us to detail, nor do they properly belong to this part of the article; they are connected with the description of the eye itself, the various modes of vision, and optical deceptions to which we are liable; and these will be considered in a succeeding part of this treatise.

§ 6. Of Optical Instruments, and Discoveries concerning them.

So little were the ancients acquainted with the invention science of Optics, that they seem to have had no instruments of the optical kind, excepting the glasses, globes and spectacles formerly mentioned, which they used in some cases for magnifying and burning. Alhazen, as we have seen, gave the first hint of the invention of spectacles, and it is probable that they were found out soon after his time. From the writings of Alhazen, together with the observations and experiments of Roger Bacon, it is not improbable that some monks gradually hit upon the construction of spectacles; to which Bacon's lesser segment, notwithstanding his mistake concerning it, was a nearer approach than Alhazen's larger one. Whoever they were that pursued the discoveries of Bacon, they probably observed, that a very small convex glass, when held at a greater distance from the book, would magnify the letters more than when it was placed close to them, in which position only Bacon seems to have used it. In the next place, they might try whether two of these small segments of a sphere placed together, or a glass convex on both sides, would not magnify more than one of them. They would then find, that two of these glasses, one for each eye, would answer the purpose of reading better than one; and lastly, they might find, that different degrees of convexity suited different persons.

It is certain that spectacles were well known in the 13th century, and not long before. It is said that Alexander Spina, a native of Pisa, who died in 1313, and who was very ingenious in executing whatever he saw or heard of as having been done by others, happened to see a pair of spectacles in the hands of a person who would not explain them to him; but that he succeeded in making a pair for himself, and immediately made the construction public, for the good of others. It is also inscribed on the tomb of Salvius Armatus, a nobleman of Florence, who died 1317, that he was the inventor of spectacles.

The use of concave glasses, to help those persons who are short sighted, was probably a discovery that followed not long after that of convex ones, for the relief of those whose sight is defective in the contrary extreme, though we find no trace of this improvement. Whoever made this discovery, it was probably the result of nothing more than a random experiment. Perhaps a person who was short sighted, finding that convex glasses did him more harm than good, had the curiosity to make trial of a contrary curvature of the glass.

From this time, though both convex and concave lenses were sufficiently common, yet no attempt was made to form a telescope by a combination of them, till the end of the 16th century. Descartes considers James Metius, a person who was no mathematician, though his father and brother had applied to those sciences, as the first constructor of a telescope; and says, that as he was amusing himself with making mirrors and burning-glasses, he casually thought of looking through two of his lenses at a time; and that happening to take one that was convex and another that was concave, and happening also to hit upon a pretty good adjustment of them, he found, that, by looking through them, distant objects appeared very large and distinct. In fact, without knowing it, he had made a telescope.

Other persons say, that this great discovery was first made by John Lipperhein, a maker of spectacles at Middleburgh, or rather by his children; who, like Metius, were diverting themselves with looking thro' two glasses at a time, and placing them at different distances from one another. But Borellus, the author of a book intitled, De vero telescopii inventore, gives this honour to Zacharias Joannides, i.e. Janfen, another maker of spectacles at the same place, who made the first telescope in 1590; and it seems now to be the general opinion, that this account of Borellus is the most probable.

Indeed, Borellus's account of the discovery of telescopes is so circumstantial, and so well authenticated, that it does not seem possible to call it in question. It is not true, he says, that this great discovery was made by a person who was no philosopher: for Zacharias Janfen was a diligent inquirer into nature; and being engaged in these pursuits, he was trying what uses could be made of lenses for those purposes, when he fortunately hit upon the construction.

This ingenious mechanic, or rather philosopher, had no sooner found the arrangement of glasses that produced the effect he desired, than he inclosed them in a tube, and ran with his instrument to prince Maurice; who, immediately conceiving that it might be of use to him in his wars, desired the author to keep it a secret. But this, though attempted for some time, was found to be impossible; and several persons in that city immediately applied themselves to the making and selling of telescopes. One of the most distinguished of these was Hans Laprey, called Lipperhein by Sirturus. By him some person in Holland being very early supplied with a telescope, he passed with many for the inventor; but both Metius above-mentioned, and Cornelius Drebell of Alcmar, in Holland, applied to the inventor himself in 1620; as also did Galileo, and many others. The first telescope made by Janfen did not exceed 15 or 16 inches in length; but Sirturus, who says that he had seen it, and made an exceedingly good use of it, thought it the best that he had ever examined.

Janfen, having a philosophical turn, presently applied his instrument to such purposes as he had in view when he hit upon the construction. Directing it towards celestial objects, he distinctly viewed the spots on the surface of the moon; and discovered many new stars, particularly seven pretty considerable ones in the Great Bear. His son Joannes Zacharias, noted the lucid circle near the limb of the moon, from whence several bright rays seem to dart in different directions; and he says, that the full moon, viewed through this instrument, did not appear flat, but was evidently spherical, the middle part being prominent. Jupiter also, he says, appeared round, and rather spherical; and sometimes he perceived two, sometimes three, and at the most four small stars, a little above or below him; and, as far as he could observe, they performed revolutions round him; but this, he says, he leaves to the consideration of astronomers. This, it is probable, was the first observation of the satellites of Jupiter, though the person who made it was not aware of the importance of his discovery.

One Francis Fontana, an Italian, also claims the honour of invention; but as he did not pretend to have made it before the year 1608, and as it is well known that the instruments were made and sold in Holland some time before, his pretensions to a second discovery are not much regarded.

There are some who say that Galileo was the inventor of telescopes; but he himself acknowledges, that he first heard of the instrument from a German; but he says, that being informed of nothing more than the effects of it, first by common report, and a few days after by a French nobleman, J. Badovere, at Paris, he himself discovered the construction, by considering the nature of refraction; and thus he had much more real merit than the inventor himself.

The account of what Galileo actually did in this business is so circumstantially related by the author of his life, prefixed to the quarto edition of his works, printed at Venice in 1744, and it contains so many particulars, which cannot but be pleasing to every person who is interested in the history of telescopes, that we shall abridge a part of it, intermixing circumstances collected from other accounts.

About April or May, in 1609, it was reported at Venice, where Galileo (who was professor of mathematics in the university of Padua) then happened to be, that a Dutchman had presented to Count Maurice, of Nassau, a certain optical instrument, by means of which, distant objects appeared as if they were near; but no farther account of the discovery had reached that. that place, though this was near 20 years after the first discovery. Struck, however, with this account, Galileo instantly returned to Padua, considering what kind of an instrument this must be. The night following, the construction occurred to him; and the day after, putting the parts of the instrument together, as he had previously conceived of it, and notwithstanding the imperfection of the glasses that he could then procure, the effect answered his expectations, as he presently acquainted his friends at Venice, to which place he six days afterwards carried another and a better instrument that he had made, and where, from several eminences, he showed to some of the principal senators of that republic a variety of distant objects, to their very great astonishment. When he had made farther improvements in the instrument, he, with his usual generosity and frankness in communicating his discoveries, made a present of one of them to the Doge, Leonardo Donati, and at the same time to all the senate of Venice; giving along with the instrument a written paper, in which he explained the structure and wonderful uses that might be made of it both by land and at sea. In return for so noble an entertainment, the republic, on the 25th of August, in the same year, more than tripled his salary as professor.

Our philosopher, having amused himself for some time with the view of terrestrial objects, at length directed his tube towards the heavens; and, observing the moon, he found that the surface of it was diversified with hills and valleys, like the earth. He found that the via lactea and nebulae consisted of a collection of fixed stars, which, on account either of their vast distance, or extreme smallness, were invisible to the naked eye. He also discovered innumerable fixed stars dispersed over the face of the heavens, which had been unknown to all the ancients; and examining Jupiter, with a better instrument than any he had made before, he found that he was accompanied by four stars, which, in certain fixed periods, performed revolutions round him, and which, in honour of the house of Medici, he called Medicean planets.

This discovery he made in January 1610, new style; and continuing his observations the whole of February following, in the beginning of March next he published an account of all his discoveries, in his Nuncius Sidereus, printed at Venice, and dedicated to Cosmo great duke of Tuscany, who, by a letter which he wrote to him on the 10th of July 1610, invited him to quit Padua, and assigned him an ample stipend, as primate and extraordinary professor at Pisa, but without any obligation to read lectures, or to reside.

The extraordinary discoveries contained in the Nuncius Sidereus, which was immediately reprinted both in Germany and France, were the cause of much speculation and debate among the philosophers and astronomers of that time; many of whom could not be brought to give any credit to Galileo's account, while others endeavoured to decry his discoveries as being nothing more than fictions or illusions. Some could not be prevailed upon even to look through a telescope; so devoted were they to the system of Aristotle, and so averse to admit any other source of knowledge besides his writings. When it was found to be in vain to oppose the evidence of sense, some did not scruple to assert that the invention was taken from Aristotle; and producing a passage from his writings, in which he attempts to give a reason why flares are seen in the daytime from the bottom of a deep well, said, that the well corresponded to the tube of the telescope, and that the vapours which arose from it gave the hint of putting glasses into it; and lastly, that in both cases the sight is strengthened by the transmission of the rays through a thick and dark medium. Galileo himself tells this story with a great deal of humour; comparing such men to alchemists, who imagine that the art of making gold was known to the ancients, but lay concealed under the fables of the poets.

In the beginning of July of the same year, 1610, Galileo being still at Padua, and getting an imperfect view of Saturn's ring, imagined that that planet consisted of three parts; and therefore, in the account which he gave of this discovery to his friends, he calls it planetarii tergeminum.

Whilst he was still at Padua, which must have been either in the same month of July, or the beginning of August following, he observed some spots on the face of the sun; but, contrary to his usual custom, he did not choose, at that time, to publish his discovery; partly for fear of incurring more of the hatred of many obstinate peripatetics; and partly in order to make more exact observations on this remarkable phenomenon, and to form some conjecture concerning the probable cause of it. He therefore contented himself with communicating his observations to some of his friends at Padua and Venice, among whom we find the name of father Paul. This delay, however, was the cause of this discovery being contested with him by the famous Scheiner, who likewise made the same observation in Oct. 1611, and we suppose had anticipated Galileo in the publication of it.

About the end of August, Galileo left Padua and went to Florence; and in November following he was satisfied, that, from the September preceding, Venus had been continually increasing in bulk, and that she changed her phases like the moon. About the end of March 1611, Galileo went to Rome, where he gratified the cardinals, and all the principal nobility, with a view of the new wonders he had discovered in the heavens, and among others the solar spots.

From these discoveries Galileo obtained the name of Named Lynceus, after one of the Argonauts, who was famous in antiquity for the acuteness of his sight; and moreover, the marquis of Monticelli instituted an academy, with the title of De Lyncei, and made him a member of it. Twenty-nine years Galileo enjoyed the use of his telescope, continually enriching astronomy with his observations; but by too close an application to that instrument, and the detriment he received from the nocturnal air, his eyes grew gradually weaker, till in 1639 he became totally blind; a calamity which, however, neither broke his spirits, nor interrupted the course of his studies.

The first telescope that Galileo constructed magnified only three times; but presently after, he made of his telescope another which magnified 18 times; and afterwards, with great trouble and expense, he constructed one that magnified 33 times; and with this it was that he discovered the satellites of Jupiter and the spots of the sun. Notwithstanding Galileo must be allowed to have considerable merit with respect to telescopes, it was neither that of the person who first hit upon the construction, nor that of him who thoroughly explained the rationale of the instrument. This important service to science was performed by John Kepler, whose name is famous on many accounts in the annals of philosophy, and especially by his discovery of the great law of motion respecting the heavenly bodies; which is, that the squares of their periodical times are as the cubes of their distances from the body about which they revolve; a proposition which, however, was not demonstrated before Sir Isaac Newton. Kepler was astronomer to several of the emperors of Germany; he was the associate of the celebrated astronomer Tycho Brahe, and the matter of Descartes.

Kepler made several discoveries relating to the nature of vision; and not only explained the rationale of the telescope which he found in use, but also pointed out methods of constructing others of superior powers and more commodious application.

It was Kepler who first gave a clear explanation of the effects of lenses, in making the rays of a pencil of light converge or diverge. He showed, that a plano-convex lens makes rays that were parallel to its axis, to meet at the distance of the diameter of the sphere of convexity; but that if both sides of the lens be equally convex, the rays will have their focus at the distance of the radius of the circle, corresponding to that degree of convexity. But he did not investigate any rule for the foci of lenses unequally convex. He only says, in general, that they will fall somewhere in the medium, between the foci belonging to the two different degrees of convexity. It is to Cavalieri that we owe this investigation. He laid down this rule: As the sum of both the diameters is to one of them, so is the other to the distance of the focus. All these rules concerning convex lenses are applicable to those that are concave; with this difference, that the focus is on the contrary side of the glass, as will be particularly shown in the second part of this treatise.

The principal effects of telescopes depend upon these plain maxims, viz. That objects appear larger in proportion to the angles which they subtend at the eye; and the effect is the same whether the pencils of rays, by which objects are visible to us, come directly from the objects themselves, or from any place nearer to the eye, where they may have been united so as to form an image of the object; because they issue again from those points where there is no real substance, in certain directions, in the same manner as they did from the corresponding points in the objects themselves.

In fact, therefore, all that is effected by a telescope is, first to make such an image of a distant object, by means of a lens or mirror; and then to give the eye some assistance for viewing that image as near as possible; so that the angle which it shall subtend at the eye, may be very large compared with the angle which the object itself would subtend in the same situation. This is done by means of an eye-glass, which to refracts the pencils of rays, so that they may afterwards be brought to their several foci by the natural humours of the eye. But if the eye was so formed as to be able to see the image with sufficient distinctness at the same distance without any eye-glass, it would appear to him as much magnified as it does to another person who makes use of a glass for that purpose, though he would not in all cases have so large a field of view.

If, instead of an eye glass, an object, or the image of an object, be looked at through a small hole in a thin plate or piece of paper, held close to the eye, it may be viewed very near to the eye, and, at the same distance, the apparent magnitude of the object will be the same in both cases. For if the hole be so small as to admit but a single ray from every distant point of the object, these rays will fall upon the retina in as many other distinct points, and make a distinct image. They are only pencils or cones of rays, which have a sensible bale, as the breadth of the pupil, that are capable, by their spreading on the retina, of producing an indistinct image. As very few rays, however, can be admitted through a small hole, there will seldom be light sufficient to view any object to advantage in this manner.

If no image be actually formed by the foci of the pencils without the eye, yet if, by the help of any eyeglass, the pencils of rays shall enter the pupil, just as they would have done from any place without the eye, the visual angle will be the same as if an image had actually been formed in that place. Objects will not appear inverted through this telescope, because the pencils which form the images of them, only cross one another once, viz. at the object glass, as in natural vision they do in the pupil of the eye.

Such is the telescope that was first discovered and used by philosophers; and it is remarkable that it telescope should be of a much more difficult construction than more diverse other kinds that have been invented since. The great inconvenience attending it is, that the field of view is exceedingly small. For since the pencils of rays enter the eye very much diverging from one another, but few of them can be intercepted by the pupil, this inconvenience increases with the magnifying power of the telescope; so that philosophers at this day cannot help wondering, that it was possible, with such an instrument, for Galileo and others to have made the discoveries they did. It must have required incredible patience and address. No other telescope, however, than this, was so much as thought of for many years after the discovery. Descartes, who wrote 30 years after, mentions no others as actually constructed, though Kepler had suggested some.

It is to this great man that we are indebted for the construction of what we now call the astronomical telescope, being the best adapted for the purpose of viewing the heavenly bodies. The rationale of this instrument is explained, and the advantages of it are clearly pointed out, by this philosopher, in his Catoptrics; but, what is very surprising, he never actually reduced his excellent theory into practice. Montucla conjectures, that the reason why he did not make trial of his new construction was, his not being aware of the great increase of the field of view; so that being engaged in other pursuits, he might not think it of much consequence to take any pains about the construction of an instrument, which could do little more than answer the same purpose with those of which he was already possessed. He must also have foreseen, that the length of this telescope must have been greater in proportion to its magnifying power, so that it might appear to him to be upon the whole not quite so good a construction as the former.

It was not long, however, before Kepler's new scheme of a telescope was executed; and the first person who actually made an instrument of this construction was Father Scheiner, who has given a description of it in his *Rosa Ursina*, published in 1630. If, says he, you insert two similar lenses (that is, both convex) in a tube, and place your eye at a convenient distance, you will see all terrestrial objects inverted, indeed, but magnified and very distinct, with a considerable extent of view. He afterwards subjoins an account of a telescope of a different construction, with two convex eye-glasses, which again reverses the images, and makes them appear in their natural position. This disposition of the lenses had also been pointed out by Kepler, but had not been reduced to practice by him, any more than the former. This construction, however, answered the end but very imperfectly; and Father Rheita presently after hit upon a better construction, using three eye glasses instead of two. This got the name of the *terrestrial telescope*, being chiefly used for terrestrial objects.

The first and last of these constructions are those which are now in common use. The proportion in which the first telescope magnifies, is as the focal length of the object-glass to that of the eye-glass. The only difference between the Galilean telescope and the other is, that the pencils by which the extremities of any object are seen in this case, enter the eye diverging; whereas, in the other, they enter it converging; but if the sphere of concavity in the eye-glass of the Galilean telescope be equal to the sphere of convexity in the eye-glass of another telescope, their magnifying power will be the same. The concave eye-glass, however, being placed between the object-glass and its focus, the Galilean telescope will be shorter than the other, by twice the focal length of the eye-glass. Consequently, if the length of the telescopes be the same, the Galilean will have the greater magnifying power.

The invention of the telescope and microscope having incited mathematicians to a more careful study of dioptrics, and this having soon become almost a perfect science, by means of the discovery of Snellius, many different constructions were offered to the public. Huygens was particularly eminent for his systematic knowledge of the subject, and is the author of the chief improvements which have been made on all the dioptrical instruments till the time of Mr Dollond's discovery. He was well acquainted with the theory of aberration arising from the spherical figure of the glasses, and has showed several ingenious methods of diminishing them by proper contractions of the eyepieces. He first showed the advantages of two eye-glasses on the astronomical telescope and double microscope, and gave rules for this construction, which both enlarges the field and shortens the instrument. Mr Dollond adapted his construction to the terrestrial telescope of De Rheita; and his five eye-glasses are nothing but the Huygenian eye-piece doubled. This construction has been too hastily given up by the artists of the present day for another, also of Mr Dollond's, of four glasses.

Vision is more distinct in the Galilean telescope than in the other, owing perhaps in part to there being no intermediate image between the eye and the object. Besides the eye-glass being very thin in the centre, the Galilean rays will be less liable to be distorted by irregularities in the substance of the glass. Whatever be the cause, we can sometimes see Jupiter's satellites very clearly in a Galilean telescope not more than twenty inches or two feet long; when one of four or five feet, of the common fort, will hardly make them visible.

The same Father Rheita, to whom we are indebted Binocular for the useful construction of a telescope for land-telescope-objects, invented a binocular telescope, which Father Cherubin, of Orleans, endeavoured to bring into use afterwards. It consists of two telescopes fastened together, and made to point to the same object. When this instrument is well fixed, the object appears larger, and nearer to the eye, when it is seen through both the telescopes, than through one of them only, though they have the very same magnifying power. But this is only an illusion, occasioned by the stronger impression that two equal images, equally illuminated, make upon the eye. This advantage, however, is counterbalanced by the inconvenience attending the use of it.

The first who distinguished themselves in grinding telescopic glasses were two Italians, Eustachio Divini at Rome, and Campani at Bologna, whose fame was much superior to that of Divini, or that of any other person of his time; though Divini himself pretended, that, in all the trials that were made with their glasses, his, of a great focal distance, performed better than those of Campani, and that his rival was not willing to try them fairly, viz. with equal eye-glasses. It is generally supposed, however, that Campani really excelled Divini, both in the goodness and the focal length of his object-glasses. It was with telescopes made by Campani that Cassini discovered the nearest satellites of Saturn. They were made by the express order of Louis XIV. and were of 86, 100, and 136 Parisian feet focal length.

Campani sold his lenses for a great price, and took every possible method to keep his art of making them a secret. His laboratory was inaccessible to all the world, till after his death; when it was purchased by Pope Benedict XIV., who made a present of it to the academy called the *Institut*, established in that city; and by the account which M. Fougeroux has given of what he could discover from it, we learn, that (except a machine, which M. Campani constructed, to work the balons on which he ground his glasses) the goodness of his lenses depended upon the cleanness of his glasses, his Venetian tripoli, the paper with which he polished his glasses, and his great skill and address as a workman. It was also the general opinion at Bologna, that he owed a great part of his reputation to the secrecy and air of mystery which he affected; and that he made a great number of object-glasses, which he rejected, showing only those that were very good. He made few lenses of a very great focal distance; and having the misfortune to break one of 141 feet in two pieces, he took incredible pains to join the two parts together, which he did at length effectually, so that it was used as if it had been entire; but it is not probable that he would have taken so much pains about it, if, as he pretended, he could very easily have made another as good. Sir Paul Neille, Dr Hooke says, made telescopes of 36 feet, pretty good, and one of 57, but not of proportional goodness. Afterwards Mr Reive first, and then Mr Cox, who were the most celebrated in England as grinders of optic glasses, made some good ones of 50 and 60 feet focal distance, and Mr Cox made one of 100; but how good, Dr Hooke could not assert.

Borelli also, in France, made object-glasses of a great focal length, one of which he presented to the Royal Society; but we do not find any particular account of their goodness.

With respect to the focal length of telescopes, these and all others were far exceeded by M. Auzout, who made one object-glass of 600 feet focus; but he was never able to manage it, so as to make any use of it. Hartsocker is even said to have made some of a still greater focal length; but this ingenious mechanic, finding it impossible to make use of object-glasses the focal distance of which was much less than this, when they were inclosed in a tube, contrived a method of using them without a tube, by fixing them at the top of a tree, a high wall, or the roof of a house.

Mr Huygens, who was also an excellent mechanic, made considerable improvements in the method of using an object-glass without a tube. He placed it at the top of a very long pole, having previously inclosed it in a short tube, which was made to turn in all directions, by means of a ball and socket. The axis of this tube he could command with a fine silk string, so as to bring it into a line with the axis of another short tube, which he held in his hand, and which contained the eye-glass. In this method he could make use of object-glasses of the greatest magnifying power, at whatever altitude his object was, and even in the zenith, provided his pole was as long as his telescope; and, to adapt it to the view of objects of different altitudes, he had a contrivance, by which he could raise or depress a stage that supported his object-glass at pleasure.

M. De la Hire made some improvement in this method of managing the object-glass, fixing it in the centre of a board, and not in a tube; but as it is not probable that this method will ever be made use of, since the discovery of both reflecting and achromatic telescopes, which are now brought to great perfection, and have even micrometers adapted to them, we shall not describe this apparatus minutely; but shall only give a drawing of M. Huygens's pole, which, with a very short explanation, will be sufficient for the purpose. In fig. 1, a represents a pulley, by the help of which a stage c, d, e, f (that supports the object-glass k, and the apparatus belonging to it), may be raised higher or lower at pleasure, the whole being counterpoised by the weight b, fastened to a string g. n, is a weight, by means of which the centre of gravity of the apparatus belonging to the object-glass is kept in the ball and socket, so that it may be easily managed by the string l, u, and its axis brought into a line with the eye-glass at o. When it was very dark, M. Huy-

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(1) Lettres de Descartes, tom. ii. printed at Paris in 1657, lett. 29. and 32. See this point discussed by two learned and candid authors, M. le Roy in the Encyclopédie, under the article Telescope, and M. Montecu in Hist. des Mathem, tom. ii. p. 644. it appears, had never perused the two letters of Descartes to Mercennus which briefly touch on that subject.

Again, as to his assertion, that Gregory's construction was not nearly so advantageous as Newton's, it may be accounted for from his having let it down early in the composition of his work, and forgetting to qualify it afterwards, when, before the publication, he had received pretty sure information to the contrary. Or perhaps he was influenced by the example of Dr Bradley, who had been a most successful observer, and yet had always preferred the Newtonian telescope to the other. But we must certainly adjudge the superiority to the latter, as that is now, and has been for several years past, the only instrument of the kind in request.

Gregory, a young man of an uncommon genius, was led to the invention, in seeking to correct two imperfections of the common telescope: the first was its too great length, which made it less manageable; the second, the incorrectness of the image. Mathematicians had demonstrated, that a pencil of rays could not be collected in a single point by a spherical lens; and also, that the image transmitted by such a lens would be in some degree incurvated. These inconveniences he believed would be obviated by substituting for the object-glass a metallic speculum, of a parabolic figure, to receive the image, and to reflect it towards a small speculum of the same metal: this again was to return the image to an eye-glass placed behind the great speculum, which for that purpose was to be perforated in its centre. This construction he published in 1663, in his Optica Promota. But as Gregory, by his own account, was endowed with no mechanical dexterity, nor could find any workman capable of realizing his invention, after some fruitless attempts in that way he was obliged to give up the pursuit: and probably, had not some new discoveries been made in light and colours, a refracting telescope would never more have been thought of, considering the difficulty of the execution, and the small advantages that could accrue from it, deducible from the principles of optics that were then known.

But Newton, whose genius for experimental knowledge was equal to that for geometry, happily interposed, and saved this noble invention from well nigh perishing in its infant-state. He likewise at an early period of life had applied himself to the improvement of the telescope; but imagining that Gregory's specula were neither very necessary, nor likely to be executed, he began with prosecuting the views of Descartes, who aimed at making a more perfect image of an object, by grinding lenses, not to the figure of a sphere, but to that of one of the conic sections. Now, whilst he was thus employed, three years after Gregory's publication, he happened to take to the examination of the colours formed by a prism, and having by the means of that simple instrument discovered the different refrangibility of the rays of light, he then perceived that the errors of telescopes, arising from that cause alone, were some hundred times greater than such as were occasioned by the spherical figure of lenses. This circumstance forced, as it were, Newton to fall into Gregory's track, and to turn his thoughts to reflectors. "The different refrangibility of the rays of light (says he, in a letter to Mr Oldenburg, secretary to the Royal Society, dated in Feb. 1672) made me take reflections into consideration; and finding them regular, so that the angle of reflection of all sorts of rays was equal to the angle of incidence, I understood that by their mediation optic instruments might be brought to any degree of perfection imaginable, providing a reflecting substance could be found which would polish as finely as glass, and reflect as much light as glass transmits, and the art of communicating to it a parabolic figure be also obtained. Amidst these thoughts I was forced from Cambridge by the intervening plague, and it was more than two years before I proceeded further."

It appears, then, that if Newton was not the first inventor of the reflecting telescope, he was the main and effectual inventor. By the force of his admirable genius, he fell upon this new property of light; and thereby found, that all losses, of whatever figure, would be affected more or less with such prismatic aberrations of the rays as would be an insuperable obstacle to the perfection of a dioptric telescope.

It was towards the end of 1668, or in the beginning of the following year, when Newton, being thus obliged to have recourse to reflectors, and not relying on any artificer for making his specula, set about the work himself, and early in the year 1672 completed two small reflecting telescopes. In these he ground the great speculum into a spherical concave; not but that he approved of the parabolic form proposed by Gregory, though he found himself unable to accomplish it. In the letter that accompanied one of these instruments which he presented to the Society, he writes, "that though he then despaired of performing that work (to wit, the parabolic figure of the great speculum) by geometrical rules, yet he doubted not but that the thing might in some measure be accomplished by mechanical devices."

Not less did the difficulty appear to find a metallic substance that would be of a proper hardness, have the fewest pores, and receive the smoothest polish: a difficulty in truth which he deemed almost insurmountable, when he considered, that every irregularity in a reflecting surface would make the rays of light stray five or six times more out of their due course, than the like irregularities in a refracting one. In another letter, written soon after, he tells the secretary, "that he was very sensible that metal reflects less light than glass transmits; but as he had found some metallic substances to be more strongly reflective than others, to polish better, and to be freer from tarnishing than others, so he hoped that there might in time be found out some substances much freer from these inconveniences than any yet known." Newton therefore laboured till he found a composition that answered in some degree, and left it to those who should come after him to find a better, and presented a reflecting telescope to the Royal Society; from whom he received such thanks as were due to so curious and valuable a present. And Huygens, one of the greatest geniuses of the age, and himself a distinguished improver of the refractor, no sooner was informed by Mr Oldenburg of the discovery, than he wrote in answer, "that it was an admirable telescope; and that Mr Newton had well considered the advantage which a concave concave speculum had above convex glasses in collecting the parallel rays, which according to his own calculation was very great; Hence that Mr Newton could give a far greater aperture to that speculum than to an object-glass of the same distance of focus, and consequently much more magnify in his way than by an ordinary telescope: Besides, that by the reflector he avoided an inconvenience inseparable from object-glasses, which was the obliquity of both their surfaces, which vitiated the refraction of the rays that pass towards the sides of the glass, and did more hurt than men were aware of: Again, that by the mere reflection of the metallic speculum there were not so many rays lost as in glasses, which reflected a considerable quantity by each of their surfaces, and besides intercepted many of them by the obscurity of their matter: That the main business would be, to find a matter for this speculum that would bear as good and even a polish as glass. Lastly, he believed that Mr Newton had not been without considering the advantage which a parabolic speculum would have over a spherical one in this construction; but had despaired, as he himself had done, of working other surfaces than spherical ones with due exactness." Huygens was not satisfied with thus expressing to the society his high approbation of the late invention; but drew up a favourable account of the new telescope, which he caused to be published in the Journal des Savants for the year 1672, and by that channel it was soon known over Europe.

But how excellent forever the contrivance was; how well forever supported and announced to the public; yet whether it was that the artists were deterred by the difficulty and labour of the work, or that the discoveries even of a Newton were not to be exempted from the general fatality attending great and useful inventions, the making a slow and vexatious progress to the authors; the fact is, that, excepting an unsuccessful attempt which the society made, by employing an artificer to imitate the Newtonian construction, but upon a larger scale, and a disguised Gregorian telescope, set up by Cassgrain abroad as a rival to Newton's, and that in theory only (for it never was put in execution by the author), no reflector was heard of for nearly half a century after. But when that period was elapsed, a reflecting telescope was at last produced to the world of the Newtonian construction by Dr Hadley, which the author had the satisfaction to find executed in such a manner as left no room to fear that the invention would any longer continue in obscurity.

This memorable event was owing to the genius, dexterity, and application, of Mr Hadley the inventor of the reflecting quadrant, another most valuable instrument. The two telescopes which Newton had made were but six inches long, were held in the hand for viewing objects, and in power were compared to a six-feet refractor; whereas Hadley's was above five feet long, was provided with a well-contrived apparatus for managing it, and equalled in performance the famous aerial telescope of Huygens of 123 feet in length. Excepting as to the manner of making the specula, we have, in the transactions of 1723, a complete description, with a figure, of this telescope, together with that of the machine for mounting it; but, by a strange omission, Newton's name is not once mentioned in that paper, so that any person not acquainted with the history of the invention, and reading that account only, might be apt to conclude that Hadley had been the sole contriver of it.

The same celebrated artist, after finishing two telescopes of the Newtonian construction, accomplished a third in the Gregorian way; but, it would seem, less successfully, by Dr Smith's declaring so strongly in favour of the other. Mr Hadley spared no pains to instruct Mr Molyneux and the reverend Dr Bradley; and when these gentlemen had made a sufficient proficiency in the art, being desirous that these telescopes should become more public, they liberally communicated to some of the principal instrument-makers of London the knowledge they had acquired from him. Now such scholars, as it is easy to imagine, soon advanced beyond their masters, and completed reflectors by other and better methods than what had been taught them.

Certain it is, at least, that Mr James Short, as early as the year 1734, had signalled himself at Edinburgh by his work of this kind. Mr Macaulay wrote that year to Dr Jurin, "that Mr Short, who had begun with making glass specula, was then applying himself to improve the metallic; and that, by taking care of the figure, he was enabled to give them larger apertures than others had done; and that upon the whole they surpassed in perfection all that he had seen of other workmen." He added, "that Mr Short's telescopes were all of the Gregorian construction; and that he had much improved that excellent invention." This character of excellence Mr Short maintained to the last; and with the more facility, as he had been well grounded both in the geometrical and philosophical principles of optics, and upon the whole was a most intelligent person in whatever related to his profession. It was supposed he had fallen upon a method of giving the parabolic figure to his great speculum; a point of perfection that Gregory and Newton had wished for, but despaired of attaining; and that Hadley had never, as far as we know, attempted, either in his Newtonian or Gregorian telescope. Mr Short indeed said he had acquired that faculty, but never would tell by what peculiar means he effected it; so that the secret of working that configuration, whatever it was, as far as it then appeared, died with that ingenious artist. Mr Mudge, however, hath lately realized the expectation of Sir Isaac Newton, who, above 100 years ago, presaged that the public would one day possess a parabolic speculum, not accomplished by mathematical rules, but by mechanical devices.

This was a defideratum, but it was not the only want supplied by this gentleman: he has taught us likewise a better composition of metals for the specula, how to grind them better, and how to give them a finer polish; and this last part, (namely, the polish), he remarks, was the most difficult and essential of the whole operation. "In a word (says Sir John Pringle), I am of opinion, there is no optician in this great city (which hath been so long and so justly renowned for ingenious and dexterous makers of every kind of mathematical instruments) so partial to his own abilities..." as not to acknowledge, that, however some parts of the mechanical process now disclosed might have been known before by individuals of the profession, yet that Mr Mudge hath opened to them all some new and important lights, and upon the whole hath greatly improved the art of making reflecting telescopes."

The late reverend and ingenious John Edwards devoted much of his time to the improvement of reflecting telescopes, and brought them to such perfection, that Dr Maskelyne, the astronomer royal, found telescopes constructed by him to surpass in brightness, and other essentials, those of the same size made by the best artists in London. The chief excellence of his telescopes arises from the composition, which, from various trials on metals and semimetals, he discovered for the specula, and from the true parabolic figure, which, by long practice, he had found a method of giving them, preferable to any that was known before him. His directions for the composition of specula, and for casting, grinding, and polishing them, were published, by order of the commissioners of longitude, at the end of the Nautical Almanack for the year 1787. To the same almanack is also annexed his account of the cause and cure of the tremors which particularly affect reflecting telescopes more than refracting ones, together with remarks on the said tremors by Dr Maskelyne. See Telescope.

But in constructing reflecting telescopes of extraordinary magnifying powers, Dr Herschel has displayed skill and ingenuity surpassing all his predecessors in this department of mechanics. He has made them from 7, 10, 20, to even 40 feet in length; and with the instrument of these latter dimensions he is now employed in making discoveries in astronomy. Of its construction, magnifying powers, and the curious collection of machinery by which it is supported and moved from one part of the heavens to another, accounts will be given under the word Telescope.

The greatest improvement in refracting telescopes hitherto made public (c) is that of Mr Dollond, of which an account has already been given in a preceding section, wherein his discoveries in the science of Optics were explained. But, besides the obligation we are under to him for correcting the aberration of the rays of light in the focus of object-glasses, arising from their different refrangibility, he made another considerable improvement in telescopes, viz. by correcting, in a great measure, both this kind of aberration, and also that which arises from the spherical form of lenses, by an expedient of a very different nature; viz. increasing the number of eye-glasses.

If any person, says he, would have the visual angle of a telescope to contain 20 degrees, the extreme pencils of the field must be bent or refracted in an angle of 10 degrees; which, if it be performed by one eye-glass, will cause an aberration from the figure, in proportion to the cube of that angle; but if two glasses are so proportioned and situated, as that the refraction may be equally divided between them, they will each of them produce a refraction equal to half the required angle; and therefore, the aberration being in proportion to the cube of half the angle taken twice over, will be but a fourth part of that which is in proportion to the cube of the whole angle; because twice the cube of 1 is but \( \frac{1}{4} \) of the cube of 2; so the aberration from the figure, where two eye-glasses are rightly proportioned, is but a fourth of what it must unavoidably be, where the whole is performed by a single eye-glass. By the same way of reasoning, when the refraction is divided between three glasses, the aberration will be found to be but the ninth part of what would be produced from a single glass; because three times the cube of 1 is but one-ninth of the cube of 3. Whence it appears, that by increasing the number of eye-glasses, the indistinctness which is observed near the borders of the field of a telescope may be very much diminished, though not entirely taken away.

The method of correcting the errors arising from the different refrangibility of light is of a different consideration from the former. For, whereas the errors from the figure can only be diminished in a certain proportion according to the number of glasses, in this they may be entirely corrected by the addition of only one glass; as we find in the astronomical telescope, that two eye-glasses, rightly proportioned, will cause the edges of objects to appear free from colours, quite to the borders of the field. Also in the day-telescope, where no more than two eye-glasses are absolutely necessary for erecting the object, we find, that by the addition of a third, rightly situated, the colours, which would otherwise make the image confused, are entirely removed. This, however, is to be understood with some limitation: for though the different colours into which the extreme pencils must necessarily be divided by the edges of the eye-glasses, may in this manner be brought to the eye in a direction parallel to each other, so as, by the humours of the eye, to be made to converge to a point on the retina; yet, if the glasses exceed a certain length, the colours may be spread too wide to be capable of being admitted through the pupil or aperture of the eye; which is the reason, that in long telescopes, constructed in the common manner, with three eye-glasses, the field is always very much contracted.

These considerations first set Mr Dollond on considering how to enlarge the field, by increasing the number of eye-glasses without any hindrance to the distinctness or brightness of the image; and though others had been about the same work before, yet, observing that some five-glass telescopes which were then made would admit of farther improvement, he endeavoured to construct one with the same number of glasses in a better

(c) Dr Blair's discovery, mentioned no 19, will undoubtedly lead to improvements superior to those of Dollond; but as his memoir on the subject is not yet published, we feel not ourselves at liberty to make longer extracts from it. The reader will see the whole in the Philosophical Transactions of the Royal Society of Edinburgh, whenever that body shall be pleased to favour the public with a third volume of its learned labours. better manner; which so far answered his expectations, as to be allowed by the best judges to be a considerable improvement on the former.

Encouraged by this success, he resolved to try if he could not make some farther enlargement of the field, by the addition of another glass, and by placing and proportioning the glasses in such a manner as to correct the aberrations as much as possible, without any detriment to the distinctness; and at last he obtained as large a field as is convenient or necessary, and that even in the longest telescopes that can be made.

These telescopes with six glasses having been well received, and some of them being gone into foreign parts, it seemed a proper time to the author to settle the date of his invention; on which account he drew up a letter, which he addressed to Mr Short, and which was read at the Royal Society, March 1, 1753 (D).

Various other attempts were made about this time to shorten and otherwise improve telescopes. Among these we must just mention that of Mr Caleb Smith, who, after giving much attention to the subject, thought that he had found it possible to rectify the errors which arise from the different degrees of refrangibility, on the principle that the fines of refraction, or rays differently refrangible, are to one another in a given proportion, when their fines of incidence are equal; and the method which he proposed for this purpose was to make the speculum of glasses instead of metal, the two surfaces having different degrees of convexity. But we do not find that his scheme was ever executed; nor is it probable, for reasons which have been mentioned, that any advantage could be made of it.

To Mr Short we are indebted for the excellent contrivance of an equatorial telescope, or, as he likewise called it, a portable observatory; for with it pretty accurate observations may be made with very little trouble, by those who have no building adapted to the purpose. The instrument consists of an ingenious piece of machinery, by the help of which a telescope mounted upon it may be directed to any degree of right ascension or declination, so that the place of any of the heavenly bodies being known, they may be found without any trouble, even in the day-time. Also, being made to turn parallel to the equator, any object is easily kept in view, or recovered, without moving the eye from its situation. By this instrument, Mr Short informs us, that most of the stars of the first and second magnitude have been seen even at midday, and the sun shining bright; as also Mercury, Venus, and Jupiter. Saturn and Mars are not so easy to be seen, on account of the faintness of their light, except when the sun is but a few hours above the horizon. This particular effect depends upon the telescope excluding almost all the light, except what comes from the object itself, and which might otherwise efface the impression made by its weaker light upon the eye. Any telescope of the same magnifying power would have the same effect, could we be sure of pointing it right. For the same reason, also, it is that stars are visible in the day-time from the bottom of a deep pit. Mr Ramsden has lately invented a portable observatory or equatorial telescope, which may perhaps supercede the use of Mr Short's. See Astronomy, n° 504.

In order to enable us to see the fixed stars in the daytime, it is necessary to exclude the extraneous fervor of the light as much as possible. For this reason the greater magnifying power of any telescope is used, the more easily a fixed star will be distinguished in the daytime; the light of the star remaining the same in all magnifying powers of the same telescope, but the ground upon which it is seen becoming darker by increasing the magnifying power; and the visibility of a star depends very much upon the difference between its own light and that of the ground upon which it is seen. A fixed star will be very nearly equally visible with telescopes of very different apertures, provided the magnifying power remains the same.

If a comet, or any other heavenly body, be viewed through this equatorial telescope, properly rectified, it is seen immediately by the help of the same machinery what is its true place in the heavens. Other astronomical problems may also be solved by it, with great ease and certainty.

M. Epiphan proposes to bend the tubes of long telescopes at right angles, fixing a plane mirror in the angle, in order to make them more commodious for viewing objects near the zenith of the observer; and bending the he gives particular instructions how to make them in this form, especially when they are furnished with micrometers. We are also informed that a little plane speculum is sometimes placed betwixt the last eye-glass and the eye in reflecting telescopes, at an angle of 45°, for the same purpose.

The invention of Microscopes was not much later than that of telescopes; and, according to Borellus, whose account we do not find to have been called in question by any person, we are indebted for them to the same author, at least to Z. Janfens, in conjunction with his son; and for this latter favour we may, perhaps, be considered as under more obligation to them than for the former, the microscope having more various and extensive uses, with respect to philosophy, than the telescope. In our ideas, however, it appears something greater, and more extraordinary, to be able to see objects too distant to be perceived by the naked eye, than those that are too near to be seen by us; and therefore there is more of the sublime in the telescope than the microscope. These two instruments, though different in their application, are notwithstanding very similar; as both of them assist us in the discovery of objects that we must otherwise have remained unacquainted with, by enlarging the angle which they subtend at the eye.

The Janfens, however, have not always enjoyed undisturbed, that share of reputation to which they seem to be intitled, with respect either to the telescope or the microscope. The discovery of the latter, in particular, has generally been considered as more uncertain than that of the former. All that many writers say we can depend upon is, that microscopes were first used in Germany about the year 1621. Others say positively, that this instrument was the contrivance of Cor-

(D) This paragraph is extracted from this paper in the Transactions; but Dollond's improvement, there described, is not accompanied by any diagram. For a minute account of it, and of eye-pieces in general, see Ludlam's Essays. Cornelius Drebell, no philosopher, but a man of curiosity and ingenuity, who also invented the thermometer.

According to Borellus, Zacharias Jansen and his son presented the first microscopes they had constructed to prince Maurice, and Albert archduke of Austria. William Borell, who gives this account in a letter to his brother Peter, says, that when he was ambassador in England, in 1619, Cornelius Drebell, with whom he was intimately acquainted, showed him a microscope, which he said was the same the archduke had given him, and had been made by Jansen himself. This instrument was not so short as they are generally made at present, but was six feet long, consisting of a tube of gilt copper, an inch in diameter, supported by three brass pillars in the shape of dolphins, on a base of ebony, on which the small objects were placed.

This microscope was evidently a compound one, or rather something betwixt a telescope and a microscope, what we should now, perhaps, choose to call a megroscope; so that it is possible that single microscopes might have been known, and in use, sometime before; but perhaps nobody thought of giving that name to single lenses; though, from the first use of lenses, they could not but have been used for the purpose of magnifying small objects. In this sense we have seen, that even the ancients were in possession of microscopes; and it appears from Jamblicus and Plutarch, quoted by Dr Rogers, that they gave such instruments as they used for this purpose the name of dioptra. As spectacles were certainly in use long before the invention of telescopes, one can hardly help concluding, that lenses must have been made smaller, and more convex, for the purpose of magnifying minute objects; especially as the application of this kind of microscope was nearly the same with that of a spectacle-glass, both of them being held close to the eye. At what time lenses were made so small as we now generally use them for magnifying in single microscopes, we have not found. But as this must necessarily have been done gradually, the only proper object of inquiry is the invention of the double or compound microscope; and this is clearly given, by the evidence of Borellus above-mentioned, to Zacharias Jansen, the inventor of the telescope, or his son.

The invention of compound microscopes is claimed by the same Fontana who claimed the discovery of telescopes; and though he did not publish any account of this invention till the year 1646 (notwithstanding he pretended to have made the discovery in 1618), Montucla, not having attended perhaps to the testimony of Borellus, is willing to allow his claim, as he thought there was no other person who seemed to have any better right to it.

Eufrasio Divini made microscopes with two common object-glasses, and two plano-convex eye-glasses joined together on their convex sides so as to meet in a point. The tube in which they were inclosed was as big as a man's leg, and the eye-glasses almost as broad as the palm of a man's hand. Mr Oldenburg, secretary to the royal society, received an account of this instrument from Rome, and read it at one of their meetings, August 6, 1668.

It was in this period that Hartsoeker improved single microscopes, by using small globules of glass, made by melting them in the flame of a candle, instead of the lenses which had before been made use of for that purpose. By this means he first discovered the animalcula in femein masculino, which gave rise to a new system of generation. A microscope of this kind, consisting of a globe of \( \frac{1}{8} \) of an inch in diameter, M. Huygens demonstrated to magnify 100 times; and since it is easy to make them of less than half a line in diameter, they may be made to magnify 300 times. Were it not for the difficulty of applying objects to these magnifiers, the want of light, and the small field of distinct vision, they would certainly have been the most perfect of all microscopes.

But no man distinguished himself so much by microscopical discoveries as the famous M. Leeuwenhoek, though he used only single lenses with short focus, preferring distinctness of vision to a large magnifying power.

M. Leeuwenhoek's microscopes were all single ones, each of them consisting of a small double convex-glass, set in a socket between two silver plates riveted together, and pierced with a small hole; and the object was placed on the point of a needle, so contrived as to be placed at any distance from the lens. If the objects were solid, he fastened them with glue; and if they were fluid, or on other accounts required to be spread upon glass, he placed them on a small piece of Muscovy tale, or glass blown very thin; which he afterwards glued to his needle. He had, however, a different apparatus for viewing the circulation of the blood, which he could fix to the same microscopes.

The greatest part of his microscopes M. Leeuwenhoek bequeathed to the Royal Society. They were contained in a small Indian cabinet, in the drawers of which were 13 little boxes, or cases, in each of which were two microscopes, neatly fitted up in silver; and both the glasses and the apparatus were made with his own hands.

The glasses of all these lenses is exceedingly clear, but none of them magnifies so much as those globules which are frequently used in other microscopes; but Mr Folkes, who examined them, thought that they showed objects with much greater distinctness, which M. Leeuwenhoek principally valued. His discoveries, however, are to be ascribed not so much to the goodness of his glasses, as to his great judgment, acquired by long experience, in using them. He also particularly excelled in his manner of preparing objects for being viewed to the most advantage.

Mr Baker, who also examined M. Leeuwenhoek's microscopes, and made a report concerning them to the Royal Society, found that the greatest magnifier among them enlarged the diameter of an object about 160 times, but that all the rest fell much short of that power; so he concluded that M. Leeuwenhoek must have had other microscopes of a much greater magnifying power for many of his discoveries. And it appears, he says, by many circumstances, that he had such microscopes.

It appears from M. Leeuwenhoek's writings, that he was not unacquainted with the method of viewing opaque objects by means of a small concave reflecting mirror, which was afterwards improved by M. Lieberkühn. For, after describing his apparatus for viewing cells in glass tubes, he adds, that he had an instrument to which he screwed a microscope set in brass, upon which microscope he fastened a little dish of brass, probably that his eye might be thereby assisted to see objects better; for he says he had filled the brass which was round his microscope as bright as he could, that the light, while he was viewing objects, might be reflected from it as much as possible. This microscope, with its dish, is constructed upon principles so similar to those which are the foundation of our single microscope by reflection (see Microscope), that it may well be supposed to have given the hint to the ingenious inventor of it, provided he ever attended to it.

In 1702, Mr Wilton made several ingenious improvements in the method of using single magnifiers, for the purpose of viewing transparent objects; and his microscope, which is also a necessary part of the solar microscope, is in very general use at this day. (See Microscope, sect. 1.)

In 1710, Mr Adams gave to the Royal Society the following account of his method of making small globules for large magnifiers. He took a piece of fine window glass, and cut it with a diamond into as many lengths as he thought proper, not exceeding \( \frac{1}{8} \) of an inch in breadth; then, holding one of them between the forefinger and thumb of each hand over a very fine flame, till the glass began to soften, he drew it out till it was as fine as a hair, and broke; then putting each of the ends into the purest part of the flame, he had two globules presently, which he could make larger or less at pleasure. If they were held a long time in the flame, they would have spots in them, so that he drew them out presently after they became round. The stem he broke off as near to the globe as he could, and lodging the remainder between the plates, in which holes were drilled exactly round, the microscope, he says, performed to admiration. Thro' these magnifiers, he says, that the same thread of very fine mullin appeared three or four times bigger than it did in the largest of Mr Wilton's magnifiers.

The ingenious Mr Grey hit upon a very easy expedient to make very good temporary microscopes, at a very little expense. They consist of nothing but very small drops of water, taken up with a point of a pin, and put into a small hole made in a piece of metal. Those globules of water do not, indeed, magnify so much as those which are made of glass of the same size, because the refractive power of water is not so great; but the same purpose will be answered nearly as well by making them somewhat smaller.

The same ingenious person, observing that small heterogeneous particles inclosed in the glass of which microscopes are made, were much magnified when those glasses were looked through, thought of making his microscopes of water that contained living animalcula, to see how they would look in this new situation; and he found his scheme to answer even beyond his utmost expectation, so that he could not even account for their being magnified so much as they were: for it was much more than they would have been magnified if they had been placed beyond the globe, in the proper place for viewing objects. But Montucla observes, that, when any object is inclosed within this small transparent globe, the hinder-part of it acts like a concave mirror, provided they be situated between that surface and the focus; and that, by this means, they are magnified above \( \frac{3}{4} \) times more than they would have been in the usual way.

After the happy execution of the reflecting telescope, it was natural to expect that attempts would be made to render a similar service to microscopes. Accordingly we find two plans of this kind. The first was that of Dr Robert Barker. His instrument differs in nothing from the reflecting telescope, excepting the distance of the two speculums, in order to adapt it to those pencils of rays which enter the microscope diverging; whereas they come to the telescope from very distant objects nearly parallel to each other.

This microscope is not so easy to manage as the common fort. For vision by reflection, as it is much more perfect, so it is far more difficult than that by refraction. Nor is this microscope so useful for any but very small or transparent objects. For the object, being between the speculum and image, would, if it were large and opaque, prevent a due reflection.

Dr Smith invented a double reflecting microscope, of which a theoretical and practical account is given in the remarks on the second volume of his System of Optics. Through some of those incidents to which the conducting of a work so multifarious as ours is always liable, this instrument was omitted under the article Microscope. As it is constructed on principles essentially different from all others, and, in the opinion of the ablest judges whom we have consulted, incomparably superior to them all, the reader will not be ill pleased with the following practical description, though it appears not perhaps in its most proper place.

Fig. 2. is a section of this microscope, where A.B.C and \( abc \) are two specula, the former concave, and the latter convex, inclosed within the tube D.E.F.G. The speculum A.B.C, is perforated like the speculum of a Gregorian telescope; and the object to be magnified is so placed between the centre and principal focus of that speculum, that the rays flowing from it to A.B.C are reflected towards an image \( pq \). But before they are united in that image they are received by the convex speculum \( abc \), and thence reflected through the hole B.C in the vertex of the concave to a second image \( rs \), to be viewed through an eye-glass l. The object may either be situated between the two specula, or, which is perhaps better, between the principal focus and vertex c of the convex speculum \( abc \), a small hole being made in its vertex for the incident rays to pass through. When the microscope is used, let the object be included between two little round plates of Mofcey-glass, fixed in a hole of an oblong brass plate \( mn \), intended to slide close to the back side of the convex speculum; which must therefore be ground flat on that side, and so thin that the object may come precisely to its computed distance from the vertex of the speculum. The slider must be kept tight to the back of the metal by a gentle spring. The distance of the object being thus determined once for all, distinct vision to different eyes, and through different eye-glasses, must be procured by a gentle motion of the little tubes that contain these glasses. These tubes must be made in the usual form of those that belong to Sir Isaac Newton's reflecting telescope, (see Telescope), having a small hole in the middle of each plate, at the ends of the tube, situated exactly in each focus of the glass; glares; the use of these holes and plates is to limit the visible area, and hinder any straggling rays from entering the eye. To the tube of the eye-glass is fastened the arm g, on which the adjusting screw turns. A similar arm u is attached to the fixed tube X, in which the neck of the screw turns; and by turning the button y, the eye tube is moved farther from or nearer to the object, by which means different sorts of eyes obtain distinct vision.

The rays which flow from the object directly through the hole in the concave speculum and through the eye-glass, by mixing with the reflected rays, would dilute the image on the retina, and therefore must be intercepted. This is done by a very simple contrivance. The little hole in the convex speculum is ground conical as in the figure; and a conical solid P, of which the base is larger than the orifice in the back of the convex speculum, supported on the slender pillar PQ, is so placed as to intercept all the direct rays from the eye-glass. All the tubes are strongly blacked on their insides, and so is the conical solid, to hinder all reflection of rays from these objects upon the convex speculum. The little base, too, of the solid should be made concave, that whatever light it may still reflect, may be thrown back upon the object; and its back-side being conical and blacked all over, will either absorb or laterally disperse any straggling rays which the concave speculum may scatter upon it, and so prevent their coming to the eye-glass.

Notwithstanding the interposition of this conical solid, yet when the eye-glass is taken out, distant objects may be distinctly seen through the microscope, by rays reflected from the metals, and diverging upon the eye from an image behind the convex speculum. But this mixture of foreign rays with those of the object, which is common to all kinds of microscopes in viewing transparent objects, is usually prevented by placing before the object a thick double convex lens L, to collect the sky-light exactly upon the object. This lens should be just so broad as to subtend the opposite angle to that which the concave speculum subtends at the object. The annular frame of the lens must be very narrow, and connected to the microscope by two or three slender wires or blades, whose planes produced may pass through the object, and intercept from it as little sky-light as possible.

This is not the place for explaining the principles of this microscope, or demonstrating its superiority over most others; nor are such explanation and demonstration necessary. Its excellence, as well as the principles upon which it is constructed, will be perceived by the reader, when he has made himself master of the laws of refraction and reflection as laid down in the ensuing part of this article.

In 1738 or 1739, M. Lieberkühn made two capital improvements in microscopes, by the invention of the solar microscope, and the microscope for opaque objects. When he was in England in the winter of 1739, he showed an apparatus of his own making, for each of these purposes, to several gentlemen of the Royal Society, as well as to some opticians, particularly Mr Cuff in Fleet-street, who took great pains to improve them.

The microscope for opaque objects remedies the inconvenience of having the dark side of an object next the eye. For by means of a concave speculum of silver, highly polished, in the centre of which a magnifying lens is placed, the object is so strongly illuminated that it may be examined with all imaginable ease and pleasure. A convenient apparatus of this kind, with four different speculums and magnifiers of different powers, was brought to perfection by Mr Cuff.

M. Lieberkühn made considerable improvements in his solar microscope, particularly in adapting it to the view of opaque objects; but in what manner this end was effected, M. Epinus, who was highly entertained with the performance, and who mentions the fact, was not able to recollect; and the death of the ingenious inventor prevented his publishing any account of it himself. M. Epinus invites those persons who came into the possession of M. Lieberkühn's apparatus to publish an account of this instrument; but it doth not appear that his method was ever published.

This improvement of M. Lieberkühn's induced M. Epinus himself to attend to the subject; and by this means he produced a very valuable improvement in this instrument. For by throwing the light upon the fore-side of any object by means of a mirror, before it is transmitted through the object-lens, all kinds of objects are equally well represented by it.

M. Euler proposed a scheme to introduce vision by reflected light into the magic lantern and solar micro-light microscope, by which many inconveniences to which those instruments are subject might be avoided. For this purpose, he says, that nothing is necessary but a large magic lantern, concave mirror, perforated as for a telescope; and that then the light be so situated, that none of it may pass directly through the perforation, so as to fall on the images of the objects upon the screen. He proposes to have four different machines, for objects of different sizes; the first for those of six feet long, the second for those of one foot, the third for those of two inches, and the fourth for those of two lines; but it is needless to be particular in the description of these, as more perfect instruments are described under the article Microscope.

Several improvements were made in the apparatus to the solar microscope, as adapted to view opaque objects, by M. Zeiher, who made one construction for the larger kind of objects, and another for the small ones.

Mr Marin having constructed a solar microscope of Mr Marin's larger size than common, for his own use, the illuminating lens being 4½ inches in diameter, and all the other parts of the instrument in proportion, found, that by the help of an additional part, which he does not describe, he could see even opaque objects very well. If he had made the lens any larger, he was aware that the heat produced at the focus would have been too great for the generality of objects to bear. The expense of this instrument, he says, does not much exceed the price of the common solar microscope.

The smallest globules, and consequently the greatest Di Torre's magnifiers, for microscopes, that have yet been executed, were made by T. Di Torre of Naples, who, in 1765, lent four of them to the Royal Society. The largest of them was only two Paris points in diameter, and it was said to magnify the diameter of an object 640 times. The second was the size of one Paris point, and and the third was no more than half of a Paris point, or the 1/13th part of an inch in diameter, and was said to magnify the diameter of an object 2560 times. One of these globules was wanting when they came into the hands of Mr Baker, to whose examination they were referred by the Royal Society. This gentleman, so famous for his skill in microscopes, and his extraordinary expertise in managing them, was not able to make any use of these. With that which magnifies the least, he was not able to see any object with satisfaction; and he concludes his account with expressing his hopes only, that, as his eyes had been much used to microscopes, they were not injured by the attention he had given to them, though he believed there were few persons who would not have been blinded by it.

The construction of a telescope with six eye-glasses led M. Euler to a similar construction of microscopes, by introducing into them six lenses, one of which admits of so small an aperture, as to serve, instead of a diaphragm, to exclude all foreign light, though, as he says, it neither lessens the field of view, nor the brightness of objects.

The improvement of all dioptric instruments is greatly impeded by inequalities in the substance of the attending glasses of which they are made; but though many attempts have been made to make glasses without that degree of imperfection, none of them have been hitherto quite dioptric in effectual. M. A. D. Merklein, having found some glasses which had been melted when a building was on fire, and which proved to make excellent object glasses for telescopes, concluded that its peculiar goodness arose from its not having been disturbed when it was in a fluid state; and therefore he proposed to take the metal out of the furnace in iron vessels, of the same form that was wanted for the glasses; and after it had been perfectly fluid in those vessels, to let it stand to cool, without any disturbance. But this is not always found to answer.

**Part I. Theory of Optics.**

This part of the science contains all that hath been discovered concerning the various motions of the rays of light, either through different mediums, or when reflected from different substances in the same medium. It contains also the rationale of every thing which hath been discovered with regard to vision; the optical deceptions to which we are liable; and, in short, ought to give the reason of all the known optical phenomena.—The science is commonly divided into three parts, viz. dioptrics, which contains the laws of refraction, and the phenomena depending upon them; catoptrics, which contains the laws of reflection, and the phenomena which depend on them; and, lastly, chromatics, which treat of the phenomena of colour. But this definition is of no use in a treatise of Optics, as most of the phenomena depend both on refraction and reflection, colour itself not excepted. For this reason, though we have given detached articles under the words Dioptrics, Catoptrics, and Chromatics; we have referred to this place the explanation of the laws of reflection and refraction, by which all optical phenomena may be accounted for.

**Sect. I. Of the properties of Light in general.**

Under the article Light we have given some account of the controversies concerning its nature. The opinions of philosophers may, in general, be arranged under these two: 1. That the phenomena of vision and illumination are produced by the undulations of an elastic fluid, much in the same manner as sound is produced by the undulations of air. This opinion was first offered to the public by Des Cartes, and afterwards by Mr Huyghens, and has lately been revived by Mr Euler, who has endeavoured to explain the phenomena upon mechanical principles.—2d. That the phenomena of vision are produced by the motion and action of matter emitted from the shining body with immense velocity, moving uniformly in straight lines, and acted on by other bodies; so as to be reflected, refracted, or inflected, in various ways, by means of forces which act on it in the same manner as on other inert matter. Sir Isaac Newton has shown, in the most incontrovertible manner, the total dissimilarity between the phenomena of vision and the legitimate consequences of the undulations of an elastic fluid. All Mr Euler's ingenious and laborious discussions have not removed Newton's objections in the smallest degree. Sir Isaac adopts the vulgar opinion, therefore, making light of the difficulties objected to it, because none of them are inconsistent with the established principles of mechanics, and are merely difficulties of conception to limited faculties like ours. We need not despair of being able to decide, by experiment, which of these opinions is nearest to the truth; because there are phenomena where the result should be sensibly different in the two hypotheses. At present, we shall content ourselves with giving some account of the legitimate consequences of the vulgar opinion as modified by Sir Isaac Newton, viz. that light consists of small particles emitted with very great velocity, and attracted or repelled by other bodies at very small distances.

Every visible body emits or reflects inconceivably small particles of matter from each point of its surface, in straight lines, which issue from it continually (not unlike sparks from lines from a coal) in straight lines and in all directions. These particles entering the eye, and striking upon the retina (a nerve expanded on the back part of the eye to receive their impulses), excite in our minds the idea of light. And as they differ in substance, density, velocity, or magnitude, they produce in us the ideas of different colours; as will be explained in its proper place.

That the particles which constitute light are exceedingly small, appears from hence, viz. that if a hole be made through a piece of paper with a needle, rays of light from every object on the farther side of it are capable of passing through it at once without the least confusion; for any one of those objects may as clearly be seen through it, as if no rays passed through it from any of the rest. Further, if a candle is lighted, and there be no obstacle in the way to obstruct the progres... Part I.

Refraction. Grefs of its rays, it will fill all the space within two miles of it every way with luminous particles, before it has lost the least sensible part of its substance thereby.

That these particles proceed from every point of the surface of a visible body, and in all directions, is clear from hence, viz. because wherever a spectator is placed with regard to the body, every point of that part of the surface which is turned towards him is visible to him. That they proceed from the body in right lines, we are assured, because just so many and no more will be intercepted in their passage to any place by an interposed object, as that object ought to intercept, supposing them to come in such lines.

The velocity with which they proceed from the surface of the visible body is no less surprising than their minutefness: the method whereby philosophers estimate their swiftness, is by observations made on the eclipses of Jupiter's satellites; which eclipses to us appear about seven minutes sooner than they ought to do by calculation, when the earth is placed between the sun and him, that is, when we are nearest to him; and as much later, when the sun is between him and us, at which time we are farthest from him; from whence it is concluded, that they require about seven minutes to pass over a space equal to the distance between the sun and us, which is about 95,000,000 of miles.

A stream of these particles issuing from the surface of a visible body in one and the same direction, is called a ray of light.

As rays proceed from a visible body in all directions, they necessarily become thinner and thinner, continually spreading themselves as they pass along into a larger space, and that in proportion to the squares of their distances from the body; that is, at the distance of two spaces, they are four times thinner than they are at one; at the distance of three spaces, nine times thinner, and so on: the reason of which is, because they spread themselves in a twofold manner, viz. upwards and downwards, as well as sidewise.

The particles of light are subject to the laws of attraction of cohesion, like other small bodies; for if a ray of light be made to pass by the edge of a knife, it will be diverted from its natural course, and be inflected towards the edge of the knife. The like inflection happens to a ray when it enters obliquely into a denser or rarer substance than that in which it was before, in which case it is said to be refracted; the laws of which refraction are the subject of the following section.

Sect. II. Of Refraction.

Light, when proceeding from a luminous body, without being reflected from any opaque substance, or inflected by passing very near one, is invariably found to proceed in straight lines, without the least deviation. But if it happens to pass obliquely from one medium to another, it always leaves the direction it had before, and assumes a new one; and this change of course is called its refraction. After having taken this new direction, it then proceeds invariably in a straight line till it meets with a different medium, when it is again turned out of its course. It must be observed, however, that though by this means we may cause the rays of light make any number of angles in their course, it is impossible for us to make them describe a curve, except in one single case, namely, where they pass through a medium, the density of which uniformly either increases or decreases. This is the case with the light of what the celestial bodies, which passes downwards through our atmosphere, and likewise with that which is reflected upwards through it by terrestrial objects. In both these cases, it describes a curve of the hyperbolic kind; but at all other times it proceeds in straight lines, or in what may be taken for straight lines without any sensible error.

§ 1. The cause of Refraction, and the law by which it is performed.

The phenomena of refraction are explained by an attractive power in the medium through which light passes, in the following manner: All bodies being endowed with an attractive force, which is extended to active some distance beyond their surfaces; when a ray of power in light passes out of a rarer into a denser medium (if this latter has a greater attractive force than the former, as is commonly the case), the ray, just before its entrance, will begin to be attracted towards the denser medium; and this attraction will continue to act upon it, till some time after it has entered the medium; and therefore, if a ray approaches a denser medium in a direction perpendicular to its surface, its velocity will be continually accelerated during its passage through the space in which that attraction exerts itself; and therefore, after it has passed that space, it will move on, till it arrives at the opposite side of the medium, with a greater degree of velocity than it had before it entered. So that in this case its velocity only will be altered. Whereas, if a ray enters a denser medium obliquely, it will not only have its velocity augmented thereby, but its direction will become less oblique to the surface. Just as when a stone is thrown downwards obliquely from a precipice, it falls to the surface of the ground in a direction nearer to a perpendicular one, than that with which it was thrown from the hand. From hence we see a ray of light, in passing out of a rarer into a denser medium, is refracted towards the perpendicular; that is, supposing a line drawn perpendicularly to the surface of the medium, through the point where the ray enters, and extended both ways, the ray in passing through the surface is refracted or bent towards the perpendicular line; or, which is the same thing, the line which it describes by its motion after it has passed through the surface, makes a less angle with the perpendicular, than the line it described before. All which may be illustrated in the following manner.

Let us suppose first, that the ray passes out of a vacuum into the denser medium ABCD (fig. 3.), and that the attractive force of each particle in the medium is extended from its respective centre to a distance equal to that which is between the lines AB and EF, or AB and GH; and let KL be the path described by a ray of light in its progress towards the denser medium. This ray, when it arrives at L, will enter the attractive forces of those particles which lie in AB the surface of the denser medium, and will therefore cease to proceed any longer in the right line KLM, but will be diverted from its course by being attracted towards the the line AB, and will begin to describe the curve LN, passing through the surface AB in some new direction, as OQ; thereby making a less angle with a line, as PR, drawn perpendicularly through the point N, than it would have done had it proceeded in its first direction KLM.

Farther: Whereas, we have supposed the attractive force of each particle to be extended through a space equal to the distance between AB and EF, it is evident that the ray, after it has entered the surface, will still be attracted downwards, till it has arrived at the line EF; for, till that time, there will not be so many particles above it which will attract it upwards, as below, that will attract it downwards. So that after it has entered the surface at N, in the direction OQ, it will not proceed in that direction, but will continue to describe a curve, as NS; after which it will proceed straight on towards the opposite side of the medium, being attracted equally every way; and therefore will at last proceed in the direction XST, still nearer the perpendicular PR than before.

Now if we suppose AEZY not to be a vacuum, but a rarer medium than the other, the case will still be the same; but the ray will not be so much refracted from its rectilineal course, because the attraction of the particles of the upper medium being in a contrary direction to that of the attraction of those in the lower one, the attraction of the denser medium will in some measure be destroyed by that of the rarer.

On the contrary, when a ray passes out of a denser into a rarer medium, if its direction be perpendicular to the surface of the medium, it will only lose somewhat of its velocity, in passing through the spaces of attraction of that medium (that is, the space wherein it is attracted more one way than it is another). If its direction be oblique, it will continually recede from the perpendicular during its passage, and by that means have its obliquity increased, just as a stone thrown up obliquely from the surface of the earth increases its obliquity all the time it rises. Thus, supposing the ray TS passing out of the denser medium ABCD into the rarer ABZY, when it arrives at S it will begin to be attracted downwards, and so will describe the curve SNL, and then proceed in the right line LK; making a larger angle with the perpendicular PR, than the line TSX in which it proceeded during its passage through the other medium.

We may here make a general observation on the forces which produce this deviation of the rays of light from their original path. They arise from the joint action of all the particles of the body which are sufficiently near the particle of light; that is, whose distance from it is not greater than the line AE or GA; and therefore the whole force which acts on a particle in its different situations between the planes GH and EF, follows a very different law from the force exerted by one particle of the medium.

The space through which the attraction of cohesion of the particles of matter is extended is so very small, that in considering the progress of a ray of light out of one medium into another, the curvature it describes in passing through the space of attraction is generally neglected; and its path is supposed to be bent, or, in the usual terms, the ray is supposed to be refracted only in the point where it enters the denser medium.

Now the line which a ray describes before it enters a denser or a rarer medium, is called the incident ray; that which it describes after it has entered, is the refracted ray.

The angle comprehended between the incident ray and the perpendicular, is the angle of incidence; and that between the refracted ray and the perpendicular, is the angle of refraction.

There is a certain and immutable law or rule, by which refraction is always performed; and that is this: Whatever inclination a ray of light has to the surface of any medium before it enters it, the degree of refraction will always be such, that the proportion between the sine of the angle of its incidence, and that of the angle of its refraction, will always be the same in that medium.

To illustrate this: Let us suppose ABCD (fig. 4.) CCCLIV, to represent a rarer, and ABEF a denser medium: let GH be a ray of light passing through the first and entering the second at H, and let HI be the refracted ray: then supposing the perpendicular PR drawn through the point H, on the centre H, and with any radius, describe the circle APBR; and from G and I, where the incident and refracted rays cut the circle, let fall the lines GK and IL perpendicularly upon the line PR; the former of these will be the sine of the angle of incidence, the latter of refraction. Now if in this case the ray GH is so refracted at H, that GK is double or triple, &c. of IL, then whatever other inclination the ray GH might have had, the sine of its angle of incidence would have been double or triple, &c. to that of its angle of refraction. For instance, had the ray passed in the line MH before refraction, it would have passed in some line as HN afterwards, so situated that MO should have been double or triple, &c. of NO.

When a ray passes out of a vacuum into air, the sine of the angle of incidence is found to be to that of refraction as 100036 to 100000.

When it passes out of air into water, as about 4 to 3.

When out of air into glass, as about 17 to 11.

When out of air into a diamond, as about 5 to 2.

This relation of the sine of the angle of incidence to that of refraction, which is a proposition of the most extensive use in explaining the optical phenomena on physical or mechanical principles, may be demonstrated in the following easy and familiar manner.

Lemma I. The augmentations or diminutions of the squares of the velocities produced by the uniform action of accelerating or retarding forces, are proportional to the forces, and to the spaces along which they act, jointly, or are proportional to the products of the forces multiplied by the spaces.

Let two bodies be uniformly accelerated from a state of rest in the points A a, along the spaces AB, ac, fig. 5., by the accelerating forces FF', and let AC, ac, be spaces described in equal times; it is evident, from what has been said under the articles Gravitation and Acceleration, that because these spaces are described with motions uniformly accelerated, AC and ac are respectively the halves of the spaces which would be uniformly described during the same time with the velocities acquired at C and c, and are therefore Cause of these velocities. And as these velocities are uniformly acquired in equal times, they are measures of the accelerating forces. Therefore \( AC : ac = F : f \). Also, from the nature of uniformly accelerated motion, the spaces are proportional to the squares of the acquired velocities. Therefore, using the symbols \( \sqrt{C} \), \( \sqrt{c} \), &c., to express the squares of the velocities at \( C \), &c., we have

\[ \begin{align*} \sqrt{B} : \sqrt{b} &= AB : AC \\ \sqrt{C} : \sqrt{c} &= AC^2 : ac^2 \\ \sqrt{a} : \sqrt{b} &= ac : ab \end{align*} \]

Therefore, by equality of compound ratios

\[ \sqrt{B} : \sqrt{b} = AB \times AC : ab \times ac = AB \times F : ab \times f \]

And in like manner \( \sqrt{D} : \sqrt{d} = AD \times F : ad \times f \); and \( \sqrt{B} : \sqrt{D} = BD \times F : bd \times f \).

Q. E. D.

Corol. If the forces are as the spaces inversely, the augmentations or diminutions of the squares of the velocities are equal.

Remark. If DB, db, be taken extremely small, the products BD \(\times\) F and bd \(\times\) f may be called the momentary actions of the forces, or the momentary increments of the squares of the velocities. It is usually expressed, by the writers on the higher mechanics, by the symbol \( f' \), or \( f'd \), where \( f \) means the accelerating force, and \( s \) or \( d \) means the indefinitely small space along which it is uniformly exerted. And the proposition is expressed by the fluxionary equation \( f' = v \), because \( v \) is half the increment of \( v^2 \), as is well known.

Lemma 2. (being the 39th proposition of the first book of Newton's Principia.) If a particle of matter, moving with any velocity along the line AC, be impelled by an accelerating or retarding force, acting in the same or in the opposite direction, and if the intensity of the force in the different points B, F, H, C, &c., be as the ordinates BD, FG, &c., to the line DGE, the areas BFGD, BHKD, &c., will be as the changes made on the square of the velocity at B, when the particle arrives at the points F, H, &c.

For let BC be divided into innumerable small portions, of which let FH be one, and let the force be supposed to act uniformly, or to be of invariable intensity during the motion along FH; draw GI perpendicular to HK. It is evident that the rectangle FHIG will be as the product of the accelerating force by the space along which it acts, and will therefore express the momentary increment of the square of the velocity. (Lemma 1.) The same may be said of every such rectangle. And if the number of the portions, such as FH, be increased, and their magnitude diminished without end, the rectangles will ultimately occupy the whole curvilinear area, and the force will be continually varying in its intensity. The curvilinear areas will therefore be as the finite changes made on the square of the velocity, and the proposition is demonstrated.

Corol. The whole change made on the square of the velocity, is equal to the square of that velocity which the accelerating force would communicate to the particle by impelling it along BC from a state of rest in B. For the area BCED will still express the square of this velocity, and it equally expresses the change made on the square of any velocity wherewith the particle may pass through the point B, and is independent on the magnitude of that velocity.

Remark. The figure is adapted to the case where the forces all conspire with the initial motion of the particle, or all oppose it, and the area expresses an augmentation or a diminution of the square of the initial velocity. But the reasoning would have been the same, although, in some parts of the line BC, the forces had conspired with the initial motion, and in other parts had opposed it. In such a case, the ordinates which express the intensity of the forces must lie on different sides of the abscissa BC, and that part of the area which lies on one side must be considered as negative with respect to the other, and be subtracted from it. Thus, if the forces are represented by the ordinates of the dotted curve line DH, which crosses the abscissa in H, the figure will correspond to the motion of a particle, which, after moving uniformly along AB, is subjected to the action of a variable accelerating force during its motion along BH, and the square of its initial velocity is increased by the quantity BHD; after which it is retarded during its motion along HC, and the square of its velocity in H is diminished by a quantity HCe. Therefore the square of the initial velocity is changed by a quantity BHD—HCe, or HCe—BHD.

This proposition is perhaps the most important in the whole science of mechanics, being the foundation of every application of mechanical theory to the explanation of natural phenomena. No traces of it are to be found in the writings of philosophers before the publication of Newton's Principia, although it is assumed by John Bernoulli and other detractors from Newton's greatness as an elementary truth, without any acknowledgment of their obligations to its author. It is usually expressed by the equation \( f' = v \) and \( f'f = v^2 \); i.e., the sum of the momentary actions is equal to the whole or finite increment of the square of the velocity.

**PROPOSITION.**

When light passes obliquely into or out of a transparent substance, it is refracted so that the fine angle of incidence is to the fine of the angle of refraction in the constant ratio of the velocity of the refracted light to that of the incident light.

Let ST, KR (fig. 7.), represent two planes (parallel to, and equidistant from, the refracting surface XY) which bound the space in which the light, during its passage, is acted on by the refracting forces, as explained in no 125. The intensity of the refracting forces being supposed equal at equal distances from the bounding planes, though any how different at different distances from them, may be represented by the ordinates Ta, ng, pr, cR, &c., of the curve abnpce, of which the form must be determined from observation, and may remain for ever unknown. The phenomena of reflected light show us that it is attracted by the refracting substance at some distances, and repelled at others.

Let the light, moving uniformly in the direction AB, enter the refracting stratum at B. It will not proceed in that direction, but its path will be incurved upwards, while acted on by a repulsive force, and downwards, while impelled by an attractive force. It will describe some curvilinear path Bdo CDE, which AB touches in B, and will finally emerge from the refracting stratum at E, and move uniformly in a straight line EF, which touches the curve in E. If, through b, the intersection of the curve of forces with its abscissa, we draw bo, cutting the path of the light in o, it is evident that this path will be concave upwards between B and o, and concave downwards between o and E. Also, if the initial velocity of the light has been sufficiently final, its path may be so much bent upwards, that in some point d its direction may be parallel to the bounding planes. In this case it is evident, that being under the influence of a repulsive force, it will be more bent upwards, and it will describe df, equal and similar to db, and emerge in an angle gfs equal to ABG. In this case it is reflected, making the angle of reflection equal to that of incidence. By which it appears how reflection, refraction, and inflection, are produced by the same forces and performed by the same laws.

But let the velocity be supposed sufficiently great to enable the light to penetrate through the refracting stratum, and emerge from it in the direction EF; let AB and EF be supposed to be described in equal times: They will be proportional to the initial and final velocities of the light. Now, because the refracting forces must act in a direction perpendicular to the refracting surface (since they arise from the joint action of all the particles of a homogeneous substance which are within the sphere of mutual action), they cannot affect the motion of the light estimated in the direction of the refracting surface. If, therefore, AG be drawn perpendicular to ST, and FK to KR, the lines GB, EK, must be equal, because they are the motions AB, EF, estimated in the direction of the planes. Draw now EL parallel to AB. It is also equal to it. Therefore EL, EF, are as the initial and final velocities of the light. But EF is to EL as the sine of the angle ELK to the sine of the angle EFK; that is, as the sine of the angle ABH to the sine of the angle FEI; that is, as the sine of the angle of incidence to the sine of the angle of refraction.

By the same reasoning it will appear that light, moving in the direction and with the velocity FE, will describe the path EDB, and will emerge in the direction and with the velocity BA.

Let another ray enter the refracting stratum perpendicularly at B, and emerge at Q. Take two points N, P, in the line BQ, extremely near to each other, so that the refracting forces may be supposed to act uniformly along the space NP; draw NC, PD, parallel to ST, CM perpendicular to DP, and MO perpendicular to CD, which may be taken for a straight line. Then, because the forces at C and N are equal, by supposition they may be represented by the equal lines CM and NP. The force NP is wholly employed in accelerating the light along NP; but the force CM being transverse to the motion BD, is but partly so employed, and may be conceived as arising from the joint action of the forces CO, OM, of which CO only is employed in accelerating the motion of the light, while OM is employed in incursating its path. Now it is evident, from the similarity of the triangles DCM, MCO, that DC : CM = CM : CO, and that DC × CO = CM × CM = NP × NP. But DC × CO and NP × NP are as the products of the spaces by the accelerating forces, and express the momentary increments of the squares of the velocities at C and N. (Lemma 1.) These increments, therefore, are equal. And as this must be said of every portion of the paths BCE and BNQ, it follows that the whole increment of the square of the initial velocity produced in the motion along BCE, is equal to the increment produced in the motion along BNQ. And, because the initial velocities were equal in both paths, their squares were equal. Therefore the squares of the final velocities are also equal in both paths, and the final velocities themselves are equal. The initial and final velocities are therefore in a constant ratio, whatever are the directions; and the ratio of the sines of the angles of incidence and refraction being the ratio of the velocities of the refracted and incident light, by the former case of prop. 1, is also constant.

Remark. The augmentation of the square of the initial velocity is equal to the square of the velocity which a particle of light would have acquired, if impelled from a state of rest at B along the line BQ. (Corol. of the Lemma 2.), and is therefore independent on the initial velocity. As this augmentation is expressed by the curvilinear area aTbnpcR, it depends both on the intensity of the refracting forces, expressed by the ordinates, and on the space through which they act, viz. TR. These circumstances arise from the nature of the transparent substance, and are characteristic of that substance. Therefore, to abbreviate language, we shall call this the specific velocity.

This specific velocity is easily determined for any substance in which the refraction is observed, by drawing Li perpendicular to EL, meeting in i the circle described with the radius EF. For EI being equal to EF, will represent the velocity of the refracted light, and EL represent the velocity of the incident light, and EI² = EL² + Li², and therefore Li² is the augmentation of the square of the initial velocity, and Li is the specific velocity.

It will now be proper to deduce some corollaries from these propositions, tending to explain the chief phenomena of refraction.

1. When light is refracted towards the perpendicular to the refracting surface, it is accelerated; and it of light acts retarded when it is refracted from the perpendicular. In the first case, therefore, it must be considered as refraction, having been acted on by forces conspiring (in part at least) with its motion, and vice versa. Therefore, because we see that it is always refracted towards the perpendicular, when passing from a void into any transparent substance, we must conclude that it is, on the whole, attracted by that substance. We must draw the same conclusion from observing, that it is refracted from the perpendicular in its passage out of any transparent substance whatever into a void. It has been attracted backwards by that substance.

This acceleration of light in refraction is contrary to the opinion of those philosophers who maintain, that illumination is produced by the undulation of an elastic medium. Euler attempts to prove, by mechanical laws, that the velocities of the incident and refracted light are proportional to the sines of incidence and refraction, while our principles make them in this. ratio inversely. Boscovich proposed a fine experiment for deciding this question. The aberration of the fixed stars arises from the combination of the motion of light with the motion of the telescope by which it is observed. Therefore this aberration should be greater or less when observed by means of a telescope filled with water, according as light moves slower or swifter through water than through air. He was mistaken in the manner in which the conclusion should be drawn from the observation made in the form prescribed by him; and the experiment has not yet been made in a convincing manner; because no fluid has been found of sufficient transparency to admit of the necessary magnifying power. It is an experiment of the greatest importance to optical science.

2. If the light be moving within the transparent substance, and if its velocity (estimated in a direction perpendicular to the surface) do not exceed the specific velocity of that substance, it will not emerge from it, but will be reflected backwards in an angle equal to that of its incidence. For it must be observed, that in the figure of last proposition, the excess of the square of EF above the square of EL, is the same with the excess of the square of KF above the square of KL. Therefore the square of the specific velocity is equal to the augmentation or diminution of the square of the perpendicular velocity. If therefore the initial perpendicular velocity FK (fig. 8.) be precisely equal to the specific velocity, the light will just reach the farther side of the attracting stratum, as at B, where its perpendicular velocity will be completely extinguished, and its motion will be in the direction BT. But it is here under the influence of forces tending towards the plane KR, and its motion will therefore be still incurvated towards it; and it will describe a curve BD equal and similar to EB, and finally emerge back from the refracting stratum into the transparent substance in an angle RDA equal to KEF.

If the direction of the light be still more oblique, so that its perpendicular velocity is less than the specific velocity, it will not reach the plane ST, but be reflected as soon as it has penetrated so far that the specific velocity of the part penetrated (estimated by the compounding part of the area of forces) is equal to its perpendicular velocity. Thus the ray fE will describe the path EdDa penetrating to bd, so that the corresponding area of forces abce is equal to the square of fk; its perpendicular velocity.

The extreme brilliancy of dew drops and of jewels had often excited the attention of philosophers, and it always appeared a difficulty how light was reflected at all from the posterior surface of transparent bodies. It afforded Sir Isaac Newton his strongest argument against the usual theory of reflection, viz. that it was produced by impact on solid elastic matter. He was the first who took notice of the total reflection in great obliquities; and very properly asked how it can be said that there is any impact in this case, or that the reflecting impact should cease at a particular obliquity?

It must be acknowledged that it is a very curious circumstance, that a body which is perfectly transparent should cease to be so at a certain obliquity; that a great obliquity should not hinder light from passing from a void into a piece of glass; but that the same obliquity should prevent it from passing from the glass Cause of into a void. The finest experiment for illustrating the Refraction fact is, to take two pieces of mirror-glass, not silvered, and put them together with a piece of paper between them, forming a narrow margin all round to keep them apart. Plunge this apparatus into water. When it is held nearly parallel to the surface of the water, every thing at the bottom of the vessel will be seen clearly through the glasses; but when they are turned so as to be inclined about 30 degrees, they will intercept the light as much as if they were plates of iron. It will be proper to soak the paper in varnish, to prevent water from getting between the glasses.

What is called the brilliant cut in diamonds, is such a disposition of the posterior facets of the diamond, that the light is made to fall upon them so obliquely that none of it can go through, but all is reflected. To produce this effect in the greatest possible degree, it is a matter of calculation, and merits the attention of the lapidary. When diamonds are too thin to admit of this form, they are cut in what is called the rose-fashion. This has a plain back, and the facets are all on the front, and so disposed as to reflect the rays into sufficient obliquities, to be strongly reflected from the posterior plane. Doublets are made by cutting one thin diamond rose-fashion, and another similar one is put behind it, with their plane surfaces joined. Or, more frequently, the outside diamond has the anterior facets of the brilliant, and the inner has the form of the inner part of a brilliant. If they be joined with very pure and strongly refracting varnish, little light is reflected from the separating plane, and their brilliancy is very considerable, though still inferior to a true and deep brilliant. If no varnish be used, much of the light is reflected from the flat side, and the effect of the posterior facets is much diminished. But doublets might be constructed, by making the touching surfaces of a spherical form (of which the curvature should have a due proportion to the size of the stone), that would produce an effect nearly equal to that of the most perfect brilliant.

3. Since the change made on the square of the velocity of the incident light is a constant quantity, it follows, that the refraction will diminish as the velocity of the incident light increases. For if L in fig. 7. be a constant quantity, and EL be increased, it is evident that the ratio of E, or its equal EF, to EL will be diminished, and the angle LEF, which constitutes the refraction, will be diminished. The physical cause of this is easily seen: When the velocity of the incident light is increased, it employs less time in passing through the refracting stratum or space between the planes SI and KR, and is therefore less influenced by the refracting forces. A similar effect would follow if the transparent body were moving with great velocity towards the luminous body.

Some naturalists have accounted for the different refrangibility of the differently coloured rays, by supposing that the red rays move with the greatest rapidity, and they have determined the difference of original velocity which would produce the observed difference of refraction. But this difference would be observed in the eclipses of Jupiter's satellites. They should be ruddy at their emersions, and be some seconds before they attain their pure whiteness; and they should become bluish immediately before they vanish in immersions. This is not observed. Besides, the difference in refrangibility is much greater in flint-glass than in crown-glass, and this would require a proportionally greater difference in the original velocities. This explanation therefore must be given up.

It should follow, that the refraction of a star which is in our meridian at fix o'clock in the evening should be greater than that of a star which comes on the meridian at six in the morning; because we are moving away from the first, and approaching to the last. But the difference is but \( \frac{1}{35} \) of the whole, and cannot be observed with sufficient accuracy in any way yet practised. A form of observation has been proposed by Dr Blair professor of practical astronomy in the university of Edinburgh, which promises a very sensible difference of refraction. It is also to be expected, that a difference will be observed in the refraction of the light from the east and western ends of Saturn's ring. Its diameter is about 26 times that of the earth, and it revolves in 10h. 32'; so that the velocity of its edge is about \( \frac{1}{3000} \) of the velocity of the sun's light. If therefore the light be reflected from it according to the laws of perfect elasticity, or in the manner here explained, that which comes to us from the western extremity will move more slowly than that which comes from the eastern extremity in the proportion of 2500 to 2501. And if Saturn can be seen distinctly after a refraction of 30° through a prism, the diameter of the ring will be increased one half in one position of the telescope, and will be as much diminished by turning the telescope half round its axis; and an intermediate position will exhibit the ring of a distorted shape. This experiment is one of the most interesting to optical science, as its result will be a severe touchstone of the theories which have been attempted for explaining the phenomena on mechanical principles.

If the tail of a comet be impelled by the rays of the sun, as is with great probability supposed by Euler and others, the light by which its extreme parts are seen by us must have its velocity greatly diminished, being reflected by particles which are moving away from the sun with immense rapidity. This may perhaps be discovered by its greater aberration and refrangibility.

As common day-light is nothing but the sun's light reflected from terrestrial bodies, it is reasonable to expect that it will suffer the same refraction. But nothing but observation could assure us that this would be the case with the light of the stars; and it is rather surprising that the velocity of their light is the same with that of the sun's light. It is a circumstance of connection between the solar system and the rest of the universe. It was as little to be looked for on the light of terrestrial luminaries. If light be conceived as small particles of matter emitted from bodies by the action of accelerating forces of any kind, the vast diversity which we observe in the constitution of sublunar bodies should make us expect differences in this particular. Yet it is found, that the light of a candle, of a glowworm, &c. suffers the same refraction, and consists of the same colours. This circumstance is adduced as an argument against the theory of emission. It is thought more probable that this sameness of velocity is owing to the nature of the medium, which determines the frequency of its undulations and the velocity of their propagation.

4. When two transparent bodies are contiguous, the law of refraction in its passage out of the one into the other will fraction be refracted towards or from the perpendicular, according as the refracting forces of the second are greater or less than those of the first, or rather according as the area expressing the square of the specific velocity is dy into a greater or less. And as the difference of these areas together is a determined quantity, the difference between the rigorous velocity in the medium of incidence and the velocity in the medium of refraction, will also be a determined quantity. Therefore the sine of the angle of incidence will be in a constant ratio to the line of the angle of refraction; and this ratio will be compounded of the ratio of the sine of incidence in the first medium to the sine of refraction in a void; and the ratio of the sine of incidence in a void to the sine of refraction in the second medium. If therefore a ray of light, moving through a void in any direction, shall pass through any number of media bounded by parallel planes, its direction in the last medium will be the same as if it had come into it from a void.

5. It also follows from these propositions, that if the obliquity of incidence on the posterior surface of a transparent body be such, that the light should be reflected back again, the placing a mat of the same or of another medium in contact with this surface, will cause it to be transmitted, and this the more completely, as the added medium is more dense or more refractive; and the reflection from the separating surface will be the more vivid in proportion as the posterior substance is less dense or of a smaller refractive power. It is not even necessary that the other body be in contact; it is enough if it be so near that those parts of the refracting strata which are beyond the bodies interfere with or coincide with each other.

All these consequences are agreeable to experience. The brilliant reflection from a dew-drop ceases when it touches the leaf on which it rests: The brilliancy of a diamond is greatly damaged by moisture getting behind it: The opacity of the combined mirror plates, mentioned in the second corollary, is removed by letting water get between them: A piece of glass is distinctly or clearly seen in air, more faintly when immersed in water, still more faintly amidst oil of olives, and it is hardly perceived in spirits of turpentine. These phenomena are incompatible with the notion that reflection is occasioned by impact on solid matter, whether of the transparent body, or of any ether or other fancied fluid behind it; and their perfect coincidence with the legitimate consequences of the assumed principles, is a strong argument in favour of the truth of those principles.

It is worth while to mention here a fact taken notice of by Mr Beguelin, and proposed as a great difficulty in the Newtonian theory of refraction. In order to get the greatest possible refraction, and the retraction simplest measure of the refracting power at the anterior surface of any transparent substance, Sir Isaac Newton enjoins us to employ a ray of light falling on the surface quam obliqui. But Mr Beguelin found, that when the obliquity of incidence in glass was about no light was refracted, but that it was wholly reflected. He also observed, that when he gradually increased the obliquity of incidence on the posterior surface of the glass, the light which emerged last of all did not skim along the surface, making an angle of 90° with the perpendicular, as it should do by the Newtonian theory, but made an angle of more than ten minutes with the posterior surface. Also, when he began with very great obliquities, so that all the light was reflected back into the glass, and gradually diminished the obliquity of incidence, the first ray of light which emerged did not skim along the surface, but was raised about 10 or 15 minutes.

But all these phenomena are necessary consequences of our principles, combined with what observation teaches us concerning the forces which bodies exert on the rays of light. It is evident, from the experiments of Grimaldi and Newton, that light is both attracted and repelled by solid bodies. Newton's famous analysis of these experiments discovered several alternations of actual inflection and deflection; and he gives us the precise distance from the body when some of these attractions end and repulsion commences; and the most remote action to be observed in his experiments is repulsion. Let us suppose this to be the case, although it be not absolutely necessary. Let us suppose that the forces are represented by the ordinates of a curve \(a n p c\) (see fig. 7.) which crosses the abscissa in \(b\). Draw \(b o\) parallel to the refracting surface. When the obliquity of incidence of the ray \(A B\) has become so great, that its path in the glass, or in the refracting stratum, does not cut, but only touches the line \(o b\), it can penetrate no further, but is totally reflected; and this must happen in all greater obliquities. On the other hand, when the ray \(L E\), moving within the glass, has but a very small perpendicular velocity, it will penetrate the refracting stratum no further than till this perpendicular velocity is extinguished, and its path becomes parallel to the surface, and it will be reflected back. As the perpendicular velocity increases by diminishing the obliquity of incidence, it will penetrate farther; and the last reflection will happen when it penetrates so far that its path touches the line \(o b\). Now diminish the obliquity by a single second; the light will get over the line \(o b\), will describe an arch \(o d b\) concave upwards, and will emerge in a direction \(B A\), which does not skim the surface, but is sensibly raised above it. And thus the facts observed by M. Beguelin, instead of being an objection against this theory, afford an argument in its favour.

7. Those philosophers who maintain the theory of undulation, are under the necessity of connecting the dispersive powers of bodies with their mean refractive powers. Mr Euler has attempted to deduce a necessary difference in the velocity of the rays of different colours from the different frequency of the undulations, which he assigns as the cause of their different colorific powers. His reasoning on this subject is of the most delicate nature, and unintelligible to such as are not completely master of the infinitesimal calculus of partial differences, and is unsatisfactory to such as are able to go through its intricacies. It is contradicted by fact. He says, and indeed to be consistent he must say it, that musical sounds which differ greatly in acuteness are propagated through the air with different velocities: but one of the smallest bells in the chimes of St Giles's church in Edinburgh was struck against the rim of the very deep-toned bell on which the hours are struck. When the sound was listened to by a nice observer at the distance of more than two miles, no interval whatever could be observed. A similar experiment was exhibited to Mr Euler himself, by means of a curious musical instrument (if it can be so called) used at St Petersburg, and which may be heard at three or four miles distance. But the experiment with the bells is unexceptionable, as the two sounds were produced in the very same instant. This connection between the refrangibility in general and the velocity must be admitted, in its full extent, in every attempt to explain refraction by undulation; and Euler was forced by it to adopt a certain consequence which made a necessary connection between the mean refraction and the dispersion of heterogeneous rays. Confident of his analysis, he gave a deaf ear to all that was told him of Mr Dollond's improvements on telescopes, and asserted, that they could not be such as were related; for an increase of mean refraction must always be accompanied with a determined increase of dispersion. Newton had said the same thing, being misled by a limited view of his own principles; but the dispersion assigned by him was different from that assigned by Euler. The dispute between Euler and Dollond was confined to the decision of this question only; and when some glances made by a German chemist at St Petersburg convinced Euler that his determination was erroneous, he had not the candour to give up the principle which had forced him to this determination of the dispersion, but immediately introduced a new theory of the achromatic telescopes of Dollond; a theory which took the artists out of the track marked out by mathematicians, and in which they had made considerable advances, and led them into another path, proposing maxims of construction hitherto untried, and inconsistent with real improvements which they had already made. The leading principle in this theory is to arrange the different ultimate images of a point which arise either from the errors of a spherical figure or different refrangibility, in a straight line passing through the centre of the eye. The theory itself is specious; and it requires great mathematical skill to accomplish this point, and hardly lets to decide on the propriety of the construction which it recommends. It is therefore but little known. But that it is a false theory, is evident from one simple consideration. In the most indistinct vision arising from the worst construction, this rectilineal arrangement of the images obtains completely in that pencil which is situated in the axis, and yet the vision is indistinct. But, what is to our present purpose, this new theory is purely mathematical, suiting any observed dispersive power, and has no connection with the physical theory of undulations, or indeed with any mechanical principles whatever. But, by admitting any dispersive power, whatever may be the mean refraction, all the physical doctrines in his Nova Theoria Lucis et Colorum are overlooked, and therefore never once mentioned, although the effects of Mr Zeiller's glass are taken notice of as inconsistent with that mechanic... proposition of Newton's which occasioned the whole dispute between Euler and Dollond.

They are indeed inconsistent with the universality of that proposition. Newton advances it in his optics merely as a mathematical proposition highly probable, but says that it will be corrected if he shall find it false. The ground on which he seems (for he does not expressly say so) to rely its probability is a limited view of his own principle, the action of bodies on light. He (not knowing any cause to the contrary) supposed that the action of all bodies was similar on the different kinds of light, that is, that the specific velocities of the differently coloured rays had a determined proportion to each other. This was gratuitous; and it might have been doubted by him who had observed the analogy between the chemical actions of bodies by elective attractions and repulsions, and the similar actions on light. Not only have different menstrua unequal actions on their solids, but the order of their affinities is also different. In like manner, we might expect not only that some bodies would attract light in general more than others, but also might differ in the proportion of their actions on the different kinds of light, and this so much, that some might even attract the red more than the violet. The late discoveries in chemistry show us some very distinct proofs, that light is not exempted from the laws of chemical action, and that it is susceptible of chemical combination. The changes produced by the sun's light on vegetable colours, shows the necessity of illumination to produce the green fecula; and the aromatic oils of plants, the irritability of their leaves by the action of light, the curious effects of it on the mineral acids, on manganese, and the calces of bismuth and lead, and the imbibition and subsequent emission of it by phosphorescent bodies, are strong proofs of its chemical affinities, and are quite inexplicable on the theory of undulations.

All these considerations taken together, had they been known to Sir Isaac Newton, would have made him expect differences quite anomalous in the dispersive powers of different transparent bodies; at the same time that they would have afforded to his sagacious mind the strongest arguments for the actual emission of light from the luminous body.

Having in this manner established the observed law of refraction on mechanical principles, showing it to be a necessary consequence of the known action of bodies on light, we proceed to trace its mathematical consequences through the various cases in which it may be exhibited to our observation. These constitute that part of the mathematical branch of optical science which is called dioptries.

We are quite unacquainted with the law of action of bodies on light, that is, with the variation of the intensity of the attractions and repulsions exerted at different distances. All that we can say is, that from the experiments and observations of Grimaldi, Newton, and others, light is deflected towards a body or is attracted by it, at some distances, and repelled at others, and this with a variable intensity. The action may be extremely different, both in extent and force, in different bodies, and change by a very different law with the same change of distance. But, amidst all this variety, there is a certain similarity arising from the joint action of many particles, which should be noticed, because it tends both to explain the similarity observed in the refractions of light, and also its connection with the phenomena of reflection.

The law of variation in the joint action of many particles adjoining to the surface of a refracting medium, is extremely different from that of a single particle; but when this last is known, the other may be found out. We shall illustrate this matter by a very simple case. Let DE (fig. 9.) be the surface of a medium, and let us suppose that the action of a particle of the medium on a particle of light extends to the distance EA, and that it is proportional to the ordinates ED, EF, EG, EH, &c. of the line A B C D; that is, that the action of the particle E of the medium on a particle of light in F, is to its action on a particle in H as EF to HB, and that it is attracted at F but repelled at H, as expressed by the situation of the ordinates with respect to the abscissa. In the line AE produced to B, make EB, EX, EY, EZ, &c. respectively equal to EA, EH, EC, EG, EF, &c.

It is evident that a particle of the medium at B will exert no action on the particle of light in E, and that the particles of the medium in X Y Z E, will exert on it actions proportional to HB, GZ, FZ, &c. Therefore, supposing the matter of the medium continuous, the whole action exerted by the row of particles EB will be represented by the area ABCD; and the action of the particles between B and E will be represented by the area ABCF, and that of the particles between E and B by the area EFDE.

Now let the particle of light be in F, and take FZ = AE. It is no less evident that the particle of light in F will be acted on by the particles in E alone, and that it will be acted on in the same manner as a particle in E is acted on by the particles in B. Therefore the action of the whole row of particles EB on a particle in F will be represented by the area ABCF. And thus the action on a particle of light in any point of AE will be represented by the area which lies beyond it.

But let us suppose the particles of light to be within the medium, as at s, and make sd = AE. It is again evident that it is acted on by the particles of the medium between s and d with a force represented by the area A CDE, and in the opposite direction by the particles in E with a force represented by the area FDE. This balances an equal quantity of action, and there remains an action expressed by the area ABCF. Therefore, if an equal and similar line to ABCDE be described on the abscissa EB, the action of the medium on a particle of light in e will be represented by the area yzB, lying beyond it.

If we now draw a line AKLMNBP whose ordinates CK, FQ, R, &c. are as the areas of the other curve, estimated from A and B; these ordinates will represent the whole forces which are exerted by the particles in EB, on a particle of light moving from A to B. This curve will cut the axis in points L, N such, that the ordinates drawn through them intercept areas of the first curve, which are equal on each side of the axis; and in these points the particle of light suffers no action from the medium. These points are very Cause of very different from the similar points of the curve expressing the action of a single particle. These last are in the very places where the light sustains the greatest repulsive action of the whole row of particles. In the same manner may a curve be constructed, whose ordinates express the united action of the whole medium.

From these observations we learn in general, that a particle of light within the space of action is acted on with equal forces, and in the same direction, when at equal distances on each side of the surface of the medium.

Of the focal distance of rays refracted by passing out of one medium into another of different density and through a plane surface.

Lemma. The indefinitely small variation of the angle of incidence is to the simultaneous variation of the angle of refraction as the tangent of incidence is to the tangent of refraction; or, the contemporaneous variations of the angles of incidence and refraction are proportional to the tangents of these angles.

Let \( RVF_r \), \( rVf \) (fig. 10.) be the progress of the rays refracted at \( V \) (the angle \( rVR \) being considered in its nascent or evanescent state), and \( VC \) perpendicular to the refracting surface \( VA \). From \( C \) draw \( CD \), \( CB \) perpendicular to the incident and refracted rays \( RV \), \( VF \), cutting \( rV \), \( Vf \) in \( \beta \) and \( \beta_2 \) and let \( Cd \), \( Cb \) be perpendicular to \( rV \), \( Vf \).

Because the fines of incidence and refraction are in a constant ratio, their simultaneous variations are in the same constant ratio. Now the angle \( rVR \) is to the angle \( FVf \) in the ratio of \( \frac{B\beta}{DV} \) to \( \frac{D\beta}{BV} \); that is, of \( \frac{BC}{BV} \) to \( \frac{DC}{DV} \); that is, of \( \frac{\text{fin. incid.}}{\text{cot. incid.}} \) to \( \frac{\text{fin. refr.}}{\text{cot. refr.}} \); that is, of \( \tan. \text{incid.} \) to \( \tan. \text{refr.} \).

Corollary. The difference of these variations is to the greatest or least of them as the difference of the tangents to the greatest or least tangent.

PROBLEM.

Let two rays \( RV \), \( RP \) diverge from, or converge to, a point \( R \) (figs. 1, 2, 3, 4.), and pass through the plane surface \( PV \) separating two refracting mediums \( AB \), of which let \( B \) be the most refracting, and let \( RV \) be perpendicular to the surface. It is required to determine the point of dispersion or convergence, \( F \), of the refracted rays \( VD \), \( PE \).

Make \( VR \) to \( VG \) as the fine of refraction to the fine of incidence, and draw \( GIK \) parallel to the surface, cutting the incident ray in \( I \). About the centre \( P \), with the radius \( PI \), describe an arch of a circle \( IF \), cutting \( VR \) in \( F \); draw \( PL \) tending from or towards \( F \). We say \( PE \) is the refracted ray, and \( F \) the point of dispersion or convergence of the rays \( RV \), \( RP \), or the conjugate focus to \( R \).

For since \( GI \) and \( PV \) are parallel and \( PF \) equal to \( PI \), we have \( PF : PR = PI : PR = VG : VR = \sin. \text{incid.} : \sin. \text{refr.} \). But \( PF : PR = \sin. \text{PRV} : \sin. \text{Refraction} \), \( PFV \), and \( RRV \) is equal to the angle of incidence at \( P \); therefore \( PFV \) is the corresponding angle of refraction, \( FPE \) is the refracted ray, and \( F \) the conjugate focus to \( R \).

Corol. 1. If diverging or converging rays fall on the surface of a more refracting medium, they will diverge or converge less after refraction, \( F \) being farther from the surface than \( R \). The contrary must happen when the diverging or converging rays fall on the surface of a less refracting medium, because, in this case, \( F \) is nearer to the surface than \( R \).

Corol. 2. Let \( Rp \) be another ray, more oblique than \( RP \), the refracting point \( p \) being farther from \( V \), and let \( fpe \) be the refracted ray, determined by the same construction. Because the arches \( FI \), \( fi \), are perpendicular to their radii, it is evident that they will converge to some point within the angle \( RIK \), and therefore will not cross each other between \( F \) and \( I \); therefore \( RF \) will be greater than \( RF \), as \( RF \) is greater than \( RG \), for similar reasons. Hence it follows, that all the rays which tended from or towards \( R \), and were incident on the whole of \( VPp \), will not diverge from or converge to \( F \), but will be diffused over the line \( GEf \). This diffusion is called aberration from the focus, and is so much greater as the rays are more oblique. No rays flowing from or towards \( R \) will have point of concourse with \( RV \) nearer to \( R \) than \( F \) is: But if the obliquity be inconsiderable, so that the ratio of \( RP \) to \( FP \) does not differ sensibly from that of \( RV \) to \( FV \), the point of concourse will not be sensibly removed from \( G \). \( G \) is therefore usually called the conjugate focus to \( R \). It is the conjugate focus of an indefinitely slender pencil of rays falling perpendicularly on the surface. The conjugate focus of an oblique pencil, or even of two oblique rays, whose dispersion on the surface is considerable, is of more difficult investigation. See Gravesham's Natural Philosophy for a very neat and elementary determination (§).

In a work of this kind, it is enough to have pointed out, in an easy and familiar manner, the nature of optical aberration. But as this is the chief cause of the imperfection of optical instruments, and as the only method of removing this imperfection is to diminish this aberration, or correct it by a subsequent aberration in the opposite direction, we shall here give a fundamental and very simple proposition, which will (with obvious alterations) apply to all important cases. This is the determination of the focus of an infinitely slender pencil of oblique rays \( RP \), \( Rp \).

"Retaining the former construction for the ray \( PF \) (fig. 1.) suppose the other ray \( Rp \) infinitely near to \( RP \). Draw \( PS \) perpendicular to \( PV \), and \( Rx \) perpendicular to \( RP \), and make \( Pr : PS = VR : VF \). On \( Pr \) describe the semicircle \( rRP \), and on \( PS \) the semicircle \( SxP \), cutting the refracted ray \( PF \) in \( s \), draw \( pr, ps, pf \)." It follows

(§) We refer to Gravesham, because we consider it as of importance to make such a work as ours serve as a general index to science and literature. At the same time we take the liberty to observe, that the focus in question is virtually determined by the construction which we have given: for the points \( P \), \( F \) of the line \( PF \) are determined, and therefore its position is also determined. The same is true of the position of \( G \), and therefore the intersection \( p \) of the two lines is likewise determined. Refraction follows from the lemma, that if \( p \) be the focus of refracted rays, the variation \( Pp \) of the angle of refraction is to the corresponding variation \( PR \) of the angle of incidence as the tangent of the angle of refraction \( VFP \) to the tangent of the angle of incidence \( VRP \). Now \( Pp \) may be considered as coinciding with the arch of the semicircles. Therefore the angles \( PRP, PRP \) are equal, as also the angles \( Pp, PS \). But \( PS \) is to \( PR \) as \( PR \) to \( PS \); that is, as \( VR \) to \( VF \); that is, as the cotangent of the angle of incidence to the cotangent of the angle of refraction; that is, as the tangent of the angle of refraction to the tangent of the angle of incidence. Therefore the point \( p \) is the focus.

**Of Refraction by Spherical Surfaces.**

**General Problem.**

To find the focus of refracted rays, the focus of incident rays being given.

Let \( PV \) (figs. 5, 6, 7, 8, 9, 10, 11, 12, 13, 14.) be a spherical surface whose centre is \( C \), and let the incident light diverge from or converge to \( R \).

**Solution.** Draw the ray \( RC \) through the centre, cutting the surface in the point \( V \), which we shall designate the vertex, while \( RC \) is called the axis. This ray passes on without refraction, because it coincides with the perpendicular to the surface. Let \( RP \) be another incident ray, which is refracted at \( P \), draw the radius \( PC \). In \( RP \) make \( RE \) to \( RP \) as the fine of incidence \( m \) to the fine of refraction \( n \); and about the centre \( R \), with the distance \( RE \), describe the circle \( EK \), cutting \( PC \) in \( K \); draw \( RK \) and \( PF \) parallel to it, cutting the axis in \( F \). \( PF \) is the refracted ray, and \( F \) is the focus.

For the triangles \( PCF, KCR \) are similar, and the angles at \( P \) and \( K \) are equal. Also \( RK \) is equal to \( RE \), and \( RPD \) is the angle of incidence. Now \( m : n = RK : RP = \sin DPR : \sin DPK = \sin CPF \). Therefore \( CPF \) is the angle of refraction corresponding to the angle of incidence \( RPD \), and \( PF \) is the refracted ray, and \( F \) the focus. Q.E.D.

**Cor. 1.** \( CK : CP = CR : CF \), and \( CF = \frac{CP \times CR}{CK} \).

Now \( CP \times CR \) is a constant quantity; and therefore \( CF \) is reciprocally as \( CK \), which evidently varies with a variation of the arch \( VP \). Hence it follows, that all the rays flowing from \( R \) are not collected at the conjugate focus \( F \). The ultimate situation of the point \( F \), as the point \( P \) gradually approaches to, and at last coincides with, \( V \), is called the conjugate focus of central rays, and the distance between this focus and the focus of a lateral ray is called the aberration of that ray, arising from the spherical figure.

There are, however, two situations of the point \( R \) such, that all the rays which flow from it are made to diverge from one point. One of those is \( C \) (fig. 5.), because they all pass thro' without refraction, and therefore still diverge from \( C \); the other is when rays in the rare medium with a convex surface flow from a point \( R \), so situated beyond the centre that \( CV \) is to \( CR \) as the fine of incidence in the rare medium is to the fine of refraction in the denser, or when rays in the rare medium fall on the convex surface of the denser, converging to \( F \), so situated that \( CF : CV = m : n \). In this case they will all be dispersed from refraction \( F \), so situated that \( CV : CF = n : m = CR : CV \) by spherical surfaces for fine \( RKC = n : m = CR : CP \), fine \( RPC : fine PRC \). Therefore the angle \( PRC \) is equal to \( RKC \), or to \( FPC \) (by construction of the problem), and the angle \( C \) is common to the triangles \( PRC, FPC \); they are therefore similar, and the angles \( PRC, FPC \) are equal, and \( n : m = CP : CF = CK : CR = CR : CP \); therefore \( CP : CK = CP^2 : CR^2 \); but \( CP \) and \( CR \) are constant quantities, and therefore \( CK \) is a constant quantity, and (by the corollary) \( CF \) is a constant quantity, and all the rays flowing from \( R \) are dispersed from \( F \) by refraction. In like manner rays converging to \( F \) will by refraction converge to \( R \). This was first observed by Huygens.

2. If the incident ray \( R'P \) (fig. 5.) is parallel to the axis \( RC \), we have \( PO \) to \( CO \) as the fine of incidence to the fine of refraction. For the triangles \( R'PK', PCO \) are similar, and \( PO : CO = R'K' : R'P' = m : n \).

3. In this case, too, we have the focal distance of central parallel rays reckoned from the vertex \( = \frac{m}{m-n} \times VC \). For since \( PO \) is ultimately \( VO \), we have \( m : n = VO : CO \), and \( m-n : m = VO-CO : VO = VC : VO \), and \( VO = \frac{m}{m-n} \times VC \). This is called the principal focal distance, or focal distance of parallel rays. Also \( CO \), the principal focal distance reckoned from the centre, \( = \frac{n}{m-n} \times VC \).

**N.B.** When \( m \) is less than \( n \), \( m-n \) is a negative quantity.—Also observe, that in applying symbols to this computation of the focal distances, those lines are to be accounted positive which lie from their beginnings, that is, from the vertex, or the centre, or the radiant point, in the direction of the incident rays. Thus when rays diverge from \( R \) on the convex surface of a medium, \( VR \) is accounted negative and \( VC \) positive. If the light passes out of air into glass, \( m \) is greater than \( n \); but if it passes out of glass into air, \( m \) is less than \( n \). If, therefore, parallel rays fall on the convex surface of glass out of air, in which case \( m : n = 3 : 2 \) very nearly, we have for the principal focal distance \( \frac{3}{3-2} VC, \) or \( +\frac{3}{2} VC \). But if it passes out of glass into the convex surface of air, we have \( VO = \frac{2}{2-3} VC, \) or \( -\frac{2}{3} VC \); that is, the focus \( O \) will be in the same side of the surface with the incident light. In like manner, we shall have for these two cases \( CO = +\frac{2}{3} VC \) and \( -\frac{2}{3} VC \).

4. By construction we have \( RK : RP = m : n \) by similarity of triangles \( PF : RK = CF : CR \), therefore \( mPR \times CF = nCR \times PF \) and \( mPR : nCR = PF : CF \) and \( mPR - nCR : mPR = PF - CF : PF \) ultimately \( mVR - nCR : mVR = VC : VF \).

This is a very general optical theorem, and affords an easy method for computing the focal distance of refracted rays.

For this purpose let \( VR \), the distance of the radiant point, In its application due attention must be paid to the qualities of \( r \) and \( a \), whether they be positive or negative, according to the conditions of last corollary.

5. If \( Q \) (fig. 8.) be the focus of parallel rays coming from the opposite side, we shall have \( RQ : QC = RV : VF \). For draw \( Cq \) parallel to \( PF \), cutting \( RP \) in \( q \); then \( Rq : qC = RP : PF \). Now \( q \) is the focus of the parallel rays \( FP \), \( Cq \). And when the point \( P \) ultimately coincides with the point \( V \), \( q \) must coincide with \( Q \), and we have \( RQ : QC = RV : VF \).

This is the most general optical theorem, and is equally applicable to lenses, or even to a combination of them, as to simple surfaces. It is also applicable to reflections, with this difference, that \( Q \) is to be assumed the focus of parallel rays coming the same way with the incident rays. It affords us the most compendious methods of computing symbolically and arithmetically the focal distances in all cases.

6. We have also \( Rq : RP = RV : RF \), and ultimately for central rays \( RQ : RV = RV : RF \), and \( RF = RV - r \).

7. Also \( Rq : RC = RP : RF \), and ultimately \( RQ : RV = RC : RF \), and \( RF = \frac{RV \times RC}{RQ} \).

\( N.B. \) These four points \( Q, V, C, F \), either lie all one way from \( R \), or two of them forward and two backward.

8. Also, making \( O \) the principal focus of rays coming the same way, we have \( Rq : qC = CO : OF \), and ultimately \( RQ : QC = CO : OF \), and \( OF = \frac{QC \times CO}{RQ} \), and therefore reciprocally proportional to \( RQ \), because \( QC \times CO \) is a constant quantity.

These corollaries or theorems give us a variety of methods for finding the focus of refracted rays, or the other points related to them; and each formula contains four points, of which any three being given the fourth may be found. Perhaps the last is the most simple, as the quantity \( QC \times CO \) is always negative, because \( O \) and \( Q \) are on different sides.

9. From this construction we may also derive a very easy and expeditious method of drawing many refracted rays. Draw through the centre \( C \) (fig. 15, 16.) a line to the point of incidence \( P \), and a line \( CA \) parallel to the incident ray \( RP \). Take \( VO \) to \( VC \) as the fine of incidence to the fine of refraction, and about \( A \), with the radius \( VO \), describe an arch of a circle cutting \( PC \) produced in \( B \). Join \( AB \); and \( PF \) parallel to \( AB \) is the refracted ray. When the incident light is parallel to \( RC \), the point \( A \) coincides with \( V \), and a circle described round \( V \) with the distance \( VO \) will cut the lines \( PC, pC, \&c. \) in the points \( Bb \). The demonstration is evident.

Having thus determined the focal distance of refracted rays, it will be proper to point out a little more particularly its relation to its conjugate focus of incident rays. We shall consider the four cases of light incident on the convex or concave surface of a Refraction by Spherical Surfaces or a rarer medium.

1. Let light moving in air fall on the convex surface of glass (fig. 5. to fig. 14.). Let us suppose it tending to a point beyond the glass infinitely distant. It will be collected to its principal focus \( O \) beyond the vertex \( V \). Now let the incident light converge a little, so that \( R \) is at a great distance beyond the surface. The focus of refracted rays \( F \) will be a little within \( O \) or nearer to \( V \). As the incident rays are made to converge more and more, the point \( R \) comes nearer to \( V \), and the point \( F \) also approaches it, but with a much slower motion, being always situated between \( O \) and \( C \) till it is overtaken by \( R \) at the centre \( C \), when the incident light is perpendicular to the surface in every point, and therefore suffers no refraction. As \( R \) has overtaken \( F \) at \( C \), it now passes it, and is again overtaken by it at \( V \). Now the point \( R \) is on the side from which the light comes, that is, the rays diverge from \( R \). After refraction they will diverge from \( F \) a little without \( R \); and as \( R \) recedes farther from \( V \), \( F \) recedes still farther, and with an accelerated motion, till, when \( R \) comes to \( Q \), \( F \) has gone to an infinite distance, or the refracted rays are parallel. When \( R \) still recedes, \( F \) now appears on the other side, or beyond \( V \); and as \( R \) recedes back to an infinite distance, \( F \) has come to \( O \); and this completes the series of variations, the motion of \( F \) during the whole changes of situation being in the same direction with the motion of \( R \).

2. Let the light moving in air fall on the concave surface of glass; and let us begin with parallel incident rays, conceiving, as before, \( R \) to lie beyond the glass at an infinite distance. The refracted rays will move as if they came from the principal focus \( O \), lying on that side of the glass from which the light comes. As the incident rays are made gradually more converging, and the point of convergence \( R \) comes toward the glass, the conjugate focus \( F \) moves backward from \( O \); the refracted rays growing less and less diverging, till the point \( R \) comes to \( Q \), the principal focus on the other side. The refracted rays are now parallel, or \( F \) has retreated to an infinite distance. The incident light converging still more, or \( R \) coming between \( Q \) and \( V \), \( F \) will appear on the other side, or beyond the surface, or within the glass, and will approach it with a retarded motion, and finally overtake \( R \) at the surface of the glass. Let \( R \) continue its motion backwards (for it has all the while been moving backwards, or in a direction contrary to that of the light); that is, let \( R \) now be a radiant point, moving backwards from the surface of the glass. \( F \) will at first be without it, but will be overtaken by it at the centre \( C \), when the rays will suffer no refraction. \( R \) still receding, will get without \( F \); and while \( R \) recedes to an infinite distance, \( F \) will recede to \( O \), and the series will be completed.

3. Let the light moving in glass fall on the convex surface of air; that is, let it come out of the concave surface of glass, and let the incident rays be parallel, or tending to \( R \), infinitely distant: they will be dispersed by refraction from the principal focus \( O \) within the glass. As they are made more converging, \( R \) comes nearer, and \( F \) retreats backward, till \( R \) comes to Q, the principal focus without the glass; when F is now at an infinite distance within the glass, and the refracted rays are parallel. R still coming nearer, F now appears before the glass, overtakes R at the centre C, and is again overtaken by it at V. R now becoming a radiant point within the glass, F follows it backwards, and arrives at O, when R has receded to an infinite distance, and the series is completed.

4. Let the incident light, moving in glass, fall on the concave surface of air, or come out of the convex surface of glass. Let it tend to a point R at an infinite distance without the glass. The refracted rays will converge to O, the principal focus without the glass. As the incident light is made more converging, R comes towards the glass, while F, setting out from v, also approaches the glass, and R overtakes it at the surface V. R now becomes a radiant point within the glass, receding backwards from the surface. F recedes slower at first, but overtakes R at the centre C, and passes it with an accelerated motion to an infinite distance; while R retreats to Q, the principal focus within the glass. R still retreating, F appears before the glass; and while R retreats to an infinite distance, F comes to V, and the series is completed.

§ 2. Of Glasses.

Glass for optical purposes may be ground into nine different shapes. Glasses cut into five of those shapes are called lenses, which together with their axes are described in vol. 6, page 33. (See Dioptrics). The other four are,

1. A plane-glass, which is flat on both sides, and of equal thickness in all its parts, as EF, fig. 1.

2. A flat plano-convex, whose convex side is ground into several little flat surfaces, as A, fig. 2.

3. A prism, which has three flat sides, and when viewed endwise appears like an equilateral triangle, as B.

4. A concavo-convex glass, as C, which has hitherto received no name, and is seldom, if ever, made use of in optical instruments.

A ray of light Gb (fig. 1.) falling perpendicularly on a plane glass EF, will pass through the glass in the same direction hi, and go out of it into the air in the same straight line ih.

A ray of light AB falling obliquely on a plane glass, will go out of the glass in the same direction, but not in the same straight line: for in touching the glass, it will be refracted in the line BC; and in leaving the glass, it will be refracted in the line CD.

Lemma. There is a certain point E within every double convex or double concave lens, through which every ray that passes will have its incident and emergent parts QA, ag parallel to each other: but in a plano-convex or plano-concave lens, that point E is removed to the vertex of the concave or convex surface; and in a meniscus, and in that other concavo-convex lens, it is removed a little way out of them, and lies next to the surface which has the greatest curvature.

For let REr be the axis of the lens joining the centres R, r of its surfaces A, a. Draw any two of their semidiameters RA, ra parallel to each other, and join the points A, a, and the line Aa will cut the axis in the point E above described. For the triangles REA, rEa being equiangular, RE will be to Er in the given ratio of the semidiameters RA, ra; and consequently the point E is invariable in the same lens. Now supposing a ray to pass both ways along the line Aa, it being equally inclined to the perpendiculars to the surfaces, will be equally bent, and contrariwise in going out of the lens; so that its emergent parts AQ, ag will be parallel. Now any of these lenses will become plano convex or plano concave, by conceiving one of the semidiameters RA, ra to become infinite, and consequently to become parallel to the axis of the lens, and then the other semidiameter will coincide with the axis; and so the points A, E or a, E will coincide. Q. E. D.

Corol. Hence when a pencil of rays falls almost perpendicularly upon any lens, whose thickness is inconsiderable, the course of the ray which passes through E, above described, may be taken for a straight line passing through the centre of the lens, without sensible error in sensible things. For it is manifest from the length of Aa and from the quantity of the refractions at its extremities, that the perpendicular distance of AQ, ag when produced, will be diminished both as the thickness of the lens and the obliquity of the ray is diminished.

PROPOSITION I.

To find the focus of parallel rays falling almost perpendicularly upon any given lens.

Let E be the centre of the lens, R and r the centres of its surfaces, Rr its axis, gE:G a line parallel to the incident rays upon the surface B, whose centre is R. Parallel to gE draw a semicircle BR, in which produce let V be the focus of the rays after their first refraction at the surface B, and joining VR, let it perpendicular cut gE produced in G, and G will be the focus of the rays that emerge from the lens.

For since V is also the focus of the rays incident upon the second surface A, the emergent rays must have their focus in some point of that ray which passes straight through this surface; that is, in the line VR, drawn through its centre r: and since the whole course of another ray is reckoned a straight line gEG *, its Corol. interference G with VR determines the focus of them all. Q. E. D.

Corol. 1. When the incident rays are parallel to the axis rR, the focal distance EF is equal to EG. For let the incident rays that were parallel to gE be gradually more inclined to the axis till they become parallel to it; and their first and second focuses V and G will describe circular arches VT and GF whose centres are R and E. For the line RV is invariable; being in proportion to RB in a given ratio of the lesser of the lines of incidence and refraction to their difference *; consequently the line EG is also invariable, being in proportion to the given line RV in the given ratio of rE to rR, because the triangles EGR, RVr are equiangular.

Corol. 2. The last proportion gives the following rule for finding the focal distance of any thin lens. As RR, the interval between the centres of the surfaces, is to rE, the semidiameter of the second surface, so is RV or RT, the continuation of the first semidiameter to the first focus, to EG or EF, the focal distance of the lens; which, according as the lens is thicker or thinner in the middle than at its edges, must lie on the Of Glasses, the same side as the emergent rays, or on the opposite side.

Corol. 3. Hence when rays fall parallel on both sides of any lens, the focal distances \(EF, Ef\) are equal. For let \(rt\) be the continuation of the semidiameter \(Er\) to the first focus \(t\) of rays falling parallel upon the surface \(A\); and the same rule that gave \(rR\) to \(rE\) as \(RT\) to \(EF\), gives also \(rR\) to \(RE\) as \(rt\) to \(Ef\). Whence \(Ef\) and \(EF\) are equal, because the rectangles under \(RE\), \(RT\) and also under \(RE\), \(rt\) are equal. For \(rE\) is to \(rt\) and also \(RE\) to \(RT\) in the same given ratio.

Corol. 4. Hence in particular in a double-convex or double-concave lens made of glass, it is as the sum of their semidiameters (or in a meniscus as their difference) to either of them, so is double the other, to the focal distance of the glass. For the continuations \(RT, rt\) are severally double their semidiameters; because in glass \(ET\) is to \(TR\) and also \(Et\) to \(tr\) as 3 to 2.

Corol. 5. Hence if the semidiameters of the surfaces of the glasses be equal, its focal distance is equal to one of them; and is equal to the focal distance of a plano-convex or plano-concave glass whose semidiameter is as short again. For considering the plane surface as having an infinite semidiameter, the first ratio of the last mentioned proportion may be reckoned a ratio of equality.

**PROPOSITION II.**

The focus of incident rays upon a single surface, of emergent sphere, or lens being given, it is required to find the focus of the emergent rays.

Let any point \(Q\) be the focus of incident rays upon a spherical surface, lens, or sphere, whose centre is \(E\); and let other rays come parallel to the line \(QEg\) the contrary way to the given rays, and after refraction let them belong to a focus \(F\); then taking \(Ef\) equal to \(EF\) in the lens or sphere, but equal to \(FC\) in the single surface, say as \(QF\) to \(FE\) to \(Ef\) to \(fQ\); and placing \(fQ\) the contrary way from \(f\) to that of \(FQ\) from \(F\), the point \(q\) will be the focus of the refracted rays, without sensible error; provided the point \(Q\) be not too remote from the axis, nor the surfaces too broad, so as to cause any of the rays to fall too obliquely upon them.

For with the centre \(E\) and semidiameters \(EF\) and \(Ef\) describe two arches \(FG, fg\) cutting any ray, \(QAag\) in \(G\) and \(g\), and draw \(EG\) and \(Eg\). Then supposing \(G\) to be a focus of incident rays (as \(GA\)), the emergent rays (as \(agq\)) will be parallel to \(GE\); and on the other hand supposing \(g\) another focus of incident rays (as \(ga\)), the emergent rays (as \(AGQ\)) will be parallel to \(gE\). Therefore the triangles \(QGE, Egq\) are equiangular, and consequently \(QG\) is to \(GE\) as \(Eg\) to \(gg\); that is, when the ray \(QAag\) is the nearest to \(QEG\), \(QF\) is to \(FE\) as \(Ef\) to \(fg\). Now when \(Q\) accedes to \(F\) and coincides with it, the emergent rays become parallel, that is \(g\) recedes to an infinite distance; and consequently when \(Q\) passes to the other side of \(F\), the focus \(q\) will also pass through an infinite space from one side of \(f\) to the other side of it.

Q. E. D.

Corol. 1. In a sphere or lens the focus \(q\) may be found by this rule: As \(QF\) to \(QE\) so \(QE\) to \(Qq\), to be placed the same way from \(Q\) as \(QF\) lies from \(Q\).

For let the incident and emergent ray \(QA, qa\) be produced till they meet in \(e\); and the triangles \(QGE, Qeg\) being equiangular, we have \(QG\) to \(QE\) as \(Qe\) to \(Qq\); and when the angles of these triangles are vanishing, the point \(e\) will coincide with \(E\); because in the sphere the triangle \(Aea\) is equiangular at the base \(Aa\), and consequently \(Ae\) and \(ae\) will at last become semidiameters of the sphere. In a lens the thickness \(Aa\) is inconsiderable.

The focus may also be found by this rule:

\[QF : FE :: QE : Eg,\] for \(QG : GE :: QA : Ag.\)

And then the rule formerly demonstrated for single surfaces holds good for the lenses.

Corol. 2. In all cases the distance \(fQ\) varies reciprocally as \(FQ\) does; and they lie contrarywise from \(f\) and \(F\); because the rectangle or the square under \(EF\) and \(Ef\), the middle terms in the foregoing proportions, is invariable.

The principal focal distance of a lens may not only be found by collecting the rays coming from the sun, considered as parallel, but also (by means of this proposition) it may be found by the light of a candle or window. For, because \(Qg : qA :: QE : EG\), we have (when \(A\) coincides with \(E\)) \(Qg : qE = QE : EF\); that is, the distance observed between the radiant object and its picture in the focus is to the distance of the lens from the focus as the distance of the lens from the radiant is to its principal focal distance. Multiply therefore the distances of the lens from the radiant and focus, and divide the product by their sum.

Corol. 3. Convex lenses of different shapes that have equal focal distances, when put into each other's places, have equal powers upon any pencil of rays to refract them to the same focus. Because the rules above-mentioned depend only upon the focal distance of the lens, and not upon the proportion of the semidiameters of its surfaces.

Corol. 4. The rule that was given for a sphere of an uniform density, will serve also for finding the focus of a pencil of rays refracted through any number of concentric surfaces, which intercede uniform mediums of any different densities. For when rays come parallel to any line drawn through the common centre of these mediums, and are refracted through them all, the distance of their focus from that centre is invariable, as in an uniform sphere.

Corol. 5. When the focuses \(Q, q\) lie on the same side of the refracting surfaces, if the incident rays flow from \(Q\), the refracted rays will also flow from \(q\); and if the incident rays flow towards \(Q\), the refracted will also flow towards \(q\); and the contrary will happen when \(Q\) and \(q\) are on contrary sides of the refracting surfaces. Because the rays are continually going forwards.

From this proposition we also derive an easy method of drawing the progress of rays through any number of lenses ranged on a common axis.

Let \(A, B, C\) (fig. 7.) be the lenses, and \(RA\) a ray incident on the first of them. Let \(a, b, c\) be their foci for parallel rays coming in the opposite direction; draw the perpendicular \(ad\), cutting the incident ray in \(d\), and draw \(da\) through the centre of the lens: \(AB\) parallel to \(da\) will be the ray refracted by the first lens. Through the focus of the second lens draw the perpendicular \(ec\), cutting \(AB\) in... § 3. Of Vision.

Having described how the rays of light, flowing from objects, and passing through convex glasses, are collected into points, and form the images of the objects; it will be easy to understand how the rays are affected by passing through the humours of the eye, and are thereby collected into innumerable points on the bottom of the eye, and thereon form the images of the objects which they flow from. For, the different humours of the eye, and particularly the crystalline humour, are to be considered as a convex glass; and the rays in passing through them to be affected in the same manner as in passing through a convex glass. A description of the coats and humours, &c. has been given at large in Anatomy; but for the reader's convenience in this place, we shall repeat in a few words as much of the description as will be sufficient for our present purpose.

The eye is nearly globular. It consists of three coats and three humours. The part DHHG of the outer coat, is called the sclerotic; the rest, DEFG, the cornea. Next within this coat is that called the choroides, which serves as it were for a lining to the eye, other, and joins with the iris, mn, mn. The iris is composed of two sets of muscular fibres; the one of a circular form, which contracts the hole in the middle called the pupil, when the light would otherwise be too strong for the eye; and the other of radial fibres, tending everywhere from the circumference of the iris towards the middle of the pupil; which fibres, by their contraction, dilate and enlarge the pupil when the light is weak, in order to let in the more of its rays. The third coat is only a fine expansion of the optic nerve L, which spreads like net-work all over the inside of the choroides, and is therefore called the retina; upon which are painted (as it were) the images of all visible objects, by the rays of light which either flow or are reflected from them.

Under the cornea is a fine transparent fluid like water, which is therefore called the aqueous humour. It gives a protuberant figure to the cornea, fills the two cavities mm and nn, which communicate by the pupil P; and has the same limpidity, specific gravity, and refractive power, as water. At the back of this lies the crystalline humour II, which is shaped like a double convex glass; and is a little more convex on the back than the fore-part. It converges the rays, which pass through it from every visible object to its focus at the bottom of the eye. This humour is transparent like crystal, is much of the consistence of hard jelly, and exceeds the specific gravity of water in the proportion of 11 to 10. It is inclosed in a fine transparent membrane, from which proceed radial fibres oo, called the ligamentum ciliare, all around its edge; and join to the circumference of the iris.

At the back of the crystalline, lies the vitreous humour KK, which is transparent like glass, and is largest of all in quantity, filling the whole orb of the eye, and giving it a globular shape. It is much of the consistence with the white of an egg, and very little exceeds the specific gravity and refractive power of water.

As every point of an object ABC, (ibid.) sends the objects out rays in all directions, some rays, from every point on the retina next the eye, will fall upon the cornea between E and F; and by passing on through the humours and pupil of the eye, they will be converged to as many points on the retina or bottom of the eye, and will thereon form a distinct inverted picture cba of the object. Thus, the pencil of rays qrs that flows from the point A of the object, will be converged to the point a on the retina; those from the point B will be converged to the point b; those from the point C will be converged to the point c; and so of all the intermediate points; by which means the whole image abc is formed, and the object made visible. Although it must be owned, that the method by which this sensation is carried from the eye by the optic nerve to the common sensory in the brain, and there discerned, is above the reach of our comprehension.

But that vision is effected in this manner, may be demonstrated experimentally. Take a bullock's eye whilst it is fresh; and having cut off the three coats from the back-part, quite to the vitreous humour, put a piece of white paper over that part, and hold the eye towards any bright object, and you will see an inverted picture of the object upon the paper.

Since the image is inverted, many have wondered why the object appears upright. But we are to consider, 1. That inverted is only a relative term; and, right. 2. That there is a very great difference between the real object and the means or image by which we perceive it. When all the parts of a distant prospect are painted upon the retina, they are all right with respect to one another, as well as the parts of the prospect itself; and we can only judge of an object's being inverted, when it is turned reverse to its natural position with respect to other objects which we see and compare it with.—If we lay hold of an upright stick in the dark, we can tell which is the upper or lower part of it, by moving our hand downward or upward; and know very well that we cannot feel the upper end by moving our hand downward. Just so we find by experience, that upon directing our eyes towards a tall object, we cannot see its top by turning our eyes downward, nor its foot by turning our eyes upward; but must trace the object the same way by the eye to see it from head to foot, as we do by the hand to feel it; and as the judgment is informed by the motion of the hand in one case, so it is also by the motion of the eye in the other.

In fig. g, is exhibited the manner of seeing the same object ABC, by both the eyes D and E at once.

When any part of the image cba falls upon the optic nerve L, the corresponding part of the object becomes invisible. On which account, nature has wisely placed the optic nerve of each eye, not in the middle of the bottom of the eye, but towards the side near the nose; so that whatever part of the image falls before the optic nerve of one eye, may not fall upon the optic nerve of the other. Thus the point a of the image of light. The nearer that any object is to the eye, the larger is the angle under which it is seen, and the magnitude under which it appears. Thus to the eye D, the object ABC is seen under the angle APC; and its image cba is very large upon the retina: but to the eye E, at a double distance, the same object is seen under the angle Apc, which is equal only to half the angle APC, as is evident by the figure. The image cba is likewise twice as large in the eye D, as the other image cba is in the eye E.

In both these representations, a part of the image falls on the optic nerve, and the object in the corresponding part is invisible.

As the sense of seeing is allowed to be occasioned by the impulse of the rays from the visible object upon the retina of the eye, and forming the image of the object thereon, and that the retina is only the expansion of the optic nerve all over the choroides; it should seem surprising, that the part of the image which falls on the optic nerve should render the like part of the object invisible; especially as that nerve is allowed to be the instrument by which the impulse and image are conveyed to the common sensory in the brain.

That the part of the image which falls upon the middle of the optic nerve is lost, and consequently the corresponding part of the object is rendered invisible, is plain by experiment. For if a person fixes three patches, A, B, C, (fig. 2.) upon a white wall, at the height of the eye, and at the distance of about a foot from each other, and places himself before them, shutting the right eye, and directing the left towards the patch C, he will see the patches A and C, but the middle patch B will disappear. Or, if he shuts his left eye, and directs the right towards A, he will see both A and C, but B will disappear; and if he directs his eye towards B, he will see both B and A, but not C. For whatever patch is directly opposite to the optic nerve N, vanishes. This requires a little practice; after which he will find it easy to direct his eye so as to lose the sight of whichever patch he pleases.

This experiment, first tried by M. Marriotte, occasioned a new hypothesis concerning the seat of vision, which he supposed not to be in the retina, but in the choroides. An improvement was afterwards made upon it by M. Picard, who contrived that an object should disappear when both the eyes were kept open. He fastened upon a wall a round white paper, an inch or two in diameter; and by the side of it he fixed two marks, one on the right hand, and the other on the left, each at about 2 feet distance from the paper, and somewhat higher. He then placed himself directly before the paper, at the distance of 9 or 10 feet, and putting the end of his finger over against both his eyes, so that the left-hand mark might be hid from the right eye, and the right-hand mark from the left eye. Remaining firm in this posture, and looking steadily, with both eyes, on the end of his finger, the paper which was not at all covered by it would totally disappear. This, he says, is the more surprising, because, without this particular encounter of the optic nerves, where no vision is made, the paper will appear double, as is the case when the finger is not rightly placed.

M. Marriotte observes, that this improvement on his experiment, by M. Picard, is ingenious, but difficult to execute, since the eyes must be considerably strained in looking at any object so near to them as four inches; and proposes another not less surprising, and more easy. Place, says he, on a dark ground, two round pieces of white paper, at the same height, and three feet from one another; then place yourself opposite to them, at the distance of 12 or 13 feet, and hold your thumb before your eyes, at the distance of about eight inches, so that it may conceal from the right eye the paper that is to the left hand, and from the left eye the paper to the right hand. Then, if you look at your thumb steadily with both eyes, you will lose sight of both the papers; the eyes being so disposed, that each of them receives the image of one of the papers upon the base of the optic nerve, while the other is intercepted by the thumb.

M. Le Cat pursued this curious experiment a little farther than M. Marriotte had done. In the place of the second paper, he fixed a large white board, and observed, that at a proper distance he lost sight of a circular space in the centre of it. He also observed the size of the paper which is thus concealed from the sight, corresponding to several distances, which enabled him to ascertain several circumstances relating to this part of the structure of the eye more exactly than had been done before.

The manner in which this curious experiment is now generally made, and which is both the easiest with respect to the eye, and the most indisputable with respect to the fact, is the following. Let three pieces of paper be fastened upon the side of a room, about two feet asunder; and let a person place himself opposite to the middle paper, and, beginning near to it, retire gradually backwards, all the while keeping one of his eyes shut, and the other turned obliquely towards that outside paper which is towards the covered eye, and he will find a situation (which is generally at about five times the distance at which the papers are placed from one another), where the middle paper will entirely disappear, while the two outermost continue plainly visible; because the rays which come from the middle paper will fall upon the retina where the optic nerve is inserted.

It will not surprise any person, even those who are the strongest advocates for the retina being the place at which the pencils of rays are terminated, and consequently the proper seat of vision, that M. Marriotte was led by this remarkable observation to suspect the contrary. He not only did so; but, in consequence of attentively considering the subject, a variety of other arguments in favour of the choroides occurred to him, particularly his observation, that the retina is transparent, as well as the crystalline and other humours of the eye, which he thought could only enable it to transmit the rays farther; and he could not persuade himself that any substance could be considered as being the termination of the pencils, and the proper seat of vision, at which the rays are not stopped in their progress.

He was farther confirmed in his opinion of the small degree Of Vision. degree of sensibility in the retina, and of the greater sensibility of the choroides, by observing that the pupil dilates itself in the shade, and contracts itself in a great light; which involuntary motion, he thought, was a clear proof that the fibres of the iris are extremely sensible to the action of light; and this part of the eye is only a continuation of the choroides. He also thought that the dark colour of the choroides was intended to make it more susceptible of the impression of light.

M. Pecquet, in answer to M. Marriotte's observation concerning the transparency of the retina, says, that it is very imperfectly so, resembling only oiled paper, or the horn that is used for lanterns; and besides, that its whiteness demonstrates it to be sufficiently opaque for stopping the rays of light, as much as is necessary for the purpose of vision; whereas, if vision be performed by means of those rays which are transmitted through such a substance as the retina, it must be very indistinct.

As to the blackness of the choroides, which M. Marriotte thought to be necessary for the purpose of vision, M. Pecquet observes, that it is not the same in all eyes; and that there are very different shades of it among the individuals of mankind, as also among birds, and some other animals, whose choroides is generally black; and that in the eyes of lions, camels, bears, oxen, flags, sheep, dogs, cats, and many other animals, that part of the choroides which is the most exposed to light, very often exhibits colours as vivid as those of mother-of-pearl, or of the iris (r). He admits that there is a defect of vision at the insertion of the optic nerve; but he thought that it was owing to the blood-vessels of the retina, the trunks of which are so large in that place as to obstruct all vision.

To M. Pecquet's objection, founded on the opacity of the retina, M. Marriotte observes, that there must be a great difference between the state of that substance in living and dead subjects; and as a farther proof of the transparency of the retina, and the power of the choroides beyond it to reflect light, he says, that if a lighted candle be held near to a person's eyes, and a drop, at the distance of eight or ten steps, be made to look at him, he would see a bright light in the dog's eyes, which he thought to proceed from the reflection of the light of the candle from the choroides of the dog, since the same appearance cannot be produced in the eyes of men, or other animals, whose choroides is black.

To M. Pecquet's remark concerning the blood vessels of the retina, M. Marriotte observes, that they are not large enough to prevent vision in every part of the base of the nerve, since the diameter of each of the two vessels occupy no more than 1/8th part of it. Besides, if this were the cause of this want of vision, it would vanish gradually; and the space to which it is confined would not be so exactly terminated as it appears to be.

We must add, that M. Pecquet also observed, that notwithstanding the insensibility of the retina at the insertion of the optic nerve when the light is only moderate; yet that luminous objects, such as a bright candle placed at the distance of four or five paces, do not absolutely disappear, in the same circumstances in which a white paper would; for that this strong light may be perceived though the picture fall on the base of the nerve. "I cannot help suspecting, however," (says Dr Priestley), "that M. Pecquet did not make this observation with sufficient care. A large candle makes no impression on that part of my eye, though it is by no means able to bear a strong light."

The common opinion was also favoured by the anatomical description of several animals by the members of the French academy, and particularly their account of the sea-calf and porcupine; in both of which the optic nerve is inserted in the very axis of the eye, exactly opposite to the pupil, which was thought to leave no room to doubt, but that in these animals the retina is perfectly sensible to the impression of light at the insertion of the nerve. But this observation may deserve to be reconsidered.

M. De la Hire took part with M. Pecquet, arguing in favour of the retina from the analogy of the senses, in all of which the nerves are the proper seat of sensation. This philosopher, however, supposed that the choroides receives the impressions of images, in order to transmit them to the retina.

M. Perrault also took the part of M. Pecquet against M. Marriotte, and in M. Perrault's works we have several letters that passed between these two gentlemen upon this subject.

This dispute about the immediate instrument of vision was revived upon the occasion of an odd experiment of M. Mery, recorded in the memoirs of the French academy for 1704. He plunged a cat in water, and exposing her eye to the strong light of the sun, observed that the pupil was not at all contracted by it; from which he concluded, that the contraction of the iris is not produced by the action of the light, but by some other circumstance. For he contended that the eye receives more light in this situation than in the open air. At the same time he thought he observed that the retina of the cat's eye was transparent, and that he could see the opaque choroides beyond it; from which he concludes, that the choroides is the substance intended to receive the rays of light, and to be the chief instrument of vision. But M. De la Hire replies to this argument of M. Mery, in a memoir for the year 1709, p. 119; in which he endeavours to show that fewer rays enter the eye under water, and that in those circumstances it is not so liable to be affected by them. Besides, it is obvious to be remarked, that the cat must be in great terror in this situation; and being an animal that has a very great voluntary power over the muscles of the iris, and being now extremely attentive to every thing about her, she might keep her eye open notwithstanding.

(r) M. Muschenbroeck says, that in many animals, as the lion, camel, bear, ox, flag, sheep, dog, cat, and many birds, the choroides is not black, but blue, green, yellow, or some other colour. Introductio, Vol. II. p. 748. standing the action of the light upon it, and though it might be very painful to her. We are informed, that when a cat is placed in a window through which the sun is shining, and consequently her iris nearly closed, if she hear a rustling, like that which is made by a mouse, on the outside of the window, she will immediately open her eyes to their greatest extent, without in the least turning her face from the light.

M. Le Cat took part with M. Marriotte in this controversy, it being peculiarly agreeable to his general hypothesis, viz. that the pia mater, of which the choroides is a production, and not the nerves themselves, is the proper instrument of sensation. He thought that the change which takes place in the eyes of old people (the choroides growing less black with age) favoured his hypothesis, as they do not see with that distinctness with which young persons do. M. Le Cat supposed that the retina answers a purpose similar to that of the scarf-skin, covering the papillae pyramidales, which are the immediate organ of feeling, or that of the porous membrane which covers the glandulous papillae of the tongue. The retina, he says, receives the impression of light, moderates it, and prepares it for its proper organ, but is not itself sensible of it.

It must be observed, that M. Le Cat had discovered that the pia mater, after closely embracing and confining the optic nerve at its entrance into the eye, divides into two branches, one of which closely lines the cornea, and at length is lost in it, while the second branch makes what is called the choroides, or uva. He also showed that the sclerotica is an expansion of the dura mater; and he sent dissections of the eye to the royal Academy of Sciences in 1739, to prove these assertions, and several others which he had advanced in his Traité de Sens, that were contrary to the opinions of the celebrated Winslow.

To these arguments in favour of the choroides, alleged by those gentlemen among whom the subject was first discussed, Dr Priestley in his history adds the following that had escaped their notice, but which were suggested to him by his friend Mr Michell.

In order that vision be distinct, the pencils of rays which issue from the several points of any object, must be collected either accurately, or at least very nearly, to corresponding points in the eye, which can only be done upon some uniform surface. But the retina being of a considerable thickness, and the whole of it being uniformly nervous, and at least nearly, if not perfectly, transparent, presents no particular surface; so that, in whatever part of it the pencils are supposed to have their foci, the rays belonging to them will be separated from one another, either before or after they arrive there, and consequently vision would be confused.

If we suppose the seat of vision to be at the nearer surface of the retina, and the images of objects to be formed by direct rays, a considerable degree of confusion could not but arise from the light reflected by the choroides, in those animals in which it is white, or coloured. On the other hand, it would be impossible that vision should be performed at this place by light reflected from the choroides, because in many animals it is perfectly black, and reflects no light at all; and yet such animals see even more distinctly than others. And we cannot but suppose that, in whatever manner vi-

sion is effected, it is the same in the eyes of all animals.

If the seat of vision be at the farther surface of the retina, and it be performed by direct rays, a white choroides could be of no use; and if it were by reflected rays, a black one could not answer the purpose.

It is likewise an argument in favour of the choroides being the organ of vision, that it is a substance which receives a more distinct impression from the rays of light than any other membrane in any part of the animal system, excepting (and perhaps not excepting) that white cuticle which lies under the scales of fishes; whereas the retina is a substance on which the light makes an exceedingly faint impression, and perhaps no impression at all; since light, in passing out of one transparent medium into another immediately contiguous to it, suffers no refraction or reflection, nor are any of the rays absorbed, unless there is some difference in the refracting power of the two media, which probably is not the case between the retina and the vitreous humour, which is in contact with it. And wherever the light is not affected by the medium it falls upon, we can hardly suppose the medium to receive any impression from the light, the action being probably always mutual and reciprocal.

Besides, the retina is so situated, as to be exposed to many rays besides those which terminate in it, and which, therefore, cannot be subservient to vision, if it be performed there. Now this is not the case with the choroides, which is in no shape transparent, and has no reflecting substance beyond it.

It is, moreover, peculiarly favourable to the hypothesis of the seat of vision being in the choroides, that we can then see a sufficient reason for the diversity of its colour in different animals, according as they are circumstanced with respect to vision. In all terrestrial animals, which have occasion to make use of their eyes by night, the choroides is either of a bright white, or of some very vivid colour, which reflects the light very strongly. On this account vision may be performed with less light, but it cannot be with great distinctness, the reflection of the rays doubling their effect, since it must extend over some space, all reflection being made at a distance from the reflecting body. Besides, the choroides in brutes is not in general perfectly white, but a little inclined to blue; and is therefore, probably, better adapted to see by the fainter coloured light, which chiefly prevails in the night; and we would add, is on the same account more liable to be strongly impressed by the colours to which they are chiefly exposed.

On the other hand, the choroides of birds in general, especially eagles, hawks, and other birds of prey, is black; by which means they are able to see with the greatest distinctness, but only in bright day-light. The owl, however, seeking her food by night, has the choroides white, like that of a cat. Lastly, in the eyes of man, which are adapted to various uses, the choroides is neither so black as that of birds, nor so white as that of those animals who make the greatest use of their eyes in the night.

As to a third hypothesis, which is in effect that of M. De la Hire, which makes both the retina and the choroides equally necessary to vision, and supposes it to be performed by the impression of light on the choroides. Of vision, communicated to the retina; Mr Michell observes, that the perceptions can hardly be supposed to be so acute, when the nerves, which are the chief instruments of sensation, do not receive the impressions immediately, but only after they have been communicated to another substance. Besides, it must be more natural to suppose, that, when the principal impression is made upon the choroides, it is communicated to the brain by its own proper nerves, which are abundantly sufficient for the purpose.

The dimensions and precise form of the spot in the eye in which there is no vision, were more accurately calculated by Daniel Bernoulli, in the following manner. He placed a piece of money O (fig. 3.) upon the floor; and then shutting one of his eyes, and making a pendulum to swing, so that the extremity of it might be nearly in the line AO, he observed at what place C it began to be invisible, and where it again emerged into view at A. Raising the pendulum higher and lower, he found other points, as H, N, P, G, B, at which it began to be invisible; and others, as M, L, E, A, at which it began to be visible again; and drawing a curve through them, he found that it was elliptical; and, with respect to his own eye, the dimensions of it were as follows; OC was 23, AC 10, BD 3, DH 13, and EG 14; so that the centre being at F, the greater axis was to the less as 8 to 7.

From these data, the plane on which the figure was drawn being obliquely situated with respect to the eye, he found, that the place in the eye that corresponded to it was a circle, the diameter of which was a seventh part of the diameter of the eye, the centre of it being 27 parts of the diameter from the point opposite to the pupil, a little above the middle. He concludes with observing, that, in order that this space in which there is no vision may be as small as possible, it is necessary that the nerve should enter the eye perpendicularly, and that both this end, and also its entering the eye at a distance from its axis, are gained by the particular manner in which the two optic nerves unite and become separate again, by crossing one another.

In favour of one of the observations of Mr Michell, concerning the use of the choroides in vision, Dr Priestley observes, that Aquapendente mentions the case of a person at Pifa, who could see very well in the night, but very little or none at all in the daytime. This is also said to be the case with those white people among the blacks of Africa, and the inhabitants of the isthmus of America, who, from this circumstance, are called moon-eyed. Our author thinks it probable that their choroides is not of a dark colour, as it is in others of the human species; but white or light-coloured, as in those animals which have most occasion for their eyes in the night. See ALBINOS.

The following considerations in favour of the retina being the proper seat of vision, are worthy of remark.

Dr Portersfield observes, that the reason why there is no vision at the entrance of the optic nerve into the eye, may be, that it wants that softness and delicacy which it has when it is expanded upon the choroides; and that, in those animals in which that nerve is inserted in the axis of the eye, it is observed to be equally delicate, and therefore probably equally susceptible, in that place as in any other part of the retina.

N° 248.

In general, the nerves, when constricted by their coats, have but little sensibility in comparison of what they are endowed with when they are divested of them, and unfolded in a soft and pulpy substance.

Haller observes, that the choroides cannot be the universal instrument of vision, because that sometimes in men and birds, but especially in fishes, it is covered internally with a black mucus, through which the rays cannot penetrate. This writer speaks of a fibrous membrane in the retina distinct from its pulpy substance. On these fibres, he conjectures, that the image of objects are painted.

M. De la Hire's argument in favour of the retina, from the analogy of the senses, is much strengthened by considering that the retina is a large nervous apparatus, immediately exposed to the impression of light; whereas the choroides receives but a slender supply of nerves, in common with the sclerotica, the conjunctiva, and the eyelids, and that its nerves are much less exposed to the light than the naked fibres of the optic nerve. Indeed, from anatomical considerations, one might imagine that any other part of the body was as sensible of the impression of light as the choroides.

That the optic nerve is of principal use in vision, is farther probable from several phenomena attending some of the diseases in which the sight is affected. When an amaurosis has affected one eye only, the optic nerve of that eye has been found manifestly altered from its sound state. Dr Prickley was present when Mr Hey examined the brain of a young girl, who had been blind of one eye, and saw that the optic nerve belonging to it was considerably smaller than the other; and he informed him, that upon cutting into it, he found it to be much harder, and cineritious. Morgagni, indeed, mentions two cases, in one of which he found the optic nerves smaller than usual, and of a cineritious colour, when, upon inquiry, he was informed that the person had not been blind, though there might have been some defect in the sight of one of the eyes. In the other case, only one of the optic nerves was affected in that manner, and the eye itself was in other respects very perfect. Here, also, he was expressly told, that the person was not blind of that eye: but it appears that he himself had not been acquainted with the persons whom he dissected; and there have been many cases of persons being blind of one eye, without knowing it themselves, for a considerable time.

Moreover, as the optic nerve is solely spent in forming the retina, so no function of the eye, not immediately subservient to vision, is affected by an amaurosis. On the contrary, those nerves which go to the choroides are found to retain, in this disease, their natural influence. The iris will contract in a recent gutta serena of one eye, if the other remains sound, and is suddenly exposed to a strong light. The sclerotica, conjunctiva, and eyelids, which receive their nerves from the same branches as the choroides, retain their sensibility in this disorder.

The manner in which persons recover from an amaurosis, favours the supposition of the seat of vision being in the retina; since those parts which are the most distant from the insertion of the nerve recover their sensibility the soonest, being in those places the most most pulpy and softer; whereas there is no reason to think that there is any difference in this respect in the different parts of the choroides. Mr Hey has been repeatedly informed, by persons labouring under an imperfect amaurosis, or gutta serena, that they could not, when looking at any object with one eye, see it distinctly when it was placed directly opposite to the pupil, as when it was situated somewhat obliquely. And those persons whom he had known to recover from a perfect amaurosis, first discovered the objects whose images fell upon that part of the retina which is at the greatest distance from the optic nerve.

We shall conclude these remarks with observing, that if the retina be as transparent as it is generally represented to be, so that the termination of the pencils must necessarily be either upon the choroides, or some other opaque substance interposed between it and the retina, the action and reaction occasioned by the rays of light being at the common surface of this body and the retina, both these mediums (supposing them to be equally sensible to the impression of light) may be equally affected; but the retina, being naturally much more sensible to this kind of impression, may be the only instrument by which the sensation is conveyed to the brain, though the choroides, or the black substance with which it is sometimes lined, may also be absolutely necessary for the purpose of vision. Indeed, when the reflection of the light is made at the common boundary of any two mediums, it is with no propriety that this effect is ascribed to one of them rather than the other; and the strongest reflections are often made back into the densest mediums, when they have been contiguous to the rarest, or even to a vacuum. This is not far from the hypothesis of M. de la Hire, and will completely account for the entire defect of vision at the inferior of the optic nerve.

Vision is distinguished into bright and obscure, distinct and obscure, and confused.—It is said to be bright, when a sufficient number of rays enter the pupil at the same time; obscure, when too few. It is distinct when each pencil of rays is collected into a focus exactly upon the retina; confused, when they meet before they come at it, or when they would pass before they meet; for, in either of these last cases, the rays flowing from different parts of the object will fall upon the same part of the retina, which must necessarily render the image confused and indistinct.—Now, that objects may appear with a due brightness, whether more or fewer rays proceed from them, we have a power of contracting or dilating the pupil, by means of the muscular fibres of the iris, in order to take in more or fewer rays as occasion requires. But this power has its limits. In some animals it is much greater than in others; particularly in such as are obliged to seek their food by night as well as by day, as in cats, &c.

That the rays may be collected into points exactly upon the retina, that is, that objects may appear distinct, whether they be nearer or farther off, i.e. whether the rays proceeding from them diverge more or less, we have a power of contracting or relaxing the ligamenta ciliaria, and thereby altering the form of the crystalline humour, and with it the focal distance of the rays. Thus, when the object we view is far off, and the rays fall upon the pupil with a very small degree of divergency, we contract the ligamenta ciliaria, which, being concave towards the vitreous humour, do thereby compress it more than otherwise they would do: by this means it is made to press harder upon the backside of the crystalline humour, which is thereby rendered flatter; and thus the rays proceed farther before they meet in a focus, than otherwise they would have done. Add to this, that we dilate the pupils of our eyes (unless in cases where the light is so strong that it offends the eye), and thereby admit rays into them that are more diverging than those which would otherwise enter. And, when the rays come from an object that is very near, and therefore diverge too much to be collected into their respective foci upon the retina, by relaxing the ligamenta ciliaria, we give the crystalline a more convex form, by which means the rays are made to suffer a proportionably greater degree of refraction in passing through it. Some philosophers are of opinion that we do this by a power of altering the form of the eye; and others, by removing the crystalline forwards or backwards as occasion requires; but neither of these opinions is probable; for the coats of the eye are too hard, in some animals, for the first; and, as to moving the crystalline out of its place, the cavities of the eye seem to be too well filled with the other humours to admit of such removal.

Besides this, in the case above-mentioned, by contracting the pupils of our eyes, we exclude the more diverging rays, and admit only such as are more easily refracted into their respective foci (c). But vision is not distinct at all distances, for our power of contracting and relaxing the ligamenta ciliaria is also circumscribed within certain limits.

In those eyes where the tunica cornea is very protuberant and convex, the rays of light suffer a very slighted and considerable refraction at their entrance into the aqueous humour, and are therefore collected into a focus before they fall upon the retina, unless the object be placed very near, so that the rays which enter the eye may have a considerable degree of divergency. People that have such eyes are said to be purblind. Now, the nearer an object is to the eye, the greater is the image of it therein, as explained above: these people, therefore can see much smaller objects than others, as seeing much nearer ones with the same distinctness; and their sight continues good longer than that of other people, because the tunica cornea of their eyes, as they grow old, becomes plainer, for want of that redundancy of humours with which they were filled before. On the contrary, old men having the cornea of their eyes too flat for want of a sufficient quantity of the aqueous humour to fill them out, if the rays diverge

(c) Accordingly it is observed, that if we make a small hole with the point of a needle through a piece of paper, and apply that hole close to the eye, making use of it, as it were, instead of a pupil, we shall be able to see an object distinctly through it, though the object be placed within half an inch of the eye. too much before they enter the eye, they cannot be brought to a focus before they reach the retina; on which account those people cannot see distinctly, unless the object be situated at a greater distance from the eye than is required for those whose eyes are of a due form. The latter require the assistance of convex glasses to make them see objects distinctly; the former of concave ones. For if either the cornea \(abc\) (fig. 4.), or crystalline humour \(c\), or both of them, be too flat, as in the eye A, their focus will not be on the retina as at A, where it ought to be, in order to render vision distinct; but beyond the eye, as at \(f\). This is remedied by placing a convex glass \(g/b\) before the eye, which makes the rays converge sooner, and imprints the image duly on the retina at \(d\). Again, if either the cornea, or crystalline humour, or both of them, be too convex, as in the eye B, the rays that enter it from the object C will be converged to a focus in the vitreous humour, as at \(f\); and by diverging from thence to the retina, will form a very confused image thereon; and, of course, the observer will have as confused a view of the object as if his eye had been too flat. This inconvenience is remedied by placing a concave glass \(g/b\) before the eye; which glass, by causing the rays to diverge between it and the eye, lengthens the focal distance so, that if the glass be properly chosen, the rays will unite at the retina, and form a distinct image of the object upon it.

Such eyes as are of a due convexity, cannot see any object distinctly at less distance than five inches; and there are numberless objects too small to be seen at that distance, because they cannot appear under any sensible angle.—Concerning the least angle under which any object is visible, there was a debate between Dr Hooke and Hevelius. The former asserted, that no object could well be seen if it subtended an angle less than one minute; and, if the object be round, as a black circular spot upon a white ground, or a white circle upon a black ground, it follows, from an experiment made by Dr Smith, that this is near the truth; and from thence he calculates, that the diameter of the picture of such least visible point upon the retina is the 8600th part of an inch; which he therefore calls a sensible point of the retina. On the other hand, Mr Courtivron concluded from his experiments, that the smallest angle of vision was 40 seconds. According to Dr Jurin, there are cases in which a much smaller angle than one minute can be discerned by the eye; and in order to throw light upon the subject, he observes, that, in order to our perceiving the impression made by any object upon our senses, it must either be of a certain degree of force, or of a certain degree of magnitude. For this reason, a star, which appears only as a lucid point through a telescope, subtending not so much as an angle of one second, is visible to the eye; though a white or black spot, of 25 or 30 seconds, is not to be perceived. Also a line of the same breadth with the circular spot will be seen under smaller angles than spots, and why.

A black spot upon a white ground, or a white spot upon a black ground, he says, can hardly be perceived by the generality of eyes when it subtends a less angle than one minute. And if two black spots be made upon white paper, with a space between them equal in breadth to one of their diameters, that space is not to be distinguished, even within the limits of perfect vision, under so small an angle as a single spot of the same size can be distinguished. To see the two spots distinctly, therefore, the breadth of the space between them must subtend an angle of more than a minute. It would be very difficult, he says, to make this experiment accurately, within the limits of perfect vision; because the objects must be extremely small: but by a rude trial, made with square bits of white paper, placed upon a black ground, he judged, that the least angle under which the interval of two objects could be perceived, was at least a fourth part greater than the least angle under which a single object can be perceived. So that an eye which cannot perceive a single object under a smaller angle than one minute, will not perceive the interval between two such objects under a less angle than 75 seconds.

Without the limits of perfect vision, the distance at which a single object ceases to be perceivable will be much greater in proportion than the distance at which a space of equal breadth between two such objects ceases to be perceivable. For, without these limits, the image of each of the objects will be attended with a penumbra, and the penumbra of the two near objects will take up part of the space between them, and thereby render it less perceivable; but the penumbra will add to the breadth of the single object, and will thereby make it more perceivable, unless its image be very faint. Upon the same principles he likewise accounts for the radiation of the stars, whereby the light seems to project from them different ways at the same time.

Mr Mayer made many experiments in order to ascertain the smallest angle of vision in a variety of respects. He began with observing at what distance a black spot was visible on white paper; and found, that when it could barely be distinguished, it subtended an angle of about 34 seconds. When black lines were disposed with intervals broader than themselves, they were distinguished at a greater distance than they could be when the objects and the intervals were of an equal breadth. In all these cases it made no difference whether the objects were placed in the shade or in the strong light of the sun; but when the degrees of light were small, their differences had a considerable effect, though by no means in proportion to the differences of the light. For if an object was illuminated to such a degree as to be just visible at the distance of nine feet, it would be visible at the distance of four feet, though the light was diminished above 160 times. It appeared in the course of these experiments, that common day light is, at a medium, equal to that of 25 candles placed at the distance of one foot from the object.

As an image of every visible object is painted on the retina of each of our eyes, it thence becomes a natural question, Why do we not see everything double? It was the opinion of Sir Isaac Newton and others, that objects appear single because the optic nerves unite before they reach the brain. But Dr Porterfield shows, from the observation of several anatomists, that the optic nerves do not mix, or confound their substance, being only united by a close cohesion; and objects have appeared single where the optic nerves were found to be disjoined.

Dr Briggs supposed that single vision was owing to the equal tension of the corresponding parts of the optic nerves, whereby they vibrated in a synchronous manner. But, besides several improbable circumstances in this account, Dr Porterfield shows that facts do by no means favour it.

To account for this phenomenon, this ingenious writer supposes, that by an original law in our natures, we imagine objects to be situated somewhere in a right line drawn from the picture of it upon the retina, through the centre of the pupil. Consequently, the field, same object appearing to both eyes to be in the same place, the mind cannot distinguish it into two. In answer to an objection to this hypothesis, from objects appearing double when one eye is distorted, he says the mind mistakes the position of the eye, imagining that it had moved in a manner corresponding to the other, in which case the conclusion would have been just. In this he seems to have recourse to the power of habit, though in words he disclaims that hypothesis.

This principle, however, has generally been thought to be sufficient to account for this appearance. Originally, every object making two pictures, one in each eye, is imagined to be double; but by degrees, we find, that when two corresponding parts of the retina are impressed, the object is but one; but if those corresponding parts be changed, by the distortion of one of the eyes, the object must again appear double as at the first. This seems to be verified by Mr Chefselden; who informs us, that a gentleman, who from a blow on his head had one eye distorted, found every object to appear double; but by degrees the most familiar ones came to appear single again, and in time all objects did so, without any amendment of the distortion. A case similar to this is mentioned by Dr Smith.

On the other hand, Dr Reid is of opinion, that the correspondence of the centres of the two eyes, on which single vision depends, does not arise from custom, but from some natural constitution of the eye and of the mind. He makes several just objections to the case of Mr Foster, recited by Dr Smith and others; and thinks that the case of the young man couched by Chefselden, who saw singly with both eyes immediately upon receiving his sight, is nearly decisive in proof of his supposition. He also found that three young gentlemen, whom he endeavoured to cure of squinting, saw objects singly, as soon as ever they were brought to direct the centres of both their eyes to the same object, though they had never been used to do so from their infancy; and he observes, that there are cases, in which, notwithstanding the fullest conviction of an object being single, no practice of looking at it will ever make it appear so, as when it is seen through a multiplying glass.

To all these solutions of the difficulty respecting single vision by both eyes, objections have been lately made which seem insurmountable. By experiments judiciously conceived and accurately conducted, Dr Wells has shown, that it is neither by custom alone, nor by an original property of the eyes alone, that objects appear single; and having demolished the theories of others, he thus accounts for the phenomenon himself.

"The visible place of an object being composed of its visible distance and visible direction, to show how it may appear the same to both eyes, it will be necessary (says he) to explain in what manner the distance and direction, which are perceived by one eye, may coincide with those which are perceived by the other." With respect to visible distance, the author's opinion seems not to differ from that which we have stated elsewhere (see Metaphysics, no 49, 50); and therefore we have to attend only to what he says of visible direction.

When a small object is so placed with respect to either eye, as to be seen more distinctly than in any other situation, our author says that it is then in the optic axis, or the axis of that eye. When the two optic axes are directed to a small object not very distant, they may be conceived to form two sides of a triangle, of which the base is the interval between the points of the corners where the axes enter the eyes. This base he called the visual base; and a line drawn from the middle of it to the point of intersection of the optic axes he calls the common axis. He then proceeds to show, that objects really situated in the optic axis do not appear to be in that line, but in the common axis.

Every person (says he) knows, that if an object be viewed through two small holes, one applied to each eye, the two holes appear but as one. The theories hitherto invented afford two explanations of this fact. According to Aguilonius, Dechales, Dr Porterfield, and Dr Smith, the two holes, or rather their borders, will be seen in the same place as the object viewed through them, and will consequently appear united, for the same reason that the object itself is seen single. But whoever makes the experiment will distinctly perceive, that the united hole is much nearer to him than the object; not to mention, that any fallacy on this head might be corrected by the information from the sense of touch, that the card or other substance in which the holes have been made is within an inch or less of our face. The other explanation is that furnished by the theory of Dr Reid. According to it, the centres of the retinas, which in this experiment receive the pictures of the holes, will, by an original property, represent but one. This theory, however, though it makes the two holes appear one, does not determine where this one is to be seen. It cannot be seen in only one of the perpendiculars to the images upon the retinas, for no reason can be given why this law of visible direction, which Dr Reid thinks established beyond dispute, if it operates at all, should not operate upon both eyes at the same time; and if it be seen by both eyes in such lines, it must appear where those lines cross each other, that is, in the same place with the object viewed through the holes, which, as I have already mentioned, is contrary to experience. Nor is it seen in any direction, the consequence of a law affecting both eyes considered as one organ, but suspended when each eye is used separately. For when the two holes appear one, if we pay attention to its situation, and then close one eye, the truly single hole will be seen by the eye remaining open in exactly the same direction as the apparently single hole was by both eyes.

"Hitherto I have supposed the holes almost touching the face. But they have the same unity of appearance, in whatever parts of the optic axes they are placed; whether both be at the same distance from the eyes, or one be close to the eye i.e., the axis of which it is, and the other almost contiguous to the object seen through them. If a line, therefore, be drawn from the object to one of the eyes, it will represent all the real or tangible positions of the hole, which allow the object to be seen by that eye, and the whole of it will coincide with the optic axis. Let a similar line be drawn to the other eye, and the two must appear but as one line; for if they do not, the two holes in the optic axes will not, at every distance, appear one, whereas experiments prove that they do. This united line will therefore represent the visible direction of every object situated in either of the optic axes. But the end of it, which is toward the face, is seen by the right eye to the left, and by the left eye as much to the right. It must be seen then in the middle between the two, and consequently in the common axis. And as its other extremity coincides with the point where the optic axes intersect each other, the whole of it must lie in the common axis. Hence the truth of the proposition is evident, that objects situated in the optic axis, do not appear to be in that line, but in the common axis."

He then proves by experiments, for which we must refer to his work, that objects situated in the common axis do not appear to be in that line, but in the axis of the eye by which they are not seen: that is, an object situated in the common axis appears to the right eye in the axis of the left, and vice versa. His next proposition, proved likewise by experiments, is, that "objects situated in any line drawn through the mutual intersection of the optic axes to the visual base, do not appear to be in that line, but in another, drawn through the same intersection, to a point in the visual base distant half this base from the similar extremity of the former line towards the left, if the objects be seen by the right eye, but towards the right if seen by the left eye."

From these propositions he thus satisfactorily accounts for single vision by both eyes. "If the question be concerning an object at the concourse of the optic axes, it is seen single, because its two similar appearances, in regard to size, shape, and colour, are seen by both eyes in one and the same direction, or, if you will, in two directions, which coincide with each other through the whole of their extent. It therefore matters not whether the distance be truly or falsely estimated; whether the object be thought to touch our eyes, or to be infinitely remote. And hence we have a reason, which no other theory of visible direction affords, why objects appeared single to the young gentleman mentioned by Mr Chefelden, immediately after his being conched, and before he could have learned to judge of distance by sight.

"When two similar objects are placed in the optic axes, one in each, at equal distances from the eyes, they will appear in the same place, and therefore one, for the same reason that a truly single object, in the concourse of the optic axes, is seen single.

"To finish this part of my subject, it seems only necessary to determine, whether the dependence of visible direction upon the actions of the muscles of the..." eyes be established by nature, or by custom. But facts are here wanting. As far as they go, however, they serve to prove that it arises from an original principle of our constitution. For Mr Chefelden's patient saw objects single, and consequently in the same directions with both eyes, immediately after he was couched; and persons affected with squinting from their earliest infancy see objects in the same directions with the eye they have never been accustomed to employ, as they do with the other they have constantly used."

The author removes many difficulties, and obviates the objections to which his theory may seem most liable. The whole work deserves to be attentively studied by every optician; and we therefore recommend it to the perusal of our readers.

We are indebted to Dr Jurin for the following curious experiments, to determine whether an object seen by both eyes appears brighter than when seen with only one.

He laid a slip of clean white paper directly before him on a table, and applying the side of a book close to his right temple, so as that the book was advanced considerably more forward than his face, he held it in such a manner, as to hide from his right eye that half of the paper which lay to his left hand, while the left half of the paper was seen by both eyes, without any impediment.

Then looking at the paper with both eyes, he observed it to be divided, from the top to the bottom, by a dark line, and the part which was seen with one eye only was manifestly darker than that which was seen with both eyes; and, applying the book to his left temple, he found, by the result of the experiment, that both his eyes were of equal goodness.

He then endeavoured to find to what degree this excess of brightness amounted; and comparing it with the appearance of an object illuminated partly by one candle and partly by two, he was surprised to find that an object seen with two eyes is by no means twice as luminous as when it is seen with one only; and after a number of trials, by which he made the proportion less and less continually, he found, that when one paper was illuminated by a candle placed at the distance of three feet, and another paper by the same candle at the same distance, and by another candle at the distance of 11 feet, the former seen by both eyes, and the latter with one eye only, appeared to be of equal whiteness; so that an object seen with both eyes appears brighter than when it is seen with one only by about a 13th part. But he acknowledges, that it is difficult to make this experiment exactly.

He then proceeded to inquire, whether an object seen with both eyes appears anything larger than when seen with one only; but he concluded that it did not, except on account of some particular circumstances, as in the case of the binocular telescope and the concave speculum.

M. du Tour maintains, that the mind attends to no more than the image made in one eye at a time; and produces several curious experiments in favour of this hypothesis, which had also been maintained by Kepler and almost all the first opticians. But, as M. Buffon observes, it is a sufficient answer to this hypothesis, how ingeniously soever it may be supported, that we see more distinctly with two eyes than with one; and that when a round object is near us, we plainly see more of the surface in one case than in the other. There are also other facts which clearly prove the contrary of what is maintained by M. du Tour.

With respect to single vision with two eyes, Dr Hartley observes, that it deserves particular attention, that the optic nerves of men, and such other animals as look the same way with both eyes, unite in the cella turcica in a ganglion, or little brain, as one may call it, peculiar to themselves; and that the affections between synchronous impressions on the two retinas must be made sooner and cemented stronger on this account; also that they ought to have a much greater power over one another's images, than in any other part of the body. And thus an impression made on the right eye alone, by a single object, may propagate itself into the left, and there raise up an image almost equal in vividness to itself; and consequently when we see with one eye only, we may, however, have pictures in both eyes.

A curious deception in vision, arising from the use of both eyes, was observed and accounted for by Dr Smith. It is a common observation, he says, that objects seen with both eyes appear more vivid and stronger than they do to a single eye; especially when both of them are equally good. A person not short-sighted may soon be convinced of this fact, by looking attentively at objects that are pretty remote, first with one eye, and then with both. This observation gave occasion to the construction of the binocular telescope, in the use of which the phenomenon is still more striking.

Besides this, our author observes, that there is another phenomenon observable with this instrument, which is very remarkable. In the foci of the two telescopes there are two equal rings, as usual, which terminate the pictures of the objects there formed, and consequently the visible area of the objects themselves. These equal rings, by reason of the equal eye-glasses, appear equal, and equally distant when seen separately by each eye; but when they are seen with both eyes, they appear much larger, and more distant also; and the objects seen through them do also appear much larger, though circumscribed by their united rings, in the same places as when they were seen separately.

He observes, that the phenomenon of the enlarged circle of the visible area in the binocular telescope, may be seen very plainly in looking at distant objects through a pair of spectacles, removed from the eyes about four or five inches, and held steady at that distance. The two innermost of the four apparent rings, which hold the glasses, will then appear united in one larger and more distant ring than the two outermost, which will hardly be visible unless the spectacles be farther removed.

A curious circumstance relating to the effect of one eye upon the other, was noticed by M. Æpinus, who observed, that, when he was looking through a hole made in a plate of metal, about the 10th part of a line in diameter, with his left eye, both the hole itself appeared larger, and also the field of view seen through it was more extended, whenever he shut his right eye; and both these effects were more remarkable when that eye was covered with his hand. He found considerable difficulty in measuring this augmentation of the apparatus. apparent diameter of the hole, and of the field of view; but at length he found, that, when the hole was half an inch, and the tablet which he viewed through it was three feet from his eye, if the diameter of the field when both his eyes were open was 1, it became 1½ when the other eye was shut, and nearly 2 when his hand was laid upon it.

Upon examining this phenomenon, it presently appeared to depend upon the enlargement of the pupil of one eye when the other is closed, the physical or anatomical cause of which he did not pretend to assign; but he observes, that it is wisely appointed by divine Providence, in order that when one eye fails, the field of view in the other may be extended. That this effect should be more sensible when the eye is covered with the hand, is owing, he observes, to the eye lids not being impervious to the light. But the enlargement of the pupil does not enlarge the field of view, except in looking through a hole, as in this particular case; and therefore persons who are blind of one eye can derive no advantage from this circumstance. Before we applaud the wisdom of Providence in any part of the constitution of nature, we should be very sure that we do not mistake concerning the effects of that constitution.

A great deal has been written by Gassendi, Le Clerc, Musschenbroek, and Du Tour, concerning the place to which we refer an object viewed by one or both eyes. But the most satisfactory account of this matter that we have met with, the reader will find in Dr Wells's Essay above quoted, which will teach any person how to satisfy himself by experiment with respect to visible position and visible motion.

§ 4. Of the Appearance of Objects seen through Media of different Forms.

For the more easy apprehension of what relates to this subject, we shall premise the five following particulars, which either have been already mentioned, or follow from what has been before laid down.

1. That as each point of an object, when viewed by the naked eye, appears in its proper place, and as that place is always to be found in the line in which the axis of a pencil of rays flowing from it enters the eye, or else in the line which Dr Wells calls the common axis; we from hence acquire an habit of judging the point to be situated in that line; and, because the mind is unacquainted with what refractions the rays suffer before they enter the eye, therefore, in cases where they are diverted from their natural course, by passing through any medium, it judges the point to be in that line produced back in which the axis of a pencil of rays flowing from it is situated the instant they enter the eye, and not in that it was in before refraction. We shall therefore, in what follows, suppose the apparent place of an object, when seen thro' a refracting medium, to be somewhene in that line produced back in which the axis of a pencil of rays flowing from it proceeds after they have passed through the medium.

2. That we are able to judge, though imperfectly, of the distance of an object by the degree of divergency, wherein the rays flowing from the same point of the object enter the pupil of the eye, in cases where that divergency is considerable; but because in what follows it will be necessary to suppose an object, when seen through a medium whereby its apparent distance is altered, to appear in some determinate situation, in those cases where the divergency of the rays at their entrance into the eye is considerable, we will suppose the object to appear where those lines which they describe in entering, if produced back, would cross each other: though it must not be affirmed, that this is the precise distance; because the brightness, distinctness, and apparent magnitude of the object, on which its apparent distance in some measure depends, will also suffer an alteration by the refraction of the rays in passing through that medium.

3. That we estimate the magnitude of an object by that of the optic angle.

4. That vision is the brighter, the greater the number of rays which enter the pupil. And,

5. That, in some cases, the apparent brightness, distinctness, and magnitude of an object, are the only means whereby our judgment is determined in estimating the distance of it.

Prof. I. An object placed within a medium terminated by a plane surface on that side which is next the eye, if the medium be denser than that in which the eye is (as we shall always suppose it to be, unless where the contrary is expressed), appears nearer to the surface of the medium than it is.

Thus, if A be a point of an object placed within the medium BDCE (fig. 5.), and Ab Ac be two rays proceeding from thence, these rays passing out of a denser into a rarer medium, will be refracted from their respective perpendiculars bd, ce, and will enter the eye at H, suppose in the directions b'f', c'g': let then these lines be produced back till they meet in F; this will be the apparent place of the point A: and because the refracted rays b'f', c'g' will diverge more than the incident ones Ab, Ac, it will be nearer to the points b and c than the point A; and as the same is true of each point in the object, the whole will appear to an eye at H, nearer to the surface BC than it is.

From hence it is, that when one end of a straight stick is put under water, and the stick is held in an oblique position, it appears bent at the surface of the water; viz. because each point that is under water appears nearer the surface, and consequently higher than it is.

From hence likewise it is, that an object at the bottom of a vessel may be seen when the vessel is filled with water, though it be so placed with respect to the eye, that it cannot be seen when the vessel is empty. To explain this, let ABCD (fig. 6) represent a vessel, and let E be an object lying at the bottom of it. This object, when the vessel is empty, will not be seen by an eye at F, because HB, the upper part of the vessel, will obstruct the ray EH; but when it is filled with water to the height GH, the ray EX being refracted at the surface of the water into the line KF, the eye at F shall see the object by means of that.

In like manner, an object situated in the horizon appears above its true place, upon account of the refraction of the rays which proceed from it in their passage through the atmosphere of the earth. For, first, if the object be situated beyond the limits of the atmosphere, its rays in entering it will be refracted towards the the perpendicular; that is, towards a line drawn from the point where they enter, to the centre of the earth, which is the centre of the atmosphere; and as they pass on, they will be continually refracted the same way, because they are all along entering a denser part, the centre of whose convexity is still the same point; upon which account the line they describe will be a curve bending downwards; and therefore none of the rays that come from that object can enter an eye upon the surface of the earth, except what enter the atmos- phere higher than they need to do if they could come in a right line from the object; consequently the ob- ject must appear above its proper place. Secondly, if the object be placed within the atmosphere, the case is still the same; for the rays which flow from it must continually enter a denser medium whose centre is below the eye; and therefore being refracted to- wards the centre, that is, downwards as before, those which enter the eye must necessarily proceed as from some point above the object; therefore the object will appear above its proper place.

From hence it is, that the sun, moon, and stars, ap- pear above the horizon, when they are just below it; and higher than they ought to do, when they are above it: Likewise distant hills, trees, &c. seem to be high- er than they are.

Further, the lower these objects are in the horizon, the greater is the obliquity with which the rays which flow from them enter the atmosphere, or pass from the rare into the denser parts of it; and therefore they appear to be the more elevated by refraction: upon which account the lower parts of them are apparently more elevated than the other. This makes their upper and under parts seem nearer than they are; as is evi- dent from the sun and moon, which appear as an oval form when they are in the horizon, their horizontal diameters appearing of the same length they would do if the rays suffered no refraction, while their vertical ones are shortened thereby.

Prop. II. An object seen through a medium ter- minated by plane and parallel surfaces, appears nearer, brighter, and larger, than with the naked eye.

For instance, let AB (fig. 7.) be the object, CDEF the medium, and GH the pupil of an eye, which is here drawn large to prevent confusion in the figure. And, 1st, let RK, RL, be two rays proceeding from the point R, and entering the denser medium at K and L; these rays will here by refraction be made to diverge less, and to proceed afterwards, suppose in the lines KA, LB; at a and b, where they pass out of the denser medium, they will be as much refrac- ted the contrary way, proceeding in the lines ac, bd, parallel to their first directions. Produce these lines back till they meet in e; this will be the apparent place of the point R; and it is evident from the fig- ure, that it must be nearer the eye than that point; and because the same is true of all other pencils flow- ing from the object AB, the whole will be seen in the situation fG, nearer to the eye than the line AB. 2d, As the rays RK, RL, would not have entered the eye, but have passed by it in the directions KR, LR, had they not been refracted in passing through the medium, the object appears brighter. 3d, The rays Ah, Bi, will be refracted at b and i into the less con- verging lines bh, il, and at the other surface into kM, parallel to Ah and Bi produced; so that the ex- tremities of the object will appear in the lines Mk, produced, viz. in f and g, and under as large an angle fMg, as the angle AgB under which an eye at q would have seen it had there been no medium in- terposed to refract the rays; and therefore it appears larger to the eye at GH, being seen through the in- terposed medium, than otherwise it would have done. But it is here to be observed, that the nearer the point e appears to the eye on account of the refraction of the rays RK, RL, the shorter is the image fg, be- cause it is terminated by the lines Mf and Mg, upon which account the object is made to appear less; and therefore the apparent magnitude of an object is not much augmented by being seen through a medium of this form.

Farther, it is apparent from the figure, that the ef- fect of a medium of this form depends wholly upon its thickness; for the distance between the lines Rr and ee, and consequently the distance between the points e and R, depends upon the length of the line Ka: Again, the distance between the lines AM and fM depends on the length of the line hk; but both Ka and hk depend on the distance between the surfaces CE and DF, and therefore the effect of this medium de- pends upon its thickness.

Prop. III. An object seen through a convex lens, appears larger, brighter, and more distant, than with the naked eye.

To illustrate this, let AB (fig. 8.) be the object; seen thro' CD the lens, and EF the eye. 1. From A and B, the convex extremities of the object, draw the lines AYr, BXr, lens, crossing each other in the pupil of the eye; the angle larger, ArB comprehended between these lines, is the angle and more under which the object would be seen with the naked distant eye. But by the interposition of a lens of this form, whose property it is to render converging rays more fo, the rays AY and BX will be made to cross each other before they reach the pupil. There the eye at E will not perceive the extremities of the object by means of these rays (for they will pass it without entering), but by some others which must fall without the points Y and X, or between them; but if they fall between them, they will be made to concur sooner than they themselves would have done; and therefore, if the ex- tremities of the object could not be seen by them, it will much less be seen by these. It remains therefore, that the rays which will enter the eye from the points A and B after refraction, must fall upon the lens without the points Y and X; let then the rays AQ and BP be such. These after refraction entering the eye at r, the extremities of the object will be seen in the lines rQ, rT, produced, and under the optic angle QrT, which is larger than ArB, and therefore the apparent magnitude of the object will be increased.

2. Let GHI be a pencil of rays flowing from the point G; as it is the property of this lens to render diver- ging rays less diverging, parallel, or converging, it is evident that some of those rays, which would pro- ceed on to F and E, and miss the eye, were they to suffer no refraction in passing through the lens, will now enter it; by which means the object will appear brighter. 3. As to the apparent distance of the ob- ject, that will vary according to the situation of it with respect to the focus of parallel rays of the lens. Then, let us suppose the object placed so much nearer the lens than its focus of parallel rays, that the refracted rays KE and LF, though rendered less diverging by passing through it, may yet have a considerable degree of divergency, so that we may be able to form a judgment of the distance of the object thereby. In this case, the object ought to appear where EK, FL, produced back concur; which, because they diverge less than the rays GH, GI, will be beyond G, that is, at a greater distance from the lens than the object is. But because both the brightness and magnitude of the object will at the same time be augmented, prejudice will not permit us to judge it quite so far off as the point where those lines meet, but somewhere between that point and its proper place.

2. Let the object be placed in the focus of parallel rays, then will the rays KE and LF become parallel; and though in this case the object would appear at an immense distance, if that distance were to be judged by the direction of the rays KE and LF, yet upon account of the brightness and magnitude of it, we shall not think it much farther from us than if it were seen by the naked eye.

3. If the object be situated beyond the focus of parallel rays, as in BA (fig. 9.), the rays flowing from thence and falling upon the lens CD, will be collected into their respective foci at a and b, and the intermediate points m, n, &c. and will there form an image of the object AB; and after crossing each other in the several points of it, as expressed in the figure, will pass on diverging as from a real object. Now if an eye be situated at c, where AC, BC, rays proceeding from the extreme points of the object make not a much larger angle ACB, than they would do if there were no lens interposed, and the rays belonging to the same pencil do not converge so much as those which the eye would receive if it were placed nearer to a or b, the object upon them accounts appearing very little larger or brighter than with the naked eye, is seen nearly in its proper place; but if the eye recedes a little way towards a b, the object then appearing both brighter and larger, seems to approach the lens: which is an evident proof of what has been so often asserted, viz. that we judge of the distance of an object in some measure by its brightness and magnitude; for the rays converge the more the farther the eye recedes from the lens; and therefore if we judged of the distance of the object by the direction of the rays which flow from it, we ought in this case to conceive it at a greater distance, than when the rays were parallel, or diverged at their entrance into the eye.

That the object should seem to approach the lens in this case, was a difficulty that exceedingly puzzled the learned Barrow, and which he pronounces insuperable, and not to be accounted for by any theory we have of vision. Molineux also leaves it to the solution of others, as that which will be inexplicable, till a more intimate knowledge of the visive faculty, as he expresses it, be obtained by mortals.

They imagined, that seeing an object appears farther off, the less the rays diverge which fall upon the eye, if they should proceed parallel to each other, it ought to appear exceeding remote; and if they should converge, it should then appear more distant still: the reason of this was, because they looked upon the apparent place of an object, as owing only to the direction of the rays whatever it was, and not at all to its apparent magnitude or splendor.

Perhaps it may proceed from our judging of the distance of an object in some measure by its magnitude, that that deception of sight commonly observed by travellers may arise; viz. that upon the first appearance of a building larger than usual, as a cathedral church, or the like, it generally seems nearer to them, than they afterwards find it to be.

Prop. IV. If an object be placed farther from a convex lens than its focus of parallel rays, and the eye circumstances be situated farther from it on the other side than the object seen place where the rays of the several pencils are collected through a lens, the object appears inverted and pendulous in the air, between the eye and the lens.

To explain this, let AB (fig. 9.) represent the object, CD the lens; and let the rays of the pencil ACD be collected in a, and those of BCD in b, forming there an inverted image of the object AB, and let the eye be placed in F: it is apparent from the figure, that some of the refracted rays which pass through each point of the image will enter the eye as from a real object in that place; and therefore the object AB will appear there, as the proposition affirms. But we are so little accustomed to see objects in this manner, that it is very difficult to perceive the image with one eye; but if both eyes are situated in such a manner, that rays flowing from each point of the image may enter both, as at G and H, and we direct our optic axes to the image, it is easy to be perceived.

If the eye be situated in a or b, or very near them on either side, the object appears exceedingly confused, viz. if at d, the rays which proceed from the same point of the object converge so very much, and if at e, they diverge so much, that they cannot be collected together upon the retina, but fall upon it as if they were the axes of so many distinct pencils coming thro' every point of the lens; wherefore little more than one single point of the object is seen at a time, and that appears all over the lens; from whence nothing but confusion arises.

If the lens be so large that both eyes may be applied to it, as in b and k, the object will appear double; for it is evident from the figure, that the rays which enter the eye at b from either extremity of the object A or B, do not proceed as from the same point with that from whence those which enter the other at k seem to flow; the mind therefore is here deceived, and looks upon the object as situated in two different places, and therefore judges it to be double.

Prop. V. An object seen through a concave lens appears nearer, smaller, and less bright, than with the naked eye.

Thus, let AB (fig. 10.) be the object, CD the pupil of an eye, and EF the lens. Now, as it is the property of a lens of this form to render diverging rays more so, and converging ones less so, the diverging rays GH, GI, proceeding from the point G, will be made to diverge more, and so to enter the eye as from some nearer point g; and the rays AH, BI, which converge, will be made to converge less, and to enter the eye as from the points a and b; wherefore the objects will appear in the situation agb, less and and nearer than without the lens. Farther, as the rays which proceed from G are rendered more diverging, some of them will be made to pass by the pupil of the eye, which otherwise would have entered it; and therefore each point of the object will appear less bright.

Prop. VI. An object seen through a polygonous glass, that is, such as is terminated by several plain surfaces, is multiplied thereby.

For instance, let A (fig. 11), be an object, and BC a polygonal glass terminated by the plain surfaces BD, DE, &c. and let the situation of the eye F be such, that the rays AB being refracted in passing through the glass, may enter it in the direction BF, and the rays AC in the direction CF. Then will the eye, by means of the former, see the object in G, and by the latter in H; and by means of the rays AI, the object will appear also in its proper situation A.

Sect. III. Of the Reflection of Light.

When a ray of light falls upon any body, however transparent, the whole of it never passes through the body, but some part is always driven back or reflected from it; and it is by this reflected light that all bodies transparent which have no light of their own become visible to us.

Of that part of the ray which enters, another part is also reflected from the second surface, or that which is farthest from the luminous body. When this part arrives again at the first surface, part of it is reflected back from that surface; and thus it continues to be reflected between the two surfaces, and to pass backwards and forwards within the substance of the medium, till some part is totally extinguished and lost.

Besides this inconsiderable quantity, however, which is lost in this manner, the second surface often reflects much more than the first; insomuch that, in certain positions, scarce any rays will pass through both sides of the medium. A very considerable quantity is also unaccountably lost or extinguished at each reflecting surface; insomuch that no body, however transparent, can transmit all the rays which fall upon it; neither, tho' it be ever so well fitted for reflection, will it reflect them all.

§ I. Of the Cause of Reflection.

The reflection of light is by no means so easily accounted for as the refraction of the same fluid. This property, as we have seen in the last section, may be accounted for in a satisfactory manner by the supposition of an attractive power diffused throughout the medium, and extending a very little way beyond it; but with regard to the reflection of light, there seems to be no satisfactory hypothesis hitherto invented. Of the principal opinions on this subject Mr Rowning hath given us the following account.

I. It was the opinion of philosophers, before Sir Isaac Newton discovered the contrary, that light is reflected by impinging upon the solid parts of bodies. But that it is not so, is clear for the following reasons.

And first, it is not reflected at the first surface of a body by impinging against it.

For it is evident, that, in order to the due and regular reflection of light, that is, that the reflected rays should not be dispersed and scattered one from another, there ought to be no rashes or unevenness in the reflecting surface large enough to bear a sensible proportion to the magnitude of a ray of light; because if the surface abounds with such, the reflected rays will rather be scattered like a parcel of pebbles thrown upon a rough pavement, than reflected with that regularity with which light is observed to be from a well polished surface. Now those surfaces, which to our senses appear perfectly smooth and well polished, are far from being so; for to polish, is no other than to grind off the larger eminences and protuberances of the metal with the rough and sharp particles of sand, emery, or putty, which must of necessity leave behind them an infinity of rashes and scratches, which, though inconsiderable with regard to the former roughnesses, and too minute to be discerned by us, must nevertheless bear a large proportion to, if not vastly exceed, the magnitude of the particles of light.

Secondly, it is not reflected at the second surface by impinging against any solid particles.

That it is not reflected by impinging upon the solid particles which constitute this second surface, is sufficiently clear from the foregoing argument; the second surfaces of bodies being as incapable of a perfect polish as the first; and it is farther confirmed from hence, viz. that the quantity of light reflected differs according to the different density of the medium behind the body. And that it is not reflected by impinging upon the particles which constitute the surface of the medium behind it, is evident, because the strongest reflection of all at the second surface of a body, is when there is a vacuum behind it. This therefore wants no farther proof.

II. It has been thought by some, that it is reflected at the first surface of a body, by a repulsive force equally diffused over it; and at the second, by an attractive force.

1. If there be a repulsive force diffused over the Objected surface of bodies that repels rays of light at all times, then, since by increasing the obliquity of a ray we diminish its perpendicular force (which is that only whereby it must make its way through this repulsive force), however weakly that force may be supposed to act, rays of light may be made to fall with so great a degree of obliquity on the reflecting surface, that there shall be a total reflection of them there, and not one particle of light be able to make its way through; which is contrary to observation; the reflection of light at the first surface of a transparent body being never total in any obliquity whatever. The hypothesis therefore in this particular must be false.

2. As to the reflection at the second surface by the attractive force of the body; this may be considered in two respects: first, when the reflection is total; secondly, when it is partial.

And first, in cases where the reflection is total, the cause of it is undoubtedly that same attractive force by which light would be refracted in passing out of the same body. This is manifest from that analogy which is observable between the reflection of light at this second surface, and its refraction there. For otherwise, what can be the reason that the total reflection should begin just when the obliquity of the incident incident ray, at its arrival at the second surface, is such that the refracted angle ought to be a right one; or when the ray, were it not to return in reflection, ought to pass parallel to the surface, without going from it? For in this case it is evident, that it ought to be returned by this very power, and in such manner that the angle of reflection shall be equal to the angle of incidence; just as a stone thrown obliquely from the earth, after it is so far turned out of its course by the attraction of the earth, as to begin to move horizontally, or parallel to the surface of the earth, is then by the same power made to return in a curve similar to that which is described in its departure from the earth, and so falls with the same degree of obliquity that it was thrown with.

But, secondly, as to the reflection at the second surface, when it is partial; an attractive force uniformly spread over it, as the maintainers of this hypothesis conceive it to be, can never be the cause thereof. Because it is inconceivable, that the same force, acting in the same circumstances in every respect, can sometimes reflect the violet coloured rays and transmit the red, and at other times reflect the red and transmit the violet.

We have stated this objection, because it is our business to conceal no plausible opinions: but it is not valid; for in each colour, the reflection takes place at that angle, and no other, where the refraction of that ray would make it parallel to the posterior surface.

This partial reflection and refraction is a great difficulty in all the attempts which have been made to give a mechanical explanation of the phenomena of optics. It is equally a defection in that explanation which was propounded by Huygens, and, since his time, revived by Euler, by means of the undulations of an elastic fluid, although a vague consideration of undulatory motions seems to offer a very specious analogy. But a rigid application of such knowledge as we have acquired of such motions, will convince any unprejudiced mathematician, that the phenomena of undulation are essentially dissimilar to the phenomena of light. The inflection of light, and its refraction, equally demonstrate that light is acted on by moving forces in a direction perpendicular to the surface; and it is equally demonstrable that such forces must, in proper circumstances, produce reflections precisely such as we observe. The only difficulty is to show how there can be forces which produce both reflection and refraction, in circumstances which are similar. The fact is, that such effects are produced: the first logical inference is, that with respect to the light which is reflected and that which is refracted, the circumstances are not similar; and our attention should be directed to the discovery of that dissimilarity. All the phenomena of combined reflection and refraction should be examined and classified according to their generality, not doubting but that these points of resemblance will lead to the discovery of their causes.

Now the experiments of Mr Bouguer show that bodies differ extremely in their powers of thus separating light by reflection and refraction, some of them reflecting much more at a given angle than others. It is not therefore a general property of light to be partly reflected and partly refracted, but a distinctive property of different bodies; and since we see that they possess it in different degrees, we are authorised to conclude that some bodies may want it altogether. We may therefore expect some success by considering how bodies are affected by light, as well as how light is affected by bodies. Now, in all the phenomena of the material world we find bodies connected by mutual forces. We know no case where a body A tends towards a body B, or, in common language, is attracted by it, without, at the same time, the body B tending towards A. This is observed in the phenomena of magnetism, electricity, gravitation, corpuscular attraction, impulse, &c. We should therefore conclude from analogy, that as bodies change the motion of light, light also changes the motion of bodies; and that the particles near the surface are put into vibration by the passage of light through among them. Suppose a parcel of cork-balls all hanging as pendulums in a symmetrical order, and that an electrified ball passes through the midst of them; it is very easy to show that it may proceed through this assemblage in various directions with a situated motion, and without touching any of them, and that its ultimate direction will have a certain inclination to its primary direction, depending on the outline of the assemblage, just as is observed in the motion of light; and, in the mean time, the cork-balls will be variously agitated. Just so must it happen to the particles of a transparent body, if we suppose that they act on the particles of light by mutual attractions and repulsions.

An attentive consideration of what happens here will show us that the superficial particles will be much more agitated than the rest; and thus a stratum be produced, which, in any instant, will act on those particles of light which are then approaching them in a manner different from that in which they will act on similarly situated particles of light, which come into the place of the first in the following moment, when these acting particles of the body have (by their motion of vibration) changed their own situation. Now it is clearly understood, that in all motions of vibration, such as the motions of pendulums, there is a moment when the body is in its natural situation, as when the pendulum is in the vertical line. This may happen in the same instant in each atom of the transparent body. The particles of light which then come within the sphere of action may be wholly reflected; in the next moment, particles of light in the very situation of the first may be refracted.

Then will arise a separation of light; and as this will depend on the manner in which the particles of bodies are agitated by it during its passage, and as this again will depend on the nature of the body, that is on the law of action of those forces which connect the particles with each other, and with the particles of light, it will be different in different bodies. But in all bodies there will be this general resemblance, that the separation will be most copious in great obliquities of incidence, which gives the repulsive forces more time for action, while it diminishes the perpendicular force of the light. Such a resemblance between the phenomena and the legitimate consequences of the assumption (the agitation of the parts of the body), gives us some authority for assigning this as the cause; nor can the assumption tion be called gratuitous. To suppose that the particles of the transparent body are not thus agitated, would be a most gratuitous contradiction of a law of nature to which we know no other exception.

Thus the objection raised in n° 132. is obviated, because the reflection and refraction is not here conceived as simultaneous, but as successive.

III. Some, being apprehensive of the insufficiency of a repulsive and attractive force diffused over the surfaces of bodies and acting uniformly, have supposed, that, by the action of light upon the surface of bodies, the matter of these bodies is put into an undulatory motion; and that where the surface of it is subsiding light is transmitted, and in those places where it is rising light is reflected. But to overlook the objections which we have just made to this theory of undulation, we have only to observe, that, were it admitted, it seems not to advance us one jot farther; for in those cases, suppose where red is reflected and violet transmitted, how comes it to pass that the red impinges only on those parts when the waves are rising, and the violet when they are subsiding?

IV. The next hypothesis that we shall take notice of, is that remarkable one of Sir Isaac Newton's fits of easy reflection and transmission, which we shall now explain and examine.

That author, as far as we can apprehend his meaning in this particular, is of opinion, that light in its passage from the luminous body, is disposed to be alternately reflected by and transmitted through any refracting surface it may meet with; that these dispositions (which he calls fits of easy reflection and easy transmission) return successively at equal intervals; and that they are communicated to it at its first emission out of the luminous body it proceeds from, probably by some very subtle and elastic substance diffused through the universe, and that in the following manner. As bodies falling into water, or passing through the air, cause undulations in each, so the rays of light may excite vibrations in this elastic substance. The quickness of which vibrations depending on the elasticity of the medium (as the quickness of the vibrations in the air, which propagate sound, depend solely on the elasticity of the air, and not upon the quickness of those in the founding body), the motion of the particles of it may be quicker than that of the rays: and therefore, when a ray at the instant it impinges upon any surface, is in that part of a vibration of this elastic substance which confines with its motion, it may be easily transmitted; and when it is in that part of a vibration which is contrary to its motion, it may be reflected. He farther supposes, that when light falls upon the surface of a body, if it be not in a fit of easy transmission, every ray is there put into one, so that when they come at the other side (for this elastic substance, easily pervading the pores of bodies, is capable of the same vibrations within the body as without it), the rays of one colour shall be in a fit of easy transmission, and those of another in a fit of easy reflection, according to the thickness of the body, the intervals of the fits being different in rays of a different kind. This seems to account for the different colours of the bubble and thin plate of air and water, as is obvious enough; and likewise for the reflection of light at the second surface of a thicker body; for the light reflected from thence is also observed to be coloured, and to form rings according to the different thicknesses of the body, when not intermixed and confounded with other light, as will appear from the following experiment. If a piece of glass be ground concave on one side and convex on the other, both its concavity and convexity having one common centre; and if a ray of light be made to pass through a small hole in a piece of paper held in that common centre, and be permitted to fall on the glass; besides those rays which are regularly reflected back to the hole again, there will be others reflected to the paper, and form coloured rings surrounding the hole, not unlike those occasioned by the reflection of light from thin plates.

It is ever with extreme reluctance that we venture untenable, to call in question the doctrines of Newton; but to his theory of reflection there is this insuperable objection, that it explains nothing, unless the cause of the fits of more easy reflection and transmission be held as legitimate, namely, that they are produced by the undulations of another elastic fluid, incomparably more subtile than light, acting upon it in the way of impulse. The fits themselves are matters of fact, and no way different from what we have endeavoured to account for: but to admit this theory of them would be to transgress every rule of philosophy, as we have shown them to be susceptible of explanation from acknowledged optical laws.

§ 2. Of the Laws of Reflection.

The fundamental law of the reflection of light, is, that in all cases the angle of reflection is equal to the angle of incidence. This is found by experiment to be the case, and besides may be demonstrated mathematically from the laws of percussion in bodies perfectly elastic. The axiom therefore holds good in every case of reflection, whether it be from plane surfaces or spherical ones, and that whether they are convex or concave; and hence the seven following propositions relating to the reflection of light from plane and spherical surfaces may be deduced.

I. Rays of light reflected from a plane surface have the same degree of inclination to one another that their respective incident ones have.—For the angle of reflection of each ray being equal to that of its respective incident one, it is evident, that each reflected ray will have the same degree of inclination to that portion of the surface from whence it is reflected that its incident one has: but it is here supposed, that all those portions of surface from whence the rays are reflected, are situated in the same plain; consequently the reflected rays will have the same degree of inclination to each other that their incident ones have, from whatever part of the surface they are reflected.

II. Parallel rays reflected from a concave surface are rendered converging.—To illustrate this, let AF, CD, EB, (fig. 1.) represent three parallel rays falling upon the concave surface FB, whose centre is C. To the points F and B draw the lines CF, CB; these being drawn from the centre, will be perpendicular to the surface at those points. The incident ray from a concave surface, and also passing through the centre, will be perpendicular to the surface, and therefore will return after reflection in the same line; but the oblique rays AF and and EB will be reflected into the lines FM and BM, situated on the contrary side of their respective perpendiculars CF and CB. They will therefore proceed converging after reflection towards some point, as M, in the line CD.

III. Converging rays falling on the like surface, are made to converge more. For, every thing remaining as above, let GF, HB, be the incident rays. Now, because these rays have larger angles of incidence than the parallel ones AF and EB in the foregoing case, their angles of reflection will also be larger than those of the others; they will therefore converge after reflection, suppose in the lines FN and BN, having their point of concurrence N farther from the point C than M, that to which the parallel rays AF and EB converged to in the foregoing case; and their precise degree of convergency will be greater than that wherein they converged before reflection.

IV. Diverging rays falling upon the like surface, are, after reflection, parallel, diverging, or converging. If they diverge from the focus of parallel rays, they then become parallel; if from a point nearer to the surface than that, they will diverge, but in a less degree than before reflection; if from a point between that and the centre, they will converge after reflection, and that to some point on the contrary side of the centre, but situated farther from it than the point from which they diverged. If the incident rays diverge from a point beyond the centre, the reflected ones will converge to one on the other side of it, but nearer to it than the point they diverged from; and if they diverge from the centre, they will be reflected thither again.

1. Let them diverge in the lines MF, MB, proceeding from M, the focus of parallel rays; then, as the parallel rays AF and EB were reflected into the lines FM and BM (by Prop. II.), these rays will now on the contrary be reflected into them.

2. Let them diverge from N, a point nearer to the surface than the focus of parallel rays, they will then be reflected into the diverging lines FG and BH which the incident rays GF and HB described that were shewn to be reflected into them in the foregoing proposition; but the degree wherein they diverge will be less than that wherein they diverged before reflection.

3. Let them proceed diverging from X, a point between the focus of parallel rays and the centre; they then make less angles of incidence than the rays MF and MB, which became parallel by reflection; they will consequently have less angles of reflection, and proceed therefore converging towards some point, as Y; which point will always fall on the contrary side of the centre, because a reflected ray always falls on the contrary side of the perpendicular with respect to that on which its incident one falls; and of consequence it will be farther distant from the centre than X.

4. If the incident ones diverge from Y, they will, after reflection, converge to X; those which were the incident rays in the former case being the reflected ones in this. And lastly,

5. If the incident rays proceed from the centre, they fall in with their respective perpendiculars; and for that reason are reflected thither again.

V. Parallel rays reflected from a convex surface are rendered diverging.—For, let AB, GD, EF, (fig. 2.) be three parallel rays falling upon the convex surface BF, whose centre of convexity is C, and let one of them, viz. GD, be perpendicular to the surface. Through B, D, and F, the points of reflection, draw the lines CV, CG, and CT; which, because they pass through the centre, will be perpendicular to the surface at these points. The incident ray GD being perpendicular to the surface, will return after reflection in the same line, but the oblique ones AB and EF in the lines BK and FL, situated on the contrary side of their respective perpendiculars BV and FT. They will therefore diverge, after reflection, as from some point M in the line GD produced; and this point will be in the middle between D and C.

VI. Diverging rays reflected from the like surface are rendered more diverging.—For, every thing remaining as above, let GB, GF, be the incident rays. These having larger angles of incidence than the parallel ones AB and EF in the preceding case, their angles of reflection will also be larger than theirs: they will therefore diverge after reflection, suppose in the lines BP and FQ, as from some point N, farther from C than the point M; and the degree wherein they will diverge will be greater than that wherein they diverged before reflection.

VII. Converging rays reflected from the like surface, are parallel, converging, or diverging. If they tend towards the focus of parallel rays, they then become parallel; if to a point nearer the surface than that, they converge, but in a less degree than before reflection; if to a point between that and the centre, they will diverge after reflection, as from some point on the contrary side of the centre, but situated farther from it than the point they converged to: if the incident rays converge to a point beyond the centre, the reflected ones will diverge as from one on the contrary side of it, but nearer to it than the point to which the incident ones converged; and if the incident rays converge towards the centre, the reflected ones will proceed as from thence.

1. Let them converge in the lines KB and LF, tending towards M, the focus of parallel rays; then, as the parallel rays AB, EF were reflected into the lines BK and FL (by Prop. V.), those rays will now on the contrary be reflected into them.

2. Let them converge in the lines PB, QF, tending towards N a point nearer the surface than the focus of parallel rays, they will then be reflected into the converging lines BG and FG, in which the rays GB, GF proceeded that were shown to be reflected into them by the last proposition: but the degree wherein they will converge will be less than that wherein they converged before reflection.

3. Let them converge in the lines RB and SF proceeding towards X, a point between the focus of parallel rays and the centre: their angles of incidence will then be less than those of the rays KB and LF, which became parallel after reflection: their angles of reflection will therefore be less; on which account they must necessarily diverge, suppose in the lines BH and FI, from some point, as Y; which point (by Prop. IV.) will fall on the contrary side of the centre. centre with respect to X, and will be farther from it than that.

4. If the incident rays tend towards Y, the reflected ones will diverge as from X; those which were the incident ones in one case being the reflected ones in the other.

5. Lastly, if the incident rays converge towards the centre, they fall in with their respective perpendiculars; on which account they proceed after reflection as from the centre.

We have already observed, that in some cases there is a very great reflection from the second surface of a transparent body. The degree of inclination necessary to cause a total reflection of a ray at the second surface of a medium, is that which requires that the refracted angle (supposing the ray to pass out there) should be equal to or greater than a right one; and consequently it depends on the refractive power of the medium through which the ray passes, and is therefore different in different media. When a ray passes through glass surrounded with air, and is inclined to its second surface under an angle of 42 degrees or more, it will be wholly reflected there. For, as 11 is to 17 (the ratio of refraction out of glass into air), so is the sine of an angle of 42 degrees to a fourth number that will exceed the sine of a right angle. From hence it follows, that when a ray of light arrives at the second surface of a transparent substance with a great or a greater degree of obliquity than that which is necessary to make a total reflection, it will there be all returned back to the first; and if it proceeds towards that with as great an obliquity as it did towards the other (which it will do if the surfaces of the medium be parallel to each other), it will there be all reflected again, &c. and will therefore never get out, but pass from side to side, till it be wholly suffocated and lost within the body.—From hence may arise an obvious inquiry, how it comes to pass, that light falling very obliquely upon a glass window from without, should be transmitted into the room. In answer to this it must be considered, that however obliquely a ray falls upon the surface of any medium whose sides are parallel (as those of the glass in a window are), it will suffer such a degree of refraction in entering there, that it shall fall upon the second with a less obliquity than that which is necessary to cause a total reflection. For instance, let the medium be glass, as supposed in the present case; then, as 17 is to 11 (the ratio of refraction out of air into glass), so is the sine of the largest angle of incidence with which a ray can fall upon any surface to the sine of a less angle than that of total reflection. And therefore, if the sides of the glass be parallel, the obliquity with which a ray falls upon the first surface, cannot be so great, but that it shall pass the second without suffering a total reflection there.

When light passes out of a denser into a rarer medium, the nearer the second medium approaches the first in density (or more properly in its refractive power), the less of it will be refracted in passing from one to the other; and when their refracting powers are equal, all of it will pass into the second medium.

The above propositions may be all mathematically demonstrated in the following manner.

Proposition I. Of the reflection of rays from a plane surface.

"When rays fall upon a plane surface, if they diverge, the focus of the reflected rays will be at the same distance behind the surface, that the radiant point is before it: if they converge, it will be at the same distance before the surface, that the imaginary focus of the incident rays is behind it."

This proposition admits of two cases.

Case 1. Of diverging rays.

Dem. Let AB, AC, (fig. 3.) be two diverging rays incident on the plain surface DE, the one perpendicularly, the other obliquely: the perpendicular one AB will be reflected to A, proceeding as from some point in the line AB produced; the oblique one AC will be reflected into some line as CF, such that the point G, where the line FG produced intersects the line AB produced also, shall be at an equal distance from the surface DE with the radiant A. For the perpendicular CH being drawn, ACH and HCF will be the angles of incidence and reflection; which being equal, their complements ACB and FCE are too; but the angle BCG is equal to FCE, as being vertical to it: therefore in the triangles ABC and GBC the angles at C are equal, the side BC is common, and the angles at B are also equal to each other, as being right ones; therefore the lines AB and BG, which reflect the equal angles at C, are also equal; and consequently the point G, the focus of the incident rays AB, AC, is at the same distance behind the surface, that the point A is before it. Q.E.D.

Case 2. Of converging rays.

This is the converse of the former case. For supposing FC and AB to be two converging incident rays, CA and BA will be the reflected ones (the angles of incidence in the former case being now the angles of reflection, and vice versa), having the point A for their focus; but this, from what was demonstrated above, is at an equal distance from the reflecting surface with the point G, which in this case is the imaginary focus of the incident rays FC and AB.

Obs. It is not here, as in the refraction of rays in passing through a plane surface, where some of the refracted rays proceed as from one point, and some as from another: but they all proceed after reflection as from one and the same point, however obliquely they may fall upon the surface; for what is here demonstrated of the ray AC holds equally of any other, as AI, AK, &c.

The case of parallel rays incident on a plane surface is included in this proposition: for in that case we are to suppose the radiant to be at an infinite distance from the surface, and then by the proposition the focus of the reflected rays will be so too; that is, the rays will be parallel after reflection, as they were before.

Prop. II. Of the reflection of parallel rays from a spherical surface.

"When parallel rays are incident upon a spherical surface, the focus of the reflected rays will be the middle point between the centre of convexity and the surface."

This proposition admits of two cases.

Case 1. Of parallel rays falling upon a convex surface.

Dem. Let AB, DH, (fig. 4.) represent two parallel rays incident on the convex surface BH, the one perpendicularly, the other obliquely; and let C be the centre of convexity; suppose HE to be the reflected ray of the oblique incident one DH proceeding as from F. a point in the line AB produced. Through the point H draw the line CI, which will be perpendicular to the surface at that point; and the angles DHI and HIE, being the angles of incidence and reflection, will be equal. To the former of these, the angle HCF is equal, the lines AC and DH being parallel; and to the latter the angle CHF, as being vertical; wherefore the triangle CFH is isosceles, and consequently the sides CF and FH are equal: but supposing BH to vanish, FH is equal to FB; and therefore upon this supposition FC and FB are equal, that is, the focus of the reflected rays is the middle point between the centre of convexity and the surface. Q.E.D.

Case 2. Of parallel rays falling upon a concave surface.

Dem. Let AB, DH, (fig. 5.) be two parallel rays incident, the one perpendicularly, the other obliquely, on the concave surface BH, whose centre of concavity is C. Let BF and HF be the reflected rays meeting each other in F; this will be the middle point between B and C. For drawing through C the perpendicular CH, the angles DHC and FHC, being the angles of incidence and reflection, will be equal, to the former of which the angle HCF is equal, as alternate; and therefore the triangle CFH is isosceles. Wherefore CF and FH are equal: but if we suppose BH to vanish, FB and PH are also equal, and therefore CF is equal to FB; that is, the focal distance of the reflected rays is the middle point between the centre and the surface. Q.E.D.

Obs. It is here observable, that the farther the line DH, either in fig. 4. or 5. is taken from AB, the nearer the point F falls to the surface. For the farther the point H recedes from B, the larger the triangle CFH will become; and consequently, since it is always an isosceles one, and the base CH, being the radius, is everywhere of the same length, the equal legs CF and FH will lengthen; but CF cannot grow longer unless the point F approach towards the surface. And the farther H is removed from B, the faster F approaches to it.

This is the reason, that whenever parallel rays are considered as reflected from a spherical surface, the distance of the oblique one from the perpendicular one is taken so small with respect to the focal distance of that surface, that without any physical error it may be supposed to vanish.

From hence it follows, that if a number of parallel rays, as AB, CD, EG, &c. fall upon a convex surface, (as fig. 6.) and if BA, DK, the reflected rays of the incident ones AB, CD, proceed as from the point F, those of the incident ones CD, EG, viz. DK, GL, will proceed as from N, those of the incident ones EG, HI, as from O, &c. because the farther the incident ones CD, EG, &c. are from AB, the nearer to the surface are the points F, f, f, in the line BF, from which they proceed after reflection; so that properly the foci of the reflected rays BA, DK, GL, &c. are not in the line AB produced, but in a curve line passing through the points F, N, O, &c.

The same is applicable to the case of parallel rays reflected from a concave surface, as expressed by the pricked lines on the other half of the figure, where PQ, RS, TV, are the incident rays; QF, SF, VF, the reflected ones, intersecting each other in the points X, Y, and F; so that the foci of those rays are not in the line FB, but in a curve passing through those points.

Had the surface BH in fig. 4. or 5. been formed by the revolution of a parabola about its axis having its seedling focus in the point F, all the rays reflected from the one convex surface would have proceeded as from the point and F, and those reflected from the concave would have fallen upon it, however distant their incident ones concave AB, DH, might have been from each other. For in surface are the parabola, all lines drawn parallel to the axis make all reflected angles with the tangents to the points where they cut the parabola (that is, with the surface of the parabola) equal to those which are made with the same tangents by lines drawn from thence to the focus; therefore, if the incident rays describe those parallel lines, the reflected ones will necessarily describe these other, and so will all proceed as from, or meet in, the same point.

Prop. III. Of the reflection of diverging and converging rays from a spherical surface.

When rays fall upon any spherical surface, if they Prop., diverge, the distance of the focus of the reflected rays from the surface is to the distance of the radiant point of the same (or, if they converge, to that of the rays imaginary focus of the incident rays), as the distance from the focus of the reflected rays from the centre is to spherical the distance of the radiant point (or imaginary focus surface. of the incident rays) from the same.

This proposition admits of ten cases.

Case 1. Of diverging rays falling upon a convex surface.

Dem. Let RB, RD (fig. 7.) represent two diverging rays flowing from the point R as from a radiant, and falling the one perpendicularly, the other obliquely, on the convex surface BD, whose centre is C. Let DE be the reflected ray of the incident one RD, produce ED to F, and through R draw the line RH parallel to FE till it meets CD produced in H. Then will the angle RHD be equal to EDH the angle of reflection, as being alternate to it, and therefore equal also to RDH which is the angle of incidence; wherefore the triangle DRH is isosceles, and consequently DR is equal to RH. Now the lines FD and RH being parallel, the triangles FDC and RHC are similar, (or, to express it in Euclid's way, the sides of the triangle RHC are cut proportionally, 2 Elem. 6.), and therefore FD is to RH, or its equal RD, as CF to CR; but BD vanishing, FD and RD differ not from FB and RB: wherefore FB is to RB also, as CF to CR; that is, the distance of the focus from the surface is to the distance of the radiant point from the same, as the distance of the focus from the centre is to the distance of the radiant from thence. Q.E.D.

Case 2. Of converging rays falling upon a concave surface.

Dem. Let KD and CB be the converging incident rays having their imaginary focus in the point R, which was the radiant in the foregoing case. Then as RD was in that case reflected into DE, KD will in this be reflected into DF; for, since the angles of incidence in both cases are equal, as they are by being vertical, the angles of reflection will be so too; so that F will be the focus of the reflected rays: but it was there demonstrated, that FB is to RB as CF to CR; that is, the distance of the focus from the surface is to the distance (in this case) of the imaginary focus of the incident rays, as the distance of the focus from the centre is to the distance of the imaginary focus of the incident rays from the same.

CASE 3. Of converging rays falling upon a convex surface, and tending to a point between the focus of parallel rays and the centre.

DEM. Let BD (fig. 8.) represent a convex surface whose centre is C, and whose focus of parallel rays is P; and let AB, KD, be two converging rays incident upon it, and having their imaginary focus at R, a point between P and C. Now because KD tends to a point between the focus of parallel rays and the centre, the reflected ray DE will diverge from some point on the other side the centre, suppose F; as explained above (p. 308.) under prop. 7. Through D draw the perpendicular CD, and produce it to H; then will KDH and HDE be the angles of incidence and reflection, which being equal, their vertical ones RDC and CDF will be so too, and therefore the vertex of the triangle RDF is bisected by the line DC; wherefore (3 El. 6.) FD and DR, or, BD vanishing, FB and BR are to each other as FC to CR; that is, the distance of the focus of the reflected rays is to that of the imaginary focus of the incident ones, as the distance of the former from the centre is to the distance of the latter from the same.

CASE 4. Of diverging rays falling upon a concave surface, and proceeding from a point between the focus of parallel rays and the centre.

DEM. Let RB, RD, (fig. 8.) be the diverging rays incident upon the concave surface BD, having their radiant point in the point R, the imaginary focus of the incident rays in the foregoing case. Then as KD was in that case reflected into DE, RD will now be reflected into DF. But it was there demonstrated, that FB and RB are to each other as CF to CR; that is, the distance of the focus is to that of the radiant as the distance of the former from the centre is to the distance of the latter from the same.

CASE 5. Of converging rays falling upon a convex surface, and tending to a point nearer the surface than the focus of parallel rays.

DEM. Let ED, RB (fig. 7.) be the converging rays incident upon the convex surface BD whose centre is C, and focus of parallel rays is P; and let the imaginary focus of the incident rays be at E, a point between P and B; and let DR be the reflected ray. From C and R draw the lines CH, RH, the one passing through D, the other parallel to FE. Then will the angle RHD be equal to HDE the angle of incidence, as alternate to it; and therefore equal to HDR, the angle of reflection: wherefore the triangle HDR is isosceles, and consequently DR Reflection is equal to RH. Now the lines FD and RH being parallel, the triangles FDC and RHC are similar; and therefore RH, or RD, is to FD as CR to CF; but BD vanishing, RD and FD coincide with RB and FB, wherefore RB is to FB as CR to CF; that is, the distance of the focus from the surface is to the distance of the imaginary focus of the incident rays, as the distance of the focus from the centre is to the distance of the imaginary focus of the incident rays from the same.

CASE 6. Of diverging rays falling upon a concave surface, and proceeding from a point between the focus of parallel rays and the surface.

DEM. Let FD and FB represent two diverging rays flowing from the point F as a radiant, which was the imaginary focus of the incident rays in the foregoing case. Then as ED was in that case reflected into DR, FD will be reflected into DK (for the reason mentioned in Case 2.), so that the reflected ray will proceed as from the point R; but it was demonstrated in the case immediately foregoing, that RB is to FB as CR to CF; that is, the distance of the focus from the surface is to that of the radiant from the same, as the distance of the former from the centre is to that of the latter from the same.

CASE 7. Of converging rays falling upon a convex surface, and tending towards a point beyond the centre.

DEM. Let AB, ED (fig. 8.) be the incident rays tending to F, a point beyond the centre C, and let DK be the reflected ray of the incident one ED. Then because the incident ray ED tends to a point beyond the centre, the reflected ray DK will proceed as from one on the contrary side, suppose R; as explained above under Prop. VII. Through D draw the perpendicular CD, and produce it to H. Then will EDH and HDK be the angles of incidence and reflection; which being equal, their vertical ones CDI' and CDR will be so too: consequently the vertex of the triangle FDR is bisected by the line CD; wherefore, RD is to DF, or (3 Elem. 6.) BD vanishing, RB is to BF as RC to CF; that is, the distance of the focus of the reflected rays is to that of the imaginary focus of the incident rays, as the distance of the former from the centre is to the distance of the latter from the same.

CASE 8. Of diverging rays falling upon a concave surface, and proceeding from a point beyond the centre.

DEM. Let FB, FD, be the incident rays having their radiant in F, the imaginary focus of the incident rays in the foregoing case. Then as ED was in that case reflected into DK, FD will now be reflected into DR; so that R will be the focus of the reflected rays. But it was demonstrated in the foregoing case, that RB is to FB as RC to CF; that is, the distance of the focus of the reflected rays from the surface is to the distance of the radiant from the same, as the distance of the focus of the reflected rays from the centre is to the distance of the radiant from thence.

The two remaining cases may be considered as the converse of those under Prop. II. (p. 309, 310.), because the incident rays in these are the reflected ones in them; or they may be demonstrated in the same manner with the foregoing, as follows.

**Case 9.** Converging rays falling upon a convex surface, and tending to the focus of parallel rays, become parallel after reflection.

**Dem.** Let ED, RB (fig. 7.), represent two converging rays incident on the convex surface BD, and tending towards F, which we will now suppose to be the focus of parallel rays; and let DR be the reflected ray, and C the centre of convexity of the reflecting surface. Through C draw the line CD, and produce it to H, drawing RH parallel to ED produced to F. Now it has been demonstrated (Case 5. where the incident rays are supposed to tend to the point F), that RB is to FB as RC to CF; but F in this case being supposed to be the focus of parallel rays, it is the middle point between C and B (by Prop. II.), and therefore FB and FC are equal; and consequently the two other terms in the proportion, viz. RB and RC, must be so too; which can only be upon the supposition that R is at an infinite distance from B; that is, that the reflected rays BR and DR be parallel.

**E. D.**

**Case 10.** Diverging rays falling upon a concave surface, and proceeding from the focus of parallel rays, become parallel after reflection.

**Dem.** Let RD, RB (fig. 8.), be two diverging rays incident upon the concave surface BD, as supposed in Case 4., where it was demonstrated that FB is to RB as CF to CR. But in the present case RB and CR are equal, because R is supposed to be the focus of parallel rays; therefore FB and FC are so too; which cannot be unless F be taken at an infinite distance from B; that is, unless the reflected rays BF and DF be parallel.

**Obs.** It is here observable, that in the case of diverging rays falling upon a convex surface (see fig. 7.), the farther the point D is taken from B, the nearer the point F, the focus of the reflected rays, approaches to B, while the radiant R remains the same. For it is evident from the curvature of a circle, that the point D (fig. 9.) may be taken so far from B, that the reflected ray DE shall proceed as from F, G, H, or even from B, or from any point between B and R; and the farther it is taken from B, the faster the point from which it proceeds approaches towards R: as will easily appear if we draw several incident rays with their respective reflected ones, in such manner that the angles of reflection may be all equal to their respective angles of incidence, as is done in the figure. The like is applicable to any of the other cases of diverging or converging rays incident upon a spherical surface. This is the reason, that when rays are considered as reflected from a spherical surface, the distance of the oblique rays from the perpendicular one is taken so small, that it may be supposed to vanish.

From hence it follows, that if a number of diverging rays are incident upon the convex surface BD at the several points B, D, D, &c. they shall not proceed after reflection as from any point in the line RB produced, but as from a curve line passing through the several points F, f, f, &c. The same is applicable in all the other cases.

**No 248.**

Had the curvature BD (fig. 7.) been hyperbolic, having its foci in R and F; then R being the radiant (or the imaginary focus of incident rays), F would have been the focus of the reflected ones, and vice versa; however distant the points B and D might be taken from each other. In like manner, had the curve BD (fig. 8.) been elliptical, having its foci in F and R, the one of these being made the radiant (or imaginary focus of incident rays), the other would have been the focus of reflected ones, and vice versa. For both in the hyperbola and ellipse, lines drawn from each of their foci through any point make equal angles with the tangent to that point. Therefore, if the incident rays proceed to or from one of their foci, the reflected ones will all proceed as from or to the other. So that, in order that diverging or converging rays may be accurately reflected to or from a point, the reflecting surface must be formed by the revolution of an hyperbola about its longer axis, when the incident rays are such, that their radiant or imaginary focus of incident rays shall fall on one side the surface, and the focus of the reflected ones on the other: when they are both to fall on the same side, it must be formed by the revolution of an ellipse about its longer axis. However, upon account of the great facility with which spherical surfaces are formed in comparison of that with which surfaces formed by the revolution of any of the conic sections about their axes are made, the latter are very rarely used. Add to this another inconvenience, viz. that the foci of these curves being mathematical points, it is but one point of the surface of an object that can be placed in any of them at a time; so that it is only in theory that surfaces formed by the revolution of these curves about their axes render reflection perfect.

Now, because the focal distance of rays reflected from a spherical surface cannot be found by the analogy finding the laid down in the third proposition, without making use of the quantity sought; we shall here give an in-rays reflection whereby the method of doing it in all others fed from a convex surface will readily appear.

**Prob.** Let it be required to find the focal distance of diverging rays incident upon a convex surface, whose radius of convexity is 5 parts, and the distance of the radiant from the surface is 20.

**Sol.** Call the focal distance sought x; then will the distance of the focus from the centre be \(5-x\), and that of the radiant from the same \(25\); therefore by prop. 3., we have the following proportion, viz. \(x : 20 :: 5 - x : 25\); and multiplying extremes together and means together, we have \(25x = 100 - 20x\), which, after due reduction, gives \(x = \frac{100}{45}\).

If in any case it should happen that the value of \(x\) should be a negative quantity, the focal point must then be taken on the contrary side of the surface to that on which it was supposed that it would fall in stating the problem.

If letters instead of figures had been made use of in the foregoing solution, a general theorem might have been raised, to have determined the focal distance of reflected rays in all cases whatever. See this done in Suppl. to Gregory's Optics, 2d edit. p. 112.

Because it was, in the preceding section, observed, that different incident rays, though tending to or from one point, would after refraction proceed to or from different different points, a method was there inserted of determining the distinct point which each separate ray entering a spherical surface converges to, or diverges from, after refraction: the same has been observed here with regard to rays reflected from a spherical surface (see Obs. in Cafe 2. and Cafe 10.) But the method of determining the distinct point to or from which any given incident ray proceeds after reflection, is much more simple. It is only necessary to draw the reflected ray such, that the angle of reflection may be equal to the angle of incidence, which will determine the point it proceeds to or from in any case whatever.

§ 3. Of the Appearance of Bodies seen by Light reflected from Plane and Spherical Surfaces.

Whatever has been said concerning the appearance of bodies seen by refracted light through lenses, respects also the appearance of bodies seen by reflection. But besides these, there is one thing peculiar to images by reflection, viz. that each point in the representation of an object made by reflection appears situated somewhere in an infinite right line that passes through its correspondent point in the object, and is perpendicular to the reflecting surface.

The truth of this appears sufficiently from the propositions formerly laid down: in each of which, rays flowing from any radiant point, are shown to proceed after reflection to or from some point in a line that passes through the said radiant, and is perpendicular to the reflecting surface. For instance (fig. 1.), rays flowing from Y are collected in X, a point in the perpendicular CD, which, being produced, passes through Y; again (fig. 2.), rays flowing from G, proceed, after reflection, as from N, a point in the perpendicular CD, which, being produced, passes through G; and so of the rest.

This observation, however, except where an object is seen by reflection from a plain surface, relates only to those cases where the representation is made by means of such rays as fall upon the reflecting surface with a very small degree of obliquity; because such as fall at a considerable distance from the perpendicular, proceed not after reflection as from any point in that perpendicular, but as from other points situated in a certain curve, as hath already been explained; upon which account these rays are neglected, making a confused and deformed representation. And therefore it is to be remembered, that however the situation of the eye with respect to the object and reflecting surface may be represented in the following figures, it is to be supposed as situated in such a manner with respect to the object, that rays flowing from thence and entering it after reflection, may be such only as fall with a very small degree of obliquity upon the surface; that is, the eye must be supposed to be placed almost directly behind the object, or between it and the reflecting surface. The reason why it is not always so placed, is only to avoid confusion in the figures.

I. When an object is seen by reflection from a plane surface, the image of it appears at the same distance behind the surface that the object is placed before it, of the same magnitude therewith, and directly opposite to it.

To explain this, let AB (fig. 10) represent an object. The object seen by reflection from the plain surface SV; and let the rays AF, AG, be so inclined to the surface, that they shall enter an eye at H after reflection; and let AE be perpendicular to the surface: then, by the observation just mentioned, the point A will appear in some part of the line AE produced, suppose I; that is, the oblique rays AF and AG will proceed after reflection as from that point; and further, because the reflected rays FH, GK, will have the same degree of inclination to one another that their incident ones have, that point must necessarily be at the same distance from the surface that the point A is; the representation therefore of the point A will be at the same distance behind the surface that the point itself is before it, and directly opposite to it: consequently, since the like may be shown of the point B, or of any other, the whole image IM will appear at the same distance behind the surface that the object is before it, and directly opposite to it; and because the lines AI, BM, which are perpendicular to the plain surface, are for that reason parallel to each other, it will also be of the same magnitude therewith.

II. When an object is seen by reflection from a convex surface, its image appears nearer to the surface, vex surfaces; and less than the object.

Let AB (fig. 12.) represent the object, SV a reflecting surface whose centre of convexity is C; and let the rays AF, AG, be so inclined to the surface, that after reflection therefrom, they shall enter the eye at H; and let AE be perpendicular to the surface; then will the oblique rays AF, AG, proceed after reflection as from some point in the line AE produced, suppose from I; which point, because the reflected rays will diverge more than the incident ones, must be nearer to the surface than the point A. And since the same is also true of the rays which flow from B, or any other point, the representation IM will be nearer to the surface than the object; and because it is terminated by the perpendiculars AE and BF, which incline to each other, as concurring at the centre, it will also appear less.

III. When an object is seen by reflection from a concave surface, the representation of it is various, both according to its magnitude and situation, according as the distance of the object from the reflecting surface is greater or less.

1. When the object is nearer to the surface than its focus of parallel rays, the image falls on the opposite side of the surface, is more distant from it, and larger than the object.

Thus, let AB (fig. 13.) be the object, SV the reflecting surface, F the focus of parallel rays, and C its centre. Through A and B, the extremities of the object, draw the lines CE, CR, which will be perpendicular to the surface; and let the rays AR, AG, be incident upon such points of it that they shall be reflected into an eye at H. Now, because the radiant points A and B are nearer the surface than F the focus of parallel rays, the reflected rays will diverge, and will therefore proceed as from some points on the opposite side of the surface; which points, by the observation laid down at the beginning of this section, will be in the perpendiculars AE, BR, produced, suppose in I and M: but they will diverge in a less degree. degree than their incident ones (see the proposition just referred to); and therefore the said points will be farther from the surface than the points A and B. The image therefore will be on the opposite side of the surface with respect to the object; it will be more distant than it; and consequently being terminated by the perpendiculars CI and CM, it will also be larger.

2. When the object is placed in the focus of parallel rays, the reflected rays enter the eye parallel; in which case the image ought to appear at an infinite distance behind the reflecting surface: but the representation of it, for the like reasons that were given in the foregoing case, being large and distinct, we judge it not much farther from the surface than the image.

3. When the object is placed between the focus of parallel rays and the centre, the image falls on the opposite side of the centre, is larger than the object, and in an inverted position.

Thus let AB (fig. 14.) represent the object, SV the reflecting surface, F its focus of parallel rays, and C its centre. Through A and B, the extremities of the object, draw the lines CE and CN, which will be perpendicular to the surface; and let AR, AG, be a pencil of rays flowing from A. These rays proceeding from a point beyond the focus of parallel rays, will after reflection converge towards some point on the opposite side of the centre, which will fall upon the perpendicular EC produced, but at a greater distance from C than the radiant A from which they diverged. For the same reason, rays flowing from B will converge to a point in the perpendicular NC produced, which shall be farther from C than the point B; from whence it is evident, that the image IM is larger than the object AB, that it falls on the contrary side of the centre, and that their positions are inverted with respect to each other.

4. If the object be placed beyond the centre of convexity, the image is then formed between the centre and the focus of parallel rays, is less than the object, and its position is inverted.

This proposition is the converse of the foregoing: for as in that case rays proceeding from A were reflected to I, and from B to M; so rays flowing from I and M will be reflected to A and B; if therefore an object be supposed to be situated beyond the centre in IM, the image of it will be formed in AB between that and the focus of parallel rays, will be less than the object, and inverted.

5. If the middle of the object be placed in the centre of convexity of the reflecting surface, the object and its image will be coincident; but the image will be inverted with respect to the object.

That the place of the image and the object should be the same in this case, needs little explication; for the middle of the object being in the centre, rays flowing from thence will fall perpendicularly upon the surface, and therefore necessarily return thither again; so that the middle of the image will be coincident with the middle of the object. But that the image should be inverted is perhaps not so clear. To explain this, let AB (fig. 15.) be the object, having its middle point C in the centre of the reflecting surface SV; through the centre and the point R draw the line CR, which will be perpendicular to the appearance of Bodies seen by Reflection from different Surfaces.

The Appearances of Bodies seen by Reflection from different Surfaces.

Plate CCCLIX.

The Appearances of Bodies seen by Reflection from different Surfaces.

In this proposition it is to be supposed, that the object AB is so situated with respect to the reflecting surface, that the angle ACR may be right; for otherwise the angles ARC and BRC will not be equal, and part of the image will therefore fall upon the object and part off.

6. If in any of the three last cases, in each of which the image is formed on the same side of the reflecting surface with the object, the eye be situated farther from the surface than the place where the image falls, the rays of each pencil, crossing each other in the several points of the image, will enter the eye as from a real object situated there; so that the image will appear pendulous in the air between the eye and the reflecting surface, and in the position wherein it is formed, viz. inverted with respect to the object, in the same manner that an image formed by refracted light appears to an eye placed beyond it; which was fully explained under Prop. IV. (p. 304.), and therefore need not be repeated.

But as what relates to the appearance of the object when the eye is placed nearer to the surface than the image, was not there fully inquired into, that point shall now be more strictly examined under the following case, which equally relates to refracted and reflected light.

7. If the eye be situated between the reflecting surface and the place of the image, the object is then seen beyond the surface; and the farther the eye recedes from the surface towards the place of the image, the more confused, larger, and nearer, the object appears.

To explain this, let AB (fig. 16.) represent the object; IM its image, one of whose points M is formed by the concurrence of the reflected rays DM, EM, &c., which before reflection came from B; the other, I, by the concurrence of DI, EI, &c., which came from A; and let ab be the pupil of an eye, situated between the surface DP and the image. This pupil will admit the rays Ha, Kb; which, because they are tending towards I, are such as came from A, and therefore the point A will appear diffused over the space RS. In like manner the pupil will also receive into it the reflected rays Ka and Lb, which, because they are tending towards M, by supposition came from B; and therefore the point B will be seen spread as it were over the space TV, and the object will seem to fill the space RV; but the representation of it will be confused, because the intermediate points of the object being equally enlarged in appearance, there will not be room for them between the points S and T, but they will coincide in part one with another: for instance, the appearance of that point in the object, whose representation falls upon c in the image, will fill the space mn; and so of the rest. Now, if the same pupil... pupil be removed into the situation ef, the reflected rays Ee and Gf will then enter the eye, and therefore one extremity of the object will appear to cover the space XY; and because the rays Of and Le will also enter it in their progress towards M, the point B, from whence they came, will appear to cover ZV; the object therefore will appear larger and more confused than before. And when the eye recedes quite to the image, it sees but one single point of the object, and that appears diffused all over the reflecting surface: for instance, if the eye recedes to the point M, then rays flowing from the point B enter it upon whatever part of the surface they fall; and so for the rest. The object also appears nearer to the surface the farther the eye recedes from it towards the place of the image; probably because, as the appearance of the object becomes more and more confused, its place is not so easily distinguished from that of the reflecting surface itself; till at last when it is quite confused (as it is when the eye is arrived at M) they both appear as one, the surface affuming the colour of the object.

As to the precise apparent magnitude of an object seen after this manner, it is such that the angle it appears under shall be equal to that which the image of the same object would appear under were we to suppose it seen from the same place: that is, the apparent object (for such we must call it to distinguish it from the image of the same object) and the image subtend equal angles at the eye.

D. M. Here we must suppose the pupil of the eye to be a point only, because the magnitude of that causes small alteration in the apparent magnitude of the object; as we shall see by and by. Let then the point a represent the pupil, then will the extreme rays that can enter it be Ha and Ka; the object therefore will appear under the angle HaK, which is equal to its vertical one MaI, under which the image IM would appear were it to be seen from a. Again, if the eye be placed in f, the object appears under the angle GfO equal to IfM, which the image subtends at the same place, and therefore the apparent object and image of it subtend equal angles at the eye. Q.E.D.

Now if we suppose the pupil to have any sensible magnitude, such, suppose, that its diameter may be ab; then the object seen by the eye in that situation will appear under the angle HXL, which is larger than the angle HaK, under which it appeared before; because the angle at X is nearer than the angle at a, to the line IM, which is a subtense common to them both.

From this proposition it follows, that, were the eye close to the surface at K, the real and apparent object would be seen under equal angles (for the real object appears from that place under the same angle that the image does, as will be shown at the end of this section): therefore, when the eye is nearer to the image than that point, the image will subtend a larger angle at it than the object does; and consequently, since the image and apparent object subtend equal angles at the eye, the apparent object must necessarily be seen under a larger angle than the object itself, wherever the eye be placed, between the surface and the image.

As each point in the representation of an object made by reflection is situated somewhere in a right line that passes through its correspondent point in the object, and is perpendicular to the reflecting surface, as was shown in the beginning of this section; we may from hence deduce a most easy and expeditious method of determining both the magnitude and situation of the image in all cases whatever. Thus,

Through the extremities of the object AB and the centre C (fig. 17, 18, or 19.) draw the lines AC BC, and produce them as the case requires; these lines will be perpendicular to the reflecting surface, and therefore the extremities of the image will fall upon them. Through F the middle point of the object and the centre, draw the line FC, and produce it till it passes through the reflecting surface; this will also be perpendicular to the surface. Through G, the point where this line cuts the surface, draw the lines AG and BG, and produce them this way or that, till they cross the former perpendiculars; and where they cross, there I and M the extremities of the image will fall. For supposing AG to be a ray proceeding from the point A and falling upon G, it will be reflected to B; because FA is equal to FB, and FG is perpendicular to the reflecting surface; and therefore the representation of the point A will be in BG produced as well as in AC; consequently it will fall on the point I, where they cross each other. Likewise the ray BG will for the same reason be reflected to A; and therefore the representation of the point B will be in AG produced, as well as in some part of BC, that is, in M where they cross. From whence the proposition is clear.

If it happens that the lines will not cross which way forever they are produced, as in (fig. 20.), then is the object in the focus of parallel rays of that surface, and has no image formed in any place whatever. For in this case the rays AH, AG, flowing from the point A, become parallel after reflection in the lines HC, GB, and therefore do not flow as to or from any point: in like manner, rays flowing from B are reflected into the parallel lines KB and GA; so that no representation can be formed by such reflection.

From hence we learn another circumstance relating to the magnitude of the image made by reflection; viz. that it subtends the same angle at the vertex of the reflecting surface that the object does. This appears by inspection of the 17th, 18th, or 19th figure, in each of which the angle IGM, which the image subtends at G the vertex of the reflecting surface, is equal to the angle AGB, which the object subtends at the same place; for in the two first of those figures they are vertical, in the third they are the same.

And,

Farther, the angle ICM, which the image subtends at the centre, is also equal to the angle ACB which the object subtends at the same place; for in the two first figures they are the same, in the last they are vertical to each other.

From whence it is evident, that the object and its image are to each other in diameter, either as their respective distances from the vertex of the reflecting surface, or as their distances from the centre of the same. IV. As objects are multiplied by being seen thro' transparent media, whose surfaces are properly disposed, so they may also by reflecting surfaces. Thus,

1. If two reflecting surfaces be disposed at right angles, as the surfaces AB, BC, (fig. 21.), an object at D may be seen by an eye at E, after one reflection at F, in the line EF produced; after two reflections, the first at G, the second at H, in the line EH produced; and also, after one reflection made at A, in the line EA produced.

2. If the surfaces be parallel, as AB, CD, (fig. 22.), and the object be placed at E and the eye at F, the object will appear multiplied an infinite number of times: thus, it may be seen in the line FG produced, after one reflection at G; in the line FH produced, after two reflections, the first at I, the second at H; and also in FP produced, after several successive reflections of the ray EL, at the points L, M, N, O, and P: and so on in infinitum. But the greater the number of reflections are, the weaker their representation will be.

Sect. IV. Of the different Refrangibility of Light.

As this property of light solves a great number of the phenomena which could not be understood by former opticians, we shall give an account of it in the words of Sir Isaac Newton, who first discovered it; especially as his account is much more full, clear, and perspicuous, than those of succeeding writers.

"In a very dark chamber, at a round hole F (fig. 1.), about one third of an inch broad, made in the shut 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 (that is, the line passing through the middle of the prism, from one end of it to the other end, parallel to the edge of the refracting angle) was in this and the following experiments perpendicular to the incident rays. About this axis I turned the prism slowly, and saw the refracted light on the wall, 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.

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 sides it was bounded pretty distinctly; but on its ends very confusedly and indistinctly, the light there decaying and vanishing by degrees. At the distance of 18½ feet from the prism the breadth of the image was about 2½ inches, but its length was about 10½ inches, and the length of its rectilinear sides about eight inches; and ACB, the refracting angle of the prism, whereby so great a length was made, was 64 degrees. With a less angle the length of the image was less, the breadth remaining the same. It is farther to be observed, that the rays went on in straight lines from the prism to the image, and therefore at their going out of the prism had all that inclination to one another from which the length of the image proceeded. This image PT was colourable, and the more eminent colours lay in this order from the bottom at T to the top at P; red, orange, yellow, green, blue, indigo, violet; together with all their intermediate degrees in a continual succession perpetually varying."

Our author concludes from this experiment, and many more to be mentioned hereafter, "that the light fills the whole of the sun consists of a mixture of several sorts of coloured rays, some of which at equal incidences are more refracted than others, and therefore are called easily refrangible. The red at T, being nearest to the place Y, where the rays of the sun would go directly if the prism was taken away, is the least refracted of all the rays; and the orange, yellow, green, blue, indigo, and violet, are continually more and more refracted, as they are more and more diverted from the course of the direct light. For by mathematical reasoning he has proved, that when the prism is fixed in the posture above-mentioned, so that the place of the image shall be the lowest possible, or at the limit between its descent and ascent, the figure of the image ought then to be round like the spot at Y, if all the rays that tended to it were equally refracted. Therefore, seeing by experience it is found that this image is not round, but about five times longer than broad, it follows that all the rays are not equally refracted. And this conclusion is farther confirmed by the following experiments.

"In the sun-beam SF (fig. 2.), which was propagated into the room thro' the hole in the window-shut EG, at the distance of some feet from the hole, I held the prism ABC in such a posture, that its axis might be perpendicular to that beam: then I looked through the prism upon the hole F, and turning the prism to and fro about its axis to make the image PT of the hole ascend and descend, when between its two contrary motions it seemed stationary, I stopped the prism; in this situation of the prism, viewing through it the said hole F, I observed the length of its refracted image PT to be many times greater than its breadth; and that the most refracted part thereof appeared violet at p; the least refracted red, at t; and the middle parts indigo, blue, green, yellow, and orange, in order. The same thing happened when I removed the prism out of the sun's light, and looked through it upon the hole shining by the light of the clouds beyond it. And yet if the refractions of all the rays were equal according to one certain proportion of the fines of incidence and refraction, as is vulgarly supposed, the refracted image ought to have appeared round, by the mathematical demonstration above-mentioned. So then by these two experiments it appears, that in equal incidences there is a considerable inequality of refractions."

For the discovery of this fundamental property of light, which has opened the whole mystery of colours, we see our author was not only beholding to the experiments themselves, which many others had made before him, but also to his skill in geometry; which was absolutely necessary to determine what the figure of the refracted image ought to be upon the old principle of an equal refraction of all the rays; but ha- ving thus made the discovery, he contrived the following experiment to prove it at sight.

In the middle of two thin boards, DE (fig. 3.), I made a round hole in each, at G and g, a third part of an inch in diameter; and in the window-shut 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 shut 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 12 feet from the first board, I fixed the other board, d e, in such manner 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 d e, 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 a b c, 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 d e, 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 a b c, 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 d e, 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.

Our author shows also, by experiments made with convex glasses, that lights (reflected from natural bodies) which differ in colour, differ also in degrees of refrangibility; and that they differ in the same manner as the rays of the sun do.

The sun's light consists of rays differing in reflectivity; and those rays are more reflexible than others which are more refrangible. A prism, ABC (fig. 4.), whose two angles, at its base BC, were equal to one another and half right ones, and the third at A a right one, I placed in a beam FM of the sun's light, let into a dark chamber through a hole F one third part of an inch broad. And turning the prism slowly about its axis until the light which went through one of its angles ACB, and was refracted by it to G and H, began to be reflected into the line MN by its base BC, formerly at which till then it went out of the glass; I observed that those rays, as MH, which had suffered the greatest refraction, were sooner reflected than the rest. To make it evident that the rays which vanished at H were reflected into the beam MN, I made this beam pass through another prism VXY, and being refracted by it to fall afterwards upon a sheet of white paper p t placed at some distance behind it, and there by that refraction to paint the usual colours at p t. Then causing the first prism to be turned about its axis according to the order of the letters ABC, I observed, that when those rays MH, which in this prism had suffered the greatest refraction, and appeared blue and violet, began to be totally reflected, the blue and violet light on the paper which was most refracted in the second prism received a sensible increase at p t, above that of the red and yellow at t; and afterwards, when the rest of the light, which was green, yellow, and red, began to be totally reflected and vanished at G, the light of those colours at t, on the paper p t, received as great an increase as the violet and blue had received before. Which puts it past dispute, that those rays became first of all totally reflected at the base BC, which before at equal incidences with the rest upon the base BC had suffered the greatest refraction. I do not here take notice of any refractions made in the sides AC, AB, of the first prism, because the light enters almost perpendicularly at the first side, and goes out almost perpendicularly at the second; and therefore suffers none, or so little, that the angles of incidence at the base BC are not sensibly altered by it; especially if the angles of the prism at the base BC be each about 40 degrees. For the rays FM begin to be totally reflected when the angle CMF is about 50 degrees, and therefore they will then make a right angle of 90 degrees with AC.

It appears also from experiments, that the beam of light MN, reflected by the base of the prism, being augmented first by the more refrangible rays and afterwards by the less refrangible, is composed of rays differently refrangible.

The light whose rays are all alike refrangible, I call simple, homogeneous, and similar; and that whose rays are some more refrangible than others, I call compound, heterogeneous, and dissimilar. The former light I call homogeneous, not because I would affirm it to in all respects; but because the rays which agree in refrangibility agree at least in all their other properties which I consider in the following discourse.

The colours of homogeneal lights I call primary, homogeneal, and simple; and those of heterogeneal lights, simple or heterogeneal and compound. For these are always compounded of homogeneal lights, as will appear in the following discourse.

The homogeneal light and rays which appear red, or rather make objects appear so, I call rubrific or red-making; those which make objects appear yellow, green, blue, and violet, I call yellow-making, green-making, blue-making, violet-making; and so of the rest. And if at any time I speak of light and rays as coloured or endowed with colours, I would be understood to speak not philosophically and properly, but grossly, and according to such conceptions as vulgar people people in seeing all these experiments would be apt to frame. For the rays, to speak properly, are not coloured. In them there is nothing else than a certain power and disposition to stir up a sensation of this or that colour. For as sound, in a bell or musical string or other sounding body, is nothing but a trembling motion, and in the air nothing but that motion propagated from the object, and in the sensorium it is a sense of that motion under the form of sound; so colours in the object are nothing but a disposition to reflect this or that sort of rays more copiously than the rest: in rays they are nothing but their dispositions to propagate this or that motion into the sensorium; and in the sensorium they are sensations of those motions under the forms of colours. See Chromatics.

"By the mathematical proposition above-mentioned, it is certain that the rays which are equally refrangible do fall upon a circle answering to the sun's apparent disk, which will also be proved by experiment by and by. Now let AG (fig. 5.) represent the circle which all the most refrangible rays, propagated from the whole disk of the sun, would 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 sorts 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 gl, with the same distances 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, dk, el 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 into 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 light difference of the rays, we are to diminish the diameter 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 a 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, nor yet beyond the window. And therefore, instead of that hole, I used the lens in the window-frame as follows.

"In the sun's light let into my darkened chamber through a small round hole in my window-frame, at about 10 or 12 feet from the window, I placed a lens MN (fig. 6.), by which the image of the hole F might be distinctly cast upon a sheet of white paper placed at I. Then immediately after the lens I placed a prism ABC, by which the trajectories of light might be refracted either upwards or sideways, and thereby the round image which the lens alone did cast upon the paper at I, might be drawn out into a long one with parallel sides, as represented at pt. This oblong image I let fall upon another paper at about the same distance from the prism as the image at I, moving the paper either towards the prism or from it, until I found the just distance where the rectilinear sides of the image 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 I; 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 oblong 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.

"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. This will appear by the experiments which will follow. 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 paper I caused the spectrum of homogeneous light, described in the former article, so to fall that some part of the light, circumscribed by the hole in the paper. This transmitted part of the light I refracted with a prism placed behind the paper; and letting this 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 shows 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 in the beginning of this article.

"In the homogeneous light I placed flies and such like 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 objects 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 refractions, so neither were they by reflections. For all white, grey, red, yellow, green, blue, violet bodies, as paper, ashes, red lead, orpiment, indigo, tince, gold, silver, copper, glass, blue flowers, violets, bubbles of water tinged with various colours, peacocks 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 reflections and refractions; and by consequence, that the variety of colours depends upon the composition of light.

"The solar image p, formed by the separated rays, in the fifth experiment, did in the progress from its end, p, on which the most refrangible rays fell, unto its end t, on which the least refrangible rays fell, appear tinged with this series of colours; violet, indigo, blue, green, yellow, orange, red, together with all their intermediate degrees in a continual succession perpetually varying; so that there appeared as many degrees of colours as there were sorts of rays differing in refrangibility. And since these colours could not be changed by refractions nor by reflections, it follows, that all homogeneous light has its proper colour answering to its degree of refrangibility.

"Every homogeneous ray considered apart is refracted according to one and the same rule; so that its homogeneous fine of incidence is to its fine of refraction in a given ratio: that is, every different coloured ray has a different ratio belonging to it. This our author has once proved by experiment, and by other experiments has the same determined by what numbers those 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, and AD (fig. 15.) their common fine of incidence in glass be divided into 50 equal parts, then EF and GH, the fines of refraction into air, of the least and most refrangible rays, will be 77 and 78 such parts respectively. And since every colour has several degrees, the fines of refraction of all the degrees of red will have all intermediate degrees of magnitude from 77 to 77½, of all the degrees of orange from 77½ to 77¾, of yellow from 77¾ to 77½, of green from 77½ to 77¾, of blue from 77¾ to 77½, of indigo from 77½ to 77¾, and of violet from 77¾ to 78. Sect. I. The Application of the foregoing Theory to several natural Phenomena.

§ 1. Of the Rainbow.

This beautiful phenomenon hath engaged the attention of all ages. By some nations it hath been deified; though the more sensible part always looked upon it as a natural appearance, and endeavoured, however imperfectly, to account for it. The observations of the ancients and philosophers of the middle ages concerning the rainbow were such as could not have escaped the notice of the most illiterate husbandmen who gazed at the sky; and their various hypotheses deserve no notice. It was a considerable time even after the dawn of true philosophy in this western part of the world, before we find any discovery of importance on this subject. Maurolycus was the first who pretended to have measured the diameters of the two rainbows with much exactness; and he reports, that he found that of the inner bow to be 45 degrees, and that of the outer bow 56; from which Descartes takes occasion to observe, how little we can depend upon the observations of those who were not acquainted with the cause of the appearances.

One Clichovensis (the same, it is probable, who distinguished himself by his opposition to Luther, and who died in 1543) had maintained, that the second bow is the image of the first, as he thought was evident from the inverted order of the colours. For, said he, when we look into the water, all the images that we see reflected by it are inverted with respect to the objects themselves; the tops of the trees, for instance, that stand near the brink, appearing lower than the roots.

That the rainbow is opposite to the sun, had always been observed. It was, therefore, natural to imagine, that the colours of it were produced by some kind of reflection of the rays of light from drops of rain, or vapour. The regular order of the colours was another circumstance that could not have escaped the notice of any person. But, notwithstanding mere reflection had in no other case been observed to produce colours, and it could not but have been observed that refraction is frequently attended with that phenomenon, yet no person seems to have thought of having recourse to a proper refraction in this case, before one Fletcher of Bretten, who, in a treatise which he published in 1571, endeavoured to account for the colours of the rainbow by means of a double refraction and one reflection. But he imagined that a ray of light, after entering a drop of rain, and suffering a refraction both at its entrance and exit, was afterwards reflected from another drop, before it reached the eye of the spectator. He seems to have overlooked the reflection at the farther side of the drop, or to have imagined that all the bendings of the light within the drop would not make a sufficient curvature to bring the ray of the sun to the eye of the spectator. That he should think of two refractions, was the necessary consequence of his supposing that the ray entered the drop at all. This supposition, therefore, was all the light that he threw upon the subject. B. Porta supposed that the rainbow is produced by the refraction of light in the whole body of rain or vapour, but not in the separate drops.

After all, it was a man whom no writers allow to have had any pretensions to philosophy, that hit upon this curious discovery. This was Antonio De Dominis, bishop of Spalatro, whose treatise De Radiis Vitae, made by Antonio Fius et Lucius, was published by J. Bartolus in 1611. He first advanced, that the double refraction of Fletcher, with an intervening reflection, was sufficient to produce the colours of the bow, and also to bring the rays that formed them to the eye of the spectator, without any subsequent reflection. He distinctly describes the progress of a ray of light entering the upper part of the drop, where it suffers one refraction, and after being thereby thrown upon the back part of the inner surface, is from thence reflected to the lower part of the drop; at which place undergoing a second refraction, it is thereby bent, so as to come directly to the eye. To verify this hypothesis, this person (no philosopher as he was) proceeded in a very sensible and philosophical manner. For he procured a small globe of solid glass, and viewing it when it was exposed to the rays of the sun, in the same manner in which he had supposed that the drops of rain were situated with respect to them, he actually observed the same colours which he had seen in the true rainbow, and in the same order.

Thus the circumstances in which the colours of the rainbow were formed, and the progress of a ray of light through a drop of water, were clearly understood; but philosophers were a long time at a loss when they endeavoured to assign reasons for all the particular colours, and for the order of them. Indeed nothing but the doctrine of the different refrangibility of the rays of light, which was a discovery reserved for the great Sir Isaac Newton, could furnish a complete solution of this difficulty. De Dominis supposed that the red rays were those which had traversed the least space in the inside of a drop of water, and therefore retained more of their native force, and consequently, striking the eye more briskly, gave it a stronger sensation; that the green and blue colours were produced by those rays, the force of which had been, in some measure, obtunded in passing through a greater body of water; and that all the intermediate colours were composed (according to the hypothesis which generally prevailed at that time) of a mixture of these three primary ones. That the different colours were caused by some difference in the impulse of light upon the eye, and the greater or less impression that was thereby made upon it, was an opinion which had been adopted by many persons, who had ventured to depart from the authority of Aristotle.

Afterwards the same De Dominis observed, that all the rays of the same colour must leave the drop of water in a part similarly situated with respect to the eye, in order that each of the colours may appear in a circle, the centre of which is a point of the heavens, in a line drawn from the sun through the eye of the spectator. The red rays, he observed, must issue from the drop nearest to the bottom of it, in order that the circle of red may be the outermost, and therefore the most elevated in the bow.

Notwithstanding De Dominis conceived so justly of the manner in which the inner rainbow is formed, he was far from having as just an idea of the cause of the exterior bow. This he endeavoured to explain in the very same manner in which he had done the interior, viz. by one reflection of the light within the drop, preceded and followed by a refraction; supposing only that the rays which formed the exterior bow were returned to the eye by a part of the drop lower than that which transmitted the red of the interior bow. He also supposed that the rays which formed one of the bows came from the superior part of the sun's disk, and those which formed the other from the inferior part of it. He did not consider, that upon those principles, the two bows ought to have been contiguous; or rather, that an indefinite number of bows would have had their colours all intermixed; which would have been no bow at all.

When Sir Isaac Newton discovered the different refrangibility of the rays of light, he immediately applied his new theory of light and colours to the phenomena of the rainbow, taking this remarkable object of philosophical inquiry where De Dominis and Descartes, for want of this knowledge, were obliged to leave their investigations imperfect. For they could give no good reason why the bow should be coloured, and much less could they give any satisfactory account of the order in which the colours appear.

If different particles of light had not different degrees of refrangibility, on which the colours depend, the rainbow, besides being much narrower than it is, would be colourless; but the different refrangibility of differently coloured rays being admitted, the reason is obvious, both why the bow should be coloured, and also why the colours should appear in the order in which they are observed. Let \(a\) (fig. 8.) be a drop of water, and \(S\) a pencil of light; which, on its leaving the drop of water, reaches the eye of the spectator. This ray, at its entrance into the drop, begins to be decomposed into its proper colours; and upon leaving the drop, after one reflection and a second refraction, it is farther decomposed into as many small differently-coloured pencils as there are primitive colours in the light. Three of them only are drawn in this figure, of which the blue is the most, and the red the least refracted.

The doctrine of the different refrangibility of light enables us to give a reason for the size of a bow of each particular colour. Newton, having found that the fines of refraction of the most refrangible and least refrangible rays, in passing from rain-water into air, are in the proportion of 185 to 182, when the fine of incidence is 138, calculated the size of the bow; and he found, that if the sun was only a physical point, without sensible magnitude, the breadth of the inner bow would be 2 degrees; and if to this 30' was added for the apparent diameter of the sun, the whole breadth would be 2\(\frac{1}{2}\) degrees. But as the outermost colours, especially the violet, are extremely faint, the breadth of the bow will not in reality appear to exceed two degrees. He finds, by the same principles, that the breadth of the exterior bow, if it was everywhere equally vivid, would be 4° 20'. But in this case there is a greater deduction to be made, on account of the faintness of the light of the exterior bow; so that, in fact, it will not appear to be more than 3 degrees broad.

The principal phenomena of the rainbow are all explained on Sir Isaac Newton's principles in the following propositions.

When the rays of the sun fall upon a drop of rain and enter into it, some of them, after one reflection and two refractions, may come to the eye of a spectator who has his back towards the sun, and his face towards the drop.

If XY (fig. 9.) is a drop of rain, and the sun shines upon it in any lines \(s_f, s_d, s_a,\) &c. most of the rays will enter into the drop; some few of them only will be reflected from the first surface; those rays on the prism which are reflected from thence do not come under our present consideration, because they are never refracted at all. The greatest part of the rays then enter the drop, and those falling on to the second surface, will most of them be transmitted through the drop; but neither do those rays which are thus transmitted fall under our present consideration, since they are not reflected. For the rays, which are described in the proposition, are such as are twice refracted and once reflected. However, at the second surface, or hinder part of the drop, at \(p_g\), some few rays will be reflected, whilst the rest are transmitted; those rays proceed in some such lines as \(n_r, n_q\); and coming out of the drop in the lines \(r_v, q_t\), may fall upon the eye of a spectator, who is placed anywhere in those lines, with his face towards the drop, and consequently with his back towards the sun, which is supposed to shine upon the drop in the lines \(s_f, s_d, s_a,\) &c. These rays are twice refracted and once reflected; they are refracted when they pass out of the air into the drop; they are reflected from the second surface, and are refracted again when they pass out of the drop into the air.

When rays of light reflected from a drop of rain come to the eye, these are called effectual, which are able to excite a sensation.

When rays of light come out of a drop of rain, they will not be effectual, unless they are parallel and contiguous.

There are but few rays that can come to the eye at all: for since the greatest part of those rays which enter the drop XY (fig. 9.) between X and \(a\), pass out of the drop through the hinder surface \(p_g\); only few are reflected from thence, and come out through the nearer surface between \(a\) and \(y\). Now, such rays as emerge, or come out of the drop, between \(a\) and \(Y\), will be ineffectual, unless they are parallel to one another, as \(r_v\) and \(q_t\) are; because such rays as come out diverging from one another will be so far asunder when they come to the eye, that all of them cannot enter the pupil; and the very few that can enter it will not be sufficient. Of the Rainbow.

Sufficient to excite any sensation. But even rays, which are parallel, as \( r \) and \( q \), will not be effectual, unless there are several of them contiguous or very near to one another. The two rays \( r \) and \( q \) alone will not be perceived, though both of them enter the eye; for very few rays are not sufficient to excite a sensation.

When rays of light come out of a drop of rain after one reflection, those will be effectual which are reflected from the same point, and which entered the drop near to one another.

Any rays, as \( s \) and \( c \), (fig. 10.) when they have passed out of the air into a drop of water, will be refracted towards the perpendiculars \( b \), \( d \); and as the ray \( s \) falls farther from the axis \( a \) than the ray \( c \), \( s \) will be more refracted than \( c \); so that these rays, though parallel to one another at their incidence, may describe the lines \( b \) and \( d \) after refraction, and be both of them reflected from one and the same point \( e \). Now all rays which are thus reflected from one and the same point, when they have described the lines \( e \), \( g \), and after reflection emerge at \( f \) and \( g \), will be so refracted, when they pass out of the drop into the air, as to describe the lines \( f \), \( h \), \( g \), parallel to one another. If these rays were to return from \( e \) in the lines \( e \), \( d \), and were to emerge at \( b \) and \( d \), they would be refracted into the lines of their incidence \( b \), \( d \). But if these rays, instead of being returned in the lines \( e \), \( d \), are reflected from the same point \( e \) in the lines \( e \), \( g \), \( f \), the lines of reflection \( e \) and \( f \) will be inclined both to one another, and to the surface of the drop; just as much as the lines \( e \), \( b \), and \( d \) are. First \( e \) and \( g \) make just the same angle with the surface of the drop: for the angle \( b \), \( e \), which \( e \) makes with the surface of the drop, is the complement of incidence, and the angle \( g \), \( e \), which \( e \) makes with the surface, is the complement of reflection; and these two are equal to one another. In the same manner we might prove, that \( e \) and \( f \) make equal angles with the surface of the drop. Secondly, The angle \( b \), \( e \), is equal to the angle \( f \), \( g \); or the reflected rays \( e \), \( g \), and the incident rays \( b \), \( d \), are equally inclined to each other. For the angle of incidence \( b \), \( e \) is equal to the angle of reflection \( g \), \( e \), and the angle of incidence \( d \), \( e \) is equal to the angle of reflection \( f \), \( e \); consequently the difference between the angles of incidence is equal to the difference between the angles of reflection, or \( b \), \( e \) \( = \) \( d \), \( e \) \( = \) \( g \), \( e \) \( = \) \( f \), \( e \). Since therefore either the lines \( e \), \( g \), \( f \), or the lines \( e \), \( b \), \( d \), are equally inclined both to one another and to the surface of the drop; the rays will be refracted in the same manner, whether they were to return in the lines \( e \), \( b \), \( d \), or are reflected in the lines \( e \), \( g \), \( f \). But if they were to return in the lines \( e \), \( b \), \( d \), the refraction, when they emerge at \( b \) and \( d \), would make them parallel. Therefore, if they are reflected from one and the same point \( e \) in the lines \( e \), \( g \), \( f \), the refraction, when they emerge at \( g \) and \( f \), will likewise make them parallel.

But though such rays as are reflected from the same point in the hinder part of a drop of rain, are parallel to one another when they emerge, and so have one condition that is requisite towards making them effectual, yet there is another condition necessary; for rays, that are effectual, must be contiguous as well as parallel. And though rays, which enter the drop in different places, may be parallel when they emerge, those only will be contiguous which enter it nearly at the same place.

Let \( XY \) (fig. 9.) be a drop of rain, \( a \), \( g \) the axis or diameter of the drop, and \( s \), \( a \) a ray of light that comes from the sun and enters the drop at the point \( a \). This ray \( s \), \( a \), because it is perpendicular to both the surfaces, will pass straight through the drop in the line \( a \), \( g \) without being refracted; but any collateral rays, such as those that fall about \( s \), \( b \), as they pass through the drop, will be made to converge to their axis, and passing out at \( n \) will meet the axis at \( b \): rays which fall farther from the axis than \( s \), \( b \), such as those which fall about \( s \), \( c \), will likewise be made to converge; but then their focus will be nearer to the drop than \( b \). Suppose therefore \( i \) to be the focus to which the rays that fall about \( s \), \( c \) will converge, any ray \( s \), \( e \), when it has described the line \( e \) within the drop, and is tending to the focus \( i \), will pass out of the drop at the point \( o \). The rays that fall upon the drop about \( s \), \( d \), more remote still from the axis, will converge to a focus still nearer than \( i \), as suppose at \( k \). These rays therefore go out of the drop at \( p \). The rays, that fall still more remote from the axis, as \( s \), \( e \), will converge to a focus nearer than \( k \), as suppose at \( l \); and the ray \( s \), \( e \), when it has described the line \( e \) within the drop, and is tending to \( l \), will pass out at the point \( o \). The rays that fall still more remote from the axis will converge to a focus still nearer. Thus the ray \( s \), \( f \) will after refraction converge to a focus at \( m \), which is nearer than \( l \); and having described the line \( f \), \( n \) within the drop, it will pass out to the point \( n \). Now here we may observe, that as any rays \( s \), \( b \) or \( s \), \( c \), fall farther above the axis \( s \), \( a \), the points \( n \), or \( o \), where they pass out behind the drop, will be farther above \( g \); or that, as the incident ray rises from the axis \( s \), the arc \( g \), \( n \), increases, till we come to some ray \( s \), \( d \), which passes out of the drop at \( p \); and this is the highest point where any ray that falls upon the quadrant or quarter \( ax \) can pass out: for any rays \( s \), \( e \), or \( s \), \( f \), that fall higher than \( s \), \( d \), will not pass out in any point above \( p \), but at the points \( o \) or \( n \), which are below it. Consequently, tho' the arc \( g \), \( n \) increases, whilst the distance of the incident ray from the axis \( s \), \( a \) increases, till we come to the ray \( s \), \( d \); yet afterwards, the higher the ray falls above the axis \( s \), \( a \), this arc \( g \), \( n \) will decrease.

We have hitherto spoken of the points on the hinder part of the drop, where the rays pass out of it; but this was for the sake of determining the points from whence those rays are reflected, which do not pass out behind the drop. For, in explaining the rainbow, we have no farther reason to consider those rays which go through the drop; since they can never come to the eye of a spectator placed anywhere in the lines \( r \), \( v \) or \( q \), with his face towards the drop. Now, as there are many rays which pass out of the drop between \( g \) and \( p \), so some few rays will be reflected from thence; and consequently the several points between \( g \) and \( p \), which are the points where some of the rays pass out of the drop, are likewise the points of reflection for the rest which do not pass out. Therefore, in respect of those rays which are reflected, we may call \( g \), \( p \) the arc of reflection; and may say, that this arc... of reflection increases, as the distance of the incident ray from the axis sa increases, till we come to the ray sd; the arc of reflection is gn for the ray sb, it is go for the ray sc; and gp for the ray sd. But after this, as the distance of the incident ray from the axis sa increases, the arc of reflection decreases; for og less than pg is the arc of reflection for the ray sc, and ng is the arc of reflection for the ray sf.

From hence it is obvious, that some one ray, which falls above sd, may be reflected from the same point with some other ray which falls below sd. Thus, for instance, the ray sb will be reflected from the point n, and the ray sf will be reflected from the same point; and consequently, when the reflected rays nr, nq, are refracted as they pass out of the drop at r and q, they will be parallel, by what has been shown in the former part of this proposition. But since the intermediate rays, which enter the drop between sf and sb, are not reflected from the same point n, these two rays alone will be the parallel to one another when they come out of the drop, and the intermediate rays will not be parallel to them. And consequently these rays rv, qt, though they are parallel after they emerge at r and q, will not be contiguous, and for that reason will not be effectual; the ray sd is reflected from p, which has been shown to be the limit of the arc of reflection; such rays as fall just above sd, and just below sd, will be reflected from nearly the same point p, as appears from what has been already shown. These rays therefore will be parallel, because they are reflected from the same point p; and they will likewise be contiguous, because they all of them enter the drop at one and the same place very near to d. Consequently, such rays as enter the drop at d, and are reflected from p, the limit of the arc of reflection, will be effectual; since, when they emerge at the fore part of the drop between a and y, they will be both parallel and contiguous.

If we can make out hereafter that the rainbow is produced by the rays of the sun which are thus reflected from drops of rain as they fall whilst the sun shines upon them, this proposition may serve to show us, that this appearance is not produced by any rays that fall upon any part, and are reflected from any part of those drops; since this appearance cannot be produced by any rays but those which are effectual; and effectual rays must always enter each drop at one certain place in the fore-part of it, and must likewise be reflected from one certain place in the hinder surface.

When rays that are effectual emerge from a drop of rain after one reflection and two refractions, those which are most refrangible will, at their emergence, make a less angle with the incident rays than those do which are least refrangible; and by this means the rays of different colours will be separated from one another.

Let fb and gi (fig. 10.) be effectual violet rays emerging from the drop at sf; and fn, gf, effectual red rays emerging from the same drop at the same place. Now, though all the violet rays are parallel to one another, because they are supposed effectual, and though all the red rays are likewise parallel to one another for the same reason; yet the violet rays will not be parallel to the red rays. These rays, as they have different colours, and different degrees of refrangibility, will diverge from one another; any violet ray gi, which emerges at g, will diverge from any red ray gp, which emerges at the same place. Now, both the violet ray gi, and the red ray gp, as they pass out of the drop of water into the air, will be refracted from the perpendicular go. But the violet ray is more refrangible than the red one; and for that reason gi, or the refracted violet ray, will make a greater angle with the perpendicular than gp the refracted red ray; or the angle ig will be greater than the angle pgo. Suppose the incident ray sb to be continued in the direction sk, and the violet ray ig to be continued backward in the direction ik, till it meets the incident ray at k. Suppose likewise the red ray pg, to be continued backwards in the same manner, till it meets the incident ray at wo. The angle iks is that which the violet ray, or most refrangible ray at its emergence, makes with the incident ray; and the angle pws is that which the red ray, or least refrangible ray at its emergence, makes with the incident ray. The angle iks is less than the angle pws. For, in the triangle, gws, gws, or pws, is the external angle at the base, and gkw or iks is one of the internal opposite angles; and either internal opposite angle is less than the external angle at the base. (Eucl. b. I. prop. 16.) What has been shown to be true of the rays gi and gp might be shown in the same manner of the rays fb and fn, or of any other rays that emerge respectively parallel to gi and gp. But all the effectual violet rays are parallel to gi, and all the effectual red rays are parallel to gp. Therefore the effectual violet rays, at their emergence make a less angle with the incident ones than the effectual red ones. And for the same reason, in all the other sorts of rays, those which are most refrangible, at their emergence from a drop of rain after one reflection, will make a less angle with the incident rays, than those do which are least refrangible.

Or otherwise: When the rays gi and gp emerge at the same point g, as they both come out of water into air, and consequently are refracted from the perpendicular, instead of going straight forwards in the line eg continued, they will both be turned round upon the point g from the perpendicular go. Now it is easy to conceive, that either of these lines might be turned in this manner upon the point g as upon a centre, till they became parallel to sb the incident ray. But if either of these lines or rays were refracted so much from go as to become parallel to sb, the ray so much refracted, would, after emergence, make no angle with sk, because it would be parallel to it. And consequently that ray which is most turned round upon the point g, or that ray which is most refrangible, will after emergence be nearest parallel to the incident ray, or will make the least angle with it. The same may be proved of all other rays emerging parallel to gi and gp respectively, or of all effectual rays; those which are most refrangible will after emergence make a less angle with the incident rays, than those do which are least refrangible.

But since the effectual rays of different colours make different angles with sk at their emergence, they will be separated from one another: so that if the eye was placed in the beam fg hi, it would receive only rays of one colour from the drop xagv; and if it was placed Of the Rainbow.

In the beam \( f g n p \), it would receive only rays of some other colour.

The angle \( s w p \), which the least refrangible or red rays make with the incident ones when they emerge so as to be effectual, is found by calculation to be 42 degrees 2 minutes. And the angle \( s k i \), which the most refrangible rays make with the incident ones when they emerge so as to be effectual, is found to be 40 degrees 17 minutes. The rays which have the intermediate degrees of refrangibility, make with the incident ones intermediate angles between 42 degrees 2 minutes, and 40 degrees 17 minutes.

If a line is supposed to be drawn from the centre of the sun through the eye of the spectator, the angle which any effectual ray, after two refractions and one reflection, makes with the incident ray, will be equal to the angle which it makes with that line.

Let the eye of the spectator be at \( i \) (fig. 10.) and let \( q t \) be the line supposed to be drawn from the centre of the sun through the eye of the spectator; the angle \( g i t \), which any effectual ray makes with this line, will be equal to the angle \( i k i \), which the same ray makes with the incident ray \( s b \) or \( s k \). If \( s b \) is a ray coming from the centre of the sun, then since \( q t \) is supposed to be drawn from the same point, these two lines, upon account of the remoteness of the point from whence they are drawn, may be looked upon as parallel to one another. But the right line \( k i \) crossing these two parallel lines will make the alternate angles equal. (Eucl. b. I. prop. 29.) Therefore \( k i \) or \( g i t \) is equal to \( s k i \).

When the sun shines upon the drops of rain as they are falling, the rays that come from those drops to the eye of a spectator, after one reflection and two refractions, produce the primary rainbow.

If the sun shines upon the rain as it falls, there are commonly seen two bows, as \( A F B \), \( C H D \), (fig. 11.); or if the cloud and rain does not reach over that whole side of the sky where the bows appear, then only a part of one or of both bows is seen in that place where the rain falls. Of these two bows, the innermost \( A F B \) is the more vivid of the two, and this is called the primary bow. The outer part \( T F Y \) of the primary bow is red, the inner part \( V E X \) is violet; the intermediate parts, reckoning from the red to the violet, are orange, yellow, green, blue, and indigo. Suppose the spectator's eye to be at \( O \), and let \( L O P \) be an imaginary line drawn from the centre of the sun through the eye of the spectator: if a beam of light \( S \) coming from the sun falls upon any drop \( F \); and the rays that emerge at \( F \) in the line \( F O \), so as to be effectual, make an angle \( F O P \) of 42° 2' with the line \( L P \); then these effectual rays make an angle of 42° 2' with the incident rays, by the preceding proposition, and consequently these rays will be red, so that the drop \( F \) will appear red. All the other rays, which emerge at \( F \), and would be effectual if they fell upon the eye, are refracted more than the red ones, and consequently will pass above the eye. If a beam of light \( S \) falls upon the drop \( E \); and the rays that emerge at \( E \) in the line \( E O \), so as to be effectual, make an angle \( E O P \) of 40° 17' with the line \( L P \); then these effectual rays make likewise an angle of 40° 17' with the incident rays, and the drop \( E \) will appear of a violet colour. All the other rays, which emerge at \( E \), and would be effectual if they came to the eye, are refracted less than the violet ones, and therefore pass below the eye. The intermediate drops between \( F \) and \( E \) will for the same reasons be of the intermediate colours.

Thus we have shown why a set of drops from \( F \) to \( E \), as they are falling, should appear of the primary colours, red, orange, yellow, green, blue, indigo, and violet. It is not necessary that the several drops, which produce these colours, should all of them fall at exactly the same distance from the eye. The angle \( F O P \), for instance, is the same whether the distance of the drop from the eye is \( O F \), or whether it is in any other part of the line \( O F \) something nearer to the eye. And whilst the angle \( F O P \) is the same, the angle made by the emerging and incident rays, and consequently the colour of the drop, will be the same. This is equally true of any other drop. So that although in the figure the drops \( F \) and \( E \) are represented as falling perpendicularly one under the other, yet this is not necessary in order to produce the bow.

But the coloured line \( F E \), which we have already accounted for, is only the breadth of the bow. It still remains to be shown, why not only the drop \( F \) should appear red, but why all the other drops quite from \( A \) to \( B \) in the arc \( A T F Y B \) should appear of the same colour. Now it is evident, that wherever a drop of rain is placed, if the angle which the effectual rays make with the line \( L P \) is equal to the angle \( F O P \), that is, if the angle which the effectual rays make with the incident rays is 42° 2', any of those drops will be red, for the same reason that the drop \( F \) is of this colour.

If \( F O P \) was to turn round upon the line \( O P \), so that one end of this line should always be at the eye, and the other be at \( P \) opposite to the sun; such a motion of this figure would be like that of a pair of compasses turning round upon one of the legs \( O P \) with the opening \( F O P \). In this revolution the drop \( F \) would describe a circle, \( P \) would be the centre, and \( A T F Y B \) would be an arc in this circle. Now since, in this motion of the line and drop \( O F \), the angle made by \( F O \) with \( O P \), that is, the angle \( F O P \), continues the same; if the sun was to shine upon this drop as it revolves, the effectual rays would make the same angle with the incident rays, in whatever part of the arc \( A T F Y B \) the drop was to be. Therefore, whether the drop is at \( A \), or at \( T \), or at \( Y \), or at \( B \), or wherever else it is in this whole arc, it would appear red, as it does at \( F \). The drops of rain, as they fall, are not indeed turned round in this manner; but then, as innumerable of them are falling at once in right lines from the cloud, whilst one drop is at \( F \), there will be others at \( Y \), at \( T \), at \( B \), at \( A \), and in every other part of the arc \( A T F Y B \); and all these drops will be red for the same reason that the drop \( F \) would have been red, if it had been in the same place. Therefore, when the sun shines upon the rain as it falls, there will be a red arc \( A T F Y B \) opposite to the sun. In the same manner, because the drop \( E \) is violet, we might prove that any other drop, which, whilst it is falling, is in any part of the arc \( A V E X B \), will be violet; and consequently, at the same time that the red arc \( A T F Y B \) appears, there will likewise appear a violet arc. wife be a violet arc AVEXB below or within it. FE is the distance between these two coloured arcs; and from what has been said, it follows, that the intermediate space between these two arcs will be filled up with arcs of the intermediate colours, orange, yellow, blue, green, and indigo. All these coloured arcs together make up the primary rainbow.

The primary rainbow is never a greater arc than a semicircle.

Since the line LOP is drawn from the sun through the eye of the spectator, and since P (fig. 9) is the centre of the rainbow; it follows, that the centre of the rainbow is always opposite to the sun. The angle FOP is an angle of $42^\circ 2'$, as was observed, or F the highest part of the bow is $42^\circ 2'$ from P the centre of it. If the sun is more than $42^\circ 2'$ high, P the centre of the rainbow, which is opposite to the sun, will be more than $42^\circ 2'$ below the horizon; and consequently F the top of the bow, which is only $42^\circ 2'$ from P, will be below the horizon; that is, when the sun is more than $42^\circ 2'$ high, no primary rainbow will be seen. If the sun is something less than $42^\circ 2'$ high, then P will be something less than $42^\circ 2'$ below the horizon; and consequently F, which is only $42^\circ 2'$ from P, will be just above the horizon; that is, a small part of the bow at this height of the sun will appear close to the ground opposite to the sun. If the sun is $20^\circ$ high, then P will be $20^\circ$ below the horizon; and F the top of the bow, being $42^\circ 2'$ from P, will be $22^\circ 2'$ above the horizon; therefore, at this height of the sun, the bow will be an arc of a circle whose centre is below the horizon; and consequently that arc of the circle which is above the horizon, or the bow, will be less than a semicircle. If the sun is in the horizon, then P, the centre of the bow, will be in the opposite part of the horizon; F, the top of the bow, will be $42^\circ 2'$ above the horizon; and the bow itself, because the horizon passes through the centre of it, will be a semicircle. More than a semicircle can never appear; because if the bow was more than a semicircle, P the centre of it must be above the horizon; but P is always opposite to the sun, therefore P cannot be above the horizon, unless the sun is below it; and when the sun is set, or is below the horizon, it cannot shine upon the drops of rain as they fall; and consequently, when the sun is below the horizon, no bow at all can be seen.

When the rays of the sun fall upon a drop of rain, some of them, after two reflections and two refractions, may come to the eye of a spectator, who has his back towards the sun and his face towards the drop.

If HGW (fig. 12.) is a drop of rain, and parallel rays coming from the sun, as $xv$, $yw$, fall upon the lower part of it, they will be refracted towards the perpendiculars $vl$, $wo$, as they enter into it, and will describe some such lines as $wb$, $wi$. At b and i great part of these rays will pass out of the drop; but some of them will be reflected from thence in the lines $bf$, $ig$. At f and g again, great part of the rays that were reflected thither will pass out of the drop. But these rays will not come to the eye of a spectator at o. However, here again all the rays will not pass out; but some few will be reflected from f and g, in some such lines as $fd$, $gb$; and there, when they emerge out of the drop of water into the air at b and d, will be refracted from the perpendiculars, and, describing the lines $dt$, $bo$, may come to the eye of the spectator who has his back towards the sun and his face towards the drop.

These rays, which are parallel to one another after they have been once refracted and once reflected in a drop of rain, will be effectual when they emerge after two refractions and two reflections.

No rays can be effectual, unless they are contiguous and parallel. From what was said, it appears, that when rays come out of a drop of rain contiguous to one another, either after one or after two reflections, they must enter the drop nearly at one and the same place. And if such rays as are contiguous are parallel after the first reflection, they will emerge parallel, and therefore will be effectual. Let $zw$ and $yw$ be contiguous rays which come from the sun, and are parallel to one another when they fall upon the lower part of the drop, suppose these rays to be refracted at $w$ and $wo$, and to be reflected at $b$ and $i$; if they are parallel to one another, as $bf$, $gi$, after this first reflection, then, after they are reflected a second time from $f$ and $g$, and refracted a second time as they emerge at $d$ and $b$, they will go out of the drop parallel to one another in the lines $dt$ and $bo$, and will therefore be effectual.

The rays $zw$, $yw$, are refracted towards the perpendiculars $wl$, $wo$, when they enter the drop, and will be made to converge. As these rays are very oblique, their focus will not be far from the surface $wo$. If this focus is at $k$, the rays, after they have passed the focus, will diverge from thence in the directions $kb$, $ki$; and if $ki$ is the principal focal distance of the concave reflecting surface $bi$, the reflected rays $bf$, $ig$, will be parallel. These rays $ef$, $ig$, are reflected again from the concave surface $fg$, and will meet in a focus at $e$, so that $ge$ will be the principal focal distance of this reflecting surface $fg$. And because $bi$ and $fg$ are parts of the same sphere, the principal focal distances $ge$ and $ki$ will be equal to one another. When the rays have passed the focus $e$, they will diverge from thence in the lines $ed$, $eb$: and we are to show, that when they emerge at $d$ and $b$, and are refracted there, they will become parallel.

Now if the rays $vk$, $wk$, when they have met at $k$, were to be turned back again in the directions $kv$, $kw$, and were to emerge at $v$ and $w$, they would be refracted into the lines of their incidence, $vw$, $wy$, and therefore would be parallel. But since $ge$ is equal to $ik$, as has already been shown, the rays $ed$, $eb$, that diverge from $e$, fall in the same manner upon the drop at $d$ and $b$, as the rays $kv$, $kw$, would fall upon it at $v$ and $w$; and $ed$, $eb$, are just as much inclined to the refracting surface $db$, as $kv$, $kw$, would be to the surface $vw$. From hence it follows, that the rays $ed$, $eb$, emerging at $d$ and $b$, will be refracted in the same manner, and will have the same direction in respect of one another, as $kv$, $kw$, would have. But $kv$ and $kw$ would be parallel after refraction. Therefore $ed$ and $eb$ will emerge in lines $dp$, $bo$, so as to be parallel to one another, and consequently so as to be effectual. When rays that are effectual emerge from a drop of rain after two reflections and two refractions, those which are most refrangible will at their emergence make a greater angle with the incident rays than those do which are least refrangible; and by this means the rays of different colours will be separated from one another.

If rays of different colours, which are differently refrangible, emerge at any point \( b \) (fig. 12.), these rays will not be all of them equally refracted from the perpendicular. Thus, if \( bo \) is a red ray, which is of all others the least refrangible, and \( bm \) is a violet ray, which is of all others the most refrangible; when these two rays emerge at \( b \), the violet ray will be refracted more from the perpendicular \( bx \) than the red ray, and the refracted angle \( xbm \) will be greater than the refracted angle \( xbo \). From hence it follows, that these two rays, after emergence, will diverge from one another. In like manner, the rays that emerge at \( d \) will diverge from one another; a red ray will emerge in the line \( dp \), a violet ray in the line \( dt \). So that though all the effectual red rays of the beam \( bdm \) are parallel to one another, and all the effectual red rays of the beam \( bdop \) are likewise parallel to one another, yet the violet rays will not be parallel to the red ones, but the violet beam will diverge from the red beam. Thus the rays of different colours will be separated from one another.

This will appear farther, if we consider what the proposition affirms, That any violet or most refrangible ray will make a greater angle with the incident rays, than any red or least refrangible ray makes with the same incident rays. Thus if \( yw \) is an incident ray, \( bm \) a violet ray emerging from the point \( b \), and \( bo \) a red ray emerging from the same point; the angle which the violet ray makes with the incident one is \( yrm \), and that which the red ray makes with it is \( yso \). Now \( yrm \) is a greater angle than \( yso \). For in the triangle \( bry \) the internal angle \( brs \) is less than \( bsy \) the external angle at the base. (Eucl. B. I. prop. 16.) But \( yrm \) is the complement of \( brs \) or of \( bry \) to two right ones, and \( yso \) is the complement of \( bsy \) to two right ones. Therefore, since \( bry \) is less than \( bsy \), the complement of \( bry \) to two right angles will be greater than the complement of \( bsy \) to two right angles; so \( yrm \) will be greater than \( yso \).

Or otherwise: Both the rays \( bo \) and \( bm \), when they are refracted in passing out of the drop at \( b \), are turned round upon the point \( b \) from the perpendicular \( bx \). Now either of these lines \( bo \) or \( bm \) might be turned round in this manner, till it made a right angle with \( yw \). Consequently, that ray which is most turned round upon \( b \), or which is most refracted, will make an angle with \( yw \) that will be nearer to a right one than that ray makes with it which is least turned round upon \( b \), or which is least refracted. Therefore that ray which is most refracted will make a greater angle with the incident ray than that which is least refracted.

But since the emerging rays, as they are differently refrangible, make different angles with the same incident ray \( yw \), the refraction which they suffer at emergence will separate them from one another.

The angle \( yrm \), which the most refrangible or violet rays make with the incident ones, is found by calculation to be \( 54^\circ 7' \); and the angle \( yso \), which the least refrangible or red rays make with the incident ones, is found to be \( 50^\circ 57' \); the angles, which the rays of the intermediate colours, indigo, blue, green, yellow, and orange, make with the incident rays, are intermediate angles between \( 54^\circ 7' \) and \( 50^\circ 57' \).

If a line is supposed to be drawn from the centre of the sun through the eye of the spectator; the angle which, after two refractions and two reflections, any effectual ray makes with the incident ray, will be equal to the angle which it makes with that line.

If \( yw \) (fig. 12.) is an incident ray, \( bo \) an effectual ray, and \( qn \) a line drawn from the centre of the sun through the eye of the spectator; the angle \( yso \), which the effectual ray makes with the incident ray, is equal to \( son \) the angle which the same effectual ray makes with the line \( qn \). For \( yw \) and \( qn \), considered as drawn from the centre of the sun, are parallel; \( bo \) crosses them, and consequently makes the alternate angles \( yso \), \( son \), equal to one another. Eucl. B. I. Prop. 29.

When the sun shines upon the drops of rain as they are falling, the rays that come from these drops to the eye of a spectator, after two reflections and two refractions, produce the secondary rainbow.

The secondary rainbow is the outermost CHD, The figure fig. 11. When the sun shines upon a drop of rain \( H \); dark rain, and the rays \( HO \), which emerge at \( H \) so as to be below produced, make an angle \( HOP \) of \( 54^\circ 7' \) with \( LOP \) aced by two lines drawn from the sun through the eye of the spectator; the same effectual rays will make likewise an refraction angle of \( 54^\circ 7' \) with the incident rays \( S \), and the rays which emerge at this angle are violet ones, by what was observed above. Therefore, if the spectator's eye is at \( O \), none but violet rays will enter it; for all the other rays make a less angle with \( OP \), they will fall above the spectator's eye. In like manner, if the effectual rays that emerge from the drop \( G \) make an angle of \( 50^\circ 57' \) with the line \( OP \), they will likewise make the same angle with the incident rays \( S \); and consequently, from the drop \( G \) to the spectator's eye at \( O \), no rays will come but red ones; for all the other rays, making a greater angle with the line \( OP \), will fall below the eye at \( O \). For the same reason, the rays emerging from the intermediate drops between \( H \) and \( G \), and coming to the spectator's eye at \( O \), will emerge at intermediate angles, and therefore will have the intermediate colours. Thus, if there are seven drops from \( H \) to \( G \) inclusively, their colours will be violet, indigo, blue, green, yellow, orange, and red. This coloured line is the breadth of the secondary rainbow.

Now, if \( HOP \) was to turn round upon the line \( OP \), like a pair of compasses upon one of the legs \( OP \) with the opening \( HOP \), it is plain from the supposition, that, in such a revolution of the drop \( H \), the angle \( HOP \) would be the same, and consequently the emerging rays would make the same angle with the incident ones. But in such a revolution the drop would describe a circle of which \( P \) would be the centre, and CNHRD an arc. Consequently, hence, when the drop is at \( N \), or at \( R \), or anywhere else in that arc, the emerging rays make the same angle with the incident ones as when the drop is at \( H \), the colour of the drop will will be the same to an eye placed at O, whether the drop is at N, or at H, or at R, or anywhere else in that arc. Now, though the drop does not thus turn round as it falls, and does not pass through the several parts of this arc, yet, since there are drops of rain falling everywhere at the same time, when one drop is at H, there will be another at R, another at N, and others in all parts of the arc; and these drops will all of them be violet-coloured, for the same reason that the drop H would have been of this colour if it had been in any of those places. In like manner, as the drop G is red when it is at G, it would likewise be red in any part of the arc CWGQD; and so will any other drop, when, as it is falling, it comes to any part of that arc. Thus as the sun shines upon the rain, whilst it falls, there will be two arcs produced, a violet coloured one CNHRD, and a red one CWGQD; and for the same reasons the intermediate space between these two arcs will be filled up with arcs of the intermediate colours. All these arcs together make up the secondary rainbow.

The colours of the secondary rainbow are fainter than those of the primary rainbow; and are ranged in the contrary order.

The primary rainbow is produced by such rays as have been only once reflected; the secondary rainbow is produced by such rays as have been twice reflected. But at every reflection some rays pass out of the drop of rain without being reflected; so that the fewer the rays are reflected, the fewer of them are left. Therefore the colours of the secondary bow are produced by fewer rays, and consequently will be fainter, than the colours of the primary bow.

In the primary bow, reckoning from the outside of it, the colours are ranged in this order; red, orange, yellow, green, blue, indigo, violet. In the secondary bow, reckoning from the outside, the colours are violet, indigo, blue, green, yellow, orange, red. So that the red, which is the outermost or highest colour in the primary bow, is the innermost or lowest colour in the secondary one.

Now the violet rays, when they emerge so as to be effectual after one reflection, make a less angle with the incident rays than the red ones; consequently the violet rays make a less angle with the lines OP (fig. 11.) than the red ones. But, in the primary rainbow, the rays are only once reflected, and the angle which the effectual rays make with OP is the distance of the coloured drop from P the centre of the bow. Therefore the violet drops or violet arc in the primary bow will be farther from the centre of the bow than the red drops or red arc; that is, the outermost colour in the primary bow will be violet, and the innermost colour will be red. And, for the same reason, through the whole primary bow, every colour will be nearer to the centre P, as the rays of that colour are more refrangible.

But the violet rays, when they emerge so as to be effectual after two reflections, make a greater angle with the incident rays than the red ones; consequently the violet rays will make a greater angle with the line OP, than the red ones. But in the secondary rainbow the rays are twice reflected, and the angle which effectual rays make with OP is the distance of the co-

§ 2. Of Coronas, Parhelia, &c.

Under the articles Corona and Parhelion a pretty full account is given of the different hypotheses concerning these phenomena, and likewise of the method by which these hypotheses are supported, from the known laws of refraction and reflection; to which therefore, in order to avoid repetition, we must refer.

§ 3. Of the apparent Place, Distance, Magnitude, and Motion of Objects.

Philosophers in general had taken for granted, that the place to which the eye refers any visible object seen by reflection or refraction, is that in which the visual ray meets a perpendicular from the object upon the reflecting or refracting plane. But this method of judging of the place of objects was called in question by Dr Barrow, who contended that the arguments brought in favour of the opinion were not conclusive. These arguments are, that the images of theory objects appear straight in a plane mirror, but curved in speaking a convex or concave one; that a straight thread, when partly immersed perpendicularly in water, does not appear crooked as when it is obliquely plunged into the fluid; but that which is within the water seems to be a continuation of that which is without. With respect to the reflected image, however, of a perpendicular right line from a convex or concave mirror, he says, that it is not easy for the eye to distinguish the curve that it really makes; and that, if the appearance of a perpendicular thread, part of which is plunged in water, be closely attended to, it will not favour the common hypothesis. If the thread is of any shining metal, as silver, and viewed obliquely, the image of the part immersed will appear to detach itself sensibly from that part which is without the water, so that it cannot be true that every object appears to be in the same place where the refracted ray meets the perpendicular; and the same observation, he thinks, may be extended to the case of reflection. According to this writer, we refer every point of an object to the place from which the pencils of light, that give us the image of it, issue, or from which they would have issued if no reflecting or refracting substance intervened. Pursuing this principle, he proceeds to investigate the place in which the rays issuing from each of the points of an object, and which reach the eye after one reflection or refraction, meet; and he found, that if the refracting surface was plane, and the refraction was made from a denser medium into a rarer, those rays would always meet in a place between the eye and a perpendicular to the point of incidence. If a convex mirror be used, the case will be the same; but if the mirror be plane, the rays will meet in the perpendicular, and beyond it if it be con- The appa- rent place, &c. of objects.

cave. He also determined, according to these principles, what form the image of a right line will take, when it is presented in different manners to a spherical mirror, or when it is seen through a refracting medium.

Probable as Dr Barrow thought the maxim which he endeavoured to establish, concerning the supposed place of visible objects, he has the candour to mention an objection to it, and to acknowledge that he was not able to give a satisfactory solution of it. It is this. Let an object be placed beyond the focus of a convex lens; and if the eye be close to the lens, it will appear confused, but very near to its true place. If the eye be a little withdrawn, the confusion will increase, and the object will seem to come nearer; and when the eye is very near the focus, the confusion will be exceedingly great, and the object will seem to be close to the eye. But in this experiment the eye receives no rays but those that are converging; and the point from which they issue is so far from being nearer than the object, that it is beyond it; notwithstanding which, the object is conceived to be much nearer than it is, though no very distinct idea can be formed of its precise distance. It may be observed, that in reality, the rays falling upon the eye in this case in a manner quite different from that in which they fall upon it in other circumstances, we can form no judgment about the place from which they issue. This subject was afterwards taken up by Berkeley, Smith, Montucla, and others.

M. de la Hire made several valuable observations concerning the distance of visible objects, and various other phenomena of vision, which are well worth our notice. He also took particular pains to ascertain the manner in which the eye conforms itself to the view of objects placed at different distances. He enumerates five circumstances, which assist us in judging of the distance of objects, namely, their apparent magnitude, the strength of the colouring, the direction of the two eyes, the parallax of the objects, and the distinctness of their small parts. Painters, he says, can only take advantage of the two first mentioned circumstances, and therefore pictures can never perfectly deceive the eye; but in the decorations of theatres, they, in some measure, make use of them all. The size of objects, and the strength of their colouring, are diminished in proportion to the distance at which they are intended to appear. Parts of the same object which are to appear at different distances, as columns in an order of architecture, are drawn upon different planes, a little removed from one another, that the two eyes may be obliged to change their direction, in order to distinguish the parts of the nearer plane from those of the more remote. The small distance of the planes serves to make a small parallax, by changing the position of the eye; and as we do not prefer a distinct idea of the quantity of parallax, corresponding to the different distances of objects, it is sufficient that we perceive there is a parallax, to be convinced that these planes are distant from one another, without determining what that distance is; and as to the last circumstance, viz. the distinctness of the small parts of objects, it is of no use in discovering the deception, on account of the false light that is thrown upon these decorations.

To these observations concerning deceptions of sight, we shall add a similar one of M. le Cat, who took notice that the reason why we imagine objects to be larger when they are seen through a mist, is the real place of dimness or obscurity with which they are then seen; this circumstance being associated with the idea of great distance. This he says is confirmed by our being surprised to find, upon approaching such objects, that they are so much nearer to us, as well as so much smaller, than we had imagined.

Among other cases concerning vision, which fellowships under the consideration of M. de la Hire, he mentions one which is of difficult solution. It is when a candle, in a dark place, and situated beyond the limits of distinct vision, is viewed through a very narrow chink in a card; in which case a considerable number of candles, sometimes so many as six, will be seen along the chink. This appearance he attributes to small irregularities in the surface of the humours of the eye, the effect of which is not sensible when rays are admitted into the eye through the whole extent of the pupil, and consequently one principal image effaces a number of small ones; whereas, in this case, each of them is formed separately, and no one of them is so confusable as to prevent the others from being perceived at the same time.

There are few persons, M. de la Hire observes, who have both their eyes perfectly equal, not only with respect to the limits of distinct vision, but also with respect to the colour with which objects appear tinged when they are viewed by them, especially if one of the eyes has been exposed to the impression of a strong light. To compare them together in this respect, he directs us to take two thin cards, and to make in each of them a round hole of a third or a fourth of a line in diameter, and applying one of them to each of the eyes, to look through the holes on a white paper, equally illuminated; when a circle of the paper will appear to each of the eyes, and, placing the cards properly, these two circles may be made to touch one another, and thereby the appearance of the same object to each of the eyes may be compared to the greatest advantage. To make this experiment with the greatest exactness, it is necessary, he says, that the eyes be kept shut some time before the cards be applied to them.

M. de la Hire first endeavoured to explain the cause of those dark spots which seem to float before the eyes, especially those of old people. They are most visible when the eyes are turned towards an uniform white object, as the snow in the open fields. If they be fixed when the eye is so, this philosopher supposed that they were occasioned by extravasated blood upon the retina. But he thought that the movable spots were occasioned by opaque matter floating in the aqueous humour of the eye. He thought the vitreous humour was not sufficiently limpid for this purpose.

By the following calculation, M. de la Hire gives us an idea of the extreme sensibility of the optic nerves. One may see very easily, at the distance of 4000 toises, the tail of a wind-mill, 6 feet in diameter; and the eye being supposed to be an inch in diameter, the picture of this tail, at the bottom of the eye, will be \( \frac{3}{8} \) of an inch, which is less than the 666th part of a line, and is about the 65th part of a common hair, or the 8th part of a single thread of silk. So small, therefore, must one of the fibres of the optic nerve be, which he says Apparent place, &c., of objects.

Berkeley's account of the judgment formed concerning distance by confusion of vision.

The person who first took much notice of Dr Barrow's hypothesis was the ingenious Dr Berkeley, bishop of Cloyne, who distinguished himself so much by the objections which he started to the reality of a material world, and by his opposition to the Newtonian doctrine of fluxions. In his Essay towards a New Theory of Vision, he observes, that the circle formed upon the retina, by the rays which do not come to a focus, produce the same confusion in the eye, whether they cross one another before they reach the retina, or tend to do it afterwards; and therefore that the judgment concerning distance will be the same in both cases, without any regard to the place from which the rays originally issued; so that in this case, as by receding from the lens, the confusion, which always accompanies the nearness of an object, increases, the mind will judge that the object comes nearer.

But, says Dr Smith, if this be true, the object ought always to appear at a less distance from the eye than that at which objects are seen distinctly, which is not the case: and to explain this appearance, as well as every other in which a judgment is formed concerning distance, he maintains, that we judge of it by the apparent magnitude of objects only, or chiefly; so that, since the image grows larger as we recede from the lens through which it is viewed, we conceive the object to come nearer. He also endeavours to show, that, in all cases in which glasses are used, we judge of distance by the same simple rule; from which he concludes universally, that the apparent distance of an object seen in a glass is to its apparent distance seen by the naked eye as the apparent magnitude to the naked eye is to its apparent magnitude in the glass.

But that we do not judge of distance merely by the angle under which objects are seen, is an observation as old as Alhazen, who mentions several instances, in which, though the angles under which objects appear be different, the magnitudes are universally and instantaneously deemed not to be so. And Mr Robins clearly shows the hypothesis of Dr Smith to be contrary to fact in the most common and simplest cases. In microscopes, he says, it is impossible that the eye should judge the object to be nearer than the distance at which it has viewed the object itself, in proportion to the degree of magnifying. For when the microscope magnifies much, this rule would place the image at a distance, of which the sight cannot possibly form any opinion, as being an interval from the eye at which no object can be seen. In general, he says, he believes, that whoever looks at an object through a convex glass, and then at the object itself without the glass, will find it to appear nearer in the latter case, though it be magnified in the glass; and in the same trial with the concave glass, though by the glass the object be diminished, it will appear nearer through the glass than without it.

But the most convincing proof that the apparent distance of the image is not determined by its apparent magnitude, is the following experiment. If a double convex glass be held upright before some luminous object, as a candle, there will be seen two images, one erect, and the other inverted. The first is made simply by reflection from the nearest surface, the second by reflection from the farther surface, the rays undergoing a refraction from the first surface both before and after the reflection. If this glass has not too short a focal distance, when it is held near the object, the inverted image will appear larger than the other, and also nearer; but if the glass be carried off from the object, though the eye remain as near to it as before, the inverted image will diminish so much faster than the other, that, at length, it will appear very much less than it, but still nearer. Here, says Mr Robins, two images of the same object are seen under one view, and their apparent distances immediately compared; and here it is evident, that those distances have no necessary connection with the apparent magnitude. He also shows how this experiment may be made still more convincing, by sticking a piece of paper on the middle of the lens, and viewing it through a short tube.

M. Bouguer adopts the general maxim of Dr Barrow, in supposing that we refer objects to the place from which the pencils of rays seemingly converge at their entrance into the pupil. But when rays issue from below the surface of a vessel of water, or any other refracting medium, he finds that there are always two different places of this seeming convergence; one of them of the rays that issue from it in the same vertical circle, and therefore fall with different degrees of obliquity upon the surface of the refracting medium; and another, of those that fall upon the surface with the same degree of obliquity, entering the eye laterally with respect to one another. Sometimes, he says, one of these images is attended to by the mind, and sometimes the other, and different images may be observed by different persons. An object plunged in water affords an example, he says, of this duplicity of images.

If BA (fig. 1.) be part of the surface of water, and the object be at O, there will be two images of CCCLXI. it in two different places; one at G, on the caustic by refraction, and the other at E, in the perpendicular AO, which is as much a caustic as the other line. The former image is visible by the rays ODM, O dm, which are one higher than the other, in their progress to the eye; whereas the image at E is made by the rays ODM, O e f, which enter the eye laterally. This, says he, may serve to explain the difficulty of Father Tacquet, Barrow, Smith, and many other authors, and which Newton himself considered as a very difficult problem, though it might not be absolutely insoluble.

G. W. Kraft has ably supported the opinion of Dr Barrow, that the place of any point, seen by reflection from the surface of any medium, is that in which rays issuing from it, infinitely near to one another, would meet; and considering the case of a distant object, viewed in a concave mirror, by an eye very near to it, when the image, according to Euclid and other writers, would be between the eye and the object, and the rule of Dr Barrow cannot be applied; he says that in this case the speculum may be considered as a plane, the effect being the same, only the image is more obscure. Dr Porterfield gives a distinct and comprehensive view of the natural methods of judging concerning the distance of objects.

The conformation of the eye, he observes, can be of no use to us with respect to objects that are placed without the limits of distinct vision. As the object, however, does then appear more or less confused, according as it is more or less removed from those limits, this confusion affects the mind in judging of the distance of the object; it being always deemed so much the nearer, or the farther off, by how much the confusion is greater. But this confusion hath its limits also, beyond which it can never extend; for when an object is placed at a certain distance from the eye, to which the breadth of the pupil bears no sensible proportion, the rays of light that come from a point in the object and pass the pupil, are so little diverging, that they may be considered as parallel. For a picture on the retina will not be sensibly more confused, though the object be removed to a much greater distance.

The most universal, and frequently the most sure means of judging of the distance of objects is, he says, the angle made by the optic axis. For our two eyes are like two different stations, by the assistance of which distances are taken; and this is the reason why those persons who are blind of one eye, so frequently miss their mark in pouring liquor into a glass, snuffing a candle, and such other actions as require that the distance be exactly distinguished. To convince ourselves of the usefulness of this method of judging of the distance of objects, he directs us to suspend a ring in a thread, so that its side may be towards us, and the hole in it to the right and left hand; and taking a small rod, crooked at the end, retire from the ring two or three paces, and having with one hand covered one of our eyes, to endeavour with the other to pass the crooked end of the rod through the ring. This, he says, appears very easy; and yet, upon trial, perhaps once in 100 times we shall not succeed, especially if we move the rod a little quickly.

Our author observes, that by persons recollecting the time when they began to be subject to the maladies above-mentioned, they may tell when it was that they lost the use of one of their eyes; which many persons are long ignorant of, and which may be a circumstance of some consequence to a physician.

The use of this second method of judging of distances De Chales limited to 120 feet; beyond which, he says, we are not sensible of any difference in the angle of the optic axis.

A third method of judging of the distance of objects, consists in their apparent magnitudes, on which so much stress was laid by Dr Smith. From this change in the magnitude of the image upon the retina, we easily judge of the distance of objects, as often as we are otherwise acquainted with the magnitude of the objects themselves; but as often as we are ignorant of the real magnitude of bodies, we can never, from their apparent magnitude, form any judgment of their distance.

From this we may see why we are so frequently deceived in our estimates of distance, by any extraordinary magnitudes of objects seen at the end of it; as, in travelling towards a large city, or a castle, or a cathedral church, or a mountain larger than ordinary, we fancy them to be nearer than we find them to be. This also is the reason why animals, and all small objects, seen in valleys, contiguous to large mountains, appear exceedingly small. For we think the mountain nearer to us than if it were smaller; and we should not be surprised at the smallness of the neighbouring animals, if we thought them farther off. For the same reason, we think them exceedingly small, when they are placed upon the top of a mountain, or a large building; which appear nearer to us than they really are, on account of their extraordinary size.

Dr Jurin clearly accounts for our imagining objects when seen from a high building, to be smaller than seen from they are, and smaller than we fancy them to be when a high building appears them at the same distance on level ground. It is, he says, because we have no distinct idea of distance in that direction, and therefore judge of things by their pictures upon the eye only; but custom will enable us to judge rightly even in this case.

Let a boy, says he, who has never been upon any high building, go to the top of the monument, and look down into the street; the objects seen there, as men and horses, will appear so small as greatly to surprise him. But 10 or 20 years after, if in the meantime he has used himself now and then to look down from that and other great heights, he will no longer find the same objects to appear so small. And if he was to view the same objects from such heights as frequently as he sees them upon the same level with himself in the streets, he supposes that they would appear to him just of the same magnitude from the top of the monument, as they do from a window one story high. For this reason it is, that statues placed upon very high buildings ought to be made of a larger size than those which are seen at a nearer distance; because all persons, except architects, are apt to imagine the height of such buildings to be much less than it really is.

The fourth method by which Dr Porterfield says that we judge of the distance of objects, is the force with which their colour strikes upon our eyes. For if we be assured that two objects are of a similar and like colour, and that one appears more bright and lively than the other, we judge that the brighter object is the nearer of the two.

The fifth method consists in the different appearance of the small parts of objects. When these parts appear distinct, we judge that the object is near; but when they appear confused, or when they do not appear at all, we judge that it is at a greater distance. For the image of any object, or part of an object, diminishes as the distance of it increases.

The sixth and last method by which we judge of the distance of objects is, that the eye does not represent to our mind one object alone, but at the same time all those that are placed betwixt us and the principal object, whose distance we are considering; and the more this distance is divided into separate and distinct parts, the greater it appears to be. For this reason, distances upon uneven surfaces appear less than upon a plane; for the inequalities of the surfaces, such as hills, and holes, and rivers, that lie low and out of sight, either do not appear, or hinder the parts that lie behind them from appearing; and so the whole apparent distance is diminished by the parts that do not not appear in it. This is the reason that the banks of a river appear contiguous to a distant eye, when the river is low and not seen.

Dr Porterfield very well explains several fallacies in vision depending upon our mistaking the distances of objects. Of this kind, he says, is the appearance of parallel lines, and long vistas consisting of parallel rows of trees; for they seem to converge more and more as they are farther extended from the eye. The reason of this, he says, is because the apparent magnitudes of their perpendicular intervals are perpetually diminishing, while, at the same time, we mistake their distance. Hence we may see why, when two parallel rows of trees stand upon an ascent, whereby the more remote parts appear farther off than they really are, because the line that measures the length of the vista now appears under a greater angle than when it was horizontal, the trees, in such a case, will seem to converge less, and sometimes, instead of converging, they will be thought to diverge.

For the same reason that a long vista appears to converge more and more the farther it is extended from the eye, the remoter parts of a horizontal walk or a long floor will appear to ascend gradually; and objects placed upon it, the more remote they are the higher they will appear, till the last be seen on a level with the eye; whereas the ceiling of a long gallery appears to descend towards a horizontal line, drawn from the eye of the spectator. For this reason, also, the surface of the sea, seen from an eminence, seems to rise higher and higher the farther we look; and the upper parts of high buildings seem to slope, or incline forwards over the eye below, because they seem to approach towards a vertical line proceeding from the spectator's eye; so that statues on the top of such buildings, in order to appear upright, must recline, or bend backwards.

Our author also shows the reason why a windmill, seen from a great distance, is sometimes imagined to move the contrary way from what it really does, by our taking the nearer end of the sail for the more remote. The uncertainty we sometimes find in the course of the motion of a branch of lighted candles, turned round at a distance, is owing, he says, to the same cause; as also our sometimes mistaking a convex for a concave surface, more especially in viewing seals and impressions with a convex glass or a double microscope; and lastly, that, upon coming in a dark night into a street, in which there is but one row of lamps, we often mistake the side of the street they are on.

Far more light was thrown upon this curious subject by M. Bouguer.

The proper method of drawing the appearance of two rows of trees that shall appear parallel to the eye, on this subject is a problem which has exercised the ingenuity of several philosophers and mathematicians. That the apparent magnitude of objects decreases with the angle under which they are seen, has always been acknowledged. It is also acknowledged, that it is only by custom and experience that we learn to form a judgment both of magnitudes and distances. But in the application of these maxims to the above-mentioned problem, all persons, before M. Bouguer, made use of the real distance instead of the apparent one; by which only the mind can form its judgment. And it is manifest, that, if any circumstances contribute to make the distance appear otherwise than it is in reality, the apparent magnitude of the object will be affected by it; for the same reason, that, if the magnitude be misapprehended, the idea of the distance will vary.

For want of attending to this distinction, Tacquet pretended to demonstrate, that nothing can give the idea of two parallel lines (rows of trees for instance) to an eye situated at one of their extremities, but two hyperbolic curves, turned the contrary way; and M. Varignon maintained, that in order to make a vista appear of the same width, it must be made narrower, instead of wider, as it recedes from the eye.

M. Bouguer observes, that very great distances, and those that are considerably less than they, make nearly the same impression upon the eye. We, therefore, always imagine great distances to be less than they are; and for this reason the ground plan of a long vista always appears to rise. The visual rays come in a determinate direction; but as we imagine that they terminate sooner than they do, we necessarily conceive that the place from which they issue is elevated. Every large plane, therefore, as AB, (fig. 2.) viewed by an eye at O, will seem to lie in such a direction as CCCLXI. A'b; and consequently lines, in order to appear truly parallel on the plane A'B, must be drawn so as that they would appear parallel on the plane A'b, and be from thence projected to the plane A'B.

To determine the inclination of the apparent ground-plan A'b to the true ground-plan A'B, our ingenious author directs us to draw upon a piece of level ground two straight lines of sufficient length (for which purpose lines fastened to small sticks are very convenient), making an angle of 3 or 4 degrees with one another. Then a person, placing himself within the angle, with his back towards the angular point, must walk backwards and forwards till he can fancy the lines to be parallel. In this situation, a line drawn from the point of the angle thro' the place of his eye, will contain the same angle with the true ground-plan which this does with the apparent one.

M. Bouguer then shows other more geometrical methods of determining this inclination; and says, that by these means he has often found it to be 4 or 5 degrees, though sometimes only 2 or 2½ degrees. The determination of this angle, he observes, is variable, depending upon the manner in which the ground is illuminated and the intensity of the light. The colour of the soil is also not without its influence, as well as the particular conformation of the eye, by which it is more or less affected by the same degree of light, and also the part of the eye on which the object is painted. When, by a slight motion of his head, he contrived, that certain parts of the soil, the image of which fell towards the bottom of his eye, should fall towards the top of the retina, he always thought that this apparent inclination became a little greater.

But what is very remarkable, and what he says he can assure his reader may be depended upon, is, that if he looks towards a rising ground, the difference between the apparent ground-plan and the true one will be much more considerable, so that they will sometimes make an angle of 25 or 30 degrees. Of this he had had made frequent observations. Mountains, he says, begin to be inaccessible when their sides make an angle from 35 to 37 degrees with the horizon, as then it is not possible to climb them but by means of stones or shrubs, to serve as steps to fix the feet on. In these cases, both he and his companions always agreed that the apparent inclination of the side of the mountain was 60 or 70 degrees.

These deceptions are represented in fig. 3, in which, when the ground plan AM, or AN, are much inclined, the apparent ground-plan Am, or An, makes a very large angle with it. On the contrary, if the ground dips below the level, the inclination of the apparent to the true ground-plan diminishes, till, at a certain degree of the slope, it becomes nothing at all; the two plans AP and Ap being the same, so that parallel lines drawn upon them would always appear so. If the inclination below the horizon is carried beyond the situation AP, the error will increase; and what is very remarkable, it will be on the contrary side; the apparent plan Ar being always below the true plan AR, so that if a person would draw upon the plan AR lines that shall appear parallel to the eye, they must be drawn converging, and not diverging, as is usual on the level ground; because they must be the projections of two lines imagined to be parallel, on the plan Ar, which is more inclined to the horizon than AR.

These remarks, he observes, are applicable to different planes exposed to the eye at the same time. For if EH, fig. 4, be the front of a building, at the distance of AB from the eye, it will be reduced in appearance to the distance Ab; and the front of the building will be bb, rather inclined towards the spectator, unless the distance be inconsiderable.

After making a great number of observations upon this subject, our author concludes, that when a man stands upon a level plane, it does not seem to rise sensibly but at some distance from him. The apparent plane, therefore, has a curvature in it, at that distance, the form of which is not very easy to determine; so that a man standing upon a level plane, of infinite extent, will imagine that he stands in the centre of a basin. This is also, in some measure, the case with a person standing upon the level of the sea.

He concludes with observing, that there is no difficulty in drawing lines according to these rules, so as to have any given effect upon the eye, except when some parts of the prospect are very near the spectator, and others very distant from him; because, in this case, regard must be had to the conical or conoidal figure of a surface. A right line passing at a small distance from the observer, and below the level of his eye, in that case almost always appears sensibly curved at a certain distance from the eye; and almost all figures in this case are subject to some complicated optical alteration to which the rules of perspective have not as yet been extended. If a circle be drawn near our feet, and within that part of the ground which appears level to us, it will always appear to be a circle, and at a very considerable distance it will appear an ellipse; but between these two situations, it will not appear to be either the one or the other, but will be like one of those ovals of Descartes, which is more curved on one of its sides than the other.

On these principles a parterre, which appears distorted when it is seen in a low situation, appears perfectly regular when it is viewed from a balcony or any other eminence. Still, however, the apparent irregularity takes place at a greater distance, while the part that is near the spectator is exempt from it. If AB, fig. 5, be the ground plane, and AA a perpendicular, under the eye, the higher it is situated, at O, to the greater distance will T, the place at which the plane begins to have an apparent ascent along Tb, be removed.

All the varieties that can occur with respect to the visible motion of objects, are thus succinctly summed up by Dr Porterfield under eleven heads.

1. An object moving very swiftly is not seen, unless it be very luminous. Thus a cannon-ball is not seen if it is viewed tranversely: but if it be viewed according to the line it describes, it may be seen, because its picture continues long on the same place of the retina; which, therefore, receives a more sensible impression from the object.

2. A live coal swung briskly round in a circle appears a continued circle of fire, because the impressions made on the retina by light, being of a vibrating, and consequently of a lasting nature, do not presently perish, but continue till the coal performs its whole circuit, and returns again to its former place.

3. If two objects, unequally distant from the eye, move with equal velocity, the more remote one will appear the slower; or, if their velocities be proportional to their distances, they will appear equally swift.

4. If two objects, unequally distant from the eye, move with unequal velocities in the same direction, their apparent velocities are in a ratio compounded of the direct ratio of their true velocities, and the reciprocal one of their distances from the eye.

5. A visible object moving with any velocity appears to be at rest, if the space described in the interval of one second be imperceptible at the distance of the eye. Hence it is that a near object moving very slowly, as the index of a clock, or a remote one very swiftly, as a planet, seems to be at rest.

6. An object moving with any degree of velocity will appear at rest, if the space it runs over in a second of time be to its distance from the eye as 1 to 1400.

7. The eye proceeding straight from one place to another, a lateral object, not too far off, whether on the right or left, will seem to move the contrary way.

8. The eye proceeding straight from one place to another, and being sensible of its motion, distant objects will seem to move the same way, and with the same velocity. Thus, to a person running eastwards, the moon on his right hand appears to move the same way, and with equal swiftness; for, by reason of its distance, its image continues fixed upon the same place of the retina, from whence we imagine that the object moves along with the eye.

9. If the eye and the object move both the same way, only the eye much swifter than the object, the last will appear to go backwards.

10. If two or more objects move with the same velocity, and a third remain at rest, the moveable ones will appear fixed, and the quiescent in motion the contrary. contrary way. Thus clouds moving very swiftly, their parts seem to preserve their situation, and the moon to move the contrary way.

11. If the eye be moved with great velocity, lateral objects at rest appear to move the contrary way. Thus to a person sitting in a coach, and riding briskly through a wood, the trees seem to retire the contrary way; and to people in a ship, &c. the shores seem to recede.

At the conclusion of these observations, our author endeavours to explain another phenomenon of motion, which, though very common and well known, had not, as far as he knew, been explained in a satisfactory manner. It is this: If a person turns swiftly round, without changing his place, all objects about will seem to move round in a circle the contrary way; and this deception continues not only while the person himself moves round, but, which is more surprising, it also continues for some time after he ceases to move, when the eye, as well as the object, is at absolute rest.

The reason why objects appear to move round the contrary way, when the eye turns round, is not so difficult to explain: for though, properly speaking, motion is not seen, as not being in itself the immediate object of sight; yet by the sight we easily know when the image changes its place on the retina, and thence conclude that either the object, the eye, or both, are moved. But by the sight alone we can never determine how far this motion belongs to the object, how far to the eye, or how far to both. If we imagine the eye at rest, we ascribe the whole motion to the object, though it be truly at rest. If we imagine the object at rest, we ascribe the whole motion to the eye, though it belongs entirely to the object; and when the eye is in motion, though we are sensible of its motion, yet, if we do not imagine that it moves so swiftly as it really does, we ascribe only a part of the motion to the eye, and the rest of it we ascribe to the object, though it be truly at rest. This last, he says, is what happens in the present case, when the eye turns round; for though we are sensible of the motion of the eye, yet we do not apprehend that it moves so fast as it really does; and therefore the bodies about appear to move the contrary way, as is agreeable to experience.

But the great difficulty still remains, viz. Why, after the eye ceases to move, objects should, for some time, still appear to continue in motion, though their pictures on the retina be truly at rest, and do not at all change their place. This, he imagined, proceeds from a mistake we are in with respect to the eye, which, though it be absolutely at rest, we nevertheless conceive as moving the contrary way to that in which it moved before; from which mistake, with respect to the motion of the eye, the objects at rest will appear to move the same way which the eye is imagined to move; and, consequently, will seem to continue their motion for some time after the eye is at rest.

This is ingenious, but perhaps not just. An account of this matter, which seems to us more satisfactory, has been lately given to the public by Dr Wells. "Some of the older writers upon optics (says this able philosopher) imagined the vivise spirits to be contained in the head, as water is in a vessel; which, therefore, when once put in motion by the rotation of our bodies, must continue in it for some time after this has ceased; and to this real circular movement of the vivise spirits, while the body is at rest, they attributed the apparent motions of objects in giddiness. Dechales saw the weakness of this hypothesis; and conjectured, that the phenomenon might be owing to a real movement of the eyes; but produced no fact in proof of his opinion. Dr Porterfield, on the contrary, supposed the difficulty of explaining it to consist in showing, why objects at rest appear in motion to an eye which is also at rest. The solution he offered of this representation of the phenomenon, is not only extremely ingenious, but is I believe the only probable one which can be given. It does not apply, however, to the fact which truly exists; for I shall immediately show, that the eye is not at rest, as he imagined. The last author I know of who has touched upon this subject is Dr Darwin. His words are, 'When any one turns round rapidly on one foot till he becomes dizzy, and falls upon the ground, the spectra of the ambient objects continue to present themselves in rotation, or appear to librate, and he seems to behold them for some time in motion.' I do not indeed pretend to understand his opinion fully; but this much seems clear, that if such an apparent motion of the surrounding objects depends in any way upon their spectra, or the illusive representations of those objects, occasioned by their former impressions upon the retinas, no similar motion would be observed, were we to turn ourselves round with our eyes shut, and not to open them till we became dizzy; for in this case, as the surrounding objects could not send their pictures to the retinas, there would consequently be no spectra to present themselves afterward in rotation. But whoever will make the experiment, will find, that objects about him appear to be equally in motion, when he has become dizzy by turning himself round, whether this has been done with his eyes open or shut. I shall now venture to propose my own opinion upon this subject.

"If the eye be at rest, we judge an object to be in motion when its picture falls in succeeding times upon different parts of the retina; and if the eye be in motion, we judge an object to be at rest, as long as there is no change in the place of its picture upon the retina. This holds a certain correspondence with the change of the at rest eye's position. Let us now suppose the eye to be in motion, while, from some disorder in the system of sensation, we are either without those feelings which indicate the various positions of the eye, or are not able to attend to them. It is evident, that in such a state of things an object at rest must appear to be in motion, since it finds in succeeding times its picture to different parts of the retina. And this seems to be what happens in giddiness. I was first led to think so from observing, that, during a slight fit of giddiness I was accidentally seized with, a coloured spot, occasioned by looking steadily at a luminous body, and upon which I happened at that moment to be making an experiment, was moved in a manner altogether independent of the positions I conceived my eyes to perform. To determine this point, I again produced the spot, by looking some time at the flame of a candle; then turning myself round till I became dizzy, I suddenly discontinued this motion, and directed my eyes..." to the middle of a sheet of paper, fixed upon the wall place, &c. of my chamber. The spot now appeared upon the paper, but only for a moment; for it immediately after seemed to move to one side, and the paper to the other, notwithstanding I conceived the position of my eyes to be in the mean while unchanged. To go on with the experiment, when the paper and spot had proceeded to a certain distance from each other, they suddenly came together again; and this separation and conjunction were alternately repeated a number of times, the limits of the separation gradually becoming less, till at length the paper and spot both appeared to be at rest, and the latter to be projected upon the middle of the former. I found also, upon repeating and varying the experiment a little, that when I had turned myself from left to right, the paper moved from right to left, and the spot consequently the contrary way; but that when I had turned from right to left, the paper would then move from left to right. These were the appearances observed while I stood erect. When I inclined, however, my head in such a manner as to bring the side of my face parallel to the horizon, the spot and paper would then move from each other, one upward and the other downward. But all these phenomena demonstrate, that there was a real motion in my eyes at the time I imagined them to be at rest; for the apparent situation of the spot, with respect to the paper, could not possibly have been altered, without a real change of the position of those organs. To have the same thing proved in another way, I desired a person to turn quickly round, till he became very giddy; then to stop himself, and look steadfastly at me. He did so, and I could plainly see, that although he thought his eyes were fixed, they were in reality moving in their sockets, first toward one side and then toward the other."

M. Le Cat well explains a remarkable deception, by which a person shall imagine an object to be on the opposite side of a board, when it is not so, and also inverted and magnified. It is illustrated by fig. 6, in which D represents the eye, and CB a large black board, pierced with a small hole. E is a large white board, placed beyond it, and strongly illuminated; and d a pin, or other small object, held betwixt the eye and the first board. In these circumstances, the pin shall be imagined to be at F, on the other side of the board, where it will appear inverted and magnified; because what is in fact perceived, is the shadow of the pin upon the retina; and the light that is flopped by the upper part of the pin coming from the lower part of the enlightened board, and that which is flopped by the lower part coming from the upper part of the board, the shadow must necessarily be inverted with respect to the object.

There is a curious phenomenon relating to vision, which some persons have ascribed to the inflection of light, but which Mr Melville explains in a very different and very simple manner.

When any opaque body is held at the distance of three or four inches from the eye, so that a part of some more distant luminous object, such as the window, or the flame of a candle, may be seen by rays passing near its edge, if another opaque body, nearer to the eye, be brought across from the opposite side, the edge of the first body will seem to swell outwards, and meet the latter; and in doing so will intercept a portion of the luminous object that was seen before.

This appearance he explains in the following manner: Let AB (fig. 7.) represent the luminous object to which the sight is directed, CD the more distant opaque body, GH the nearer, and EF the diameter of the pupil. Join ED, FD, EG, FG, and produce them till they meet AB in K, N, M, and L. It is plain that the parts AN, MB, of the luminous object cannot be seen. But taking any point a between N and K, and drawing a D d, since the portion d F of the pupil is filled with light flowing from that point, it must be visible. Any point b, between a and K, must fill f F, a greater portion of the pupil, and therefore must appear brighter. Again, any point c, between b and K, must appear brighter than b, because it fills a greater portion g F with light. The point K itself, and every other point in the space KL, must appear very luminous, since they send entire pencils of rays EKF, ELF, to the eye; and the visible brightness of every point from L towards M, must decrease gradually, as from K to N, that is, the spaces KN, LM, will appear as dim shadowy borders, or fringes, adjacent to the edges of the opaque bodies.

When the edge G is brought to touch the right line KE, the penumbras unite; and as soon as it reaches ND, the above phenomenon begins; for it cannot pass that right line without meeting some line a D d, drawn from a point between N and K, and, by intercepting all the rays that fall upon the pupil, render it invisible. In advancing gradually to the line KDE, it will meet other lines b D f, c D g, &c. and therefore render the points b, c, &c. from N to K, successively invisible; and therefore the edge of the fixed opaque body CD must seem to swell outwards, and cover the whole space NK; while GH, by its motion, covers MK. When GH is placed at a greater distance from the eye, CD continuing fixed, the space OP to be passed over in order to intercept NK is less; and therefore, with an equal motion of GH, the apparent swelling of CD must be quicker; which is found true by experience.

If ML represent a luminous object, and REFQ any plane exposed to its light, the space FQ will be entirely shaded from the rays, and the space FE will be occupied by a penumbra, gradually darker, from E to F. Let now GH continue fixed, and CD move parallel to the plane EF; and as soon as it passes the line LF, it is evident that the shadow QF will seem to swell outwards; and when CD reaches ME, so as to cover with its shadow the space RE, QF, by its extension, will cover FE. This is found to hold true likewise by experiment.

§ 4. Of the Concave Figure of the Sky.

This apparent concavity is only an optical deception founded on the incapacity of our organs of vision to take in very large distances.—Dr Smith, in his History on Complete System of Optics, hath demonstrated, that if the surface of the earth was perfectly plane, the distance of the visible horizon from the eye would scarce exceed the distance of 5000 times the height of the eye above the ground, supposing the height of the eye between five and six feet: beyond this distance, Concavity all objects would appear in the visible horizon. For, of the sky; let OP be the height of the eye above the line PA drawn upon the ground; and if an object AB, equal in height to PO, be removed to a distance PA equal to 5000 times that height, it will hardly be visible by reason of the smallness of the angle AOB. Consequently any distance AC, how great soever, beyond A, will be invisible. For since AC and BO are parallel, the ray CO will always cut AB in some point D between A and B; and therefore the angle AOC, or AOD, will always be less than AOB, and therefore AD or AC will be invisible. Consequently all objects and clouds, as CE and FG, placed at all distances beyond A, if they be high enough to be visible, or to subtend a bigger angle at the eye than AOB, will appear at the horizon AB; because the distance AC is invisible.

Hence, if we suppose a vast long row of objects, or a vast long wall ABZY (fig. 9.), built upon this plane, and its perpendicular distance OA from the eye at O to be equal to or greater than the distance Oa of the visible horizon, it will not appear straight, but circular, as if it was built upon the circumference of the horizon aczy; and if the wall be continued to an immense distance, its extreme parts YZ will appear in the horizon at yz, where it is cut by a line Oy parallel to the wall. For, supposing a ray YO, the angle YOy will become insensibly small. Imagine this infinite plane OAYy, with the wall upon it, to be turned about the horizontal line O like the lid of a box, till it becomes perpendicular to the other half of the horizontal plane LMy, and the wall parallel to it, like a vast ceiling over head; and then the wall will appear like the concave figure of the clouds overhead. But though the wall in the horizon appear in the figure of a semicircle, yet the ceiling will not, but much flatter. Because the horizontal plane was a visible surface, which suggested the idea of the same distances quite round the eye: but in the vertical plane extended between the eye and the ceiling, there is nothing that affects the sense with an idea of its parts but the common line Oy; consequently the apparent distances of the higher parts of the ceiling will be gradually diminished in ascending from that line. Now when the sky is quite overcast with clouds of equal gravities, they will all float in the air at equal heights above the earth, and consequently will compose a surface resembling a large ceiling, as flat as the visible surface of the earth. Its concavity therefore is not real, but apparent: and when the heights of the clouds are unequal, since their real shapes and magnitudes are all unknown, the eye can seldom distinguish the unequal distances of those clouds that appear in the same directions, unless when they are very near us, or are driven by contrary currents of the air. So that the visible shape of the whole surface remains alike in both cases. And when the sky is either partly overcast or partly free from clouds, it is matter of fact that we retain much the same idea of its concavity as when it was quite overcast.

The concavity of the heavens appears to the eye, which is the only judge of an apparent figure, to be a less portion of a spherical surface than a hemisphere. Dr Smith says, that the centre of the concavity is much below the eye; and by taking a medium among several observations, he found the apparent distance Blue colour of its parts at the horizon to be generally between three of the sky, and four times greater than the apparent distance of its parts overhead. For let the arch ABCD represent the apparent concavity of the sky, O the place of the eye, OA and OC the horizontal and vertical apparent distances, whose proportion is required. First observe when the sun or the moon, or any cloud or star, is in such a position at B, that the apparent arches BA, BC, extended on each side of this object towards the horizon and zenith, seem equal to the eye; then taking the altitude of the object B with a quadrant, or a cross-staff, or finding it by astronomy from the given time of observation, the angle AOB is known. Drawing therefore the line OB in the position thus determined, and taking in it any point B at pleasure, in the vertical line CO produced downwards, seek the centre E of a circle ABC, whose arches BA, BC, intercepted between B and the legs of the right angle AOC, shall be equal to each other; then will this arch ABCD represent the apparent figure of the sky. For by the eye we estimate the distance between any two objects in the heavens by the quantity of sky that appears to lie between them; as upon earth we estimate it by the quantity of ground that lies between them. The centre E may be found geometrically by constructing a cubic equation, or as quickly and sufficiently exact by trying whether the chords BA, BC, of the arch ABC drawn by conjecture are equal, and by altering its radius BE till they are so. Now in making several observations upon the sun, and some others upon the moon and stars, they seemed to our author to bisect the vertical arch ABC at B, when their apparent altitudes or the angle AOB was about 23 degrees; which gives the proportion of OC to OA as 3 to 10 or as 1 to 3 nearly. When the sun was but 30 degrees high, the upper arch seemed always less than the under one; and, in our author's opinion, always greater when the sun was about 18 or 20 degrees high.

§ 5. Of the Blue Colour of the Sky, and of Blue and Green Shadows.

The opinions of ancient writers concerning the colour of the sky merit no notice. The first who gave any rational explanation was Fromondus. By him it was supposed, that the blueness of the sky proceeded from a mixture of the white light of the sun with the black space beyond the atmosphere, where there is neither refraction nor reflection. This opinion prevailed very generally even in modern times, and was maintained by Otto Guerick and all his contemporaries, who asserted that white and black may be mixed in such a manner as to make a blue. Mr Bouguer had recourse to the vapours diffused through the atmosphere, to account for the reflection of the blue rays rather than any other. He seems however to suppose, that it arises from the constitution of the air itself, whereby the fainter coloured rays are incapable of making their way through any considerable tract of it. Hence he is of opinion, that the colour of the air is properly blue; to which opinion Dr Smith seems also to have inclined.

To this blue colour of the sky is owing the appearance of blue and green shadows in the mornings and evenings. Blue colour evenings.—These were first taken notice of by M. Buffon in the month of July 1742, when he observed that the shadows of trees which fell upon a white wall were green. He was at that time standing upon an obser- vation-eminence, and the sun was setting in the cleft of a mountain, so that he appeared considerably lower than the horizon. The sky was clear, excepting in the west, which, though free from clouds, was lightly shaded with vapours, of a yellow colour, inclining to red. Then the sun itself was exceedingly red, and was seemingly at least four times as large as he appears to be at mid-day. In these circumstances, he saw very distinctly the shadows of the trees, which were 30 or 40 feet from the white wall, coloured with a light green, inclining to blue. The shadow of an arbour, which was three feet from the wall, was exactly drawn upon it, and looked as if it had been newly painted with verdigris. This appearance lasted near five minutes; after which it grew fainter, and vanished at the same time with the light of the sun.

The next morning, at sunrise, he went to observe other shadows, upon another white wall; but instead of finding them green, as he expected, he observed that they were blue, or rather of the colour of lively indigo. The sky was serene, except a slight covering of yellowish vapours in the east; and the sun arose behind a hill, so that it was elevated above his horizon. In these circumstances, the blue shadows were only visible three minutes; after which they appeared black, and in the evening of the same day he observed the green shadows exactly as before. Six days passed without his being able to repeat his observations, on account of the clouds; but the 7th day, at sunset, the shadows were not green, but of a beautiful sky-blue. He also observed, that the sky was in a great measure free from vapours at that time; and that the sun set behind a rock, so that it disappeared before it came to his horizon. Afterwards he often observed the shadows both at sunrise and sunset; but always observed them to be blue, though with a great variety of shades of that colour. He showed this phenomenon to many of his friends, who were as much surprised at it as he himself had been; but he says that any person may see a blue shadow, if he will only hold his finger before a piece of white paper at sunrise or sunset.

The first person who attempted to explain this phenomenon was the Abbé Mazées, in a memoir of the society in Berlin for the year 1752. He observed, that when an opaque body was illuminated by the moon and a candle at the same time, and the two shadows were cast upon the same white wall, that which was enlightened by the candle was reddish, and that which was enlightened by the moon was blue. But, without attending to any other circumstances, he supposed the change of colour to be occasioned by the diminution of the light; but M. Melville, and M. Bouguer, both independent of one another, seem to have hit upon the true cause of this curious appearance, and which hath been already hinted at. The former of these gentlemen, in his attempts to explain the blue colour of the sky, observes, that since it is certain that no body assumes any particular colour, but because it reflects one sort of rays more abundantly than the rest; and since it cannot be supposed that the Blue colour constituent parts of pure air are grofs enough to separate any colours of themselves; we must conclude with Sir Isaac Newton, that the violet and blue making rays are reflected more copiously than the rest, by the finer vapours diffused through the atmosphere, whose parts are not big enough to give them the appearance of visible opaque clouds. And he shows, that in proper circumstances, the bluish colour of the sky-light may be actually seen on bodies illuminated by it, as, he says, it is objected should always happen upon this hypothesis. For that if, on a clear cloudless day, a sheet of white paper be exposed to the sun's beams, when any opaque body is placed upon it, the shadow which is illuminated by the sky only will appear remarkably bluish compared with the rest of the paper, which receives the sun's direct rays.

M. Bouguer, who has taken the most pains with this subject, observes, that as M. Buffon mentions the shadows appearing green only twice, and that at all other times they were blue, this is the colour which they regularly have, and that the blue was changed into green by some accidental circumstance. Green, he says, is only a composition of blue and yellow, so that this accidental change may have arisen from the mixture of some yellow rays in the blue shadow; and that perhaps the wall might have had that tinge, so that the blue is the only colour for which a general reason is required. And this, he says, must be derived from the colour of pure air, which always appears blue, and which always reflects that colour upon all objects without distinction; but which is too faint to be perceived when our eyes are strongly affected by the light of the sun, reflected from other objects around us.

To confirm this hypothesis, he adds some curious observations of his own, in which this appearance is agreeably diversified. Being at the village of Boucholtz in July 1764, he observed the shadows projected on the white paper of his pocket-book, when the sky was clear. At half an hour past 6 in the evening, when the sun was about four degrees high, he observed that the shadow of his finger was of a dark grey, while he held the paper opposite to the sun; but when he inclined it a little horizontally, the paper had a bluish cast, and the shadow upon it was of a beautiful bright blue.

When his eye was placed between the sun and the paper laid horizontally, it always appeared of a bluish cast; but when he held the paper thus inclined between his eye and the sun, he could distinguish, upon every little eminence occasioned by the inequality of the surface of the paper, the principal of the prismatic colours. He also perceived them upon his nails, and upon the skin of his hand. This multitude of coloured points, red, yellow, green, and blue, almost effaced the natural colour of the objects.

At three quarters past six, the shadows began to be blue, even when the rays of the sun fell perpendicularly. The colour was the most lively when the rays fell upon it at an angle of 45 degrees; but with a less inclination of the paper, he could distinctly perceive, that the blue shadow had a border of a stronger blue on that side which looked towards the sky, and a red border on that side which was turned towards the earth. To see these borders, the body that made the shadow shadow was obliged to be placed very near the paper; and the nearer it was the more sensible was the red border. At the distance of three inches, the whole shadow was blue. At every observation, after having held the paper towards the sky, he turned it towards the earth, which was covered with verdure; holding it in such a manner, that the sun might shine upon it while it received the shadows of various bodies; but in this position he could never perceive the shadow to be blue or green at any inclination with respect to the sun's rays.

At seven o'clock, the sun being still about two degrees high, the shadows were of a bright blue, even when the rays fell perpendicularly upon the paper, but were the brightest when it was inclined at an angle of 45°. At this time he was surprised to observe, that a large tract of sky was not favourable to this blue colour, and that the shadow falling upon the paper placed horizontally was not coloured, or at least the blue was very faint. This singularity, he concluded, arose from the small difference between the light of that part of the paper which received the rays of the sun and that which was in the shade in this situation. In a situation precisely horizontal, the difference would vanish, and there could be no shadow. Thus too much or too little of the sun's light produced, but for different reasons, the same effect; for they both made the blue light reflected from the sky to become insensible. This gentleman never saw any green shadows, but when he made them fall on yellow paper. But he does not absolutely say, that green shadows cannot be produced in any other manner; and supposes, that if it was on the same wall that M. Buffon saw the blue shadows, seven days after having seen the green ones, the cause of it might be the mixture of yellow rays, reflected from the vapours, which he observed were of that colour.

These blue shadows, our author observes, are not confined to the times of the sun-rising and sun-setting; on the 19th of July, when the sun has the greatest force, he observed them at three o'clock in the afternoon, but the sun shone through a mist at that time.

If the sky is clear, the shadows begin to be blue; when, if they be projected horizontally, they are eight times as long as the height of the body that produces them, that is, when the centre of the sun is 7° 8' above the horizon. This observation, he says, was made in the beginning of August.

Besides these coloured shadows, which are produced by the interception of the direct rays of the sun, our author observed others similar to them at every hour of the day, in rooms into which the light of the sun was reflected from some white body, if any part of the clear sky could be seen from the place, and all unnecessary light was excluded as much as possible. Observing these precautions, he says that the blue shadows may be seen at any hour of the day, even with the direct light of the sun; and that this colour will disappear in all those places of the shadow from which the blue sky cannot be seen.

All the observations that our author made upon the yellow or reddish borders of shadows above-mentioned, led him to conclude, that they were occasioned by the interception of the sky-light, whereby part of the shadow was illuminated either by the red rays reflected from the clouds, when the sun is near the horizon, or from some terrestrial bodies in the neighbourhood.

This conjecture is favoured by the necessity he was under of placing any body near the paper, in order to produce this bordered shadow, as he says it is easily demonstrated, that the interception of the sky-light can only take place when the breadth of the opaque body is to its distance from the white ground on which the shadow falls, as twice the fine of half the amplitude of the sky to its cosine.

At the conclusion of his observations on these blue shadows, he gives a short account of another kind of shadows, which, he does not doubt, have the same origin. These he often saw early in the spring when he was reading by the light of a candle in the morning, and consequently the twilight mixed with that of his candle. In these circumstances, the shadow that was made by intercepting the light of his candle, at the distance of about five feet, was of a beautiful and clear blue, which became deeper as the opaque body which made the shadow was brought nearer to the wall, and was exceedingly deep at the distance of a few inches only. But wherever the day-light did not come, the shadows were all black without the least mixture of blue.

§ 6. Of the Irradiations of the Sun's Light appearing through the interstices of the Clouds.

This is an appearance which every one must have observed when the sky was pretty much overcast with clouds at some distance from each other. At that time several large beams of light, something like the appearance of the light of the sun admitted into a smoky room, will be seen generally with a very considerable degree of divergence, as if the radiant point was situated at no great distance above the clouds. Dr Smith observes that this appearance is one of those which serve to demonstrate that very high and remote objects in the heavens do not appear to us in their real shapes and positions, but according to their perspective projections on the apparent concavity of the sky. He acquaints us, that though these beams are generally seen diverging, as represented in fig. 11, it is not always the case. He himself, in particular, once saw them converging towards a point diametrically opposite to the sun; for, as near as he could conjecture, the point to which they converged was situated as much below the horizon as the sun was then elevated above it. This part is represented by the line \( tD \), and the point below it in opposition to the sun is \( E \); towards which all the beams \( v \), \( v \), &c., appeared to converge.

"Observing (says our author) that the point of convergence was opposite to the sun, I began to suspect that this unusual phenomenon was but a case of the usual apparent divergence of the beams of the sun from his apparent place among the clouds, as represented in fig. 11. I say an apparent divergence; for though nothing is more common than for rays to diverge from a luminous body, yet the divergence of these beams in such large angles is not real, but apparent. Because it is impossible for the direct rays of the sun to cross one another at any point of the apparent concavity of the sky, in a greater angle than about half a degree. For

Vol. XII. Part I. Irradiations: the diameter of the earth being so extremely small, in comparison to the distance of the sun, as to subtend an angle at any point of his body of but 20 or 22 seconds at most; and the diameter of our visible horizon being extremely smaller than that of the earth; it is plain, that all the rays which fall upon the horizon from any given point of the sun, must be inclined to each other in the smallest angles imaginable: the greatest of them being as much smaller than that angle of 22 seconds, as the diameter of the visible horizon is smaller than that of the earth. All the rays that come to us from any given point of the sun may therefore be considered as parallel to each other; as the rays \(eB_g\) from the point \(e\), or \(fB_h\) from the opposite point \(f\); and consequently the rays of these two pencils that come from opposite points of the sun's real diameter, and cross each other in the sun's apparent place \(B\) among the clouds, can constitute no greater an angle with each other than about half a degree; this angle of their intersection \(eB_f\) being the same as the sun would appear under an eye placed among the clouds at \(B\), or (which is much the same) to an eye at \(O\) upon the ground. Because the sun's real distance \(OS\) is inconceivably greater than his apparent distance \(OB\). Therefore the rays of the sun, as \(B_g\), \(B_h\), do really diverge from his apparent place \(B\) in no greater angles \(gB_h\) than about half a degree. Nevertheless they appear to diverge from the place \(B\) in all possible angles, and even in opposite directions. Let us proceed then to an explanation of this apparent divergence, which is not self-evident by any means; though at first sight we are apt to think it is, by not distinguishing the vast difference between the true and apparent distances of the sun.

What I am going to demonstrate is this. Supposing all the rays of the sun to fall accurately parallel to each other upon the visible horizon, as they do very nearly, yet in both cases they must appear to diverge in all possible angles. Let us imagine the heavens to be partly overcast with a spacious bed of broken clouds, \(v_1\), \(v_2\), &c., lying parallel to the plane of the visible horizon, here represented by the line \(AOD\); and when the sun's rays fall upon these clouds in the parallel lines \(v_1\), \(v_2\), &c., let some of them pass through their intervals in the lines \(w_1\), \(w_2\), &c., and fall upon the plane of the horizon at the places \(t_1\), \(t_2\), &c. And since the rest of the incident rays \(v_3\), \(v_4\), &c., are supposed to be intercepted from the place of the spectator at \(O\) by the cloud \(x\), and from the intervals between the transmitted rays \(w_1\), \(w_2\), &c., by the clouds \(v_1\), \(v_2\), &c., a small part of these latter rays \(w_1\), \(w_2\), &c., when reflected every way from some certain kind of thin vapours floating in the air, may undoubtedly be sufficient to affect the eye with an appearance of lights and shades, in the form of bright beams in the places \(t_1\), \(t_2\), &c., and of dark ones in the intervals between them; just as the like beams of light and shade appear in a room by reflections of the sun's rays from a smoky or dusty air within it; the lights and shades being here occasioned by the transmission of the rays through some parts of the window, and by their interruption at other parts.

Now, if the apparent concavity of this bed of clouds \(v_1\), \(v_2\), to the eye at \(O\), be represented by the arch \(ABCD\), and be cut in the point \(B\) by the line \(OB_w\) drawn parallel to the beams \(t_1\); it will be evident by the rules of perspective, that these long beams will not irradiate appear in their real places, but upon the concave \(AB\) of the sun's CD diverging every way from the place \(B\), where the fun himself appears, or the cloud \(x\) that covers his body, as represented separately in full view in fig. 11.

And for the same reason, if the line \(BO\) be produced towards \(E\), below the plane of the horizon \(AOD\), and the eye be directed towards the region of the sky directly above \(E\), the lower ends of the same real beams \(w_1\), \(w_2\), will now appear upon the part \(DF\) of this concave; and will seem to converge towards the point \(E\), situated just as much below the horizon as the opposite point \(B\) is above it: which is separately represented, in full view in fig. 12.

For if the beams \(w_1\), \(w_2\) be supposed to be visible throughout their whole lengths, and the eye be directed in a plane perpendicular to them, here represented by the line \(OF\); they and their intervals will appear broadest in and about this plane, because these parts of them are the nearest to the eye; and therefore their remoter parts and intervals will appear gradually narrower towards the opposite ends of the line \(BE\).

As a farther illustration of this matter, we may conceive the spectator at \(O\) to be situated upon the top of so large a descent \(OHI\) towards a remote valley \(IK\), and the sun to be so very low, that the point \(E\), opposite to him, may be seen above the horizon of this shady valley. In this case it is manifest, that the spectators at \(O\) would now see these beams converging so far as to meet each other at the point \(E\) in the sky itself.

I do not remember to have ever seen any phenomenon of this kind by moon-light; nor so much as of light by beams diverging from her apparent place. Probably moon-light is too weak after reflections from any kind of vapours, to cause a sensible appearance of lights and shades so as to form these beams. And in the unusual phenomenon I well remember, that the converging fun-beams towards the point below the horizon were not quite so bright and strong as those usually are that diverge from him; and that the sky beyond them appeared very black (several showers having passed that way), which certainly contributed to the evidence of this appearance. Hence it is probable that the thinness and weakness of the reflected rays from the vapours opposite to the sun, is the chief cause that this appearance is so very uncommon in comparison to that other of diverging beams. For as the region of the sky round about the sun is always brighter than the opposite one, so the light of the diverging beams ought also to be brighter than that of the converging ones. For, though rays are reflected from rough unpolished bodies in all possible directions, yet it is a general observation, that more of them are reflected forward obliquely, than are reflected more directly backwards. Besides, in the present case, the incident rays upon the opposite region to the sun, are more diminished by continual reflections from a longer tract of the atmosphere, than the incident rays upon the region next the sun.

The common phenomenon of diverging beams, I think, is more frequent in summer than in winter; and diverging also when the sun is lower than when higher up; probably because the lower vapours are denser, and therefore more strongly reflective than the higher; because in winter the lower sky-light is not so bright as the upper. § 7. Of the Illumination of the Shadow of the Earth by the refraction of the Atmosphere.

The ancient philosophers, who knew nothing of the refractive power of the atmosphere, were very much perplexed to account for the body of the moon being visible when totally eclipsed. At such times she generally appears of a dull red colour, like tarnished copper, or of iron almost red-hot. This, they thought, was the moon's native light, by which she became visible when hid from the brighter light of the sun. Plutarch indeed, in his discourse upon the face of the moon, attributes this appearance to the light of the fixed stars reflected to us by the moon; but this must be far too weak to produce that effect. The true cause of it is the scattered beams of the sun bent into the earth's shadow by refractions through the atmosphere in the following manner.

Let the body of the sun, says Dr Smith, be represented by the greater circle \(ab\), and that of the earth by the lesser one \(cd\); and let the lines \(ace\) and \(bde\) touch them both on their opposite sides, and meet in \(e\) beyond the earth; then the angular space \(ced\) will represent the conic figure of the earth's shadow, which would be totally deprived of the sun's rays, were none of them bent into it by the refractive power of the atmosphere. Let this power just vanish at the circle \(hi\), concentric to the earth, so that the rays \(ah\) and \(bi\), which touch its opposite sides, may proceed unrefracted, and meet each other at \(k\). Then the two nearest rays to these that flow within them, from the same points \(a\) and \(b\), being refracted inwards through the margin of the atmosphere, will cross each other at a point \(l\), somewhat nearer to the earth than \(k\); and in like manner, two opposite rays next within the two last will cross each other at a point \(m\), somewhat nearer to the earth than \(l\), having suffered greater refractions, by passing through longer and denser tracts of air lying somewhat nearer to the earth. The like approach of the successive intersections \(k, l, m\), is to be understood of innumerable couples of rays, till you come to the intersection \(n\) of the two innermost; which we may suppose just to touch the earth at the points \(o\) and \(p\). It is plain then, that the space bounded by these rays \(on, np\) will be the only part of the earth's shadow wholly deprived of the sun's rays. Let \(fmg\) represent part of the moon's orbit when it is nearest to the earth, at a time when the earth's dark shadow \(onp\) is the longest; in this case I will show that the ratio of \(tm\) to \(tn\) is about 4 to 3; and consequently that the moon, though centrally eclipsed at \(m\), may yet be visible by means of those scattered rays above-mentioned, first transmitted to the moon by refraction through the atmosphere, and from thence reflected to the earth.

For let the incident and emergent parts \(aq, rn\), of the ray \(aqorn\), that just touches the earth at \(o\), be produced till they meet at \(u\), and let \(aqu\) produced meet the axis \(st\) produced in \(x\); and joining \(ux\) and \(um\), since the refractions of an horizontal ray passing from \(o\) to \(r\), or from \(o\) to \(q\), would be alike and equal, the external angle \(ux\) is double the quantity of the usual refraction of an horizontal ray; and the angle \(aus\) is the apparent measure of the sun's semidiameter seen from the earth; and the angle \(ust\) is that of the earth's semidiameter seen from the sun (called his horizontal parallax); and lastly, the angle \(umu\) is that of the earth's semidiameter seen from the moon (called her horizontal parallax); because the elevation of the point \(u\) above the earth is too small to make a sensible error in the quantity of these angles; whose measures by astronomical tables are as follow:

| Angle | Value | |-------|-------| | \(aus\) | 15°50' | | \(ust\) | 90°10' | | \(tux\) | 15°40' | | \(nux\) | 67°30' | | \(tnu\) | 83°10' | | \(tmu\) | 62°10' |

Therefore (by a preceding prop.) we have \(tm : tn :: (ang. \(tnu\) : ang. \(tnu\) : \(83° - 10'\) : \(62° - 10'\)) : 4 : 3, in round numbers; which was to be proved. It is easy to collect from the moon's greatest horizontal parallax of \(62° - 10'\), that her least distance \(tm\) is about \(55\frac{1}{2}\) semidiameters of the earth; and therefore the greatest length \(tn\) of the dark shadow, being three quarters of \(tm\), is about \(41\frac{1}{2}\) semidiameters.

The difference of the last mentioned angles \(tnu, tmu\), is \(mun = 21'\), that is, about two thirds of \(3° - 40'\), the angle which the whole diameter of the sun subtends at \(u\). Whence it follows, that the middle point \(m\) of the moon centrally eclipsed, is illuminated by rays which come from two thirds of every diameter of the sun's disk, and pass by one side of the earth; and also by rays that come from the opposite two thirds of every one of the said diameters, and pass by the other side of the earth. This will appear by conceiving the ray \(aqorn\) to be inflexible, and its middle point \(o\) to slide upon the earth, while the part \(rn\) is approaching to touch the point \(m\); for then the opposite part \(qa\) will trace over two thirds of the sun's diameter. The true proportion of the angles \(nmu, aus\), could not be preserved in the scheme, by reason of the sun's immense distance and magnitude with respect to the earth.

Having drawn the line \(atx\), it is observable, that all the incident rays, as \(aq, ax\), flowing from any one point of the sun to the circumference of the earth, will be collected to a focus \(x\), whose distance \(tx\) is less than \(tm\) in the ratio of 62 to 67 nearly; and thus an image of the sun will be formed at \(x\), whose rays will... will diverge upon the moon. For the angle \( t \times u \) is the difference of the angles \( x \times u \), \( u \times t \) found above; and \( t \times u \): \( t \times m \); \( t \times m \): \( t \times u \): \( 62^\circ - 10' = 67^\circ - 30' \).

"The rays that flow next above \( a \) and \( a' \), by passing through a thinner part of the atmosphere, will be united at a point in the axis \( a \times a' \) somewhat farther from the earth than the last focus \( s \); and the same may be said of the rays that pass next above these, and so on; whereby an infinite series of images of the sun will be formed, whose diameters and degrees of brightness will increase with their distances from the earth.

"Hence it is manifest why the moon eclipsed in her perigee is observed to appear always duller and darker than in her apogee. The reason why her colour is always of the copper kind between a dull red and orange, I take to be this. The blue colour of a clear sky shows manifestly that the blue-making rays are more copiously reflected from pure air than those of any other colour; consequently they are less copiously transmitted through it among the rest that come from the sun, and so much the less as the tract of air through which they pass is the longer. Hence the common colour of the sun and moon is whitest in the meridian, and grows gradually more inclined to diluted yellow, orange, and red, as they descend lower, that is, as the rays are transmitted through a longer tract of air; which tract being still lengthened in passing to the moon and back again, causes a still greater loss of the blue-making rays in proportion to the rest; and so the resulting colour of the transmitted rays must lie between a dark orange and red, according to Sir Isaac Newton's rule for finding the result of a mixture of colours. We have an instance of the reverse of this case in leaf-gold, which appears yellow by reflected and blue by transmitted rays. The circular edge of the shadow in a partial eclipse appears red; because the red-making rays are the least refracted of all others, and consequently are left alone in the conical surface of the shadow, all the rest being refracted into it.

§ 8. Of the Measures of Light.

That some luminous bodies give a stronger, and others a weaker light, and that some reflect more light than others, was always obvious to mankind; but no person, before M. Bouguer, hit upon a tolerable method of ascertaining the proportion that two or more lights bear to one another. The methods he most commonly used were the following.

He took two pieces of wood or pasteboard \( EC \) and \( CD \) (fig. 4.), in which he made two equal holes \( P \) and \( Q \), over which he drew pieces of oiled or white paper. Upon these holes he contrived that the light of the different bodies he was comparing should fall; while he placed a third piece of pasteboard \( FC \), so as to prevent the two lights from mixing with one another. Then placing himself sometimes on one side, and sometimes on the other, but generally on the opposite side of this instrument, with respect to the light, he altered their position till the papers in the two holes appeared to be equally enlightened. This being done, he computed the proportion of their light by the squares of the distances at which the luminous bodies were placed from the objects. If, for instance, the distances were as three and nine, he concluded that the light they gave were as nine and eighty-one.

Where any light was very faint, he sometimes made use of lenses, in order to condense it; and he inclosed them in tubes or not as his particular application of them required.

To measure the intensity of light proceeding from the heavenly bodies, or reflected from any part of the sky, he contrived an instrument which resembles a kind of portable camera obscura. He had two tubes, of which the inner was black, fastened at their lower extremities by a hinge \( C \), (fig. 5.) At the bottom of these tubes were two holes, \( R \) and \( S \), three or four lines in diameter, covered with two pieces of fine white paper. The two other extremities had each of them a circular aperture, an inch in diameter; and one of the tubes consisted of two, one of them sliding into the other, which produced the same effect as varying the aperture at the end. When this instrument is used, the observer has his head, and the end of the instrument \( C \), so covered, that no light can fall upon his eye, besides that which comes through the two holes \( R \) and \( S \), while an assistant manages the instrument, and draws out or shortens the tube \( DE \), as the observer directs. When the two holes appear equally illuminated, the intensity of the lights is judged to be inversely as the squares of the tubes.

In using this instrument, it is necessary that the object should subtend an angle larger than the aperture \( A \) or \( D \), seen from the other end of the tube; for, otherwise, the lengthening of the tube has no effect. To avoid, in this case, making the instrument of an inconvenient length, or making the aperture \( D \) too narrow, he has recourse to another expedient. He constructs an instrument, represented (fig. 6.), consisting of two object-glasses, \( AE \) and \( DF \), exactly equal, fixed in the ends of two tubes six or seven feet, or, in some cases, ten or twelve feet long, and having their foci at the other ends. At the bottom of these tubes \( B \), are two holes, three or four lines in diameter, covered with a piece of white paper; and this instrument is used exactly like the former.

If the two objects to be observed by this instrument be not equally luminous, the light that issues from them must be reduced to an equality, by diminishing the aperture of one of the object-glasses; and then the remaining surface of the two glasses will give the proportion of their lights. But for this purpose, the central parts of the glass must be covered in the same proportion with the parts near the circumference, leaving the aperture such as is represented (fig. 7.), because the middle part of the glass is thicker and less transparent than the rest.

If all the objects to be observed lie nearly in the same direction, our author observes, that these two long tubes may be reduced into one, the two object-glasses being placed close together, and one eye-glass sufficing for them both. The instrument will then be the same with that of which he published an account in 1748, and which he called a heliometer, or altimeter.

Our author observes, that it is not the absolute quantity, but only the intensity of the light, that is measured by these two instruments, or the number of the rays, in proportion to the surface of the luminous body; and it is of great importance that these two things be... Measures be distinguished. The intensity of light may be very great, when the quantity, and its power of illumina- ting other bodies, may be very small, on account of the smallness of its surface; or the contrary may be the case, when the surface is large.

Having explained these methods which M. Bouguer took to measure the different proportions of light, we shall subjoin in this place a few miscellaneous examples of his application of them.

It is observable, that when a person stands in a place where there is a strong light, he cannot distinguish objects that are placed in the shade; nor can he see any thing upon going immediately into a place where there is very little light. It is plain, therefore, that the action of a strong light upon the eye, and also the impression which it leaves upon it, makes it insensible to the effect of a weaker light. M. Bouguer had the curiosity to endeavour to ascertain the proportion between the intensities of the two lights in this case; and by throwing the light of two equal candles upon a board, he found that the shadow made by intercepting the light of one of them, could not be perceived by his eye, upon the place enlightened by the other, at little more than eight times the distance; from whence he concluded, that when one light is eight times eight, or 64 times less than another, its presence or absence will not be perceived. He allows, however, that the effect may be different on different eyes; and supposes that the boundaries in this case, with respect to different persons, may lie between 60 and 80.

Applying the two tubes of his instrument, mentioned above, to measure the intensity of the light reflected from different parts of the sky; he found, that when the sun was 25 degrees high, the light was four times stronger at the distance of eight or nine degrees from his body, than it was at 31 or 32 degrees. But what struck him most was to find, that when the sun is 15 or 20 degrees high, the light decreases on the same parallel to the horizon to 110 or 120 degrees, and then increases again to the place exactly opposite to the sun.

The light of the sun, our author observes, is too strong, and that of the stars too weak, to determine the variation of their light at different altitudes: but as, in both cases, it must be in the same proportion with the diminution of the light of the moon in the same circumstances, he made his observations on that luminary, and found, that its light at $19^\circ 16'$ is to its light at $66^\circ 11'$, as $1681$ to $2500$; that is, the one is nearly two thirds of the other. He chose those particular altitudes, because they are those of the sun at the two solstices at Croisiére, where he then resided.

When one limb of the moon touched the horizon of the sea, its light was 2000 times less than at the altitude of $66^\circ 11'$. But this proportion he acknowledges must be subject to many variations, the atmosphere near the earth varying so much in its density. From this observation he concludes, that at a medium light is diminished in the proportion of about $2500$ to $1681$, in traversing 7469 toises of dense air.

Lastly, our accurate philosopher applied his instrument to the different parts of the sun's disk, and found that the centre is considerably more luminous than the extremities of it. As near as he could make the ob-

servation, it was more luminous than a part of the disk $\frac{1}{4}$ths of the semidiameter from it, in the proportion of $35$ to $28$; which, as he observes, is more than in the proportion of the sines of the angles of obliquity. On the other hand, he observes, that both the primary and secondary planets are more luminous at their edges than near their centres.

The comparison of the light of the sun and moon is a subject that has frequently exercised the thoughts of philosophers; but we find nothing but random conjectures, before our author applied his accurate measures in this case. In general, the light of the moon is imagined to bear a much greater proportion to that of the sun than it really does; and not only are the imaginations of the vulgar, but those of philosophers also, imposed upon with respect to it. It was a great surprize to M. de la Hire to find that he could not, by the help of any burning mirror, collect the beams of the moon in a sufficient quantity to produce the least sensible heat. Other philosophers have since made the like attempts with mirrors of greater power, though without any greater success; but this will not surprize us, when we see the result of M. Bouguer's observations on this subject.

In order to solve this curious problem concerning M. Bouguer's comparison of the light of the sun and moon, he compared each of them to that of a candle in a dark culainum room, one in the day-time, and the other in the night. The following, when the moon was at her mean distance the moon from the earth; and, after many trials, he concluded that the light of the sun is about 300,000 times greater than that of the moon; which is such a disproportion, that, as he observes, it can be no wonder that philosophers have had so little success in their attempts to collect the light of the moon with burning-glasses. For the largest of them will not increase the light 1000 times; which will still leave the light of the moon, in the focus of the mirror, 300 times less than the intensity of the common light of the sun.

To this account of the proportion of light which we actually receive from the moon, it cannot be displeasing to the reader, if we compare it with the quantity which would have been transmitted to us from that opaque body, if it reflected all the light it receives. Dr Smith thought that he had proved, from two different considerations, that the light of the full moon would be to our day-light as 1 to about 90,900, if no rays were lost at the moon.

In the first place, he supposes that the moon, enlightened by the sun, is as luminous as the clouds are calculation, at a medium. He therefore supposed the light of the sun to be equal to that of a whole hemisphere of clouds, or as many moons as would cover the surface of the heavens. But on this Dr Priestley observes, that it is true, the light of the sun shining perpendicularly upon any surface would be equal to the light reflected from the whole hemisphere, if every part reflected all the light that fell upon it; but the light that would in fact be received from the whole hemisphere (part of it being received obliquely) would be only one-half as much as would be received from the whole hemisphere, if every part of it shone directly upon the surface to be illuminated.

In his Remarks, par. 97, Dr Smith demonstrates his method of calculation in the following manner: Let the little circle cf dg represent the moon's body half enlightened by the sun, and the great circle ab, a spherical shell concentric to the moon; and CCLXII touching the earth; ab, any diameter of that shell perpendicular to a great circle of the moon's body, represented by its diameter cd; e the place of the shell receiving full moon light from the bright hemisphere f dg. Now, because the surface of the moon is rough like that of the earth, we may allow that the sun's rays, incident upon any small part of it, with any obliquity, are reflected from it every way alike, as if they were emitted. And therefore, if the segment df thone alone, the points a, c, would be equally illuminated by it; and likewise if the remaining bright segment dg thone alone, the points b, e would be equally illuminated by it. Consequently, if the light at the point a was increased by the light at b, it would become equal to the full moon-light at e. And conceiving the same transfer to be made from every point of the hemispherical surface bbik to their opposite points in the hemisphere kaeb, the former hemisphere would be left quite dark, and the latter would be uniformly illuminated with full moon-light; arising from a quantity of the sun's light, which, immediately before its incidence on the moon, would uniformly illuminate a circular plane equal to a great circle of her body, called her disk. Therefore the quantities of light being the same upon both surfaces, the density of the sun's incident light is to the density of full moon-light, as that hemispherical surface biek is to the said disk; that is, as any other hemispherical surface whose centre is at the eye, to that part of it which the moon's disk appears to possess very nearly, because it subtends but a small angle at the eye: that is, as radius of the hemisphere to the verified fine of the moon's apparent semidiameter, or as 10,000,000 to 110623 or as 90,400 to 1; taking the moon's mean horizontal diameter to be 16'7".

Strictly speaking, this rule compares moon-light at the earth with day-light at the moon; the medium of which, at her quadratures, is the same as our day-light; but is less at her full in the duplicate ratio of 365 to 366, or thereabout, that is, of the sun's distances from the earth and full moon: and therefore full-moon light would be to our day-light as about 1 to 90,900, if no rays were lost at the moon.

Secondly, I say that full-moon light is to any other moon light as the whole disk of the moon to the part that appears enlightened, considered upon a plane surface. For now let the earth be at b, and let dl be perpendicular to f g, and gm to cd; then it is plain, that gl is equal to dm; and that gl is equal to a perpendicular section of the sun's rays incident upon the arch dg; which at b appears equal to dm; the eye being unable to distinguish the unequal distances of its parts. In like manner, conceiving the moon's surface to consist of innumerable physical circles parallel to cf dg, as represented at A, the same reason holds for every one of these circles as for cf dg. It follows then, that the bright part of the surface visible at b, when reduced to a flat as represented at B, by the crescent pdgmp, will be equal and similar to a perpendicular section of all the rays incident on that part, represented at C by the crescent pgip. Now of Aberration.

The whole disk being in proportion to this crescent, as the quantities of light incident upon them; and the light falling upon every rough particle, being equally rarified in diverging to the eye at b, considered as equidistant from them all; it follows, that full moon-light is to this moon-light as the whole disk pdgq to the crescent pdgmp.

Therefore, by compounding this ratio with that in the former remark, day-light is to moon-light as the surface of an hemisphere whose centre is at the eye, to the part of that surface which appears to be possessed by the enlightened part of the moon."

Mr Michell made his computation in a much more Mr. Mitchell's calculation. simple and easy manner, and in which there is much less danger of falling into any mistake. Considering the distance of the moon from the sun, and that the density of the light must decrease in the proportion of the square of that distance, he calculated the density of the sun's light, at that distance, in proportion to its density at the surface of the sun; and in this manner he found, that if the moon reflected all the light it receives from the sun, it would only be the 45,000th part of the light we receive from the greater luminary. Admitting, therefore, that moon-light is only a 300,000th part of the light of the sun, Mr Michell concludes, that it reflects no more than between the 6th and 7th part of what falls upon it.

Sect. IV. Of Aberration.

The great practical use of the science of optics is Theory of to aid human sight; but it has been repeatedly observed during the progress of this article, that in constructing dioptrical instruments for this purpose, great difficulties arise from the aberration of light. It has been shown, page 288, &c. how to determine the concourse of any refracted ray PF' with the ray RVCF (figs. 5, 6, &c. Plate CCCLV.) which passes through the centre C, and therefore falls perpendicularly on the spherical surface at the vertex V, and suffers no refraction. This is the conjugate focus to R for the two rays RP, RV, and for another ray flowing from R and falling on the surface at an equal distance on the opposite side to P. In short, it is the conjugate focus for all the rays flowing from R and falling on the spherical surface in the circumference of a circle described by the revolution of the point P round the axis RVCF; that is, of all the rays which occupy the conical surface described by the revolution of RP, and the refracted rays occupy the conical surface produced by the revolution of PP'.

But no other rays flowing from R are collected at F'; for it appeared in the demonstration of that proposition, that rays incident at a greater distance from the axis RC were collected at a point between C and F'; and then the rays which are incident on the whole arch PC, or the spherical surface generated by its revolution round RC, although they all cross the axis RC, are diffused over a certain portion of it, by what has been called the aberration of figure. It is called also (but improperly) the aberration from the geometrical focus, by which is meant the focus of an infinitely slender pencil of rays, of which the middle ray (or axis of the pencil) occupies the lens RC, and suffers suffers no refraction. But there is no such focus. But if we make \( mRV = nRC : mRV = VC : VF \), the point \( F \) is called the geometrical focus, and is the remotest limit from \( C \) of all the foci (equally geometrical) of rays flowing from \( R \). The other limit is easily determined by constructing the problem for the extreme point of the given arch.

It is evident from the construction, that while the point of incidence \( P \) is near to \( V \), the line \( CK \) increases but very little, and therefore \( CF \) diminishes little, and the refracted rays are but little diffused from \( F \); and therefore they are much denser in its vicinity than any other point of the axis. It will soon be evident that they are incomparably denser. It is on this account that the point \( F \) has been called the conjugate focus \( \text{κατά} \text{ξενία} \) to \( R \), and the geometrical focus and the diffusion has been called aberration. A geometrical point \( R \) is thus represented by a very small circle (or physical point as it is improperly called) at \( F \), and \( F \) has drawn the chief attention. And as, in the performance of optical instruments, it is necessary that this extended representation of a mathematical point \( R \) be very small, that may not sensibly interfere with the representations of the points adjacent to \( R \), and thus cause indistinct vision, a limit is thus set to the extent of the refracting surface which must be employed to produce this representation. But this evidently diminishes the quantity of light, and renders the vision obscure, though distinct. Artists have therefore endeavoured to execute refracting surfaces of forms not spherical, which collect accurately to one point the light issuing from another, and the mathematicians have furnished them with forms having this property; but their attempts have been fruitless. Spherical surfaces are the only ones which can be executed with accuracy. All are done by grinding the refracting substance in a mould of proper materials. When this is spherical, the two work themselves, with moderate attention, into an exact sphere; because if any part is more prominent than another, it is ground away, and the whole gets of necessity one curvature. And it is astonishing to what degree of accuracy this is done. An error of the millionth part of an inch would totally destroy the figure of a mirror of an inch focal distance, so as to make it useless for the coarsest instrument. Therefore all attempts to make other figures are given up. Indeed other reasons make them worse than spherical, even when accurately executed. They would not collect to accurate focuses the rays of oblique pencils.

It is evident from these observations, that the theory of aberrations is absolutely necessary for the successful construction of optical instruments; and it must be acceptable to the reader to have a short account of it in this place. Enough shall be said here to show the general nature and effects of it in optical instruments, and in some of the more curious phenomena of nature. Under the article Telescope the subject will be resumed, in such a manner as to enable the reader who possesses a very moderate share of mathematical knowledge, not only to understand how aberrations are increased and diminished, but also how, by a proper employment of contrary aberrations, their hurtful effects may be almost entirely removed in all important cases. And the manner in which the subject shall be treated in the present general sketch, will have the advantage of pointing out at the same time the maxims of construction of the greatest part of optical instruments, which generally produce their effects by means of pencils of rays which are either out of the axis altogether, or are oblique to it; cases which are seldom considered in elementary treatises of optics.

Let \( PV \) (fig. 1.) be a spherical surface of a refracting substance (glass for instance), of which \( C \) is the centre, and let an indefinitely slender pencil of rays \( AP \) be incident on it, in a direction parallel to a ray \( CV \) passing through the centre. It is required to determine the focus \( f \) of this pencil.

Let \( AP \) be reflected into \( PF \). Draw \( CI \), \( CR \) the radius of incidence and refraction, and \( CP \) the radius of the Draw \( RP \) perpendicular to \( CP \), and \( Bf \) parallel to \( AP \) or \( CV \). I say, first, \( f \) is the focus of the infinitely slender pencil, or, more accurately speaking, \( f \) is the remotest limit from \( P \) of the concourse of rays with \( PF \), refracted by points lying without the arch \( VP \), or the nearest limit for rays incident between \( V \) and \( P \).

Draw the radius \( CP \), the line \( PF \); and draw \( PS \) parallel to \( PF \); and \( PO \) perpendicular to \( PF \). It is evident, that if \( f \) be the focus, \( e'PF \) is the angle of refraction corresponding to the angle of incidence \( aPC \), as \( C'PF \) is the angle corresponding to \( APC \). Also \( PCP \) is the increment of the angle of incidence, and the angle \( e'PF \) is equal to the sum of the angle \( C'PF \) and \( C'PC \), and the angle \( gPF \) is equal to the angle \( P/P \). Therefore \( e'PF = C'PF + P/PC + P/PP \). Therefore \( PCP + P/PP \) is the corresponding increment of the angle of refraction. Also, because \( RP = CP \) (being right angles) the angle \( P/PO = KPC \), and \( P/PP = PR : PC \).

Therefore, by a preceding Lemma in this article, page 286, we have \( PCP + P/PP : PCP = tan. ref. : tan. incid. = &c. \)

\( T, R : T, I ; \) and \( P/PP : PCP = T, R - T, I : T, I, \)

\( = \text{diff} : T, I ; \) but \( P/PP : PCP = P/PP : P/PP : P/PP : P/PP \),

\( = PR : PF = DR : DB \) (because \( DP \) is parallel to \( Bf \) by construction) \( = \tan. CPR - \tan. CPI : \tan. CPI \).

Now \( CPI \) is the angle of incidence; and therefore \( CPR \) is the angle properly corresponding to it as an angle of refraction; and the point \( f \) is properly determined.

Hence the following rule. As the difference of the tangents of incidence and refraction is to the tangent of incidence, so is the radius of the surface multiplied by the cosine of refraction to the distance of the focus of an infinitely slender pencil of parallel incident rays.

N.B.: We here consider the cosine of refraction as a number. This was first done by the celebrated Leonhard Euler, and is one of the greatest improvements in mathematics which this century can boast of. The sines, tangents, secants, &c., are considered as fractional numbers, of which the radius is unity. Thus, \( CP \times \sin. 32^\circ \) is the same thing with \( \frac{1}{2}CP \), or \( \frac{1}{2} \). And in like manner, \( CB \), drawn perpendicular to the axis \( \times \sin. 19^\circ 58' 16'' 32'' \), is the same thing with \( \frac{1}{3}CB \).

Also \( \cos. 60^\circ \) is the same thing with twice \( CB \), &c. In this manner, \( BE = BC \times \text{fin. BCE} \), and also \( BE = CE \times \text{tan. BCE} \), and \( CB = CE \times \text{sec. BCE} \), &c. &c.

This manner of considering the lines which occur in geometrical constructions is of immense use in all parts of mixed mathematics; and nowhere more remarkably than in optics, the most beautiful example of them. Of this an important instance shall now be given.

Corol. 1. The distance \( fG \) of this lateral focus from the axis \( CV \) (that is, from the line drawn through the centre parallel to the incident light) is proportional to the cube of the semi-aperture \( PH \) of the spherical surface.

For \( fG = BE \). Now \( BE = CB \times \text{fin. BCE} = CB \times \text{fin. CPA} \); and \( CB = RC \times \text{cos. RCB} = RC \times \text{fin. CPR} \), and \( RC = CP \times \text{fin. CPR} \). Therefore \( BE = PC \times \text{fin. PCA} = PC \times \text{fin. refr. fin. incid.} \).

But \( \text{fin. refr.} = \frac{m^2}{n^2} \text{fin. incid.} \). Therefore, finally,

\[ BE, \text{or } fG = PC \times \frac{m^2}{n^2} \times \text{fin. incid.} \]

But \( PC \times \text{fin. incid.} \)

is evidently \( PH \) the semi-aperture; therefore the proposition is manifest.

Corol. 2. Now let this slender pencil of rays be incident at the vertex \( V \). The focus will now be a point \( F \) in the axis, determined by making \( CV : CF = m : n \). Let the incident pencil gradually recede from the axis \( CF \), still, however, keeping parallel to it. The focus \( F \) will always be found in a curve line \( DC'F \), so constituted that the ordinate \( G \) will be as the cube of the line \( PH \), perpendicular to the axis intercepted between the axis and that point of the surface which is cut by a tangent to the curve in \( f \).

All the refracted rays will be tangents to this curve, and the adjacent rays will cross each other in these lateral foci \( f \); and will therefore be incomparably more dense along the curve than anywhere within its area. This is finely illustrated by receiving on white paper the light of the sun refracted through a globe or cylinder of glass filled with water. If the paper is held parallel to the axis of the cylinder, and close to it, the illuminated part will be bounded by two very bright parallel lines, where it is cut by the curve; and these lines will gradually approach each other as the paper is withdrawn from the vessel, till they coalesce into one very bright line at \( F \), or near it. If the paper be held with its end touching the vessel, and its plane nearly perpendicular to the axis, the whole progress of the curve will be distinctly seen.

As such globes were used for burning-glasses, the point of greatest condensation (which is very near but not exactly in \( F \)) was called the focus. When these curves were observed by Mr Tchirnhaus, he called them caustics; and those formed by refraction he called diacaustics, to distinguish them from the catacaustics formed by reflection.

It is somewhat surprising, that these curves have been so little studied since the time of Tchirnhaus. The doctrine of aberrations has indeed been considered in a manner independent on their properties. But whoever considers the progress of rays in the eye-piece of optical instruments, will see that the knowledge of the properties of diacaustic curves determines directly, and almost accurately, the foci and images that are formed there. For, let the object-glass of a telescope or microscope be of any dimensions, the pencils incident on the eye-glasses are almost all of this evanescent bulk. These advantages will be shown in their proper places; and we proceed at present to extend our knowledge of aberrations in general, first considering the aberrations of parallel incident rays.

Abiding by the instance represented by the figure, it is evident that the caustic will touch the surface in a point \( r \), so situated that \( c : p : r = m : n \). The refracted ray \( \phi \Phi \) will touch the surface, and will cross the axis in \( \Phi \), the nearest limit of diffusion along the axis. If the surface is of smaller extent, as \( PV \), the caustic begins at \( f \), when the extreme refracted ray \( P \phi \) touches the caustic, and crosses the axis in \( F' \), and the opposite branch of the caustic in \( K \). If there be drawn an ordinate \( KO \) to the caustic, it is evident that the whole light incident on the surface \( PV \) passes through the circle whose diameter is \( Kk \), and that the circle is the smallest space which receives all the refracted light.

It is of great importance to consider the manner in which the light is distributed over the surface of this circle of smallest diffusion; for this is the representation over the small object. Each point of a planet, for instance, is represented by this little circle; and as the circles representing the different adjacent points must interfere with each other, an indistinctness must arise similar to what is observed when we view an object through a pair of spectacles which do not fit the eye. The indistinctness must be in proportion to the number of points whose circles of diffusion interfere; that is, to the area of these circles, provided that the light is uniformly diffused over them: but if it be very rare at the circumference, the impression made by the circles belonging to the adjacent points must be less sensible. Accordingly, Sir Isaac Newton, supposing it incomparably rarer at the circumference than towards the centre, affirms, that the indistinctness of telescopes arising from the spherical figure of the object-glass was some thousand times less than that arising from the unequal refrangibility of light; and therefore, that the attempts to improve them by diminishing or removing this aberration were needless, while the distinctness from unequal refrangibility remained. It is surprising, that a philosopher so eminent for sagacity and for mathematical knowledge, should have made such a mistake, and unfortunate that the authority of his great name hindered others from examining the matter, trusting to his assertion, that the light was so rare at the border of this circle. His mistake is surprising, because the very nature of a caustic should have showed him, that the light was infinitely dense at the borders of the circle of smallest diffusion. The first person who detected this oversight of the British philosopher was the Abbé Bofcovitch, who, in a dissertation published at Vienna in 1767, showed, by a very beautiful analysis, that the distribution was extremely different from what Newton had asserted, and that the superior distinctness arising from unequal refrangibility was incomparably less than he had said. We shall attempt to make this delicate and interesting matter conceivable by those who have but small mathematical preparation.

Let the curve \( DVZC \) (fig. 2.) be the caustic (magnified), \( EI \) its axis, \( I \) the focus of central rays, \( CCCLXIII \) \( B \) the focus of extreme rays, and \( IB \) the line containing... It is plain, that from the centre \( O \) there can be drawn two rays \( OV, Ov \), touching the caustic in \( V, v \). Therefore the point \( O \) will receive the ray \( EO \), which passes through the vertex of the refracting surface, and all the rays which are incident on the circumference of a circle described on the refracting surface by the extremity of the ray \( OV, or Ov \). The density of the light at \( O \) will therefore be indefinitely great.

From the point \( C \) there can be drawn two rays; one of them \( CX \) touching the caustic in \( C \), and the other \( C \), touching it at \( d \) on the opposite side. The rays which touch the caustic in the immediate vicinity of \( Cy \), both in the arch \( CV \) and the arch \( CI \) will cut \( OC \) in points indefinitely near to each other; because their distance from each other in the line \( OC \) will be to their uniform distance on the refracting surface as the distance between their points of contact with the caustic to the distance of these points from the refracting surface. Here therefore at \( C \) the density of the light will also be indefinitely great.

(From any point \( H \), lying between \( O \) and \( C \), may be drawn three rays. One of them, \( LH'T, P \), touching the arch \( CD \) of the caustic in \( T \), cutting the refracting surface in \( P \), and the axis in \( L \); another, \( TH'P \), touching the arch \( CI \) of the caustic in \( I \). The third is \( HH'P \), touching the arch \( cd \) of the opposite branch of the caustic in \( r \).

It will greatly assist our conceptions of this subject, if we consider a ray of light from the refracting surface as a thread attached at \( I \) of this figure, or at \( F \) of fig. i. and gradually unslipped from the caustic \( DVCI \) on one side, and then lapped on the opposite branch \( Icvd \); and attend to the point of its intersection with the diameter \( cOC \) of the circle of smallest diffusion.

Therefore, 1. let the ray be first supposed to pass through the refracting surface at \( F \), the right hand extremity of the aperture. The thread is then folded up on the whole right hand branch \( ICVD \) of the caustic; and if the straight part of it \( FD \) be produced, it will cut the diameter of the circle of smallest diffusion in the opposite extremity \( c \). Or suppose a ruler in place of the thread, applied to the caustic at \( D \) and to the refracting surface at \( F \), the part of it \( Dc \), which is detached from the caustic, cuts \( COc \) in the point \( c \); 2. Now suppose the ruler to revolve gradually, its extremity moving across the arch \( FA \) of the refracting surface while the edge is applied to the caustic; the point of contact with the caustic will shift gradually down the branch \( DV \) of the caustic, while its edge passes across the line \( cC \); and when the point of contact arrives at \( V \), the extremity will be at \( Y \) on the refracting surface, and the intersection of the edge will be at \( O \). 3. Continuing the motion, the point of contact shifts from \( V \) to \( Z \), the extremity from \( Y \) to \( Q' \), and the intersection from \( O \) to \( Q \), so that \( CQ^2 = \frac{OC^2}{2} \), as will presently appear. 4. After this, the point of contact will shift from \( Z \) to \( C \), the extremity from \( Q' \) to \( X \), halfway from \( F \) to \( A \), as will soon be shown, and the intersection from \( Q \) to \( C \). 5. The point of contact will now shift from \( C \) down to \( I \), the extremity will pass from \( X \) to \( A \), and the intersection will go back from \( C \) to \( O \). 6. The ruler must now be applied to the other branch of the caustic \( Icvd \), and the point of contact will ascend from \( I \) to \( c \), the extremity will pass from \( A \) to \( x \), half way to \( f \) from \( A \), and the intersection from \( O \) to \( c \). 7. The point of contact will ascend from \( C \) to \( z \), the extremity passes from \( x \) to \( g' \), and the intersection from \( C \) to \( q \), \( Og^2 \) being \( \frac{Ox^2}{2} \). 8. While the contact of the ruler and caustic shifts from \( z \) to \( v \), the extremity shifts from \( g' \) to \( y \), and the intersection from \( q \) to \( O \). 9. The contact rises from \( v \) to \( d \), the extremity passes from \( y \) to \( f \), and the intersection from \( O \) to \( C \); and then the motion across the refracting surface is completed, the point of contact shifting down from \( D \) to \( I \), along the branch \( DVZCI \), and then ascending along the other branch \( Icvd \), while the intersection passes from \( c \) to \( C \), back again from \( C \) to \( c \), and then back again from \( c \) to \( C \), where it ends, having thrice passed through every intermediate point of \( cC \).

We may form a notion of the density of the light density in any point \( H \), by supposing the incident light of uni-light, to form density at the refracting surface, and attending to the constitution of the rays in the circle of smallest diffusion. Their vicinity may be estimated both in the direction of the radii \( OH \), and in the direction of the circumference described by its extremity \( H \), during its revolution round the axis; and the density must be conceived as proportional to the number of originally equidistant rays, which are collected into a spot of given area. These have been collected from a corresponding spot or area of the refracting surface; and as the number of rays is the same in both, the density at \( H \) will be to the density of the refracting surface, as the area occupied by the refracting surface to the corresponding area at \( H \). The vicinity of the rays in the direction of the radius depends on the proportion between \( PT \) and \( TH \). For the ray adjacent to \( PTH \) may be supposed to cross it at the point of contact \( T \); and therefore the uniform distance between them at the surface of that medium is to the distance between the same rays at \( H \) as the distance of \( T \) from the refracting surface to its distance from \( H \). Therefore the number of rays which occupy a tenth of an inch, for example, of the radius \( AP \), is to the number which would occupy a tenth of an inch at \( H \) as \( TH \) to \( TP \); and the radial density at \( P \) is to the radial density at \( H \) as \( TH \) to \( TP \). In the next place, the circumferential density at \( P \) is to that at \( H \) as the radius \( AP \) to the radius \( OH \). For supposing the figure to turn round its axis \( AI \), the point \( P \) of the refracting surface will describe a circumference whose radius is \( AP \), and \( H \) will describe a circumference whose radius is \( OH \); and the whole rays which pass through the full circumference pass also thro' the last; and therefore their circumferential densities will be in the inverse proportion of the spaces into which they are collected. Now the radius \( AP \) is to the radius \( OH \) as \( AL \) to \( OL \); and circumferences have the same proportion with their radii. Therefore the circumferential density at \( P \) is to that in \( H \) as \( AL \) to \( OL \) inversely; and it was found that the radial density was as \( AN \) to \( ON \) inversely, being as \( TH \) to \( TP \), which are very nearly in this ratio. Therefore the absolute density (or number of rays collected in a given space) at \( P \) will be to that at \( H \), in the ratio compounded of these ratios; that is, in the ratio of \( ON \times OL \) to \( AN \times AL \). But as \( NL \) bears but a very small ratio to \( AN \) or \( AL \),

\[ Xx = AN \times AL \] AN×AL may be taken as equal to AO² without any sensible error. It never differs from it in telescopes tenth part, and is generally incomparably smaller. Therefore the density at H may be considered as proportional to ON×OL inversely. And it will afterwards appear that NS is = 32L. Therefore the density at H is inversely as ON×NS.

Now describe a circle on the diameter OS, and draw NT cutting the circumference N = ON×NS, and the density at H is as N² inversely. This gives us a very easy estimation of the density, viz., draw a line from the point of contact of the ray which touches the part VC of the caustic, and the density is in the inverse subduplicate ratio of the part of this line intercepted between the axis and the circumference SPO. It will afterwards appear that the density corresponding to this ray is one half of the density corresponding to all the three; or a better expression will be had for the density at H by drawing RB perpendicular to RS, and BO perpendicular to O, making R in o; then BO is as \( \frac{1}{\sqrt{N}} \), or is proportional to the density, as is evident.

When H is at O, N is at S, and PO is infinite. As H moves from O, N descends, and PO diminishes, till H comes to Q, and T to z, and PO to r, and PO to R. When H moves from Q towards C, T descends below z, PO again increases, till it is again infinite, when H is at C, T at C, and N at O.

Thus it appears, without any minute consideration, that the light has a density indefinitely great in the centre O; that the density decreases to a minimum in some intermediate point Q, and then increases again to infinity at the margin C. Hence it follows, that the indistinctness arising from the spherical figure of the refracting surfaces is incomparably greater than Newton supposed; and that the valuable discovery of Mr Dollond of achromatic lenses, must have failed of answering his fond expectations, if his very method of producing them had not, at the same time, enabled him to remove that other indistinctness by employing contrary aberrations. And now, since the discoveries by Dr Blair of substances which disperse the different colours in the same proportions, but very different degrees, has enabled us to employ much larger portions of the sphere than Mr Dollond could introduce into his object-glasses, it becomes absolutely necessary to study this matter completely, in order to discover and ascertain the amount of the errors which perhaps unavoidably remain.

This slight sketch of the most simple case of aberration, namely, when the incident rays are parallel, will serve to give a general notion of the subject; and the reader can now see how contrary aberrations may be employed in order to form an ultimate image which shall be as distinct as possible. For let it be proposed to converge parallel rays accurately to the focus F (fig. 3.) by the refraction of spherical surfaces of which V is the vertex. Let PV be a convex lens of such a form that rays flowing from F and passing through it immediately round the vertex V are collected to the conjugate focus R, while the extreme ray FP, incident on the margin of the lens P, is converged to r, nearer to V, having the longitudinal aberration RR. Let PV be a plano-concave lens, of such sphericity that a ray AP, parallel to the axis CV, and incident on the point P, as far from its vertex V as P in the other lens is from its vertex, is dispersed from r, the distance RV being equal to TV, while the central rays are dispersed from P, as far from V as R is from V. It is evident, that if these lenses be joined as in fig. 4, a ray AP, parallel to the common axis CV, will be collected at the distance VF equal to VF in the fig. 4, and that rays passing through both lenses in the neighbourhood of the axis will be collected at the same point F.

This compound lens is said to be without spherical aberration; and it is true that the central and the extreme rays are collected in the same point F; but the rays which fall on the lens between the centre and margin are a little diffused from F, and it is not possible to collect them all to one point. For in the rules for computing the aberration, quantities are neglected which do not preserve (in different apertures) the same ratio to the quantities retained. The diffusion is least when the aberration is corrected, not for the very extremity, but for a certain intermediate point (varying with the aperture, and having no known ratio to it); and when this is done the compound lens is in its state of greatest perfection, and the remaining aberration is quite insensible.

This subject will be resumed under the article Telescope, and prosecuted as far as the construction of optical instruments requires.

Sect. IV. Of Optical Instruments.

Of the mechanism of optical instruments particular accounts are given in this work under their respective denominations. These it would be improper to repeat; but as it belongs to the science of optics to explain, by the laws of refraction and reflection, the several phenomena which those instruments exhibit, we must in this place enumerate the instruments themselves, omitting entirely, or stating very briefly, such facts as are stated at large in other places. In this enumeration we shall begin with the multiplying-glass, not because it is first in importance, but that it may not intervene between instruments more useful, and which have a mutual relation to one another.

§ I. The Multiplying-glass.

The multiplying-glass is made by grinding down the round side b'k (fig. 1.) of a plano-convex glass A.B, into several flat surfaces, as b, b', d, d'. An object C will not appear magnified when seen through this glass by the eye at H; but it will appear multiplied as many different objects as the glass contains multiplying-plane surfaces. For, since rays will flow from the object C to all parts of the glass, and each plane surface will refract these rays to the eye, the same object will appear to the eye in the direction of the rays which enter it through each surface. Thus, a ray g'H, falling perpendicularly on the middle surface, will go through the glass to the eye without suffering any refraction; and will therefore show the object in its true place at C; whilst a ray ab flowing from the same object, and falling obliquely on the plane surface bb', will be refracted in the direction be, by passing through the glass; and, upon leaving it, will go on to the eye in the direction eH; which will cause cause the same object C to appear also at E, in the direction of the ray Hc, produced in the right line Hcn. And the ray cd, flowing from the object C, and falling obliquely on the plane surface dK, will be refracted (by passing through the glass) and leaving it at f to the eye at H; which will cause the same object to appear at D, in the direction Hf. If the glass be turned round the line g/H, as an axis, the object C will keep its place, because the surface b/d is not removed; but all the other objects will seem to go round C, because the oblique planes, on which the rays abcd fall, will go round by the turning of the glass.

§ 2. Mirrors.

It has been elsewhere observed, that of mirrors there are three principally used in optical experiments (See CATOPTRICS, Sect. I.); the plane mirror, the spherical convex mirror, and the spherical concave mirror. Of these the plane mirror first claims our attention, as it is more common, and undoubtedly more ancient, than the other two. It has been said (ubi supra), that the image reflected by this mirror appears as far behind the surface as the object is before it; that the image will appear of the same size, and in the same position with the object; that every such mirror will reflect an image of twice its own length and breadth; and that in certain circumstances it will reflect several images of the same object. For these phenomena it is our business in this place to account by the laws of reflection.

Let AB (fig. 2.) be an object placed before the reflecting surface g/bi of the plane mirror CD; and let the eye be at o. Let Ah be a ray of light flowing from the top A of the object, and falling upon the mirror at b, and bm be a perpendicular to the surface of the mirror at b; the ray Ah will be reflected from the mirror to the eye at o, making an angle mbo equal to the angle Abm: then will the top of the image E appear to the eye in the direction of the reflected ray ob produced to E, where the right line ApE, from the top of the object, cuts the right line obE, at E. Let Bi be a ray of light proceeding from the foot of the object at B to the mirror at i; and ni a perpendicular to the mirror from the point i, where the ray Bi falls upon it: this ray will be reflected in the line io, making an angle nio equal the angle Bin, with that perpendicular, and entering the eye at o; then will the foot P of the image appear in the direction of the reflected ray oi, produced to F, where the right line BF cuts the reflected ray produced to F. All the other rays that flow from the intermediate points of the object AB, and fall upon the mirror between b and i, will be reflected to the eye at o; and all the intermediate points of the image EF will appear to the eye in the direction of these reflected rays produced. But all the rays that flow from the object, and fall upon the mirror above b, will be reflected back above the eye at o; and all the rays that flow from the object, and fall upon the mirror below i, will be reflected back below the eye at o; so that none of the rays that fall above b, or below i, can be reflected to the eye at o; and the distance between b and i is equal to half the length of the object AB.

Hence it appears, that if a man sees his whole image in a plane looking-glass, the part of the glass that reflects his image must be just half as long and half as broad as himself; let him stand at any distance from it whatever; and that his image must appear just size of a far behind the glass as he is before it. Thus, the man looking-AB (fig. 3.) viewing himself in the plane mirror CD, glass which is just half as long as himself, sees his whole image as at EF, behind the glass, exactly equal to see his own size. For a ray AC proceeding from his eye whole at A, and falling perpendicularly upon the surface of image, the glass at C, is reflected back to his eye, in the same line CA; and the eye of his image will appear at E, in the same line produced to E, beyond the glass. And a ray BD, flowing from his foot, and falling obliquely on the glass at D, will be reflected as obliquely on the other side of the perpendicular abD, in the direction DA; and the foot of his image will appear at F, in the direction of the reflected ray AD, produced to F, where it is cut by the right line BGF, drawn parallel to the right line ACE. Just the same as if the glass were taken away, and a real man stood at F, equal in size to the man standing at B: for to his eye at A, the eye of the other man at E would be seen in the direction of the line ACE; and the foot of the man at F would be seen by the eye A, in the direction of the line ADF.

If the glass be brought nearer the man AB, as suppose to cb, he will see his image as at CDG: for the reflected ray CA (being perpendicular to the glass) will show the eye of the image as at C; and the incident ray Db, being reflected in the line ba, will show the foot of his image as at G; the angle of reflection abA being always equal to the angle of incidence Bba; and so of all the intermediate rays from A to B. Hence, if the man AB advances towards the glass CD, his image will approach towards it; and if he recedes from the glass, his image will also recede from it.

If the object be placed before a common looking-glass, and viewed obliquely, three, four, or more images of it, will appear behind the glass.

To explain this, let ABCD (fig. 11.) represent the glass; and let EF be the axis of a pencil of rays flowing from E, a point in an object situated there. The rays of this pencil will in part be reflected at F, suppose into the line FG. What remains will (after refraction at F, which we do not consider here) pass on to H; from whence (on account of the quicksilver which is spread over the second surface of glasses of this kind to prevent any of the rays from being transmitted there) they will be strongly reflected to K, where part of them will emerge and enter an eye at L. By this means one representation of the said point Why three will be formed in the line LK produced, suppose in M; or four. Again, another pencil, whose axis is EN, first reflected at N, then at O, and afterwards at P, will form in a second representation of the same point at Q: And plane mirrors, thirdly, another pencil, whose axis is ER, after reflection at the several points R, S, H, T, V, successively, will exhibit a third representation of the same point at X; and so on in infinitum. The same being true of each point in the object, the whole will be represented in the like manner; but the representations will be faint, in proportion to the number of reflections the rays suffer, and the length of their progress within the glass. We may add to these another representation of the same object in the line LO produced, made by such of the rays as fall upon O, and are from thence reflected to the eye at L.

This experiment may be tried by placing a candle before the glass at E, and viewing it obliquely, as from L.

2. Of Concave and Convex Mirrors. The effects of these in magnifying and diminishing objects have been already in general explained; but for the better understanding the nature of reflecting telescopes, it will still be proper to subjoin the following particular description of the effects of concave ones.

When parallel rays (fig. 4.), as df'a, Cmb, etc., fall upon a concave mirror AB (which is not transparent, but has only the surface ABB of a clear polish), they will be reflected back from that mirror, and meet in a point m, at half the distance of the surface of the mirror from C the centre of its concavity; for they will be reflected at as great an angle from a perpendicular to the surface of the mirror, as they fell upon it with regard to that perpendicular, but on the other side thereof. Thus, let C be the centre of concavity of the mirror ABB; and let the parallel rays df'a, Cmb, and etc., fall upon it at the points a, b, and c. Draw the lines Cia, Cmb, and Cbc, from the centre C to these points; and all these lines will be perpendicular to the surface of the mirror, because they proceed thereto like so many radii or spokes from its centre. Make the angle Cab equal to the angle daC, and draw the line aml, which will be the direction of the ray df'a, after it is reflected from the point a of the mirror; so that the angle of incidence daC is equal to the angle of reflection Cab; the rays making equal angles with the perpendicular Cia on its opposite sides.

Draw also the perpendicular Cbc to the point c, where the ray etc. touches the mirror; and having made the angle Cei equal to the angle Cce, draw the line cmi, which will be the course of the ray etc., after it is reflected from the mirror.

The ray Cmb falling through the centre of concavity of the mirror, and falling upon it at b, is perpendicular to it; and is therefore reflected back from it in the same line Amc.

All these reflected rays meet in the point m; and in that point the image of the body which emits the parallel rays da, Cb, and etc., will be formed; which point is distant from the mirror equal to half the radius bmc of its concavity.

The rays which proceed from any celestial object may be esteemed parallel at the earth; and therefore the image of that object will be formed at m, when the reflecting surface of the concave mirror is turned directly towards the object. Hence, the focus m of parallel rays is not in the centre of the mirror's concavity, but half way between the mirror and that centre.

The rays which proceed from any remote terrestrial object are nearly parallel at the mirror; not strictly so, but come diverging to it, in separate pencils, or as it were bundles of rays, from each point of the side of the object next the mirror; and therefore they will not be converged to a point at the distance of half the radius of the mirror's concavity from its reflecting face, but into separate points at a little greater distance from the mirror. And the nearer the object is to the mirror, the farther these points will be from it; and an inverted image of the object will be formed in them, which will seem to hang pendant in the air; and will be seen by an eye placed beyond it (with regard to the mirror) in all respects like the object, and by concave mirrors.

Let AceB (fig. 5.) be the reflecting surface of a mirror, whose centre of concavity is at C; and let the upright object DE be placed beyond the centre C, and send out a conical pencil of diverging rays from its upper extremity D, to every point of the concave surface of the mirror AceB. But to avoid confusion, we only draw three rays of that pencil, as DA, De, DB.

From the centre of concavity C, draw the three right lines CA, Ce, CB, touching the mirror in the same points where the foresaid rays touch it; and all these lines will be perpendicular to the surface of the mirror. Make the angle Cad equal to the angle DAC, and draw the right line Ad for the course of the reflected ray DA; make the angle Ced equal to the angle DeC, and draw the right line cd for the course of the reflected ray De; make also the angle Cbe equal to the angle DBC, and draw the right line Be for the course of the reflected ray DB. All these reflected rays will meet in the point d, where they will form the extremity d of the inverted image ed, similar to the extremity D of the upright object DE.

If the pencil of rays Ef, Eg, Eb, be also continued to the mirror, and their angles of reflection from it be made equal to their angles of incidence upon it, as in the former pencil from D, they will all meet at the point e by reflection, and form the extremity e of the image ed, similar to the extremity E of the object DE.

And as each intermediate point of the object, between D and E, sends out a pencil of rays in like manner to every part of the mirror, the rays of each pencil will be reflected back from it, and meet in all the intermediate points between the extremities e and d of the image; and so the whole image will be formed, not at i, half the distance of the mirror from its centre of concavity C, but at a greater distance, between i and the object DE; and the image will be inverted with respect to the object.

This being well understood, the reader will easily see how the image is formed by the large concave mirror of the reflecting telescope, when he comes to the description of that instrument.

When the object is more remote from the mirror than its centre of concavity C, the image will be less than the object, and between the object and mirror; when the object is nearer than the centre of concavity, the image will be more remote and bigger than the object. Thus, if ED be the object, de will be its image: For, as the object recedes from the mirror, the image approaches nearer to it; and as the object approaches nearer to the mirror, the image recedes farther from it; on account of the lesser or greater divergence of the pencils of rays which proceed from the object; for the less they diverge, the sooner they are converged to points by reflection; and the more they diverge, the farther they must be reflected before they meet.

If the radius of the mirror's concavity, and the distance of the object from it, be known, the distance of the image from the mirror is found by this rule: Divide the product of the distance and radius by double the distance made less by the radius, and the quotient is the distance required.

If the object be in the centre of the mirror's concavity, the image and object will be coincident, and equal in bulk.

If a man places himself directly before a large concave mirror, but farther from it than its centre of concavity, he will see an inverted image of himself in the air, between him and the mirror, of a less size than himself. And if he holds out his hand towards the mirror, the hand of the image will come out towards his hand, and coincide with it, of an equal bulk, when his hand is in the centre of concavity; and he will imagine he may shake hands with his image. If he reaches his hand farther, the hand of the image will pass by his hand,* and come between his hand and his body: and if he moves his hand towards either side, the hand of the image will move towards the other; so that whatever way the object moves, the image will move the contrary.

All the while a bystander will see nothing of the image, because none of the reflected rays that form it enter his eyes.

§ 3. Microscopes.

Under the word Microscope a copious detail has been given of the construction of those instruments as they are now made by the most eminent artists. In that article it fell not within our plan to treat scientifically of their magnifying powers: these can be explained only by the laws of refraction and reflection, which we shall therefore apply to a few microscopes, leaving our readers to make the application themselves to such others as they may choose to analyse by optical principles.

The first and simplest of all microscopes is nothing more than a very small globule of glass, or a convex lens whose focal distance is extremely short. The magnifying power of this microscope is thus ascertained by Dr Smith. "A minute object pq, seen distinctly through a small glass AE by the eye put close to it, appears so much greater than it would to the naked eye, placed at the least distance qL from whence it appears sufficiently distinct, as this latter distance qL is greater than the former qE. For having put your eye close to the glass EA, in order to see as much of the object as possible at one view, remove the object pq to end fro till it appear most distinctly, suppose at the distance qE. Then conceiving the glass AE to be removed, and a thin plate, with a pin-hole in it, to be put in its place, the object will appear distinct, and as large as before, when seen through the glass, only not so bright. And in this latter case it appears so much greater than it does to the naked eye at the distance qL, either with the pin-hole or without it, as the angle pEq is greater than the angle pLq, or as the latter distance qL is greater than the former qE. Since the interpolation of the glass has no other effect than to render the appearance distinct, by helping the eye to increase the refraction of the rays in each pencil, it is plain that the greater apparent magnitude is entirely owing to a nearer view than could be taken by the naked eye. As the human eye is so constructed, as, for reasons already assigned, to have distinct vision only when the rays which fall upon it are parallel or nearly so; it follows, that if the eye be so perfect as to see distinctly by pencils of parallel rays falling upon it, the distance Eg, of the object from the glass, is then the focal distance of the glass. Now, if the glass be a small round globule, of about \( \frac{1}{4} \) th of an inch diameter, its focal distance Eg, being three quarters of its diameter, is \( \frac{1}{4} \) th of an inch; and if qL be eight inches, the distance at which we usually view minute objects, this globule will magnify in the proportion of 8 to \( \frac{1}{4} \), or of 160 to 1.

2. The Double or Compound Microscope (fig. 8.) consists of an object-glass cd, and an eye-glass ef. The small object ab is placed at a little greater distance from the glass cd than its principal focus; so that the pencils of rays flowing from the different points of the object, and passing through the glass, may be made to converge, and unite in as many points between g and b, where the image of the object will be formed: which image is viewed by the eye through the eyeglass ef. For the eye-glass being so placed, that the image gb may be in its focus, and the eye much about the same distance on the other side, the rays of each pencil will be parallel after going out of the eyeglass, as at e and f, till they come to the eye at k, where they will begin to converge by the refractive power of the humours; and after having crossed each other in the pupil, and passed through the crystalline and vitreous humours, they will be collected into points on the retina, and form the large inverted image A.B thereon.

By this combination of lenses, the aberration of the light from the figure of the glass, which in a single lens, in a compound of the kind above-mentioned is very considerable, is in some measure corrected. This appeared to be the case, even to former opticians, so sensibly to be the case, even to former opticians, that they very soon began to make the addition of another lens. The instrument, however, receives a considerable improvement by the addition of a third lens. For, says Mr Martin, it is not only evident from the theory of this aberration, that the image of any point is rendered less confused by refraction through two lenses than by an equal refraction through one; but it also follows, from the same principle, that the same point has its image still less confused when formed by rays refracted through three lenses than by an equal refraction through two; and therefore a third lens added to the other two will contribute to make the image more distinct, and consequently the instrument more complete. At the same time the field of view is amplified, and the use of the microscope rendered more agreeable, by the addition of the other lens. Thus also we may allow a somewhat larger aperture to the object-lens, and thereby increase the brightness of objects, and greatly heighten the pleasure of viewing them. For the same reason, Mr Martin has proposed a four-glass microscope, which answers the purposes of magnifying and of distinct vision still more perfectly.

The magnifying power of double microscopes is ca... fly understood; thus: The glass L next the object PQ is very small, and very much convex, and consequently its focal distance LF is very short; the distance LQ of the small object PQ is but a little greater than LF:

Greater it must be, that the rays flowing from the object may converge after passing through the glass, and, crossing one another, form an image of the object; and it must be but a little greater, that the image pq may be at a great distance from the glass, and consequently may be much larger than the object itself.

This picture pq being viewed through a convex glass AE, whose focal distance is qE, appears distinct as in a telescope. Now the object appears magnified upon two accounts; first, because, if we viewed its picture pq with the naked eye, it would appear as much greater than the object, at the same distance, as it really is greater than the object, or as much as Lq is greater than LQ; and, secondly, because this picture appears magnified through the eye-glass as much as the least distance at which it can be seen distinctly with the naked eye, is greater than qE, the focal distance of the eye-glass. For example, if this latter ratio be five to one, and the former ratio of Lq to LQ be 20 to 1; then, upon both accounts, the object will appear 5 times 20, or 100 times greater than to the naked eye.

Fig. 10. represents the section of a compound microscope with three lenses. By the middle one GK the pencils of rays coming from the object-glass are refracted so as to tend to a focus at O; but being intercepted by the proper eye-glass DF, they are brought together at I, which is nearer to that lens than its proper focus at L; so that the angle DIF, under which the object now appears, is larger than DLF, under which it would have appeared without this additional glass; and consequently the object is more magnified in the same proportion. Dr Hooke tells us, that, in most of his observations, he made use of a double microscope with this broad middle-glass when he wanted to see much of an object at one view, and taking it out when he would examine the small parts of an object more accurately; for the fewer refractions there are, the more bright and clear the object appears.

Having in the historical part of this article given a practical account of the construction of Dr Smith's power of double reflecting microscope, it may not be improper Dr Smith's in this place to ascertain its magnifying power. This we shall do from the author himself, because his symbols, being general, are applicable to such microscopes of all dimensions; and though the mere practical reader may perhaps be at first sight puzzled by them, yet, if he will substitute any particular numbers for m and n, &c. he may ascertain with ease the magnifying power of such a microscope of those particular dimensions.

Between the centre E and principal focus T of a concave speculum ABC, whose axis is EQTC, place an object PQ; and let the rays flowing from it be reflected from the speculum AB towards an image pq; but before they unite in it, let them be received by a convex speculum abc, and thence be reflected, through a hole BC in the vertex of the concave, to a second image w, to be viewed through an eye-glass l.

The object may be situated between the specula C, or, which is better, between the principal focus and vertex c of the convex one, a small hole being made in its vertex for the incident rays to pass through.

In both cases we have TQ, TE, Tq, continual proportions in some given ratio, suppose of 1 to n; and also Tq, tc, tx, continual proportions in some other given ratio, suppose of 1 to m. Then if d be the usual distance at which we view minute objects distinctly with the naked eye, and x the focal distance of the least eye-glass, through which the object appears sufficiently bright and distinct, it will be magnified in the ratio of mnd to xl.

For the object PQ, and its first image pq, are terminated on one side by the common axis of the specula, and on the other by a line PE, drawn through the centre E of the concave ABC. Likewise the images pq and wx are terminated by the common axis and by the line ecw, drawn through the centre e of the convex abc. Hence, by the similar triangles wx, pq, eq, Ex. v. 12a, and also pqE, PQE, we have wx : pq :: xe : qe :: m : r, and pq : PQ :: qE : QE :: n : r; and consequently wx : PQ :: mn : 1, whence wx = mn × PQ.

Now if lx be the focal distance of the eye-glass l, the points P, Q, of the object, are seen through it by the rays of two pencils emerging parallel to the lines wl, xl respectively; that is, PQ appears under an angle equal to wxl, which is as wxl = mnPQ ; and to the naked eye at the distance d from PQ, it appears under an angle PqQ which is as PQd, and therefore is magnified in the ratio of these angles, that is, of mnd to xl.

Corol. Having the numbers m, n, d, to find an eye-glass which shall cause the microscope to magnify M times in diameter, take xl = mndM. For the apparent magnitude is to the true as M : 1 : : mnd : xl.

We shall conclude this part of our subject with the following easy method of ascertaining the magnifying power of such microscopes as are most in use.

The apparent magnitude of any object, as must appear from what hath been already delivered, is measured by the angle under which it is seen; and the most this angle is greater or smaller according as the common object is near to or far from the eye; and of consequence the less the distance at which it can be viewed the larger it will appear. The naked eye is unable to distinguish any object brought exceedingly near it; but looking through a convex lens, however near the focus of that lens be, there an object may be distinctly seen; and the smaller the lens is, the nearer will be its focus, and in the same proportion the greater will be its magnifying power. From these principles it is easy to find the reason why the first or greatest magnifiers are so extremely minute; and also to calculate the magnifying power of any convex lens employed in a single microscope: For as the proportion of the natural light is to the focus, such will be its power of magnifying. If the focus of a convex lens, for instance, be at one inch, and the natural light at eight inches, which is the common standard, an object may be seen through that lens at one inch distance from the eye, and will appear in its diameter eight times larger than than it does to the naked eye; but as the object is magnified every way, in length as well as in breadth, we must square this diameter to know how much it really is enlarged; and we then find that its superficies is magnified 64 times.

Again, suppose a convex lens whose focus is only one-tenth of an inch distant from its centre; as in eight inches, the common distance of distinct vision with the naked eye, there are 80 such tenths, an object may be seen through this glass 80 times nearer than with the naked eye. It will, of consequence, appear 80 times longer, and as much broader, than it does to common sight; and therefore is 6400 times magnified. If a convex glass be so small that its focus is only \( \frac{1}{5} \) of an inch distant, we find that eight inches contains 160 of these twentieth-parts; and of consequence the length and breadth of any object seen through such a lens will be magnified 160 times, and the whole surface 25,600 times. As it is an easy matter to melt a drop or globule of a much smaller diameter than a lens can be ground, and as the focus of a globule is no farther off than a quarter of its own diameter, it must of consequence magnify to a prodigious degree. But this excessive magnifying power is much more than counterbalanced by its admitting so little light, want of distinctness, and showing such a minute part of the object to be examined; for which reason, these globules, though greatly in vogue some time ago, are now almost entirely rejected. Mr Leeuwenhoek, as has been already observed, made use only of single microscopes consisting of convex lenses, and left to the Royal Society a legacy of 26 of those glasses. According to Mr Folke's description of these, they were all exceedingly clear, and showed the object very bright and distinct; "which (says Mr Folke) must be owing to the great care this gentleman took in the choice of his glasses, his exactness in giving it the true figure, and afterwards, among many, reserving only such for his use as upon trial he found to be most excellent. Their powers of magnifying are different, as different objects may require: and as on the one hand, being all ground glasses, none of them are so small, or consequently magnify to so great a degree, as some of those drops frequently used in other microscopes; yet, on the other hand, the distinctness of these very much exceeds what I have met with in glasses of that sort. And this was what Mr Leeuwenhoek ever proposed to himself; rejecting all those degrees of magnifying in which he could not so well obtain that end. For he informs us in one of his letters, that though he had above 40 years by him glasses of an extraordinary smallness, he had made but very little use of them; as having found, in a long course of experience, that the most considerable discoveries were to be made with such glasses as, magnifying but moderately, exhibited the object with the greatest brightness and distinctness."

In a single microscope, if you want to learn the magnifying power of any glass, no more is necessary than to bring it to its true focus, the exact place whereof will be known by an object's appearing perfectly distinct and sharp when placed there. Then, with a pair of small compasses, measure, as nearly as you can, the distance from the centre of the glass to the object you were viewing, and afterwards applying the compasses to any ruler, with a diagonal scale of the parts of an inch marked on it, you will easily find how many parts of an inch the said distance is. When that is known, compute how many times those parts of an inch are contained in eight inches, the common standard of sight, and that will give you the number of times the diameter is magnified: squaring the diameter will give the superficies; and, if you would learn the solid contents, it will be shown by multiplying the superficies by the diameter.

The superficies of one side of an object only can be seen at one view; and to compute how much that is magnified, is most commonly sufficient: but sometimes it is satisfactory to know how many minute objects are contained in a larger; as suppose we desire to know how many animalcules are contained in the bulk of a grain of sand: and to answer this, the cube, as well as the surface, must be taken into the account. For the greater satisfaction of those who are not much versed in these matters, we shall here subjoin the following ### Table of the Magnifying Powers of Convex Glasses

Employed in Single Microscopes, according to the distance of their focus; Calculated by the scale of an inch divided into 100 parts. Showing how many times the diameter, the superficies, and the cube of an object, is magnified, when viewed through such glasses, to an eye whose natural sight is at eight inches, or 800 of the 100th-parts of an inch.

| Diameter | Superficies | Cube | |----------|-------------|------| | 1/2 or 50 | 16 | 256 | | 4/5 or 40 | 20 | 400 | | 1/3 or 30 | 26 | 676 | | 1/2 or 20 | 40 | 1,600| | 15 | 53 | 2,809| | 14 | 57 | 3,249| | 13 | 61 | 3,721| | 12 | 66 | 4,356| | 11 | 72 | 5,184| | 10 | 80 | 6,400| | 9 | 88 | 7,744| | 8 | 100 | 10,000| | 7 | 114 | 12,996| | 6 | 133 | 17,689| | 5 | 160 | 25,600| | 4 | 200 | 40,000| | 3 | 266 | 70,750| | 2 | 400 | 160,000| | 1 | 800 | 640,000|

The greatest magnifier in Mr Leeuwenhoek's cabinet of microscopes, presented to the Royal Society, has its focus, as nearly as can well be measured, at one-twentieth of an inch distance from its centre; and consequently magnifies the diameter of an object 160 times, and the superficies 25,600. But the greatest magnifier in Mr Wilton's single microscopes, as they are now made, has usually its focus at no farther distance than about the 50th part of an inch; whereby it has a power of enlarging the diameter of an object 400, and its superficies 160,000 times.

The magnifying power of the polar microscope must be calculated in a different manner; for here the difference between the focus of the magnifier and the distance of the screen or sheet whereon the image of the object is cast, is the proportion of its being magnified. Suppose, for instance, the lens made use of has its focus at half an inch, and the screen is placed at the distance of five feet, the object will then appear magnified in the proportion of five feet to half an inch: and as in five feet there are 120 half inches, the diameter will be magnified 120 times, and the superficies 14,400 times; and, by putting the screen at farther distances, you may magnify the object almost as much as you please; but Mr Baker advises to regard distinctly more than bigness, and to place the screen just at that distance where the object is seen most distinct and clear.

With regard to the double reflecting microscope, Mr Baker observes, that the power of the object-lens is indeed greatly increased by the addition of two eye-glasses; but as no object-lens can be used with them of so minute a diameter, or which magnifies of itself near so much as those that can be used alone, the glasses of this microscope, upon the whole, magnify little or nothing more than those of Mr Wilton's single microscope.

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§ 4. Telescopes.

I. The Refracting Telescope.

After what has been said concerning the structure of the compound microscope, and the manner in which the rays pass through it to the eye, the nature of the common astronomical telescope will easily be understood: for it differs from the microscope only in that the object is placed at so great a distance from it, that the rays of the same pencil, flowing from thence, may be considered as falling parallel to one another upon the object-glass; and therefore the image made by that glass is looked upon as coincident with its focus of parallel rays.

1. This will appear very plain from the 12th figure, in which A.B is the object emitting the several pencils of rays A c d, B c d, &c., but supposed to be at so great a distance from the object-glass c d, that the rays of the same pencil may be considered as parallel to each other; they are therefore supposed to be collected into their respective foci at the points m and p, situated at the focal distance of the object-glass c d. Here they form an image E, and crossing each other proceed diverging to the eye-glass b g; which being placed at its own focal distance from the points m and p, the rays of each pencil, after passing through that glass, will become parallel among themselves; but the pencils themselves will converge considerably with respect to one another, even so as to cross at e, very little farther from the glass g b than its focus; because, when they entered the glass, their axes were almost parallel, as coming through the object-glass at the point k, to whose whose distance the breadth of the eye-glass in a long telescope bears very small proportion. So that the place of the eye will be nearly at the focal distance of the eye-glass, and the rays of each respective pencil being parallel among themselves, and their axes crossing each other in a larger angle than they would do if the object were to be seen by the naked eye, vision will be distinct, and the object will appear magnified.

The power of magnifying in this telescope is as the focal length of the object-glass to the focal length of the eye-glass.

Dem. In order to prove this, we may consider the angle A & B as that under which the object would be seen by the naked eye; for in considering the distance of the object, the length of the telescope may be omitted, as bearing no proportion to it. Now the angle under which the object is seen by means of the telescope is g e h, which is to the other A & B, or its equal g' h', as the distance from the centre of the object-glass to that of the eye-glass. The angle, therefore, under which an object appears to an eye assisted by a telescope of this kind, is to that under which it would be seen without it, as the focal length of the object-glass to the focal length of the eye-glass.

It is evident from the figure, that the visible area, or space which can be seen at one view when we look through this telescope, depends on the breadth of the eye-glass, and not of the object glass; for if the eye-glass be too small to receive the rays g m, p b, the extremities of the object could not have been seen at all: a larger breadth of the object-glass conduces only to the rendering each point of the image more luminous by receiving a larger pencil of rays from each point of the object.

It is in this telescope as in the compound microscope, where we see, when we look through it, not the object itself, but only an image of it at C E D: now that image being inverted with respect to the object, as it is, because the axes of the pencils that flow from the object cross each other at k, objects seen through a telescope of this kind necessarily appear inverted.

This is a circumstance not at all regarded by astronomers: but for viewing objects upon the earth, it is convenient that the telescope should represent them in their natural posture; to which use the telescope with three eye-glasses, as represented fig. 15, is peculiarly adapted, and the progress of the rays through it from the object to the eye is as follows:

A B is the object sending out the several pencils A c d, B c d, &c. which passing through the object-glass c d, are collected into their respective foci in C D, where they form an inverted image. From hence they proceed to the first eye-glass e f, whose focus being at l, the rays of each pencil are rendered parallel among themselves, and their axes, which were nearly parallel before, are made to converge and cross each other: the second eye-glass g h, being so placed that its focus shall fall upon m, renders the axes of the pencils which diverge from thence parallel, and causes the rays of each, which were parallel among themselves, to meet again at its focus E F on the other side, where they form a second image inverted with respect to the former, but erect with respect to the object. Now this image being seen by the eye at a b through the eye-glass i k, affords a direct representation of the object, and under the same angle that the first image C D

would have appeared, had the eye been placed at l, supposing the eye-glasses to be of equal convexity; and therefore the object is seen equally magnified in this as in the former telescope, that is, as the focal distance of the object-glass to that of any one of the eye-glasses, and appears erect.

If a telescope exceeds 20 feet, it is of no use in viewing objects upon the surface of the earth; for if it magnifies above 90 or 100 times, as those of that length usually do, the vapours which continually float near the earth in great plenty, will be so magnified as to render vision obscure.

2. The Galilean Telescope with the concave eye-glass is constructed as follows:

A B (fig. 1.) is an object sending forth the pencils of rays g h i, k l m, &c. which, after passing through the object-glass c d, tend towards e F f (where we will suppose the focus of it to be), in order to form an inverted image there as before; but in their way to it are made to pass through the concave glass n o, so placed that its focus may fall upon E, and consequently the rays of the several pencils which were converging towards those respective focal points e, E, f, will be rendered parallel among themselves: but the axes of those pencils crossing each other at F, and diverging from thence, will be rendered more diverging, as represented in the figure. Now these rays entering the pupil of an eye, will form a large and distinct image a b upon the retina, which will be inverted with respect to the object, because the axis of the pencils cross in F. The object of course will be seen erect, and the angle under which it will appear will be equal to that which the lines a F, b F, produced back through the eye-glass, form at F.

It is evident, that the less the pupil of the eye is, the less is the visible area seen through a telescope of this kind; for a less pupil would exclude such pencils as proceed from the extremities of the object A B, as is evident from the figure. This is an inconvenience that renders this telescope unfit for many uses; and is only to be remedied by the telescope with the convex eye-glasses, where the rays which form the extreme parts of the image are brought together in order to enter the pupil of the eye, as explained above.

It is apparent also, that the nearer the eye is placed to the eye-glass of this telescope, the larger is the area seen through it; for, being placed close to the glass, as in the figure, it admits rays that come from A and B, the extremities of the object, which it could not if it was placed farther off.

The degree of magnifying in this telescope is in the same proportion with that in the other, viz. as the focal distance of the object-glass is to the focal distance of the eye-glass.

For there is no other difference but this, viz. that as the extreme pencils in that telescope were made to converge and form the angle g e h (fig. 12.), or i n k (fig. 13.), there are now made to diverge and form the angle a F b (fig. 1.); which angles, if the concave glass in one has an equal refractive power with the convex glass in the other, will be equal, and therefore each kind will exhibit the object magnified in the same degree.

There is a defect in all these kinds of telescopes, not to be remedied in a single lens by any means whatever, which was thought only to arise from hence,

Y y

Referring vis, that spherical glasses do not collect rays to one telescope; and the same point. But it was happily discovered by Sir Isaac Newton, that the imperfection of this sort of telescope, so far as it arises from the spherical form of the glasses, bears almost no proportion to that which is owing to the different refrangibility of light. This diversity in the refraction of rays is about a 28th part of the whole; so that the object-glass of a telescope cannot collect the rays which flow from any one point in the object into a less room than the circular space whose diameter is about the 56th part of the breadth of the glass.

To show this, let AB (fig. 2.) represent a convex lens, and let CDF be a pencil of rays flowing from the point D; let H be the point at which the least refrangible rays are collected to a focus; and I, that where the most refrangible concur. Then, if IH be the 28th part of EH, IK will be a proportional part of EC (the triangles HIK and HEC being similar); consequently LK will be the 28th part of FC. But MN will be the least space into which the rays will be collected, as appears by their progress represented in the figure. Now MN is but about half of KL; and therefore it is about the 56th part of CF: so that the diameter of the space into which the rays are collected will be about the 56th part of the breadth of that part of the glass through which the rays pass; which was to be shown.

Since therefore each point of the object will be represented in so large a space, and the centres of those spaces will be contiguous, because the points in the object the rays flow from are so; it is evident, that the image of an object made by such a glass must be a most confused representation, though it does not appear so when viewed through an eye-glass that magnifies in a moderate degree; consequently the degree of magnifying in the eye-glass must not be too great with respect to that of the object-glass, lest the confusion become sensible.

Notwithstanding this imperfection, a dioptrical telescope may be made to magnify in any given degree, provided it be of sufficient length; for the greater the focal distance of the object-glass is, the less may be the proportion which the focal distance of the eye-glass may bear to that of the object-glass, without rendering the image obscure. Thus, an object-glass, whose focal distance is about four feet, will admit of an eye-glass whose focal distance shall be little more than an inch, and consequently will magnify almost 48 times; but an object-glass of 40 feet focus will admit of an eye-glass of only four inches focus, and will therefore magnify 120 times; and an object-glass of 100 feet focus will admit of an eye-glass of little more than six inches focus, and will therefore magnify almost 200 times.

The reason of this disproportion in their several degrees of magnifying is to be explained in the following manner: Since the diameter of the spaces, into which rays flowing from the several points of an object are collected, are as the breadth of the object-glass, it is evident that the degree of confusedness in the image is as the breadth of that glass; for the degree of confusedness will only be as the diameters or breadths of those spaces, and not as the spaces themselves. Now the focal length of the eye-glass, that is, its power of magnifying, must be as that degree; for, if it exceeds it, it will render the confusedness sensible; and therefore it must be as the breadth or diameter of the object-glass. The diameter of the object-glass, which is as the square root of its aperture or magnitude, must be as the square-root of the power of magnifying in the telescope; for unless the aperture itself be as the power of magnifying, the image will want light: the square root of the power of magnifying will be as the square root of the focal distance of the object-glass; and therefore the focal distance of the eye-glass must be only as the square root of that of the object-glass. So that in making use of an object-glass of a longer focus, suppose, than one that is given, you are not obliged to apply an eye-glass of a proportionally longer focus than what would suit the given object-glass, but such an one only whose focal distance shall be to the focal distance of that which will suit the given object-glass, as the square root of the focal length of the object-glass you make use of, is to the square root of the focal length of the given one. And this is the reason that longer telescopes are capable of magnifying in a greater degree than shorter ones, without rendering the object confused or coloured.

But the inconvenience of very long telescopes is so great, that different attempts have been made to remove it. Of these, the most successful have been by Dollond and Blair; and the general principles upon which these and Blair's eminent opticians proceeded have been mentioned in the historical part of this article, and in the preceding section. The public will soon be favoured with a fuller account of Dr. Blair's discovery from his own pen; and of Dollond's, it may be sufficient to observe, in addition to what has been already said, that the object-glasses of his telescopes are composed of three distinct lenses, two convex and one concave; of which the concave one is placed in the middle, as is represented in fig. 3., where a and c show the two convex lenses, and b b the concave one, which is by the British artists placed in the middle. The two convex ones are made of London crown glass, and the middle one of white flint glass; and they are all ground to spheres of different radii, according to the refractive powers of the different kinds of glass and the intended focal distance of the object-glass of the telescope. According to Bofovich, the focal distance of the parallel rays for the concave lens is one-half, and for the convex glass one-third of the combined focus. When put together, they refract the rays in the following manner. Let a b, a b (fig. 4.), be two red rays of the sun's light falling parallel on the first convex lens c. Supposing there was no other lens present but that one, they would then be converged into the lines b e, b e, and at last meet in the focus g. Let the lines g h, g h, represent two violet rays falling on the surface of the lens. These are also refracted, and will meet in a focus; but as they have a greater degree of refrangibility than the red rays, they must of consequence converge more by the same power of refraction in the glass, and meet sooner in a focus, supposing at r.—Let now the concave lens d d be placed in such a manner as to intercept all the rays before they come to their focus. Were this lens made of the same materials, and ground to the same radius with the convex one, it would would have the same power to cause the rays diverge that the former had to make them converge. In this case, the red rays would become parallel, and move on in the line \( o_1 \), \( o_2 \): But the concave lens, being made of flint glass, and upon a shorter radius, has a greater refractive power, and therefore they diverge a little after they come out of it; and if no third lens was interposed, they would proceed diverging in the lines \( o_1 p_1 \), \( o_2 p_2 \); but, by the interpolation of the third lens \( o_3 \), they are again made to converge, and meet in a focus somewhat more distant than the former, as at \( x \). By the concave lens the violet rays are also refracted, and made to diverge: but having a greater degree of refrangibility, the same power of refraction makes them diverge somewhat more than the red ones; and thus, if no third lens was interposed, they would proceed in such lines as \( l_m n \), \( l_m n \). Now as the differently coloured rays fall upon the third lens with different degrees of divergence, it is plain, that the same power of refraction in that lens will operate upon them in such a manner as to bring them all together to a focus very nearly at the same point. The red rays, it is true, require the greatest power of refraction to bring them to a focus; but they fall upon the lens with the least degree of divergence. The violet rays, though they require the least power of refraction, yet have the greatest degree of divergence; and thus all meet together at the point \( x \), or very nearly so.

But, though we have hitherto supposed the refraction of the concave lens to be greater than that of the convex ones, it is easy to see how the errors occasioned by the first lens may be corrected by it, though it should have even a less power of refraction than the convex one. Thus, let \( a b \), \( a b \) (fig. 5.), be two rays of red light falling upon the convex lens \( c \), and refracted into the focus \( q \); let also \( g h \), \( g h \), be two violet rays converged into a focus at \( r \); it is not necessary, in order to their convergence into a common focus at \( x \), that the concave lens should make them diverge: it is sufficient if the glass has a power of dispersing the violet rays somewhat more than the red ones; and many kinds of glass have this power of dispersing some kinds of rays, without a very great power of refraction. It is better, however, to have the object glass composed of three lenses; because there is then another correction of the aberration by means of the third lens; and it might be impossible to find two lenses, the errors of which would exactly correct each other. It is also easy to see, that the effect may be the same whether the concave glass is a portion of the same sphere with the others or not; the effect depending upon a combination of certain circumstances, of which there is an infinite variety.

By means of this correction of the errors arising from the different refrangibility of the rays of light, it is possible to shorten dioptric telescopes considerably, and yet leave them equal magnifying powers. The reason of this is, that the errors arising from the object-glass being removed, those which are occasioned by the eye glass are inconsiderable: for the error is always in proportion to the length of the focus in any glass; and in very long telescopes it becomes exceedingly great, being no less than \( \frac{1}{8} \)th of the whole; but in glasses of a few inches focus it becomes trifling. Refracting telescopes, which go by the name of Dollond's, are therefore now constructed in the following manner. Let \( A B \) (fig. 6.) represent an object-glass composed of three lenses as above described, and converging the rays \( 1, 2, 3, 4 \), &c. to a very distant focus as at \( x \). By means of the interposed lens \( C D \), however, they are converged to one much nearer, as at \( y \), where an image of the object is formed. The rays diverging from thence fall upon another lens \( E F \), where the pencils are rendered parallel, and an eye placed near that lens would see the object magnified and very distinctly. To enlarge the magnifying power still more, however, the pencils thus become parallel are made to fall upon another at \( G H \); by which they are again made to converge to a distant focus; but, being intercepted by the lens \( I K \), they are made to meet at the nearer one \( z \); whence diverging to \( L M \), they are again rendered parallel, and the eye at \( N \) sees the object very distinctly.

From an inspection of the figure it is evident, that Dollond's telescope thus constructed is in fact two telescopes combined together; the first ending with the lens \( E F \), and the second with \( L M \). In the first we do not perceive the object itself, but the image of it formed at \( y \); and in the second we perceive only the image of that image formed at \( z \). Nevertheless such telescopes are exceedingly distinct, and represent objects fo'clearly as to be preferred, in viewing terrestrial things, even to reflectors themselves. The latter indeed have greatly the advantage in their powers of magnifying, but they are much deficient in point of light. Much more light is lost by reflection than by refraction: and as in these telescopes the light must unavoidably suffer two reflections, a great deal of it is lost; nor is this loss counterbalanced by the greater aperture which these telescopes will bear, which enables them to receive a greater quantity of light than the refracting ones. The metals of reflecting telescopes also are very much subject to tarnish, and require much more dexterity to clean them than the glasses of refractors; which makes them more troublesome and expensive, though for making discoveries in the celestial regions they are undoubtedly the only proper instruments which have been hitherto constructed. If Dr Blair indeed shall be so fortunate as discover a vitreous substance of the same powers with the fluid in the compound object-glass of his telescope (and from his abilities and perseverance we have every thing to hope), a reflecting telescope may be constructed superior for every purpose to the best reflector.

II. The Reflecting Telescope.

The inconveniences arising from the great length of refracting telescopes, before Dollond's discovery, are sufficiently obvious; and these, together with the difficulties occasioned by the different refrangibility of light, induced Sir Isaac Newton to turn his attention to the subject of reflection, and endeavour to realize the ideas of himself and others concerning the possibility of constructing telescopes upon that principle.—The instrument which he contrived is represented, fig. 7., where \( A B C D \) is a large tube, open at \( A D \) and closed at \( B C \), and of a length at least equal to the distance of the focus from the metallic spherical concave speculum \( G H \) placed at the end \( L C \). The rays \( E G \), \( F H \), &c. proceeding from a remote object \( P R \), intersect... Reflecting intersect one another somewhere before they enter the tube, so that EG, eg are those that come from the lower part of the object, and fh, FH from its upper part; these rays, after falling on the speculum GH, will be reflected, so as to converge and meet in mn, where they will form a perfect image of the object. But as this image cannot be seen by the spectator, they are intercepted by a small plane metallic speculum KK, intersecting the axis at an angle of 45°, by which the rays tending to mn will be reflected towards a hole LL in the side of the tube, and the image of the object will thus be formed in qS; which image will be less distinct, because some of the rays which would otherwise fall on the concave speculum GH, are intercepted by the plane speculum: nevertheless it will appear in a considerable degree distinct, because the aperture AD of the tube, and the speculum GH are large. In the lateral hole LL is fixed a convex lens, whose focus is at Sg; and therefore this lens will refract the rays that proceed from any point of the image, so as at their exit they will be parallel, and those that proceed from the extreme points Sg will converge after refraction, and form an angle at O, where the eye is placed; which will see the image Sg, as if it were an object, through the lens LL; consequently the object will appear enlarged, inverted, bright, and distinct. In LL lenses of different convexities may be placed, which by being moved nearer to the image or farther from it, would represent the object more or less magnified, provided that the surface of the speculum GH be of a perfectly spherical figure. If, in the room of one lens LL, three lenses be disposed in the same manner with the three eye-glasses of the refracting telescope, the object will appear erect, but less distinct than when it is observed with one lens. On account of the position of the eye in this telescope, it is extremely difficult to direct the instrument towards any object. Huygens, therefore, first thought of adding to it a small refracting telescope, the axis of which is parallel to that of the reflector. This is called a finder, or director. The Newtonian telescope is also furnished with a suitable apparatus for the commodious use of it.

In order to determine the magnifying power of this telescope, it is to be considered that the plane speculum KK is of no use in this respect. Let us then suppose, that one ray proceeding from the object coincides with the axis GLIA (fig. 8.) of the lens and speculum; let bb be another ray proceeding from the lower extreme of the object, and passing through the focus I of the speculum KH: this will be reflected in the direction bbd, parallel to the axis GLA, and falling on the lens dLd, will be refracted to G; so that GL will be equal to LI, and dG = dI. To the naked eye the object would appear under the angle Ibi = bIA; but by means of the telescope it appears under the angle eGL = dLI = idi: and the angle idi is to the angle Ibi: bI: Id; consequently the apparent magnitude by the telescope is to that by the naked eye as the distance of the focus of the speculum from the speculum, to the distance of the focus of the lens from the lens.

The Newtonian telescope was still inconvenient. Notwithstanding the contrivance of Huygens, objects were by it found with difficulty. The telescope of Gregory, therefore, soon obtained the preference, to which for most purposes it is justly intitled, as the reader will perceive from the following construction.

Let TYYT (fig. 9.) be a tube, in which LdD is a metallic concave speculum, perforated in the mid-telescope. Elle at X; and EF a lens concave mirror, so fixed by the arm or strong wire RT, which is moveable by means of a long screw on the outside of the tube, as to be moved nearer to or farther from the larger speculum LdD, its axis being kept in the same line with that of the great one. Let AB represent a very remote object, from each part of which issue pencils of rays, e.g. cd, CD, from A the upper extreme of the object, and IJ, iJ, from the lower part B; the rays LL, CD from the extremes crossing one another before they enter the tube. These rays, falling upon the larger mirror LD, are reflected from it into the focus KH, where they form an inverted image of the object AB, as in the Newtonian telescope. From this image the rays, issuing as from an object, fall upon the small mirror EF, the centre of which is at e; so that after reflection they would meet in their foci at QQ, and there form an erect image. But since an eye at that place could see but a small part of an object, in order to bring rays from more distant parts of it into the pupil, they are intercepted by the plano-convex lens MN, by which means a smaller erect image is formed at PV, which is viewed through the meniscus SS by an eye at O. This meniscus both makes the rays of each pencil parallel, and magnifies the image PV. At the place of this image all the foreign rays are intercepted by the perforated partition ZZ. For the same reason the hole near the eye O is very narrow. When nearer objects are viewed by this telescope, the small speculum EF is removed to a greater distance from the larger LD, so that the second image may be always formed in PV; and this distance is to be adjusted (by means of the screw on the outside of the great tube) according to the form of the eye of the spectator. It is also necessary, that the axis of the telescope should pass through the middle of the speculum EF, and its centre, the centre of the speculum LL, and the middle of the hole X, the centres of the lenses MN, SS, and the hole near O. As the hole X in the speculum LL can reflect none of the rays issuing from the object, that part of the image which corresponds to the middle of the object must appear to the observer more dark and confused than the extreme parts of it. Besides, the speculum EF will also intercept many rays proceeding from the object; and therefore, unless the aperture TT be large, the object must appear in some degree obscure.

In the best reflecting telescopes, the focus of the small mirror is never coincident with the focus of the great one, where the first image KH is formed, but a little beyond it (with respect to the eye), as at n; the consequence of which is, that the rays of the pencils will not be parallel after reflection from the small mirror, but converge so as to meet in points about QgQ, where they would form a larger upright image than PV, if the glass R was not in their way; and this image might be viewed by means of a single eyeglass properly placed between the image and the eye: but then the field of view would be less, and consequently... To find the magnifying power of this telescope, multiply the focal distance of the great mirror by the distance of the small mirror from the image next the eye, and multiply the focal distance of the small mirror by the focal distance of the eye glass; then divide the product of the former multiplication by the product of the latter, and the quotient will express the magnifying power.

One great advantage of the reflecting telescope is, that it will admit of an eye-glass of a much shorter focal distance than a refracting telescope will; and consequently it will magnify so much the more: for the rays are not coloured by reflection from a concave mirror, if it be ground to a true figure, as they are by passing through a convex glass, let it be ground ever so true.

The nearer an object is to the telescope, the more its pencils of rays will diverge before they fall upon the great mirror, and therefore they will be the longer of meeting in points after reflection; so that the first image KH will be formed at a greater distance from the large mirror, when the object is near the telescope, than when it is very remote. But as this image must be formed farther from the small mirror than its principal focus n, this mirror must be always set at a greater distance from the large one, in viewing near objects, than in viewing remote ones. And this is done by turning the screw on the outside of the tube, until the small mirror be so adjusted, that the object (or rather its image) appears perfect.

In looking through any telescope towards an object, we never see the object itself, but only that image of it which is formed next the eye in the telescope. For if a man holds his finger or a stick between his bare eye and an object, it will hide part (if not the whole) of the object from his view: But if he ties a stick across the mouth of a telescope before the object-glass, it will hide no part of the imaginary object he saw through the telescope before, unless it covers the whole mouth of the tube: for all the effect will be, to make the object appear dimmer, because it intercepts part of the rays. Whereas, if he puts only a piece of wire across the inside of the tube, between the eye-glass and his eye, it will hide part of the object which he thinks he sees; which proves, that he sees not the real object, but its image. This is also confirmed by means of the small mirror EF, in the reflecting telescope, which is made of opaque metal, and stands directly between the eye and the object towards which the telescope is turned; and will hide the whole object from the eye at O, if the two glasses ZZ and SS are taken out of the tube.

Great improvements have been lately made in the construction of both reflecting and refracting telescopes, as well as in the method of applying those instruments to the purposes for which they are intended. These, however, fall not properly under the science of optics, as fitter opportunities occur of giving a full account of them, as well as of the magic lantern, camera obscura, &c., under other articles of our multifarious work. See CATOPTRICS, DIOPTRICS, SPECULUM, and TELESCOPE. We shall conclude this article with some observations

On the different Merits of Microscopes and Telescopes, compared with one another; how far we may reasonably depend on the Discoveries made by them, and what hopes we may entertain of further Improvements.

The advantages arising from the use of microscopes and telescopes depend, in the first place, upon their property of magnifying the minute parts of objects, fo and tele- that they can by that means be more distinctly viewed by the eye; and, secondly, upon their throwing more light into the pupil of the eye than what is done without them. The advantages arising from the magnifying power would be extremely limited, if they were not also accompanied by the latter: for if the same quantity of light is spread over a large portion of surface, it becomes proportionably diminished in force; and therefore the objects, though magnified, appear proportionally dim. Thus, though any magnifying glass should enlarge the diameter of the object 10 times, and consequently magnify the surface 100 times, yet if the focal distance of the glass was about eight inches (provided this was possible), and its diameter only about the size of the pupil of the eye, the object would appear 100 times more dim when we looked through the glass, than when we beheld it with our naked eyes; and this, even on a supposition that the glass transmitted all the light which fell upon it, which no glass can do. But if the focal distance of the glass was only four inches, though its diameter remained as before, the inconvenience would be vastly diminished, because the glass could then be placed twice as near the object as before, and consequently would receive four times as many rays as in the former case, and therefore we would see it much brighter than before. Going on thus, still diminishing the focal distance of the glass, and keeping its diameter as large as possible, we will perceive the object more and more magnified, and at the same time very distinct and bright. It is evident, however, that with regard to optical instruments of the microscopic kind, we must sooner or later arrive at a limit which cannot be passed. This limit is formed by the following particulars. 1. The quantity of light lost in passing through the glass. 2. The diminution of the glass itself, by which it receives only a small quantity of rays. 3. The extreme shortness of the focal distance of great magnifiers, whereby the free access of the light to the object which we wish to view is impeded, and consequently the reflection of the light from it is weakened. 4. The aberrations of the rays, occasioned by their different refrangibility.

To understand this more fully, as well as to see how far these obstructions can be removed, let us suppose the lens made of such a dull kind of glass that it transmits only one half of the light which falls upon it. It is evident that such a glass, of four inches focal distance, and which magnifies the diameter of an object twice, still supposing its own breadth equal to that of the pupil of the eye, will show it four times magnified in surface, but only half as bright as if it was seen by the naked eye at the usual distance; for the light which falls upon the eye from the object at eight inches distance, and likewise the surface of the object in its natural natural size, being both represented by 1, the surface of the magnified object will be 4, and the light which makes that magnified object visible only 2; because though the glass receives four times as much light as the naked eye does at the usual distance of distinct vision, yet one half is lost in passing through the glass. The inconvenience in this respect can therefore be removed only as far as it is possible to increase the clearness of the glass, so that it shall transmit nearly all the rays which fall upon it; and how far this can be done, hath not yet been ascertained.

The second obstacle to the perfection of microscopic glasses is the small size of great magnifiers, by which, notwithstanding their near approach to the object, they receive a smaller quantity of rays than might be expected. Thus, suppose a glass of only \( \frac{1}{5} \)th of an inch focal distance; such a glass would increase the visible diameter 80 times, and the surface 6400 times. If the breadth of the glass could at the same time be preserved as great as that of the pupil of the eye, which we shall suppose \( \frac{1}{5} \)ths of an inch, the object would appear magnified 6400 times, at the same time that every part of it would be as bright as it appears to the naked eye. But if we suppose that this magnifying glass is only \( \frac{1}{10} \)th of an inch in diameter, it will then only receive \( \frac{1}{4} \)th of the light which otherwise would have fallen upon it; and therefore, instead of communicating to the magnified object a quantity of illumination equal to 6400, it would communicate only one equal to 1600, and the magnified object would appear four times as dim as it does to the naked eye. This inconvenience, however, is still capable of being removed, not indeed by increasing the diameter of the lens, because this must be in proportion to its focal distance, but by throwing a greater quantity of light on the object. Thus, in the above-mentioned example, if four times the quantity of light which naturally falls upon it could be thrown upon the object, it is plain that the reflection from it would be four times as great as in the natural way; and consequently the magnified image, at the same time that it was as many times magnified as before, would be as bright as when seen by the naked eye. In transparent objects this can be done very effectually by a concave speculum, as in the reflecting microscope already described; but in opaque objects the case is somewhat more doubtful; neither do the contrivances for viewing these objects seem entirely to make up for the deficiencies of the light from the smallness of the lens and shortness of the focus.

When a microscopic lens magnifies the diameter of an object 40 times, it hath then the utmost possible magnifying power, without diminishing the natural brightness of the object.

The third obstacle arises from the shortness of the focal distance in large magnifiers; but in transparent objects, where a sufficient quantity of light is thrown on the object from below, the inconvenience arises at last from straining the eye, which must be placed nearer the glass than it can well bear; and this entirely supercedes the use of magnifiers beyond a certain degree.

The fourth obstacle arises from the different refrangibility of the rays of light, and which frequently causes such a deviation from truth in the appearances of things, that many people have imagined themselves to have made surprising discoveries, and have even published them to the world; when in fact they have been only as many optical deceptions, owing to the unequal refractions of the rays. For this there seems to be no remedy, except the introduction of achromatic glasses into microscopes as well as telescopes. How far this is practicable, hath not yet been tried; but when these glasses shall be introduced (if such introduction is practicable), microscopes will then undoubtedly have received their ultimate degree of perfection.

With regard to telescopes, those of the refracting Dollond's kind have evidently the advantage of all others, where the aperture is equal, and the aberrations of the rays are corrected according to Mr Dollond's method; being superior to cause the image is not only more perfect, but a much all others, greater quantity of light is transmitted than what can be reflected from the best materials hitherto known. Unluckily, however, the imperfections of the glass set a limit to these telescopes, as hath already been observed, so that they cannot be made above three feet and an half long. On the whole, therefore, the reflecting telescopes are preferable in this respect, that they may be made of dimensions greatly superior; by which means they can both magnify to a greater degree, and at the same time throw much more light into the eye.

With regard to the powers of telescopes, however, they are all of them exceedingly less than what we would be apt to imagine from the number of times which they magnify the object. Thus, when we hear of a telescope which magnifies 200 times, we are apt to imagine, that, on looking at any distant object through it, we should perceive it as distinctly as we would with our naked eye at the 200th part of the distance. But this is by no means the case; neither is there any theory capable of directing us in this matter: we must therefore depend entirely on experience.

The best method of trying the goodness of any telescope is by observing how much farther off you are able to read with it than you can with the naked eye. But that all deception may be avoided, it is proper to choose something to be read where the imagination cannot give any assistance, such as a table of logarithms, or something which consists entirely of figures; and hence the truly useful power of the telescope is easily known. In this way Mr Short's large telescope, which magnifies the diameter of objects 1200 times, is yet unable to afford sufficient light for reading at more than 200 times the distance at which we can read with our naked eye.

With regard to the form of reflecting telescopes, it is now pretty generally agreed, that when the Gregorian ones are well constructed, they have the advantage supreme of those of the Newtonian form. One advantage common evident at first sight is, that with the Gregorian telescope an object is perceived by looking directly through Newtonian it, and consequently is found with much greater ease than in the Newtonian telescope, where we must look into the side. The unavoidable imperfection of the specula common to both, also gives the Gregorian an advantage over the Newtonian form. Notwithstanding the utmost care and labour of the workmen, it is found impossible to give the metals either a perfectly spherical or a perfectly parabolical form. Hence arises some Part II.

Microscopes and telescopes compared.

Some indistinctness of the image formed by the great speculum, which is frequently corrected by the little one, provided they are properly matched. But if this is not done, the error will be made much worse; and hence many of the Gregorian telescopes are far inferior to the Newtonian ones; namely, when the specula have not been properly adapted to each other. There is no method by which the workman can know the specula which will fit one another without a trial; and therefore there is a necessity for having many specula ready made of each sort, that in fitting up a telescope those may be chosen which best suit each other.

The brightness of any object seen through a telescope, in comparison with its brightness when seen by the naked eye, may in all cases be easily found by the following formula. Let \( n \) represent the natural distance of a visible object, at which it can be distinctly seen; and let \( d \) represent its distance from the object-glas of the instrument. Let \( m \) be the magnifying power of the instrument; that is, let the visual angle subtended at the eye by the object when at the distance \( n \), and viewed without the instrument, be to the visual angle produced by the instrument as \( r \) to \( m \). Micro-

Let \( a \) be the diameter of the object-glas, and \( p \) be scopes and that of the pupil. Let the instrument be so constructed, that no parts of the pencils are intercepted for want of sufficient apertures of the intermediate glasses. Lastly, let the light lost in reflection or refraction be neglected.

The brightness of vision through the instrument will be expressed by the fraction \( \frac{a^2}{m^2} \), the brightness of natural vision being 1. But although this fraction may exceed unity, the vision through the instrument will not be brighter than natural vision. For, when this is the case, the pupil does not receive all the light transmitted through the instrument.

In microscopes, \( n \) is the nearest limits of distinct vision, nearly 8 inches. But a difference in this circumstance, arising from a difference in the eye, makes no change in the formula, because \( m \) changes in the same proportion with \( n \).

In telescopes, \( n \) and \( d \) may be accounted equal, and the formula becomes \( \frac{a^2}{m^2} \).

INDEX.

ABERRATION, theory of, n° 251. Evils of—remedy, 252. Light distributed by over the smallest circle of diffusion, 253. Contrary aberrations correct each other, 255. Adam's method of making globules for large magnifiers, 110. Aerial speculums mentioned by Mr Gray, 47. Aerial images formed by concave mirrors, 254. Aethers, supposed, do not solve the phenomena of inflection, &c. 67. Air, refractive power of, 13, 14. Strongly reflects the rays proceeding from beneath the surface of water, 37. Alhambert (M. d'), his discoveries concerning achromatic telescopes, 17. Alhazen's discoveries concerning the refraction of the atmosphere, 6. His conjectures about the cause of it, ib. He gave the first hint of the magnifying power of glasses, ib. Alkaline salt diminishes the mean refraction, but not the dispersive power of glasses, 18. Angles, refracted tables of, published by Kepler and Kircher, n° 11. Antonio de Dominis, bishop of Spalatro, discovered the nature of the rainbow, 203. Apparent place of objects seen by reflection, first discovered by Kepler, 27. Barrow's theory respecting, 210. M. de la Hire's observations, 211. Berkeley's hypothesis on distance by confused vision, 213. Objected to by Dr Smith, 214. The objection obviated by Robins, 215. M. Bouguer adopts Barrow's maxim, 216. Porterfield's view of this subject, 217. Atmosphere varies in its refractive power at different times, 20. Illumination of the shadow of the earth by the refraction of the atmosphere, § 7, p. 339, &c. Attractive force supposed to be the cause of reflection, 176. The supposition objected to, 177. Obviated, 178. Another hypothesis, 179. Sir Isaac Newton's hypothesis, 180. untenable, 181. Azout (Mr) makes an object-glas of an extraordinary focal length, 93. On the apertures of refracting telescopes, 96. B. Bacon (Roger), his discoveries, n° 6, 8. Bacon (Lord), his mistake concerning the possibility of making images appear in the air, 26. Barker's (Dr) reflecting microscope, 113. Barrow's theory respecting the apparent place of objects, 210. Adopted by Bouguer, 216. Beams of light, the phenomenon of diverging, more frequent in summer than in winter, 241. Beaune (Mr) cannot fire inflammable liquids with hot iron or a burning coal, unless those substances be of a white heat, 45. Berkeley's theory of vision, 72. His hypothesis concerning the apparent place of objects, 213. Objected to by Dr Smith, 214. The objection obviated by Mr Robins, 215. Binocular telescope invented by Father Reita, 91. Black marble in some cases reflects very powerfully, 36. Blair (Dr Robert) makes an important discovery, 19. Blair and Dollond's reflecting telescope superior to all others, n° 278. Bodies which seem to touch one another are not in actual contact, 46. Eight hundred pounds weight on every square inch necessary to bring two bodies into apparent contact, 64. Bouguer's experiments to discover the quantity of light lost by reflection, 33. His discoveries concerning the reflection of glass and polished metal, 35. His observations concerning the apparent place of objects, 216. Throws great light on the subject of fallacies of vision, 220. Explains the phenomena of green and blue shadows seen in the sky, 234, 235. Contrivances for measuring light, 244. Calculations concerning the light of the moon, 248. Boyle's experiments concerning the light of differently coloured substances, 28. Briggs' solution of single vision with two eyes, 159. Brilliant, the cut in diamonds produces total reflection, 129. Brilliant curious appearance of the shadow of one, n°56.

Buffon's experiments on the reflection of light, 34. Observed green and blue shadows in the sky, 231, 232.

Burning-glasses of the ancients described, 25.

Campani's telescope, 92.

Candles, rays of light extended from, in several directions, like the tails of comets, 51.

Cat (M. le) explains the magnifying of objects by the inflection of light, 68. Accounts for the large appearance of objects in mist, 212. Explains a remarkable deception of vision, 225.

Cheft (Mr) made the same discovery with Dollond for the improvement of refracting telescopes, 18.

Clairaut's calculations respecting telescopes, 17.

Cold, why most intense on the tops of mountains, 43.

Colours discovered to arise from refraction, 15. Supposed by Dechales to arise from the inflection of light, 50. Produced by a mixture of shadows, 58. Colours simple or compound, 196.

Concave glasses, 74. An object seen through a concave lens is seen nearer, smaller, and less bright, than with the naked eye, 170. Law of reflection from a concave surface, 183. Proved, 185. Concave mirrors, p. 344.

Convex lens, an object seen thro' appears brighter, larger, and more distant, than when seen by the naked eye, n°168. In certain circumstances it appears inverted and pendulous in the air, 169. Law of reflection from a convex surface, 184. Proved mathematically, 185. Method of finding the focal distance of rays reflected from a convex surface, 189. Convex mirrors, p. 344.

Contact of bodies in many cases apparent without being real, 46. Eight hundred pounds on every square inch necessary to produce apparent contact, 64. Real contact of bodies perhaps never observed, 66.

N°249.

Corona, p. 327.

Crystal hath some refractive properties different from other transparent substances, n°39.

Cylinders: experiments by Miraldi concerning their shadows, 54.

D.

Deception in vision; a remarkable one explained by M. le Cat, 225.

Descartes' observations on the inflection of light, 50.

Descartes: his discoveries concerning vision, 71. Account of the invention of telescopes, 75.

Diamond, the brilliant cut in, produces total reflection, 129.

Dioptric instruments: difficulties attending the construction of them, 120. Telescopes why made so long, 95.

Distance of objects, § 3. p. 327, &c. Berkeley's account of the judgment formed concerning distance by confused vision, 213. Smith's account 214. Objected to by Robin's, 215. Bouguer adopts Barrow's maxim, 216. Portefield's view of it, 217.

Divini, a celebrated maker of telescopes, 92. His microscope, 107.

Diverging beams more frequent in summer than in winter, 241.

Dollond (Mr) discovers a method of correcting the errors arising from refraction, 17. He discovers a mistake in one of Newton's experiments, ib. Discovers the different refractive and dispersive power of glass, ib. Difficulties in the execution of his plan, p. 238. His improvements in the refracting telescope, n°99, 100. Dollond and Blair's refracting telescope superior to all others, 278.

Dominis (De) discovered the cause of the colours of the rainbow, 203.

E.

Edwards (Mr) improvements in the reflecting telescope, 98.

Emergent rays, the focus of, found, 144.

Equatorial telescope or portable observatory, 102. New one invented by Ramsden, ib.

Euler (Mr) first suggested the thought of improving refracting telescopes, n°17. His controversy with Clairaut, &c. ib. His scheme for introducing vision by reflected light into the solar microscope and magic lantern, 116. His theory of undulation contrary to fact, 136: and therefore misleads artists, 137.

Eye: the density and refractive powers of its humours first ascertained by Scheiner, 70. Description of it, 145. Dimensions of the infusible spot of it, 151. Eyes seldom both equally good, 217. Seat of vision in, dispute about, 150. Arguments for the retina being the seat of vision in, 152.

Eyes, single vision with two, 158 Various hypotheses concerning it, 159, 162, 161, &c. Brightness of objects greater when seen with two eyes than only with one, 163. When one eye is closed, the pupil of the other is enlarged, 164.

F.

Fallacies, several, of vision explained, 219. Great light thrown on this subject by M. Bouguer, 220.

Focus, the, of rays refracted by spherical surfaces ascertained, 141. Focus of parallel rays falling perpendicular upon any lens, 143. Focus of emergent rays found, 144. Proportional distance of the focus of rays reflected from a spherical surface, 188. Method of finding the focal distance of rays reflected from a convex surface, 189.

Fontana claims the honour of inventing telescopes, 79.

Force, repulsive, supposed to be the cause of reflection, 174. The supposition objected to, 175. Attractive, supposed, 176. The supposition objected to, 177. The objection obviated, 178.

Funk (Baron-Alexander), his observation concerning the light in mines, 47.

G.

Galileum telescope, more difficult of construction than others, 86.

Galileo made a telescope without a pattern, n°80. An account of his discoveries with it, 81. Why called Lyncens, 82. Account of his telescopes, 83. Was not acquainted with their ratios, 84. His telescope, 269. Magnifying power of, 270.

Glass globes, their magnifying powers known to the ancients, 3. Different kinds of them, ib. Table of the different compositions of glass for correcting the errors in refracting telescopes, p. 240. Shows various colours when split into thin laminae, n°31. Table of the quantities of light reflected from glass not quicksilvered, at different angles of incidence, p. 249. Glass, multiplying, phenomena of, n°256.

Glaziers, difference in their powers of refraction and dispersion of the light, p. 239.

Globes have shorter shadows than cylinders, n°55. And more light in their shadows, 57.

Globules, used for microscopes by Hartsoeker, 108. Adam's method of making them, 111.

Gregory's invention of the reflecting telescope, 97. Gregorian telescope, 275. Magnifying power of, 276. Gregorian telescope superior for common uses to the Newtonian, 279.

Grey (Mr), observation on aerial spectacles, 47. Temporary microscopes, 112.

Grimaldi first observes that colours arise from refraction, 15. Inflection of light first discovered by him, p. 253. His discoveries concerning inflection, n°49.

H.

Hairs, remarkable appearance of their shadows, 52.

Hartsoeker's microscope, 108.

Herchel's improvements in reflecting telescopes, 99.

Hire (M. de la), his reason why rays of light seem to proceed from luminous bodies when viewed with the eyes half shut, 51. Observations on the apparent place of objects, 211.

Hook (Dr), his discoveries concerning INDEX.

Concerning the inflection of light, n° 48.

Horizon, an object situated in, appears above its true plane, 166. Extent of the visible horizon on a plane surface, 227.

Horizontal moon, Ptolemy's hypothesis concerning it, 5.

Huygens greatly improves the telecoopes of Scheiner and Rhetta, 89. Improves the Newtonian telescope, 273.

Janfien (Zacharias), the first inventor of telecoopes, 77 and 78. Made the first microscope, 105, 106.

Images, Lord Bacon's mistake concerning the possibility of making them appear in the air, 26. Another mistake on the same subject by Vitellio, ib. B. Porta's method of producing this appearance, ib. Kircher's method, ib. Images, aerial, formed by concave mirrors, 259.

Impulse, doubtful if it has ever been observed, 66.

Incidence, ratio of the fine of to that of refraction, 126.

Incident velocity, increase or diminishes refraction, 130.

Inflection of light, discoveries concerning it, p. 253. Dr Hooke's discoveries concerning it, n° 48. Grimaldi's observations, 49. Dechales's observations, 50. Newton's discoveries, 52. Maraldi's, 53, 54. Probably produced by the same forces with reflection and refraction, 63.

Inversion, a curious instance of it observed by Mr Grey, 47.

Irradiations of the sun's light appearing through the interstices of the clouds, § 6, p. 337, &c. Converging observed by Dr Smith, n° 238. Explained by him, 239. Not observed by moonlight, 245.

Jupiter's satellites discovered by Janfien, 78. By Galileo, and called by him Medicean planets, 81.

K.

Kepler first discovered the true reason of the apparent place of objects seen by reflecting mirrors, 27. His discoveries concerning vision, n° 69. Improved the construction of telecoopes, 84, 87. His method first put in practice by Scheiner, 88.

Kircher attempted a rational theory of refraction, 11.

L.

Lead increases the dispersive power of glass, 17.

Lenses, their effects first discovered by Kepler, 88. Lenses, how many, 142. The focus of parallel rays falling perpendicular upon any lens, 143. Convex, an object seen through, appears larger, brighter, and more distant than by the naked eye, 168. In some circumstances it appears inverted, and pendulous in the air, 169. An object seen through a concave lens is seen nearer, smaller, and less bright, than with the naked eye, 170.

Leeuwenhoek's microscope, 109.

Light, its phenomena difficult to be accounted for, 1. Discovered not to be homogeneous, 16. Quantity of, reflected by different substances, 42. Quantity of it absorbed by plaster of Paris, 41. By the moon, ib. Observations on the manner in which bodies are heated by it, 43. No heat produced by it on a transparent medium, unless it is reflected from the surface, ib. Newton's experiments with respect to its inflection, 52. Reflected, refracted, and inflected by the same forces, 63. Different opinions concerning the nature of, 121. It issues in straight lines from each point of a luminous surface, 122. In what case the rays of light describe a curve, 124. Its motion accelerated or retarded by refraction, 127. Light of all kinds subject to the same laws, 132. The law of refraction when light passes out of one transparent body into another contiguous to it, 133. Some portion of light always reflected from transparent bodies, 171. Light is not reflected by impinging on the solid parts of bodies at the first surface, 172. Nor at the second, 173. Light consists of several sorts of coloured rays differently refrangible, 194. Reflected light differently refrangible, 195. Bouguer's contrivances for measuring light, 244. These instruments measure only the intensity of light, 245. Great variation of the light of the moon at different altitudes, 246. Variation in different parts of the disks of the sun and planets, 247. Bouguer's calculations concerning the light of the moon, 248. Dr Smith's, 249. Mr Mihell's, 250. Density of, in different points of refraction, 254.

Lignum nephriticum, remarkable properties of its infusion, 29.

Lines can be seen under smaller angles than spots, and why, 157.

Liquid substances cannot be fired by the solar rays concentrated, 44.

Long-fingered, § 155.

M.

Magic lantern, Mr Euler's attempt to introduce vision by reflected light into, 116.

Magnitudes of objects, § 3, p. 327, &c.

Mairan (Mr.), his observations on the inflection of light, 59.

Maraldi's discoveries concerning the inflection of light, 53, 54, 55. Further pursues Grimaldi's and Sir I. Newton's experiments, 56, 57. His experiments with a mixture of coloured shadows, 58.

Martin's (Mr) improvement of the solar microscope, 117.

Maurolycus, his discoveries, 9, 69.

Maxias (Abbé), attempts an explanation of the phenomena of green and blue shadows seen in the sky, 233.

Media, the various appearances of objects through different, stated and investigated, 165. An object seen through a plane medium, appears nearer and brighter than seen by the naked eye n° 167.

Melville (Mr), his observations on the heating of bodies by light, 43. Discovers that bodies which seem to touch are not in actual contact, 46. Explains a curious phenomenon of vision, 226. Explains the phenomena of green and blue shadows in the sky, 234.

Michell's (Mr), calculation of the light of the moon, 250.

Microscopes, their history, 105. Made by Janfien, 106. By Divini, 107. By Hartsoeker, 108. By Leeuwenhoek, 109. By Wilton, 110. Adam's method of making globules for large magnifiers, 111. Temporary microscopes, by Mr Grey, 112. Dr Barker's reflecting microscope, 113. Smith's reflecting microscope superior to all others, 114. Solar microscopes and that for opaque objects, 115. Mr Euler's scheme of introducing vision by reflected light into the solar microscope and magic lantern, 116. Martin's improvement, 117. Di Torre's extraordinary magnifying microscope, 118. Could not be used by Mr Baker, 119. Microscope compound use of several lenses in, 260. Dr Smith's, magnifying power of, 261. Easy method of ascertaining the magnifying power of, 262. Further observations on the magnifying power of, 263. Table of the magnifying powers of glasses used in, ib. Solar, magnifying power of 264. Merits of, compared with the telescope, 277.

Mines better illuminated in cloudy than in clear weather, 47.

Mirrors, § 2, p. 343, &c. Size of, in which a man may see his whole image, n° 257. Why three or four images of objects are seen in plane mirrors, 268. Aerial images formed by concave mirrors, 279.

Miss, account of the largeness Z z of objects in, by M. le Cat, n° 212.

Moon, Maraldi's mistake concerning the shadow of it, 56. Why visible when totally eclipsed, 242. Why the moon appears duller when eclipsed in her perigee than in her apogee, 243. Great variation of the light of the moon at different altitudes, 246. M. Bouguer's calculations concerning the light of, 248. Dr Smith's, 249. Mr Mitchell's, 250.

Illusion produced without impulse, 65, 66. Motion of light accelerated or retarded by refraction, 127.

Multiplying glasses, § 1, p. 342, &c. Phenomena of, n° 256.

N.

Newton (Sir Isaac) his discovery concerning colours, 16. Mistaken in one of his experiments 18. His discoveries concerning the inflection of light, 52. Theory of refraction objected to, 134. These objections are the necessary consequences of the theory, and therefore confirm it, 135. Reflecting telescope, 273. Magnifying power of, 274. Inferior to Gregoryan, 279.

Nollet (Abbé) cannot fire inflammable liquids by burning glasses, 44.

Objects on the retina of the eye appear inverted, 146. Why seen upright, 147. An object when viewed with both eyes does not appear double, because the optic nerve is insensible of light, 148. Proved by experiments, 149. Seen with both eyes brighter than when seen only with one, 163. The various appearances of objects seen through different media stated and investigated, 165. An object situated in the horizon appears above its true plane, 166. An object seen through a plane medium appears nearer and brighter than seen by the naked eye, 167. Object seen through a convex lens appears larger, brighter, and more distant, 168. In some circumstances an object thro' a convex lens appears inverted and pendulous in the air, 169. Barrow's theory respecting the apparent place of objects, 210. M. de la Hire's observations, 211. M. le Cat's account of the largeness of objects in mist, 212. Why objects seen from a high building appear smaller than they are, 218. Dr Porterfield's account of objects appearing to move to a giddy person when they are both at rest, 221.

Primary rainbow never greater than a semicircle, and why, 207. Its colours stronger than those of the secondary, and ranged in contrary order, 209.

Prisms in some cases reflect as strongly as quicksilver, 309. Why the image of the sun by heterogeneous rays passing thro' a prism is oblong, 197. Ptolemy first treated of refraction scientifically, 4.

R.

Rainbow (knowledge of the nature of) a modern discovery, 201. Approach towards it by Fletcher or Brew, 202. The discovery made by Antonio de Dominis bishop of Spalatro, 203. True cause of its colours, 204. Phenomena of the rainbow explained on the principles of Sir I. Newton, 205. Two rainbows seen at once, 206. Why the arc of the primary rainbow is never greater than a semicircle, 207. The secondary rainbow produced by two reflections and two refractions, 208. Why the colours of the secondary rainbow are fainter than those of the primary, and ranged in a contrary order, 209.

Raymén's (Mr) new equatorial telescope, 102.

Rays of light extinguished at the surface of transparent bodies, 38. Why they seem to proceed from any luminous object when viewed with the eyes half shut, 51. Rays at a certain obliquity are wholly reflected by transparent substances, 128. The focus of rays refracted by spherical surfaces ascertained, 141. The focus of parallel rays falling perpendicularly upon any lens, 143. Emergent rays, the focus of, found, 144. Rays proceeding from one point and falling on a parabolic concave surface are all reflected from one point, 187. Proportional distance of the focus of rays reflected from a spherical surface, 188. Several sorts of coloured rays differently refrangible, 194. Why the image of the sun by heterogeneous rays passing through a prism is oblong, 197. Every homogenous ray is refracted according to one and the same rule, 200.

Reflected light, table of its quantity from different substances, 40.

Reflecting telescope of Newton, 273. Magnifying power of, 274. Improved by Dollond and Blair, superior to all others, 278.

Reflection of light, opinions of the ancients concerning it, 23. Bouguer's experiments concerning the quantity of light lost by it, 33. Method of ascertaining the quantity lost in all the varieties of reflection, ib. Buffon's experiments on the same subject, 34. Bouguer's discoveries concerning the reflection of glas, and of polished metal, 35. Great difference of the quantity of light reflected at different angles of incidence, 36. No reflection but at the surface of a medium, 43. Is not produced by impulse, 65, 66. Rays at a certain obliquity are wholly reflected by transparent substances, 128. Total reflection produced by the brilliant cut in diamonds, 129. Some portion of light always reflected from transparent bodies, 171. Light is not reflected by impinging on the solid parts of bodies at the first surface, 172, nor at the second, 173. Fundamental law of reflection, 182. Laws of, from a concave surface, 183. From a convex, 184. These preceding propositions proved mathematically, 185. Reflected rays from a spherical surface never proceed from the same point, 186. Rays proceeding from one point and falling on a parabolic concave surface are all reflected from one point, 187. Proportional distance of the focus of rays reflected from a spherical surface, 188. Method thod of finding the focal distance of rays reflected from a convex surface, 189. The appearance of objects reflected from plane surfaces, 190. Reflected light differently refrangible, 195.

Refracting telescopes improved by Mr Dollond, 17. By Dr Blair, 19. Magnify in proportion to their lengths, 271. Imperfections in, remedied, 272.

Refraction, known to the ancients, 2. Its law discovered by Snellius, 11. Explained by Descartes, 12. Fallacy of his hypothesis, 13. Experiments of the Royal Society for determining the refractive powers of different substances, ib. M. de la Hire's experiments on the same subject, ib. Refraction of air accurately determined, 13, 14. Mistake of the Academy of Sciences concerning the refraction of air, 13. Allowance for refraction in computing the height of mountains, first thought of by Dr Hooke, 14. Mr Dollond discovers how to correct the errors of telescopes arising from refraction, 17. The same discovery made by Mr Cheff, 18. Important discovery of Dr Blair for this purpose, 19. Refraction defined, 123. Phenomena of refraction solved by an attractive power in the medium, 125. Refraction explained and illustrated, pages 279, 280. Ratio of the fine of incidence to the fine of refraction, n° 126. Refraction accelerates or retards the motion of light, 127. Refraction diminishes as the incident velocity increases, 130. Refraction of a star greater in the evening than in the morning, 131. Laws of refraction when light passes out of one transparent body into another contiguous to it, 133. The Newtonian theory of refraction objected to, 134. Which objections, as they are the necessary consequences of that theory, confirm it, 135. Laws of refraction in plane surfaces, 140. The focus of rays refracted by spherical surfaces ascertained, 141. Light consists of several sorts of coloured rays differently refrangible, 194. Reflected light differently refrangible, 195. Every homogeneal ray is refracted according to one and the same rule, 200.

Reid's solution of single vision with two eyes, 161. Repulsive force supposed to be the cause of reflection, 174. Objected to, 175. Another hypothesis, 179. Sir Isaac Newton's, 180. untenable, 181.

Retina of the eye, objects on, inverted, 146. Why seen upright, 147. When viewed with both eyes, not seen double because the optic nerve is insensible of light, 148. Arguments for the retina's being the seat of vision, 152.

Rheita's telescope improved by Huygens, 89. His binocular telescope, 91.

Robin's (Mr) objection to Smith's account of the apparent place of objects, 215.

S

Saturn's ring discovered by Galileo, 81.

Secondary rainbow produced by two reflections and two refractions, 208. Its colours, why fainter than those of the primary, and ranged in contrary order, 209.

Scheiner completes the discoveries concerning vision, 70. puts the improvements of the telescope by Kepler in practice, 83.

Shadows of bodies, observations concerning them, 48, 49, 52. Green shadows observed by Buffon, 231. Blue ones, 232. Explained by Abbé Mazeas, 233. Explained by Melville and Bouguer, 234. Curious observations relative to this subject, 235. Blue shadows not confined to the mornings and evenings, 236. Another kind of shadows, 237. Illumination of the shadow of the earth by the refraction of the atmosphere, § 7. p. 339, &c.

Short's (Mr) equatorial telescope, n° 102.

Short-lightedness, 155.

Sky, concave figure of, § 4. p. 324. &c. Extent of the visible horizon on a plane surface, n° 227. Why a long row of objects appears circular, 228. Why the concavity of the sky appears less than a semicircle, 229. Opinions of the ancients respecting the colour of the sky, 230. Green shadows observed by M. Buffon, 231. Blue shadows observed by him, 232. The phenomena explained by Abbé Mazeas, 233. By Melville and Bouguer, 234. Curious observations relative to this subject, 235.

Smith's (Mr Caleb) proposal to shorten telescopes, 101.

Smith's (Dr) reflecting microscope superior to all others, 114. Account of the apparent place of objects, 214. Objected to, 215. Converging irradiations of the sun observed and explained by, 238, 239. He never observed them by moonlight, 240. Diverging beams more frequent in summer than in winter, 241. Calculation concerning the light of the moon, 249. His microscope magnifying power of, 261.

Solar microscope, 115. Mr Euler's attempt to introduce vision by reflected light into the solar microscope, 116. Martin's improvement, 117. Magnifying power of, 264.

Spectacles, when first invented, 73.

Spots of the sun discovered by Galileo, 81. Not seen under so small an angle as lines, 157.

Stars, twinkling of, explained by Mr Michell, 21. By Mutchenbroek, 22. By other philosophers, ib. A momentary change of colour observable in some stars, ib.

Z z 2

Why visible by day at the bottom of a well, 32. How to be observed in the daytime, 103. The refraction of a star greater in the evening than in the morning, 131.

Sun, image of, by heterogeneous rays passing through a prism, why oblong, 197. The image of, by simple and homogeneous light, circular, 193. Variation of light in different parts of the sun's disk, 247.

Surfaces of transparent bodies have the property of extinguishing light, and why, 38. Supposed to consist of small transparent planes, 40, 41, 42. Laws of refraction in plane surfaces, 140. The focus of rays refracted by spherical surfaces ascertained, 141. Reflected rays from a spherical surface never proceed from the same point, 186. The appearance of objects from plane surfaces, 192. From convex, 191. From concave, 192. The apparent magnitude of an object seen by reflection from a concave surface, 193.

Telescopes, different compositions of glasses for correcting the faults of the refracting ones, 18. Descartes's account of the invention of them, 75. Other accounts, 76. Borellus's account probably the true one, 77. The first one exceeding good, 78. Fontana claims the honour of the invention, 79. Galileo made one without a pattern, 80. His discoveries on this head, 81. From which he acquired the name of Lynceus, 82. Account of his telescopes, 83. Rationale of the telescope first discovered by Kepler, 84. Reason of the effects of telescopes, 85. Galilean telescope difficult of construction, 86. Telescopes improved by Kepler, 87. His method first practised by Scheiner, 88. Huygens improves the telescopes of Scheiner and Rheita, 89. Vision most distinct in the Galilean ones, 90.