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

HISTORY.

§ 1. Discoveries concerning the Light.

These are enumerated under the article Light so fully, 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 subtle 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, does 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 anything 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 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 spectulum, 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 farther 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 purposely 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 philosophers 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 in 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, concerning the horizontal line, 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. fraction 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.

Discoveries of Alhazen

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 the ancient opinion of crystalline orbs in the regions 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 atmospherical 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 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 anything 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, Meitlin, 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 grotis 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 philosopher 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.

Of Vitello

In 1270, Vitello, 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 the 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 shew the state of knowledge, or rather of ignorance, at that time, to observe, that both Vitello, 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 Vitello was Roger Bacon, a man of very extensive genius, and who wrote upon almost every branch of science; yet 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 assent to 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 every thing 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 optics, 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 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 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 assisted 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 unsolved. Alhazen and Vitello, indeed, had attempted it; but failed, by attempting to measure the angle itself, instead of its sine. At last, however, 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 it appeared in the writings of Descartes, who published it under a different form, without making any acknowledgement of his obligations to Snellius, whose papers Huygens affirms us, from his own knowledge, Descartes had seen.

It does not appear that, before Descartes, any person attempted to explain the cause of refraction; which he undertook to do by the revolution 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 Thouleuse, 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. Concerning 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, viz. 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 30° 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 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 36½. 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 every-where, 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 abovementioned 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 Hauksbee 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 29.7½, 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 plates 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°, 8'3; so is the sine of any other incidence, to the sine of its angle of refraction; and so is radius, or 1000000, to 999736; 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 proportionable to its density, power of 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 may be tried, by heating the condensed or rarified 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 ethereal 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 Pike of Tenerife, 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 oblong, and that colours proceed from refraction.—The way in which first he discovered this was by Vitellio'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 light dispersive 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 contrary ways, and so be returned by the latter into the same course from which he 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 window, 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. This true cause, therefore, of the length of that image was discovered to be no other, than that light is not similar, or homogenous; 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 sine of the incidence of every kind of light, considered apart, is to its sine 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 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 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 ascertained 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 pretend to controvert the experiments of Newton; but he said, that they were not contrary to his hypotheses, 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 he thought 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. Klingenschierna 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 near will the result of it approach to Newton's description.

This paper of M. Klingenschierna, being communicated to Mr Dollond by M. Mallet, made him entertain... tain 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 through this double prism. For when it appeared neither raised or 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, that 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 shewing 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 dilution 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 $\frac{3}{4}$ 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 glass, especially as experience had already shewn 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 glass, 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 glass, with respect to the divergency of colours. The yellow or straw-coloured foreign sort, commonly called Venice glass, and the English crown glass, proved to be very nearly alike in that respect; though, in general, the crown glass seemed to make the light diverge the least of the two. The common English plate-glasses made the light diverge more; and the white crystal, or English flint-glass, most of all.

It was now his business to examine the particular qualities of every kind of glass 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 glass, and the white flint glass. He therefore ground one wedge of white flint, of about 25 degrees; and another of crown glass, 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 glass to different angles, till he got one which was equal, with respect to the divergency of the light, to that in the white flint-glass: 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 glass to be to that of the crown glass, 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 ap-plied plied together, so as to refract in a contrary direction, would refract the light without any dispersion of the rays.

In a letter to M. Klingenschierna, quoted by M. Clairaut, Mr Dollond says, that the fine of incidence in crown glass is to that of its general refraction as 1 to 1.53, and in flint glass 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 glass, and the concave of white flint glass. 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 glass 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 any thing that had been produced 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 fervile 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 reflection 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 Tournieres, and the results agreed with Mr Dollond's.

(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. lond'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 Mathématiques. 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 benefits 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, 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-glasses. 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 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.

No 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, Mr 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. Zwiher of Pittsburgh 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. 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. Zeiller, dated Pittsburgh 30th of January 1764, in which he gives him a particular account of the success of his experiments on the composition of glasses; and that, having mixed minium and sand in different proportions, the result of the mean refraction and the dispersion of the 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. — 1 : 2 | 1732 : 1000 | 2207 : 1000 | | V. — 1 : 3 | 1724 : 1000 | 1800 : 1000 | | VI. — 1 : 4 | 1664 : 1000 | 1354 : 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 glasses, 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 entertained any doubt concerning the same property in flint glasses, 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 glass a constancy 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.

We shall conclude the history of the discoveries concerning refraction, with some account of the refraction of the atmosphere.—Tables 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 lowed 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 so 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 \( \frac{1}{5} \) of an inch; but afterwards, repeating the experiment on a cloudy day, when the air was rather gross 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 shews, 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 Michell 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 1000 to 1), may not exceed 3000 or 4000, and from stars of the second magnitude they may, therefore, probably not exceed 100. 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, so 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 density of light does not contribute to this effect in so great a degree as he had imagined, especially in consequence of observing that even Venus does sometimes twinkle. This he once observed her to do remarkably when she was about 6 degrees high, though Jupiter, which was then about 16 degrees high, and was sensibly less luminous, did not twinkle at all. If, notwithstanding the great number of rays which, no doubt, come to the eye from such a surface as this planet presents, its appearance be liable to be affected in this manner, it must be owing to such undulations in the atmosphere, as will probably render the effect of every other cause altogether insensible. The conjecture, however, has so much probability in it, that it well deserves to be recited.

M. Mutchchenbroek supposes, 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 frosty, 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 nearer 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 interposing 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 steadfastly 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.

§ 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 philologists 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 doth 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 sunbeams 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 light or vision, the ancient geometers contented themselves with deducing a system of optics from the 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 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 geometer.

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 fires by means of a concave speculum. It seems indeed to have been known pretty early, that there is an increase 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 geometer 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. Plate CCVI

In some drawings of this instrument frustum 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 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 surprized 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 tho' this was large, he could not in a long time set a piece of wood on fire with it; tho' a flat lens 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 whitest 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, that a virtuoso, of unsuspected credit, acquainting 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 fusion of the property of lignum nephriticum first observed by Kircher. cher. 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 coatl or tlapazatli, 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 oculist, 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 tinging 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 doliquum, 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 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, tho' 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 glas, 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.

The first distinct account of the colours exhibited by thin plates of various substances, are met with among the observations of Mr Boyle. To show the effect of thin chemists that colours may be made to appear or va-plates, nith, where there is no accession or change either of the fulphurous, 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, shewed 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 surprise 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, tho' 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 tho' 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 trajection; 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 muscovy 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 thro' 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 Bererton, at a meeting of the Royal Society in 1666, produced some 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 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 well by day. It is, says he, because the light that at the bottom 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. Mairan'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 intitled Essai d'Optique, which was received with general approbation. Afterwards, 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 circumspection, 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 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 is necessary, he observes, that the two beams of 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. Buffon, 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 on 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 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 any thing 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, he found, that when it was placed at an angle 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 56x 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 platter, placed at an angle of 75°, with the incident rays, reflected 1/3 part of the light 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 roughness 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, 1/3 at least is lost, and that probably no substances reflect more than this. The rays were received at an angle of 11½ 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 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, 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° 35' of incidence, though not perfectly polished, yet it reflected almost as well as quick-silver. 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 quick-silver. 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 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 1/2, 1/3, 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 quick- 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 32nd 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 | |---------------------|-----------------------| | ½ | 721 | | 1 | 692 | | 1 ½ | 669 | | 2 | 639 | | 2 ½ | 614 | | 5 | 501 | | 7 | 409 | | 10 | 333 | | 12 ½ | 271 | | 15 | 211 |

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 | 584 | | 5 | 543 | | 7 | 474 | | 10 | 412 | | 12 ½ | 356 | | 15 | 299 | | 20 | 222 | | 25 | 157 |

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, \( \frac{1}{100} \) 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 \( \frac{1}{500} \); 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 Mr Edwards, (see Phil. Transf. vol. 53. p. 229.) that there is a remarkably strong reflection into water, with respect to rays issuing from the water; and persons under water have seen images of things in the water 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 at light, is truly remarkable, and, as there is reason to believe, had not been noticed by any person before M. Bouguer. It had been conjectured by Sir Isaac Newton, that rays of light become extinct only by impinging upon the solid part 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 inflects the light.

One of the above-mentioned observations, viz. all Strong red light being reflected at certain angles of incidence prism, from light into denser substances, had frequently been observed, 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 \( \frac{1}{3} \) reflected. This property retains its full force as far as an incidence of 49° 49', (supposing the proportion of the sines 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 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 $41^\circ 32'$; and in every medium it depends much on the invariable proportion of the sine of the angle of refraction to the sine 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.

Refining his observations on the diminution of the quantity 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 appearances of opaque bodies to consist of very small planes, it appears from these observations, that there are fewer of them in these 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 these 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 follows:

| Inclinations of the small surfaces with respect to the large one. | 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 variations in the number of little planes, or surfaces, he expresses in the form of a curve; and afterwards he shews, 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 abovementioned principles, 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 proportionally 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 abovementioned, 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 of small surface, separately taken, is extremely irregular, bodies, 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 one another.

If it be asked what becomes of those rays that are reflected from one alperty to another, he shews 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 \(\frac{1}{3}\) of an inch, which space a particle of light describes in \(1\) second. With so rapid a motion, therefore, may the internal parts of bodies be agitated by the influence of light, as to perform \(125,000,000,000,000\) 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 subtle, and more 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, should warm it more than the common sunbeams.

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 distance 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 proportion 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 carefully compared together.

From this doctrine he thinks it reasonable to suppose, 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 inferior 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, collected 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 ascribed to it; since we know not in what high proportion 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, preserves its own colour and its own direction, in the same manner. The attempts of the Abbé Nollet to fire inflammable substances by the power of the solar rays collected in the foci of burning mirrors, have a near relation 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 February 1757, he was not able to do it either with spirit-of-wine, olive-oil, oil-of-turpentine, or ether; 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 liquor 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 these experiments, observed farther, that the same substances which were easily fired by the flame of burning bodies, could not be set on fire by the contact of the hottest bodies that did not actually flame. Neither ether 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 observations on the reflection of light, Mr Melville discovered that bodies which seem to touch one another are not always in actual contact. "It is common (says he) to admire the volatility and lustre of drops of rain that lie on the leaves of colewort, and some 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 produced by a copious reflection of light from the flattened part of its surface contiguous to the plant. He observed 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 discerned.

From these two observations put together, he concluded, 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 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, affumes 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 under-ground 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 so 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 five 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 discoveries. 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,

Plate CCVI 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 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 fainest, 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 inclination 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's discoveries. 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, EE, 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 IL, 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, supposing, 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 betwixt 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 the light 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 $\frac{4}{100}$ or $\frac{5}{100}$ part of an ancient Roman foot, and the second aperture, GH, $\frac{3}{100}$ or $\frac{3}{100}$; 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 Observations an additional observation of Dechales; who took of Dechales' 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. ches; 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 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 exercised 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 eye-lids 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 eye-lid, 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 eye-lashes, in this situation of the eye, is inflected 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 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 came 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 thro' 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 $1, \sqrt{\frac{1}{3}}, \sqrt{\frac{1}{7}}, \sqrt{\frac{1}{11}}$, 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 $1, \sqrt{\frac{1}{3}}, \sqrt{\frac{1}{7}}, \sqrt{\frac{1}{11}}, \sqrt{\frac{1}{15}}$, 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 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 pate-board, board, black on both sides; and in the middle of it he had made a hole about \( \frac{1}{4} \) 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 thro' 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 \( \frac{1}{4} \) of an inch, or \( \frac{1}{8} \) 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 narrowest 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 too 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 4th 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 \( 1^\circ 54' \). The knives being thus fixed together, he placed them in a beam of the sun's light let into his darkened chamber, thro' 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. smooth white ruler, at the distance of \( \frac{1}{2} \) 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 meet 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 millesimal parts of an inch | |--------------------------------------------------|------------------------------------------------------------------------| | \( \frac{1}{2} \) | 0.012 | | \( \frac{3}{4} \) | 0.020 | | \( \frac{8}{9} \) | 0.034 | | \( \frac{32}{9} \) | 0.057 | | \( \frac{96}{9} \) | 0.081 | | \( \frac{131}{9} \) | 0.087 |

From these observations he concluded, that the 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 fig. 2 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 eis, fkt, 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 xip, 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 eis and xip, 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, DE, 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 \( \frac{1}{12} \) of an inch, and in the full violet \( \frac{1}{12} \). 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 \( \frac{1}{12} \), and the violet \( \frac{1}{12} \) of an inch; and these distances of the fringes held the same proportion. portion 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 of Newton on reflected light is M. Maraldi; 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 lines in diameter; when its shadow, being received upon a paper held close to it, was everywhere equally black and well defined, and continued to be so to the distance of 23 inches 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 CCVII., 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 ⅓ 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 brittle, 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, that sufficiently singular, 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 CCVII. fig. 4.

The shadow of the same plate at $4\frac{1}{2}$ 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 $2\frac{1}{2}$ 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 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. 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 brittles were visible upon it, as in fig. 8. To explain this appearance, he supposed that the rays of the sun glid a little along the brittle, so as to enlighten part of that which is 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 infected 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 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 lights, 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 an 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 hypotheses, 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 hit upon the following ingenious method of making them all appear very distinct.

He took a circular board ABED (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 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 near A, 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 are 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 shews, 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 shews that the ray \(ab\), 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 violet at \(j\); 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 in which the beam of light enters, as at E, 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 shews, 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 and leaving the atmosphere, as at \(b\) and \(ru\), 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, 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 shews, by fig. 12, that the rays \(ab\) are farther separated by both the refractions than the rays \(cd\).

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 attributes 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 reflected; 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 I, and that of these rays the blue tinge observable in the shadows of some bodies are formed.

M. Le Cat has well explained a phenomenon of objects vision depending upon the inflection of light, which sometimes shews, that, in some cases, objects appear magnified by this means. Looking at a distant steeple, when a flexion of wire, of a less diameter than the pupil of his eye, was light held pretty near to it, and drawing it several times betwixt 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 reflected 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.

This discovery, depending upon the inflection of the rays of light, led him to several others depending upon the same principle. Thus he magnified small objects, as the head of a pin, by looking at them through a small hole in a card; so that the rays which formed the image must necessarily pass near the circumference of the hole, as to be attracted by it. He also observed, that, by bringing his finger near the cone of light, which transmitted to him the image of any object well insulated, as a red coal in the midst of cinders, or a black coal in the midst of the fire, the object seemed to stretch itself towards his finger, as it approached, and to follow it to a certain distance when it was withdrawn. He thought it was owing to the same cause, that the clouds which pass over the face of the sun occasion various motions in the shadow of bodies; and when these clouds are interrupted in different places, those shadows seem to dance, which is particularly visible in the shadows made by the leads in glass-windows. It was to the same inflection of light that he ascribed, in part, the prismatic colours which he saw by the means of a very fine pin, placed near to his eye, and on which he made the light of a candle to fall obliquely.

§ 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 issuing from external objects, and throws them upon the retina, where is the focus of each pencil. He did not, however, find out, that, by means of this refraction of the rays, 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. Montucia conjectures, that he was prevented from coming to this difficulty of accounting for the upright appearance of objects, as this image 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 Vitellio 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 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 discussing every possible hypothesis concerning the proper seat of vision, he demonstrates that it is in the retina, and shews that this was the opinion of Alhazen, Vitellio, Kepler, and all the most eminent philosophers. He produces many reasons of his own for this hypothesis; answers a great number of 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 description of the eye itself, the various modes of vision and optical deceptions to which we are liable, belong properly to the succeeding part of this treatise.

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

So little were the ancients acquainted with the science of Optics, that they seem to have had no instruments of the optical kind, excepting the glass globes and speculums 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 and the observations and experiments of Bacon together, 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 Spinosa, 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 Salvinius Armatus, a nobleman of Florence, who died 1317, that he was the inventor of spectacles.

The use of concave glasses, to help those persons of concave who are short-sighted, was, probably, a discovery glasses, that did not follow 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 Descartes's 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 scopes, 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 Antwerp, Middleburgh, or rather by his children; who, like Metius, were diverting themselves with looking through two glasses at a time, and placing them at different distances from one another. But Borellus, the author of a book intitled, De vero teleoptici 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 probably 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 these purposes, when he 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 included 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 Lippersheim by Sirtorus. 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 Drebbel of Alcmar, in Holland, applied to the inventor himself in 1620; as also did Galileo, and many others. The first telescope made by Janjen did not exceed 15 or 16 inches in length; but Sirtorus, who says that he had seen it, and made use of it, thought it the best that he had ever examined.

Janjen, 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, tho' the person who made it was not aware of the importance of his discovery.

One Francis Fontana, an Italian, also claims the 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 further account of the discovery had reached that place, tho' 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 shewed 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 27th 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 nebula 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 Cosimo, 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, was the cause of much speculation and debate among the philosophers and astronomers of that time; many of whom could not be brought. 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 stars 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 planetam 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 concealed 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 Lynceus, 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 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 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 master 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 explication 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 general 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 subsistence, 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 so refracts the pencils of rays, as 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 thro' 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 base, 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 eye-glass, 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 should be of a much more difficult construction than some 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; and 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 thought of for many years after the discovery. Descartes, who wrote 30 years after, mentions no other 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 Catoptries; 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 telephones be the same, the Galilean will have the greater magnifying power.

Vision is also more distinct in these telephones; owing perhaps, in part, to there being no intermediate image between the eye and the object. Besides, the Galilean telescope, the eye-glass being very thin in the centre, the 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 sort, will hardly make them visible.

The same Father Rheita, to whom we are indebted for the useful construction of a telescope for land-telephones, objects, invented a binocular telescope, which Father Cherubin, of Orleans, endeavoured to bring into use afterwards. It consists of two telephones 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, tho' 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 telecopic 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 eyeglasses. 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 Lewis XIV. and were of 86, 100, and 126 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 basins on which he ground his glasses) the goodness of his lenses depended upon the clearness of his glass, 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, shewing 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 Neville, Dr Hooke says, made telescopes of 36 feet, pretty good, and one of 50, 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 affirm.

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. Hartsoeker 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 h, 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 a. When it was very dark, M. Huygens was obliged to make his object-glass visible by a lantern, y, so constructed as to throw the rays of light in a parallel direction up to it.

The recollection of the incredible pains which philosophers of the last age took in making observations, and the great expenses they were obliged to be at for that purpose, should make us sensible of the obligations we are under to such men as Gregory, Newton, and Dollond, who have enabled us to get clearer and more satisfactory views of the remote parts of our system, with much less labour and expense; and should likewise make us more diligent and solicitous to derive all the advantages we possibly can from such capital improvements.

The reason why it is necessary to make the common dioptric telescopes so very long, is, that the length of them must be increased in no less a proportion than the duplicate of the increase of their magnifying power; so that, in order to magnify twice as much as before, with the same light and distinctness, the telescope must be lengthened four times; and to magnify thrice as much; nine times; and so on.

Before we mention the reflecting telescope, it must be observed, that M. Auzout, in a paper delivered to the Royal Society, observed, that the apertures which the object-glasses of refracting telescopes can bear with distinctness, are in about a sub-duplicate proportion to their lengths; and upon this supposition he drew up a table of the apertures proper for object-glasses of a great variety of focal lengths, from 4 inches to 400 feet. Upon this occasion, however, Dr Hooke observed, that the same glass will bear a greater or less aperture, according to the less or greater light of the object. If, for instance, he was viewing the sun, or Venus, or any of the fixed stars, he used smaller apertures; but if he wanted to view the moon by daylight; or Saturn, Jupiter, or Mars, by night; he used a larger aperture.

But the merit of all these improvements was in a manner cancelled by the discovery of the much more commodious reflecting telescope. For a refracting telescope, even of 1000 feet focus, supposing it possible to be made use of, could not be made to magnify with distinctness more than 1000 times; whereas a reflecting telescope, not exceeding 9 or 10 feet, will magnify 1200 times.

"It must be acknowledged," says Dr Smith in his Complete System of Optics, "that Mr James Gregory of Aberdeen was the first inventor of the reflecting telescope; but his construction is quite different from Sir Isaac Newton's, and not nearly so advantageous."

But, according to Dr Pringle, Mercennus was the man who entertained the first thought of a reflector. A telescope with specula he certainly proposed to the celebrated Descartes many years before Gregory's invention, though indeed in a manner so very unsatisfactory, that Descartes, who had given particular attention to the improvement of the telescope, was so far from approving the proposal, that he endeavoured to convince Mercennus of its fallacy (A). Dr Smith, 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 set 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, as he himself declares, 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 some new discoveries been made in light and colours, a reflecting 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 happy genius for experimental knowledge was equal to that for geometry, happily interpreted, 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 lenses, of whatever figure, would be affected more or less with such prismatic aberrations.

(A) Lettres de Descartes, tome ii. printed at Paris in 1657, lett. 29. and 32. See this point discussed by two learned and candid authors, M. le Roi in the Encyclopedie, under the article Telescope, and M. Montecula in Hist. des Mathem. tome ii. p. 644. 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 unsurmountable, 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 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 Scavans for the year 1672, and by that channel it was soon known over Europe.

But how excellent soever the contrivance was; how well soever 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 moving 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 those 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 signalized himself at Edinburgh by his work of this kind. Mr MacLaurin 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, prefigured that the public would one day possess a parabolic speculum, not accomplished by mathematical rules, but by mechanical devices.

This was a desideratum, 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 dextrous 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 greatest improvement in refracting telescopes 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 thro' 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 contriving 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 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: 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.

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 lines of refraction, or rays differently refrangible, are to one another in a given proportion, when their lines 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 mid-day, 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.

In order to enable us to see the fixed stars in the day-time, it is necessary to exclude the extraneous 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 day-time; 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. But a particular description of its construction and uses, will be afterwards given.

M. Epinus 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 tubes of 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 the 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, microscopes 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. Janfen, 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 Jansens, 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 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, shewed 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 telescope was evidently a compound one, or rather something betwixt a telescope and a microscope, what we should now, perhaps, choose to call a megalascope; so that it is possible that single microscopes might have been known, and in use, some time 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 improvement 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.

Eustachio 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 femine mafculino, which gave rise to a new system of generation. A microscope of this kind, consisting of a globule of $\frac{1}{2}$ 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 microscopic discoveries as the famous M. Leeuwenhoek, though he used only single lenses with short foci, 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 mucovine talc, 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, (of which an exact copy, taken from the picture in his works, may be seen, Plate CCVIII. fig. 2.) seems to like our opaque microscope, with its silver speculum, that it may well be supposed to have given the hint to the ingenious inv... ventor 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 they have been so generally approved, that hardly any other method is made use of at this day. The capital advantage attending this microscope consists in its being furnished with a pretty large lens, to throw light upon the objects, and also with a fine screw and spring, to bring them nearer, or remove them farther from the magnifier at pleasure. This microscope is also a necessary part of the solar microscope, invented afterwards.

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 ½ of an inch in breadth; then, holding one of them between the fore-finger 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 item he broke off as near to the globule as he could, and lodging the remainder between the plates, in which holes were drilled exactly round, the microscope, he says, performed to admiration. Through these magnifiers, he says, that the same thread of very fine muslin 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. These 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 globule, in the proper place for viewing objects. But Montucla observes, that, when any object is inclosed within this small transparent globule, 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 32 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 also 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 Dr Barker's from the reflecting telescope, excepting the distance of reflecting the two speculums, in order to adapt it to those pencils of rays which enter the telescope diverging; whereas they come from very distant objects nearly parallel to each other.

This microscope is not so easy to manage as the common sort. 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.

In Dr Smith's reflecting microscope there are two reflecting mirrors, one concave and the other convex, and microscope, the image is viewed by a lens.

This microscope, though far from being executed in the best manner, performed, Dr Smith says, nearly as well as the very best refracting microscopes; so that he did not doubt but that it would have excelled them, if it had been executed properly. Dr Smith's own account of this instrument may be seen in his Optics, Remarks, p. 94.

In 1738 or 1739, M. Lieberkuhn made two capital Solar improvements in microscopes, by the invention of the and that for solar microscope, and the microscope for opaque objects. When he was in England in the winter of 1739, he shewed 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 solar microscope, as made by Mr Cuff, was composed of a tube, a looking-glass, a convex lens, and a Wilton's microscope. Of this, and of another constructed by Mr Martin, a particular description will be afterwards given.

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. Lieberkuhn 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. Lieberkuhn's apparatus to publish an account of this instrument; but it doth not appear that his method was ever published.

This improvement of M. Lieberkuhn's induced M. Epinus himself to attend to the subject; and by this means produced a very valuable improvement in this instrument. For by throwing the light upon the foreside 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 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 concave mirror, perforated as for a telescope; and that 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. An idea of this contrivance is given in Plate CCVIII. fig. 3, in which OD represents the concave mirror, E the object, L the lights, and A the lens, through which the rays are transmitted to the screen.

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

Mr Martin having constructed a solar microscope of a 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 be 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 magnifiers, for microscopes, that have yet been executed, were made by T. Di Torre of Naples, who, in 1765, sent 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 the third was no more than half of a Paris point, or the 144th 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 glass of which they are made; but though many attempts have been made to make glass without that imperfection, none of them have been hitherto quite effectual. M. A. D. Merklein, having found some glass 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 glass; 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 II. 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. dioptries, which contains the laws of refraction, and the phenomena depending upon them; catoptries, which contains the laws of reflection, and the phenomena which depend on them; and lastly chromatics, which treats 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 we have not followed that method of dividing our subject; but have explained the particular laws of refraction and reflection, afterwards showing how, by these, the most remarkable optical phenomena may be accounted for.

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

Without entering into any repetition of the controversies concerning the nature of Light, which are fully set forth under that article, we shall here give a brief description of its properties considered as the subject of the optical science, and which hold good in all cases without regard to other theories.

Every visible body emits or reflects inconceivably small particles of matter from each point of its surface, which issue from it continually (not unlike sparks 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 progress 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 minuteness: 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 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 thereby 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 the celestial bodies, which passes downwards through our atmosphere, and likewise with that which is reflected upwards through it by terrestrial objects. In both 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.

§ I. 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 of refraction passes, in the following manner. All bodies being endowed with an attractive force, which is extended to some distance beyond their surfaces; when a ray of light passes out of a rarer into a denser medium (if this medium latter has a greater attractive force than the former, as is commonly the case) the rays, 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. 4.) 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. 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 line AB, and will begin to describe the curve LN, passing thro' 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 onwards 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 A BYZ 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 A BYZ, 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.

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. 5.) 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 thro' 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, 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 100026 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 may be demonstrated mathematically in the following manner.

Lemma. If from a point at M (fig. 6.) taken anywhere without the circle PNQ, a line as MP be drawn passing through L the centre of the circle, and terminated in the circumference at P, the product of MQ multiplied by MP is equal to the difference between the squares of ML and PL.

Demonstration of the Lemma. Call MQ, a; and the radius of the circle LQ or LP, b; then will the diameter QP be expressible by 2b; and the whole line MP, by a+2b; then multiplying MQ by MP, that is, a by a+2b, we have for the product of this, aa+2ab.

Now the square of the line ML, which is expressible by a+b, is aa+2ab+bb; and the square of PL is bb; but the difference between these squares, viz. aa+2ab+bb and bb, is evidently aa+2ab; and therefore the product of MQ multiplied by MP is equal to the difference between the squares of ML and PL.

Demonstration of the Proposition. When a ray of light passes through the space of attraction of any medium, it is evident that its motion will be subject to the like laws with that of projectiles, provided we suppose it to be acted upon with an equal degree of force during its whole passage through that space, as is commonly supposed to be the case in projectiles to what- ever height they are thrown from the earth. We will therefore suppose first, that the force of attraction of the denser medium is at all distances the same as far as it reaches, and that the ray proceeds out of a denser into a rarer medium; in which case it will be attracted back towards the denser medium, during its passage thro' the space of attraction, in like manner as a projectile thrown upwards is while it rises from the earth. Let then ABCD (fig. 6.) represent the denser medium, and ABEF the space of attraction; and let GH be a ray about to enter the force of attraction at H, and let GH be produced to M. Now it is evident, that, in this supposition, the ray, when at H, is in the same circumstances with a projectile about to be thrown upwards from H towards M: it will therefore describe a portion of a parabola as HI; to which the line HM will be a tangent at H; and the line IK, in which it would proceed after it had passed the space of attraction, a tangent to it at I; for after having left the attractive force at I, it goes straight on in its last direction. Let the perpendicular IR be drawn meeting GH produced in M, and let KI be produced to L. On the centre L, with the radius LI, describe the circle PNQ, let fall the perpendicular LO upon MR, and join the points L and N. Now it is demonstrated in the case of projectiles†, that the parameter of the point H is equal to $\frac{HM}{MI}$, and therefore the parameter multiplied by MI is equal to HM. And it is there further demonstrated, that the said parameter is equal to four times the height which a body must fall from, to acquire the velocity the projectile has at H. This parameter therefore is a quantity not at all depending on the direction of the projectile, but on its velocity only; and consequently, in the present supposition, it is a given quantity, the ray GH being supposed to have the same velocity, whatever is its inclination to the surface HB. Now the tangent KI being produced to L, will, by the property of the parabola, bisect the other tangent HM: wherefore the line LO being parallel to HR, MR will also be bisected in O; and adding the equal lines OI and ON to each part, MN will be equal to IR; but the line IR is also a line independent of the inclination of the ray GH, its length being determined by the breadth of the space of attraction ABEF only, and therefore MN is a given quantity. Now, whereas MI, when multiplied by the parameter of the point H, which before was shewn to be a given line, is equal to the square of HM, therefore the same line MI, when multiplied by any other given line (viz. MN) if it is not equal to, will nevertheless bear a given proportion to, the square of HM: but since MI multiplied by MN bears a given proportion (viz. a proportion that does not depend on the inclination of the ray GH) to the square of MH, its equal, viz. the product of MQ multiplied by MP, or what is equal to this, the difference between the squares of ML and PL (by the foregoing lemma), or, which is the same thing, of ML and LI, (because PL and LI are radii of the same circle), does so too. Now the square of ML bears also a given proportion to the square of MH (ML being equal to half MH); consequently there is a given proportion between the square of ML and the difference of the squares of ML and LI; and therefore there is a certain proportion between the lines themselves, viz. between ML and LI. But in every triangle the sides are proportionable to the sines of their opposite angles; therefore in the triangle MLI, the sine of the angle LMI has a given proportion to the sine of the angle LIM, or of its complement to two right ones MIK (for they have the same sine): But LMI, being an angle made by the incident ray GH produced, with the perpendicular RM, is the angle of incidence; and MIK, being made by the refracted ray IK, and the same perpendicular, is the angle of refraction; therefore in this case there is a constant ratio between the sine of the angle of incidence, and that of the angle of refraction.

We have here supposed that the force of attraction is everywhere uniform; but if it be otherwise, provided it be the same everywhere at the same distances from the surface AB, the proportion between the forementioned sines will still be a given one. For, let us imagine the space of attraction divided into parallel planes, and the attraction to be the same through the whole breadth of each plane though different in different planes, the sine of the angle of incidence out of each will, by what has been demonstrated above, be to the sine of the angle of refraction into the next in a given ratio; and therefore, since the sine of the angle of refraction out of one will be the sine of the angle of incidence into the next, it is evident that the sine of the angle of incidence into the first will be in a given ratio to the sine of the angle of refraction out of the last. Now let us suppose the thickness of these planes diminished in infinitum, and their number proportionably increased, the law of refraction will still continue the same; and therefore, whether the attraction be uniform or not, there will be a constant ratio between the sine of the angle of incidence and of refraction.

Q. E. D.

For the same reason that a ray is bent towards the perpendicular when passing from a rare medium into one that is denser, it is refracted from the perpendicular when it passes from a dense medium into one that is rarer.—From this and the foregoing proposition may be deduced the following corollaries.

I. When parallel rays fall obliquely on a plane surface of a medium of different density, they are parallel also after refraction; for, having all the same inclination to the surface, they suffer an equal degree of refraction.

II. When diverging rays pass out of a rarer into a denser medium through a plain surface, they are made thereby to diverge less.

III. When they proceed out of a denser into a rarer medium, the contrary happens, and they diverge more.

IV. When converging rays pass out of a rarer into a denser medium through a plain surface, they are made thereby to converge less.

V. When converging rays proceed out of a denser into a rarer medium, they are refracted the contrary way, and so made to converge more.

All these may be illustrated in the following manner.

1. Let AB, CD (fig. 7.) be two parallel rays falling on the plain surface EF of a medium of different density: now because they both make equal angles of incidence with their respective perpendicular GH, IK, before refraction, they will make equal angles of refrac- refraction with them afterwards, and so proceed on in the parallel lines BL, DM. 2. Let the diverging rays AB, AE, AF (fig. 8.) pass out of a rarer into a denser medium through the plain surface GH, and let the ray AB be perpendicular to that surface; the rest being refracted towards their respective perpendiculars EK, FM, and those the most that fall the farthest from B, they will proceed in the directions EN and FO, diverging in a less degree from the ray AP than they did before refraction. 3. Had they proceeded out of a denser into a rarer medium, they would have been refracted from their perpendicular DK, FM; and those the most which were the most oblique, and therefore would have diverged more than before. 4. Let the converging rays AB, CD, EF (fig. 9.) pass out of a rarer into a denser medium through the plain surface GH, and let the ray AB be perpendicular to that surface; then the other rays being refracted towards their respective perpendiculars DK, FM; and EF, for instance, more than CD; they will proceed in the directions DN, FN, converging in a less degree towards the ray AN, than they did before. 5. Lastly, had the first medium been the denser, they would have refracted the other way, and therefore converged more.

VI. When rays proceed out of a rarer into a denser medium through a convex surface of the denser, if they are parallel before refraction, they become converging afterwards.—For in this case the perpendiculars at the points where the rays enter the surface are all drawn from the centre of the convexity on the other side; and therefore, as the rays are refracted towards these perpendiculars, they are necessarily refracted towards each other, and thereby made to converge.

VII. If they enter diverging, then for the same reason they are made to diverge less, to be parallel, or to converge, according to the degree of divergency they have before they enter.

For, if they diverge very much, their being bent towards their respective perpendiculars in passing through the surface, may only diminish the divergency; whereas, if they diverge in a small degree, it may make them parallel, or even to converge.

VIII. If they converge in such a manner as to tend directly towards the centre of convexity before they enter the surface, they fall in with their respective perpendiculars, and so pass on to the centre without suffering any refraction.

IX. If they converge less than their perpendiculars, that is, if they tend to a point beyond the centre of convexity, they are made by refraction to converge more; and if they converge more than their perpendiculars, that is, if they tend towards a point between the centre and the surface, then, by being refracted towards them, they are made to converge less.

This and the three foregoing corollaries may be illustrated in the following manner. 1. Let AB, CD (fig. 10.) be two parallel rays entering a denser medium through the convex surface DB, whose centre of convexity is E; and let one of these, viz. AB, be perpendicular to the surface. This will pass on through the centre, without suffering any refraction; but the other being oblique to the surface, will be refracted towards the perpendicular ED, and will therefore be made to proceed in some line, as DG, converging towards the other ray, and meeting it in G, which point for that reason is called the focus.

2. Had the ray CD diverged from the other, suppose in the line AD, it would, by being refracted towards its perpendicular ED, have been made either to diverge less, be parallel, or made to converge. 3. Let the line ED be produced to F, and if the ray had converged, so as to have described the line FD, it would have been coincident with its perpendicular, and have suffered no refraction at all. 4. If it had proceeded from any point between C and F, as from H, or, which is the same thing, towards any point beyond E, in the line BE produced, it would have been made to converge more by being refracted towards the perpendicular DE, which converges more than it; and, had it proceeded from the same point, as I, on the other side of F, that is towards any point between B and E, it would then have converged more than its perpendicular, and so, being refracted towards it, would have been made to have converged less.

X. When rays proceed out of a denser into a rarer medium through a concave surface of the denser, the contrary happens in each case.

For, being now refracted from their respective perpendiculars, as they were before towards them, if they are parallel before refraction, they diverge afterwards; if they diverge, their divergency is increased; if they converge in the direction of their perpendiculars, they suffer no refraction; if they converge less than their respective perpendiculars, they are made to converge still less, to be parallel, or to diverge; if they converge more, their convergency is increased. All which may be clearly seen by the figure, without any further illustration; imagining the rays AD, CD, &c., bent the contrary way in their refractions to what they were in the former cases.

XI. When rays proceed out of a rarer into a denser medium through a concave surface of the denser, if they are parallel before refraction, they are made to diverge.—For in this case the perpendiculars at the points where the rays enter the surface, being drawn from a point on that side of the surface from which the rays tend; if we conceive them to pass through the surface, they will be so many diverging lines on the other side; and therefore the rays, after they have passed through the same points, must necessarily be rendered diverging in being refracted towards them.

XII. If they diverge before refraction, then, for the same reason, they are made to diverge more.

XIII. Unless they proceed directly from the centre; in which case they fall in with their perpendiculars, and suffer no refraction: Or from some point between the centre of convexity and the surface; for then they diverge more than their respective perpendiculars, and therefore being by refraction brought towards them they become less diverging.

XIV. If they converge, then, being refracted towards their perpendiculars, they are either made less converging, parallel, or diverging, according to the degree they converged in before refraction.

To illustrate this, and the three foregoing cases, 1. Let AB, CD (fig. 11.), be two parallel rays entering the concave and denser medium X, the centre of whose convexity is E, and the perpendicular to the refracting: refracting surface at the point D is EF; the ray AB, if we suppose it perpendicular to the surface at B, will proceed on directly to G; but the oblique one CD being refracted towards the perpendicular DF, will recede from the other ray AG in some line as DH.

2. If the ray CD had proceeded from A, diverging in the direction AD, it would have been bent nearer the perpendicular, and therefore have diverged more.

3. But if it had diverged from the centre E, it would have fallen in with the perpendicular EF, and not have been refracted at all: and had it proceeded from S, a point on the other side of the centre E, it would, by being refracted towards the perpendicular DF, have proceeded in some line nearer it than it would otherwise have done, and so would diverge less than before refraction.

4. If it had converged in the line LD, it would have been rendered less converging, parallel, or diverging, according to the degree of convergency which it had before it entered into the refracting surface.

XV. If the same rays proceed out of a denser into a rarer medium, through a convex surface of the denser, the contrary happens in each supposition. The parallels are made to converge; those which diverge less than their respective perpendiculars, that is, those which proceed from a point beyond the centre, are made less diverging, parallel, or converging, according to the degree in which they diverge before refraction; those which diverge more than their respective perpendiculars, that is, those which proceed from a point between the centre and the refracting surface, are made to diverge still more; and those which converge are made to converge more. All which may easily be seen by considering the situation of the rays AB, CD, &c. with respect to the perpendicular EF; and therefore requires no further illustration.

XVI. When diverging rays are by refraction made to converge, the nearer the radiant point (or point whence the rays proceed) is to the refracting surface, the farther is their focus from it on the other side, and vice versa. For the nearer the radiant point is to the refracting surface, the more the rays which fall upon the same points of it diverge before refraction, upon which account they converge the less afterwards.

XVII. When the radiant point is at that distance from the surface at which parallel rays coming through it from the other side would be collected by refraction, then rays flowing from that point become parallel on the other side, and are said to have their focus at an infinite distance. For the power of refraction in the medium is the same, whether the ray passes one way or other. For instance, if the parallel rays AB, CD, (fig. 10.) in passing through the refracting surface BD, are brought to a focus in G, then rays flowing from G as a radiant point, will afterwards proceed in the parallel lines BA and DC. And the point G, where the parallel rays AB and CD meet after refraction, is called the focus of parallel rays.

XVIII. When rays proceed from a point nearer the refracting surface than the focus of parallel rays, they continue to diverge after refraction, and their focus is then an imaginary one, and situated on the same side of the surface with the radiant. For, in this case, the divergency being greater than that which they would have if they had proceeded from the focus of parallel rays, they cannot be brought to a parallelism with one another, much less to converge; and therefore they continue to diverge, though in a less degree than before they passed through the refracting surface: upon which account they proceed after refraction as if they came from some point farther distant from the refracting surface than their radiant.

All these corollaries may be expressed more determinately, and demonstrated, in the following manner:

I. When rays pass out of one medium into another of different density through a plain surface; if they diverge, the focal distance will be to that of the radiant point; if they converge, it will be to that of the imaginary focus of the incident rays, as the sine of the angle of incidence is to that of the angle of refraction.

This proposition admits of four cases.

Case I. Of diverging rays passing out of a rarer into a denser medium.

Dem. Let X (fig. 12.) represent a rarer, and Z a denser medium, separated from each other by the plane surface AB; suppose CE and CD to be two diverging rays proceeding from the point C, the one perpendicular to the surface, the other oblique; through E draw the perpendicular PK. The ray CD being perpendicular to the surface, will proceed on in the right line CQ; but the other falling on it obliquely at E, and there entering a denser medium, will suffer a refraction towards the perpendicular EK. Let then EG be the refracted ray, and produce it back till it intersects DC produced also in F; this will be the focal point. On the centre E, and with the radius EF, describe the circle AFBO, and produce EC to H; draw HI the sine of the angle of incidence, and GK that of refraction; equal to this is FP or CM, which let be drawn. Now if we suppose the points D and E contiguous, or nearly so, then will the line HE be almost coincident with FD, and therefore FD will be to CD as HE to CE; but HE is to CE as HI to CM, because the triangles HIE and CME are similar; that is, the focal distance of the ray CE is to the distance of the radiant point, as the sine of the angle of incidence is to that of the angle of refraction.

Q. E. D.

Obs. 1. Whereas the ratio of FE to ME, or, which is the same thing, that of nD to CD, bears the exact proportion of HI to CM, and because this (being the ratio of the sine of the angle of incidence to that of the angle of refraction) is always the same, the line nE is in all inclinations of the ray CE, at the same distance from CM; consequently, had CE been coincident with CD, the point H had fallen upon n; and because the circle passes through both H and F, F would also have fallen upon n; upon which account the focus of the ray CE would have been there. But the ray CE being oblique to the surface DB, the point H is at some distance from n; and therefore the point F is necessarily so too, and the more so by how much the greater that distance is: from whence it is clear, that no two rays flowing from the radiant point C, and falling with different obliquities on the surface BD, will, after refraction there, proceed as from the same point; therefore, strictly speaking, there is no one point in the line D produced, that can more properly be called the focus of rays flowing from C, than another; for those which enter the refracting surface near D, will after refraction proceed, as has been observed, from the parts about N; those which enter near E, will flow as from the parts about F; those which enter about T, as from some points in the line DF produced, &c. And it is farther to be observed, that, when the angle DCE becomes large, the line NF increases apace; wherefore those rays which fall near T, proceed, after refraction, as from a more diffused space than those which fall at the same distance from each other near the point D. Upon which account it is usual with optical writers to suppose the distance between the points where the rays enter the plain surface of a refracting medium, to be inconsiderable with regard to the distance of the radiant point, if they diverge; or to that of their imaginary focus, if they converge: and unless there be some particular reason to the contrary, they consider them as entering the refracting medium in a direction as nearly perpendicular to its surfaces as may be.

Case 2. Of diverging rays proceeding out of a denser into a rarer medium.

Dem. Let X be the denser, Z the rarer medium, FD and FE two diverging rays proceeding from the point F; and supposing the perpendicular PK drawn as before, FP will be the sine of the angle of incidence of the oblique ray FE; which in this case being refracted from the perpendicular, will pass on in some line as ER, which being produced back to the circumference of the circle, will cut the ray FD somewhere, suppose in C; this therefore will be the imaginary focus of the refracted ray ER: draw RS the sine of the angle of refraction, to which HI will be equal: but here also FP, or its equal CM, is to HI as EC to EH, or (if the point D and E be considered as contiguous) as DC to DF; that is, the sine of the angle of incidence is to the sine of the angle of refraction, as the focal distance to that of the radiant point. Q. E. D.

Case 3. Of converging rays passing out of a denser medium into a rarer.

Dem. Let Z be the denser, X the rarer medium, and GE the incident ray; this will be refracted from the perpendicular into a line, as EH; then all things remaining as before, GK, or its equal FP, or CM, will be the sine of the angle of incidence, and HI that of refraction: but these lines, as before, are to each other as DC to DF; that is, the focal distance is to the distance of the imaginary focus, as the sine of the angle of incidence to that of the angle of refraction. Q. E. D.

Case 4. Of converging rays passing out of a rarer into a denser medium.

Dem. Let Z be the rarer, X the denser medium, and RE the incident ray; this will be refracted towards the perpendicular into a line, as EF; C will be the imaginary focus, and F the real one; HI, which is equal to RS, the sine of the angle of incidence, and FP that of the angle of refraction: but these are to each other, as DF to DC; and therefore the focal distance is to that of the imaginary focus, as the sine of the angle of incidence is to that of the angle of refraction. Q. E. D.

II. When parallel rays fall upon a spherical surface of different density, the focal distance will be to the distance of the centre of convexity, as the sine of the angle of incidence is to the difference between that sine and the sine of the angle of refraction.

This proposition admits of four cases.

Case 1. Of parallel rays passing out of a rarer into a denser medium, through a convex surface of the denser.

Dem. Let AB (fig. 13.) represent a convex surface; C its centre of convexity; HA and DB two parallel rays, passing out of the rarer medium X into the denser Z, the one perpendicular to the refracting surface, the other oblique; draw CB; this being a radius, will be perpendicular to the surface at the point B; and the oblique ray DB, being in this case refracted towards the perpendicular, will proceed in some line, as BF, meeting the other ray in F, which will therefore be the focal point: produce CB to N; then will DBN, or its equal BCA, be the angle of incidence, and FBC that of refraction. Now, whereas any angle has the same sine with its complement to two right ones, the angle FCB being the complement of ACB, which is equal to the angle of incidence, may here be taken for that angle; and therefore, as the sides of a triangle have the same relation to each other that the sines of their opposite angles have, FB being opposite to this angle, and FC being opposite to the angle of refraction, they may here be considered as the sines of the angles of incidence and of refraction. And for the same reason CB may be considered as the sine of the angle CFB; which angle being, together with the angle FBC, equal to the external one ACB, (32. El. 1.) is itself equal to the difference between those two last angles; and therefore the line FB is to CB as the sine of the angle of incidence is to the sine of an angle which is equal to the difference between the angle of incidence and of refraction. Now, because in very small angles as these are, (for we suppose in this case also the distance AB to vanish, the reason of which will be shown by-and-by,) their sines bear nearly the same proportion to each other that they themselves do, the distance FB will be to CB as the sine of the angle of incidence is to the difference between that sine and the sine of the angle of refraction; but because BA vanishes, FB and FA are equal, and therefore FA is to CA in that proportion. Q. E. D.

Obs. 2. It appears from the foregoing demonstration, that the focal distance of the oblique ray DB is such, that the line BF shall be to the line CB or CA as the sine of the angle of incidence to the sine of an angle, which angle is equal to the difference between the angle of incidence and refraction; therefore, so long as the angles BCA, &c. are small, so long the line BF is pretty much of the same length, because small angles have nearly the same relation to each other that their sines have. But when the point B is removed far from A, so that the ray DB enters the surface, suppose about O, the angles BCA, &c. becoming large, the sine of the angle of incidence begins to bear a considerably less proportion to the sine of an angle which is equal to the difference between the angle of incidence and refraction than before, and therefore the line BF begins to bear a much less proportion to BC; wherefore its length decreases apace upon which account those rays which enter the surface about. about O, not only meet nearer the centre of convexity than those which enter at A, but are collected into a more diffused space. From hence it is, that the point where those only which enter near A are collected, is reckoned the true focus; and the distance AB in all demonstrations relating to the foci of parallel rays entering a spherical surface whether convex or concave, is supposed to vanish.

Case 2. Of parallel rays passing out of a denser into a rarer medium through a concave surface of the denser.

Dem. Let X be the denser, Z the rarer medium. AB the surface by which they are separated, C the centre of convexity, and HA and DB two parallel rays, as before. Through B, the point where the oblique ray DB enters the rarer medium, draw the perpendicular CN; and let the ray DB, being in this case refracted from the perpendicular, proceed in the direction BM; produce BM back to H; this will be the imaginary focus; and DBN, or its equal ACB, will be the angle of incidence, and CBM, or its equal HBN (for they are vertical) that of refraction: produce DB to L, and draw BF such, that the angle LBF may be equal to DBH: then because NBD and DBH together are equal to NBH the angle of refraction, therefore BCA which is equal to the first, and LBF which is equal to the second, are together equal to the angle of refraction; but LBF is equal to BFA (as being alternate to it); consequently BFA and BCA together are equal to the angle of refraction; and therefore since one of them, viz. BCA, is equal to the angle of incidence, the other is the difference between that angle and the angle of refraction. Now FB, the fine of the angle FCB, or, which is the same thing, of its complement to two right ones, BCA, the angle of incidence, is to CB the fine of the angle BFC, as FB to CB, that is, as HB to CB; for the angles DBH and LBF being equal, the lines BF and BH are so too; but the distance BA vanishing, HB is to CB, as HA to CA: that is, the fine of the angle of incidence is to the fine of an angle which is the difference between the angle of incidence and refraction, or, because the angles are small, to the difference between the fine of the angle of incidence and that of refraction, as the distance of the focus from the surface is to that of the centre from the same. Q.E.D.

Case 3. Of parallel rays passing out of a rarer into a denser medium through a concave surface.

Dem. Let X be the denser medium having the concave surface AB, and let LB and FA be the incident rays. Now, whereas, when DB was the incident ray, and passed out of a rarer into a denser medium, as in Case 1, it was refracted into the line BE, this ray LB, having the same inclination to the perpendicular, will also suffer the same degree of refraction, and will therefore pass on afterwards in the line FB produced, v.g. towards P. So that, whereas in that case the point F was the real focus of the incident ray DB, the same point will in this be the imaginary focus of the incident ray LB: but it was there demonstrated, that the distance FA is to CA, as the fine of the angle of incidence is to the difference between that and the fine of the angle of refraction; therefore the focal distance of the refracted ray BP is to the distance of the centre of convexity in that proportion. Q.E.D.

Case 4. Of parallel rays passing out of a denser into a rarer medium through a convex surface of the denser.

Dem. Let Z be the denser medium, having the convex surface AB, and let LB and FA be the incident rays, as before. Now, whereas, when DB was the incident ray passing out of a denser into a rarer medium, it was refracted into BM, as in Case 2, having a point as H in the line MB produced for its imaginary focus; therefore LB, for the like reason as was given in the last case, will in this be refracted into BH, having the same point H for its real focus. So that here also the focal distance will be to that of the centre of convexity, as the fine of the angle of incidence is to the difference between that and the fine of the angle of refraction. Q.E.D.

III. When diverging or converging rays enter into a medium of different density through a spherical surface, the ratio compounded of that which the focal distance bears to the distance of the radiant point (or of the imaginary focus of the incident rays, if they converge), and of that which the distance between the same radiant point (or imaginary focus) and the centre bears to the distance between the centre and focus, is equal to the ratio which the fine of the angle of incidence bears to the fine of the angle of refraction.

This proposition admits of 16 cases.

Case 1. Of diverging rays passing out of a rarer into a denser medium, thro' a convex surface of the denser, with such a degree of divergency, that they shall converge after refraction.

Dem. Let BD (fig. 14.) represent a spherical surface, C its centre of convexity; and let there be two diverging rays AB and AD proceeding from the radiant point A, the one perpendicular to the surface, the other oblique. Thro' the centre C produce the perpendicular one to F; and draw the radius CB, and produce it to K, and let BF be the refracted ray; then will F be the focal point; produce AB to H, and through the point F draw the line FG parallel to CB. AB being the incident ray, and CK perpendicular to the surface at the point B, the angle ABK, or, which is equal to it, because of the parallel lines CB and FG, FGH, is the angle of incidence. Now, whereas, the complement of any angle to two right ones has the same fine with the angle itself, the fine of the angle FGB, that being the complement of FGH to two right ones, may be considered as the fine of the angle of incidence; which fine the line FB, as the sides of a triangle have the same relation to each other that the fines of their opposite angles have, may be taken for. Again, the angle FBC is the angle of refraction, or its equal, because alternate to it, BFG; to which BG, being an opposite side, may be looked upon as the fine. But FB is to BG in a ratio compounded of FB to BA, and of BA to BG; for the ratio that any two quantities bear to each other, is compounded of the ratio which the first bears to any other, and of the ratio which that other bears to the second. Now FB is to BA, supposing BD to vanish, as FD to DA; and BA is to BG, because of the parallel lines CB and FG, as AC to CF. That is, the ratio ratio compounded of FD the focal distance, to DA the distance of the radiant point, and of AC, the dis- tance between the radiant point and the centre, to CF, the distance between the centre and the focus, is equal to that which the fine of the angle of incidence bears to the fine of the angle of refraction. Q. E. D.

Obs. Whereas the focal distance of the oblique ray AB is such, that the compound ratio of FB to BA and of AC to CF, shall be the same, whatever be the distance between B and D; it is evident, that since AC is always of the same length, the more the line AB lengthens, the more FB must lengthen too, or else FC must shorten: but it appears by inspection of the figure, that if BF lengthens, CF will do so too, and in a greater proportion with respect to its own length than BF will; therefore the lengthening of BF will conduce nothing towards preserving the equality of the proportion; but as AB lengthens, BF and CF must both shorten, which is the only possible way wherein the proportion may be continued the same. And it is also apparent, that the farther B moves from D towards O, the faster AB lengthens; and therefore the farther the rays enter from D, the nearer to the refracting surface is the place where they meet, but the space they are collected in is the more diffused: and therefore, in this case, as well as those taken no- tice of in the two foregoing observations, different rays, tho' flowing from the same point, shall constitute dif- ferent focuses; and none are so effectual as those which enter at, or very near, the point D. And since the same is observable of converging as well as of diver- ging rays, none except those which enter very near that point are usually taken into consideration; up- on which account it is, that the distance DB, in deter- mining the focal distances of diverging or converging rays entering a convex or concave surface, is supposed to vanish.

Those who would see a method of determining the precise point, which the ray AB, whether it be paral- lel, converging, or diverging to the ray AF, con- verges to or diverges from after refraction at B or any other given point in the surface DO, may find it in the Appendix to Molineux's Optics; which, for the sake of those who have not the book, we shall subjoin at the end of this section.

Case 2. Of converging rays passing out of a rarer into a denser medium through a concave surface of the denser with such a degree of convergency, that they shall diverge after refraction.

Dem. Let the incident rays be HB and FD pas- sing out of a rarer into a denser medium thro' the con- cave surface BD, and tending towards the point A, from whence the diverging rays flowed in the other case; then the oblique ray HB, having its angle of inci- dence HBC equal to ABK the angle of incidence in the former case, will be refracted into the line BL, such, that its refracted angle KBL will be equal to FBC the angle of refraction in the former case; that is, it will proceed after refraction in the line FB pro- duced, having the same focal distance FD with the di- verging rays AB, AD, in the other case. But, by what has been already demonstrated, the ratio com- pounded of FD, the focal distance, to DA, in this case, the distance of the imaginary focus of the in- cident rays, and of AC, the distance between the same imaginary focus and the centre, to CF, the dis- tance between the centre and the focus, is equal to that which the fine of the angle of incidence bears to the fine of the angle of refraction. Q. E. D.

Case 3. Of diverging rays passing out of a rarer into a denser medium through a convex surface of the denser with such a degree of divergency as to con- tinue diverging.

Dem. Let AB, AD (fig. 15.) be the diverging plate rays, and let their divergency be so great, that the re- fracted ray BL shall also diverge from the other; pro- duce LB back to F, which will be the focal point; draw the radius CB, and produce it to K; produce BA likewise towards G, and draw FG parallel to BC. Then will ABK be the angle of incidence, whose fine BF may be taken for, as being opposite to the angle BGF, which is the complement of the other to two right ones. And LBC is the angle of refraction, or its equal KBF, or, which is equal to this, BFG, as being alternate; therefore BG, the opposite side to this, may be taken for the fine of the angle of refraction. But BF is to BG, for the like reason as was given in case the first, in a ratio compounded of BF to BA, and of BA to BG. Now BF is to BA, (DB va- nishing) as DF to DA; and because of the parallel lines FG and BC, the triangles CBA and AGF are similar; therefore BA is to AG as CA to AF; conse- quently BA is to BA together with AG, that is, to BG, as CA is to CA together with AF, that is, CF. Therefore the ratio compounded of DF the fo- cal distance to DA the distance of the radiant point, and of CA the distance between the radiant point and the centre, to CF the distance between the centre and the focus, is equal to that which the fine of the angle of incidence bears to the fine of the angle of refrac- tion. Q. E. D.

Case 4. Of converging rays passing out of a rarer into a denser medium thro' a concave surface of the den- ser in such manner that they shall continue converging.

Dem. Let HB and CD be the incident rays pas- sing out of the rarer into the denser medium thro' the concave surface BD, and tending towards A the same point from whence the diverging rays flowed in the last case. Then because the ray HB has the same incli- nation to the perpendicular CK that AB had before, it will suffer the same degree of refraction, and pass on in the line LB produced, having its focus F at the same distance from the refracting surface as that of the diverging ray AB in the other case. Therefore, &c. Q. E. D.

Case 5. Of diverging rays passing out of a den- ser into a rarer medium through a convex surface of the denser.

Dem. Let AB, AD (fig. 16.) be the incident rays passing out of a denser into a rarer medium through the concave surface BD whose centre is C; and let BL be the refracted ray, which produce back to F, and draw FG parallel to CB. Here ABK is the angle of incidence, to which its alternate one FGB being equal, FB the opposite side may be con- sidered as the fine of it. The angle of refraction is LBC or FBK; of which BFG being the complement to two right ones, BG the opposite side may be look- ed upon as its fine. But BF is to BG, in the compound ratio of BF to BA, and of BA to BG, for the reason given above. Now (BD vanishing) BF is to BA as DF to DA, and BA is to BG as CA to CF; that is, the ratio compounded of the focal distance to the distance of the radiant point, &c. Q.E.D.

Case 6. Of converging rays passing out of a denser into a rarer medium through a concave surface of the denser.

Dem. Let HB, CD, be the incident rays tending towards the point A which was the radiant in the last Case. Then, for the reason already given, the oblique ray will suffer such a degree of refraction, as to have its focus F at the same distance from the surface, as the diverging rays AB, AD, had in that case. Therefore, &c. Q.E.D.

When the mediums through which rays pass, and the refracting surfaces are such, that rays flowing from A (fig. 14.) are collected in F, then rays flowing from F through the same mediums the contrary way, will be collected in A. For when rays pass out of one medium into another, the sine of the angle of incidence bears the same proportion to the sine of the angle of refraction, as the sine of the angle of refraction does to the sine of the angle of incidence, when they pass the contrary way. This is applicable to each of the six following Cases compared respectively with the six foregoing; therefore they may be considered as the converse of them; or they may be demonstrated independently of them, as follows.

Case 7. Of diverging rays passing out of a denser into a rarer medium through a concave surface of the denser, so as to converge afterwards.

Dem. Let AB, AD (fig. 1.) be two diverging rays passing thro' the concave surface BD into a rarer medium. Let C be the centre of concavity, and BF the refracted ray. Draw CB, and produce it to K; and draw FG parallel to it, meeting AB produced in G. Then will ABC be the angle of incidence; of which FB being opposite to its alternate and equal angle FGB, may be considered as the fine. The angle of refraction is FBK; of which GB, being opposite to its complement to two right ones GFB, may be taken for the fine. Now FB is to BG in a ratio compounded of FB to BA, and of BA to BG. But (BD vanishing) FB is to BA as FD to DA; and because of the parallel lines CB and FG, BA is to BG as CA to CF. Therefore the focal distance, &c. Q.E.D.

Case 8. Of converging rays passing out of a denser into a rarer medium thro' a convex surface of the denser, so as to diverge afterwards.

Dem. Let GB and FD be the incident rays tending towards A, and produce FB to L. Then as AB in the last Case was refracted into BF, GB will in this be refracted into BL, for the reasons already given, having F for its focal point. Therefore, &c. Q.E.D.

Case 9. Of diverging rays passing out of a denser into a rarer medium thro' a concave surface of the denser, in such a manner as to continue diverging.

Dem. Let AB, AD (fig. 2.) be two rays passing out of a denser into a rarer medium, through the concave surface DB, whose centre of concavity is C. Draw CB, produce it to K, and let BL be the refracted ray; produce BL back to F, and draw FG parallel to CB meeting BC produced in G. Then will ABC be the angle of incidence, of which FB being opposite to its alternative and equal angle FGB, may be considered as the fine. The refracted angle is LBK, or its equal CBF; of which BG, being opposite to its complement to two right ones BFG, is the fine. Now BF is to BG in the compound ratio of BF to BA and of BA to BG; but BF is to BA as DF to DA; and because of the parallel lines CB and FG, the triangles BCA, AGF, are similar; therefore BA is to AG as CA to AF, and consequently BA is to BG as CA to CF. Therefore, &c. Q.E.D.

Case 10. Of converging rays passing out of a denser into a rarer medium through a convex surface of the denser, in such manner as to continue converging.

Dem. Let HB, MD, be the incident rays tending towards the point A. Then will the oblique ray HB, for the reasons already given, be refracted in BF. Therefore, &c. Q.E.D.

Case 11. Of diverging rays passing out of a denser into a rarer medium through a concave surface of the denser.

Dem. Let AB, AD (fig. 3.) be the incident rays passing out of a denser into a denser medium, through the concave surface BD, whose centre of convexity is C, and supposing the line CB drawn and produced to K, the refracted ray BL drawn and produced back to F, and also FG drawn parallel to CB, ABC will be the angle of incidence; of which FB, being opposite to its complement to two right ones GFB, is the fine. The angle of refraction will be LBK, or its equal FBC; of which BG, being opposite to its equal and alternate one BFG, is the fine. Now FB is to BG in the compound ratio of FB to BA and of BA to BG. But (BD vanishing) FB is to BA as FD to DA, and because of the parallel line FG and CB, BA is to BG as CA to CF. Therefore, &c. Q.E.D.

Case 12. Of converging rays passing out of a denser into a denser medium through a convex surface of the denser.

Dem. Let HB, MD, be the incident rays tending towards A the radiant point in the last case; then, as was explained above, BF will be the refracted ray. Therefore, &c. Q.E.D.

Case 13. Of rays passing out of a denser into a denser medium from a point between the centre of convexity and the surface.

Dem. Let AB, AD (fig. 4.) be two rays passing out of a denser into a denser medium from the point A, which let be posited between C the centre of convexity and the refracting surface BD; through B draw CK, and let BL be the refracted ray; produce BL back to F, and draw FG parallel to BC. Then will ABC be the angle of incidence; of which BF, being opposite to its complement to two right ones BGF, is the fine. LBK will be the angle of refraction, or its equal FBC; of which BG, being opposite to its alternate and equal one BFG, is the fine. But, as before, BF is to BG in a compound ratio of BF to BA and of BA to BG; and (BD vanishing) BF is to BA as DF to DA, and because the lines CB and FG are pa- Of parallel, BA is to BG as CA to CF. Therefore,

CASE 14. Of rays passing out of a denser into a rarer medium towards a point between the centre of convexity and the surface.

Dem. Let the incident rays be MD, HB, tending towards A, from whence the other proceeded in the last case. Then, as in that case the refracted ray BL being produced back passed through F, in this the refracted ray itself, for the like reasons as were given in the foregoing cases, will pass through that point. Therefore, &c. Q. E. D.

CASE 15. Of rays passing out of a rarer into a denser medium from a point between the centre of convexity and the surface.

Dem. Let AB, AD (fig. 5.) be two diverging rays passing out of a denser into a rarer medium thro' the refracting surface BD, whose centre of convexity is C, a point beyond that from whence the rays flow. Through B draw CK, and let BL be the refracted ray; produce it back to F, and draw FG parallel to BC, meeting BA produced in G. ABC will be the angle of incidence; of which BF, being opposite to its alternate and equal angle BGF, is the fine. The angle of refraction is LBK, or its equal FBC; of which BG, being opposite to its complement to two right ones BFG, is the fine. But BF is to BG in the compound ratio of BF to BA and of BA to BG; and (BD vanishing) BF is to BA as DF to DA; and because of the parallel lines CB and GF, the triangles AFG and ABC are similar. BA therefore is to AG, as CA to AF; consequently BA is to BA and AG together, that is, to BG, as CA is to CA and AF together, that is, to CF; and therefore the focal distance, &c. Q. E. D.

CASE 16. Of rays passing out of a denser into a rarer medium towards a point between the centre of convexity and the surface.

Dem. Let HB, MD, be the incident rays having for their imaginary focus the point A, which was the radiant in the last case; and let C the centre of convexity of the refracting surface be posited beyond this point. Then will HB, for the reasons already given, be refracted into BF, having the point F for its real focus, which was the imaginary one of the diverging rays AB, AD, in the former case. Therefore, as before, the ratio compounded of that which the focal distance bears to the distance of the imaginary focus of the incident rays, and of that which the distance between the same imaginary focus and the centre bears to the distance between the centre and the focus, is equal to the ratio which the fine of the angle of incidence bears to the fine of the angle of refraction. Q. E. D.

The first term in the foregoing proportion (viz. that in proposition 3d) being always an unknown quantity, those who are not well versed in the use of such propositions, may think it impossible to investigate the focal distance of any refracting surface by it; we shall therefore exemplify in the following instance, by which the manner of doing it in all others will clearly be understood. V. g. Let it be required to determine the focal distance of diverging rays passing out of air into glass through a convex surface; and let the distance of the radiant point be 20, and the radius of convexity be 5; now, because we must make use of the focal distance before we know it, let that be expressed by some symbol or character as x. Then, because by the aforeaid proposition the ratio compounded of that which the focal distance bears to the distance of the radiant point (that is, in this supposition, of x to 20), and of the ratio which the distance of the same radiant point from the centre bears to the distance between the centre and the focus (in this case, of 25 to x—5), is equal to the ratio which the fine of the angle of incidence bears to the fine of the angle of refraction (that is, of 17 to 11), we shall have, in the instance before us, the following proportion, viz.

\[ x : 20 :: 17 : 11 \]

and compounding them into

\[ 25 : x - 5 :: 17 : 11 \]

one, which is done by multiplying the two first parts together, we have \( 25x - 100 = 17x + 55 \), and multiplying the extreme terms and middle terms together \( 340x - 1700 = 275x \), which equation after due reduction gives \( x = \frac{1700}{65} \).

In some cases which might have been put, the quantity 65 would have been negative; and then the quotient arising from 1700, divided by that, would have been too: that is, x the focal distance would have been negative; in which case, the focus must have been taken on the contrary side of the surface to that on which it was supposed to fall in stating the problem; that is, it must have been taken on the same side with the radiant point; for in calling the distance between the centre and the focus \( x - 5 \), it was supposed the focus would fall on the same side with the centre, or on that which is opposite to the radiant point; because otherwise that distance must have been expressed by \( x + 5 \); as any one may see by inspection of the 13th or 14th figure, in which the focus of diverging rays entering a convex surface, is supposed to fall on the same side with the radiant point.

In like manner as this problem was performed, a general theorem may be raised to solve it in all cases whatsoever, by using characters instead of figures; as every one who is not unacquainted with algebraic operations very well knows.

See this done, and applied to the passage of rays through the surface of lenses, in the following section.

A method of determining the point which a ray, entering a spherical surface at any given distance from the vertex of it, converges to, or diverges from, after refraction at the same. From the Appendix to Molineux's Dioptrics.

"Prop. To find the focus of any parcel of rays diverging from, or converging to, a given point in the axis of a spherical lens [surface], and inclined thereto under the same angle; the ratio of the fines in refraction being known.

Let GL (fig. 6.) be the lens, P any point in its surface, V the pole [vertex] thereof, C the centre of the sphere whereof it is a segment, O the object or point in the axis to or from which the rays do proceed, OP a given ray; and let the ratio of refraction be as \( r \) to \( s \); make CR to CO as \( s \) to \( r \) for the immersion of a ray, or as \( r \) to \( s \) for the emersion, (that is, as the fines of the angles in the medium which the Of ray enters, to their corresponding fines in the medium Refraction out of which it comes); and laying CR from C towards O, the point R shall be the same for all the rays of the point O. Then draw the radius PC (if need be) continued, and with the centre R and distance OP sweep a touch of an arch, intersecting PC in Q; the line QR being drawn shall be parallel to the refracted ray, and PF being made parallel thereto shall intersect the axis in the point F; which is the focus sought. Or make it as CQ : CP :: CR : CF, and CF shall be the distance of the focus from the centre of the sphere.

Dem. Let fall the perpendiculars PX on the axis, CY on the given ray, and CZ on the refracted ray. By the construction QF and PR are parallel, whence the triangles QRC and PFC are similar, and CR to QR, as CF to PF; that is, CR to OP, as CF to PF. Now CF : PF :: CZ : PX ob similia triang.; whence CR : OP :: CZ : PX, and CR : CZ :: OP : PX. Again, CR is to CO as the fines of refraction by construction; that is, as s to r, or r to s; and as CR to CZ, so (CO =) \( \frac{r}{s} \) or \( \frac{s}{r} \) CR to \( \frac{r}{s} \) or \( \frac{s}{r} \) CZ; and so is PO to PX: But as PO to PX, so CO to CY.

Ergo, CY = \( \frac{r}{s} \) or \( \frac{s}{r} \) CZ; that is, CY to CZ is as the fines of refraction; but CY is the fine of the angle of incidence, and CZ of the refracted angle. Ergo constat propotio.

Hitherto we have considered only oblique rays; it now remains to add something concerning rays parallel to the axis: in this case the point O must be considered as infinitely distant, and consequently OP, OC, and CR are all infinite: and OP and OC are in this case to be accounted as always equal, (since they differ but by a part of the radius of the sphere GPVL, which is no part of either of them): wherefore the ratio of CR to OP will be always the same, viz. as s to r for immersing rays, and as r to s for those that emerge. And by this proposition CF is to PF in the same ratio. It remains therefore to shew on the base CP how to find all the triangles CPF, wherein CF is to PF in the ratio given by the degree of refraction. This problem has been very fully considered by the celebrated Dr Wallis in his late treatise of Algebra, p. 258, to which we refer; but we shall here repeat the construction thereof. (See fig. 7. 8.)

Let GPVL be a lens, VC or PC the radius of its sphere, and let it be required to find all the points \( f_1, f_2 \), such as C may be to P in the given ratio of s to r for immersing rays, or as r to s for the emerging. Divide CV in K, and continue CV to F, that CK may be to VK, and CF to VF, in the proposed ratio. Then divide KP equally in the point a, and with that centre sweep the circle FKF; this circle being drawn, gives readily all the foci of the parallel rays OP, OP. For having continued CP till it intersect the circle in F, PF shall be always equal to VF; the distance of the focus of each respective parcel of rays OP from the vertex or pole of the lens.

To demonstrate this, draw the pricked line VF, and by what is delivered by Dr Wallis in the above-cited place, VF and CF will be always in the same proposed ratio. Again, VF being made equal to PF, CF and CF will be likewise equal, as are CP, VC; and the angles PCf, VCF, being ad verticem, are also equal: Wherefore PF will be equal to VF, and consequently CF to PF in the same ratio as CF to VF; whence, and by what foregoes, the points \( f_1, f_2 \) are the several respective foci of the several parcels of rays OP, OP. Q. E. D.

§ 2. Of Glasses.

Glass may be ground into eight different shapes at least, for optical purposes, viz.

1. A plane-glass, which is flat on both sides, and of equal thickness in all its parts, as A. Fig. 9.

2. A plano-convex, which is flat on one side, and convex on the other, as B.

3. A double-convex, which is convex on both sides, as C.

4. A plano-concave, which is flat on one side, and concave on the other, as D.

5. A double concave, which is concave on both sides, as E.

6. A meniscus, which is concave on one side, and convex on the other, as F.

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

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

Glasses ground into any of the shapes B, C, D, E, F, are generally called lenses.

A right line LK, (fig. 9.) going perpendicularly through the middle of a lens, is called the axis of the lens.

A ray of light Gh, (fig. 10.) falling perpendicularly on a plane glass EF, will pass thro' the glass in the same direction hi, and go out of it into the air in the same right course 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 right 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.

A ray of light CD, (fig. 11.) falling obliquely on the middle of a convex glass, it will go forward in the same direction DE, as if it had fallen with the same degree of obliquity on a plane glass; and will go out of the glass in the same direction with which it entered: for it will be equally refracted at the points D and E, as if it had passed through a plane surface. But the rays CG and CI will be so refracted, as to meet again at the point F. Therefore, all the rays which flow from the point C, so as to go through the glass, will meet again at F; and if they go farther onward, as to L, they cross at F, and go forward on the opposite sides of the middle ray CDEF, to what they were in approaching it in the directions HF and KF.

When parallel rays, as ABC, (fig. 12.) fall directly upon a plano-convex glass DE, and pass through it, they will be so refracted, as to unite in a point f behind it: and this point is called the principal focus; the distance of which, from the middle of the glass, is called the focal distance, which is equal to twice the radius of the sphere of the glass's convexity.

And,

When parallel rays, as ABC, (fig. 13.) fall directly upon upon a glass DE, which is equally convex on both sides, and pass through it; they will be so refracted, as to meet in a point or principal focus f, whose distance is equal to the radius or semidiameter of the sphere of the glass's convexity. But if a glass be more convex on one side than on the other, the rule for finding the focal distance is this: As the sum of the semidiameters of both convexities is to the semidiameter of either, so is double the semidiameter of the other to the distance of the focus. Or divide the double product of the radii by their sum, and the quotient will be the distance sought.

Since all those rays of the sun which pass through a convex glass are collected together in its focus, the force of all their heat is collected into that part; and is in proportion to the common heat of the sun, as the area of the glass is to the area of the focus. Hence we see the reason why a convex glass causes the sun's rays to burn after passing through it.

All these rays cross the middle ray in the focus f, and then diverge from it, to the contrary sides, in the same manner FfGa, as they converged in the space DfE in coming to it.

If another glass FG, of the same convexity as DE, be placed in the rays at the same distance from the focus, it will refract them so, that, after going out of it, they will be all parallel, as abc; and go on in the same manner as they came to the first glass DE, thro' the space ABC; but on the contrary sides of the middle ray Bf; for the ray ADf will go on from f in the direction fGa, and the ray CEF in the direction Fc; and so of the rest.

The rays diverge from any radiant point, as from a principal focus: Therefore if a candle be placed at f, in the focus of the convex glass FG, the diverging rays in the space FfG will be so refracted by the glass, as that, after going out of it, they will become parallel, as shewn in the space cba.

If the candle be placed nearer the glass than its focal distance, the rays will diverge after passing thro' the glass more or less as the candle is more or less distant from the focus.

If the candle be placed farther from the glass than its focal distance, the rays will converge after passing thro' the glass, and meet in a point, which will be more or less distant from the glass as the candle is nearer to or farther from its focus: and where the rays meet, they will form an inverted image of the flame of the candle; which may be seen on a paper placed in the meeting of the rays.

Hence, if any object ABC (fig. 15.) be placed beyond the focus F of the convex glass def, some of the rays which flow from every point of the object, on the side next the glass, will fall upon it; and after passing through it, they will be converged into as many points on the opposite side of the glass, where the image of every point will be formed, and consequently the image of the whole object, which will be inverted.

Thus, the rays Ad, Ae, Af, flowing from the point A, will converge in the space daf, and by meeting at a, will there form the image of the point A. The rays Bb, Be, Bf, flowing from the point B, will be united at b by the refraction of the glass, and will there form the image of the point B. And the rays Cd, Ce, Cf, flowing from the point C, will be united at c, where they will form the image of the point C. And so of all the other intermediate points between A and C.

The rays which flow from every particular point of the object, and are united again by the glass, are called pencils of rays.

If the object ABC be brought nearer to the glass, the picture abc will be removed to a greater distance. For then more rays flowing from every single point, will fall more diverging upon the glass; and therefore cannot be so soon collected into the corresponding points behind it. Consequently, if the distance of the object ABC (fig. 16.) be equal to the distance eB of the focus of the glass, the rays of each pencil will be so refracted by passing through the glass, that they will go out of it parallel to each other; as dI, eH, fb, from the point C; dG, eK, fD, from the point B; and dK, eE, fL, from the point A: and therefore, there will be no picture formed behind the glass.

If the focal distance of the glass, and the distance of the object from the glass, be known, the distance of the picture from the glass may be found by this rule, viz. Multiply the distance of the focus by the distance of the object, and divide the product by their difference; the quotient will be the distance of the picture.

The picture will be as much bigger or less than the object, as its distance from the glass is greater or less than the distance of the object. For, as Be (fig. 15.) is to eB, so is AC to ca. So that if ABC be the object, cba will be the picture; or if cba be the object, ABC will be the picture.

For determining the progress of the rays after refraction by any lens, whatever be its form or matter, Mr. Rowning gives the following method. "Suppose GH (fig. 14.) to be a given lens, and E a point in its axis from whence the diverging rays EL, &c. fall upon the lens, AL, the radius of the first convexity, and CK that of the second; let LKf be the direction of the diverging ray EL after its refraction at the first surface, and KF its direction after refraction at both. Then will f be the focus of the rays after their first refraction, and F the point they will meet in after both. Let BD be the thickness of the lens, and let the proportion which the sine of the angle of incidence bears to the sine of the angle of refraction be expressed by the ratio of I to R. Call EB, d; BD, t; AB, r; CD, s; Iy, w; DF, y: Now, to find f their focus after refraction at L where they enter the first surface of the lens, comes under the third proposition abovementioned: according to which the ratio compounded of x, the focal distance sought, to d, the distance of the radiant point; and of d+r, the distance between the same point and the centre, to x-r, the distance between the centre and the focus, as I to R; compounding these two ratios therefore (that is, multiplying them together) we have dx+rx:dx-dr::1:R; which proportion being converted into an equation, and duly reduced, gives x=\frac{1}{Id-Rd-Dr}.

Thus having found the distance Bf, and consequently the point f, to which the rays converge from L, we must proceed to find F, that to which they will converge after having passed through K, where they suffer Of a second refraction: this comes under the same proposition. But, if we would use the same letters as before, to express the proportion which the sine of the angle of incidence bears to that of the angle of refraction, they must be put one for the other; because, when rays pass out of a denser into a rarer medium, the sine of the angle of incidence bears the same proportion to the sine of the angle of refraction, that the sine of the angle of refraction does to the sine of the angle of incidence, when they pass out of a rarer into a denser.

This being observed, by the aforesaid proposition, we shall have the ratio compounded of \( y \), the focal distance, to \( \frac{Idr}{Id-Rd-Rr} \), the imaginary focus in the incident rays, and of \( \frac{Idr}{Id-Rd-Rr} + s \), the distance between the imaginary focus and the centre, to \( y + s \), the distance between the centre and the focus, as \( R \) to \( I \).

Which equation, if we reduce the mixed quantities \( \frac{Idr}{Id-Rd-Rr} - t \) and \( \frac{Idr}{Id-Rd-Rr} + s \), into improper fractions, will stand thus:

\[ y : \frac{Idr}{Id-Rd-Rr} = \frac{Idr}{Id-Rd-Rr} + Rdt + Rrt \]

And, compounding these ratios, we have

\[ \frac{Idry}{Id-Rd-Rr} = \frac{Idty}{Id-Rd-Rr} + Rdt + Rrt \]

And throwing out the two equal denominators \( Id-Rd-Rr \) and \( Id-Rd-Rr \), and multiplying extremes together and means together, we have \( IIdry - IIIdty + IRdty + IRrt + IIIdy - IRdy - IRry - IRdy - IRdty + RRdty + RRrt + IIIds - IRds - IRdts + RRdts + RRrts \);

which equation being reduced, gives \( y = \frac{IIdry - IIIdty + IRdty + IRrt + IIIdy - IRdy - IRry - IRdy - IRdty + RRdty + RRrt + IIIds - IRds - IRdts + RRdts + RRrts}{IIIdr - IIIdt + RRdts + RRrts} \)

This theorem may be applied to all cases whatever; even to plane surfaces mutatis mutandis, v.g. the radius of a concave surface being negative (as lying the contrary way) with respect to that of a convex, and the radius of a plain surface being an infinite line.

If we would apply this theorem to a concave surface, we must change all the signs of those members where-in the symbol expressing the radius of that surface occurs; and if to a plane surface, all the members which involve the radius must be considered as infinite quantities: that is, all, except them, must be struck out of the equation as nothing. So, likewise, if we would have it extend to other rays besides diverging ones, the point where converging rays would meet, lying on the contrary side to that from whence the diverging ones were supposed to flow, its distance must be made negative; and the distance where parallel rays meet being infinite, it is only changing the signs of all those members in which \( d \) is found, if the rays are supposed converging; or making those members infinite, in case the rays are supposed parallel; which is done by striking out all the rest, as bearing no proportion to them.

§ 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 plate coats and three humours. The part DHHG of the CCX. outer coat, is called the sclerotica; the rest, DEFG, the cornea. Next within this coat is that called the choroides, which serves as it were for a lining to the other, and joins with the iris, mm, mm. The iris is of the eye, 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 every where 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 a 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 out out rays in all directions, some rays, from every point on the side next the eye, will fall upon the cornea between E and F; and by passing on thro' 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 q r s 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, 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. 2, 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 next the nose; so that whatever part of the image falls upon the optic nerve of one eye, may not fall upon the optic nerve of the other. Thus the point a of the image cba falls upon the optic nerve of the eye D, but not of the eye E; and the point c falls upon the optic nerve of the eye E, but not of the eye D; and therefore, to both eyes taken together, the whole object ABC is visible.

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, (fig. 3.) 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 AEC, 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. 4.) upon a white wall, at the height of the eye, and 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. Mariotte, occasioned a new hypothesis concerning the seat of vision, concerning which he supposed not to be in the retina, but in the seat of 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 around 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. Mariotte observes, that this improvement on his Of 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. Mariotte 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. Mariotte 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 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 (A). 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. Mariotte'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. Mariotte 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, stags, 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. 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. Mariotte observes, that there must be a great difference betwixt 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 dog, 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. Mariotte 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 a 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

(A) M. Muschenbroeck says, that in many animals, as the lion, camel, bear, ox, stag, sheep, dog, cat, and many birds, the choroides is not black, but blue, green, yellow, or some other colour. Introductio, Vol. II. p. 748. 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 1729, p. 119; in which he endeavours to shew 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 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 this 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 constringing 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 uvea. He also shewed 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 des 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 be 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 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 vision 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 circumcincted 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 daylight. 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 suppose it be performed by the impression of light on the choroides 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. 1, 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 was 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 Pisa, 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.

The following considerations in favour of the retina being the proper seat of vision are worthy of remark. Dr Porterfield 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 inferred in the axis of the eye, it is observed to be equally delicate, and therefore probably equally sensible, in that place as in any other part of the retina. 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 images of objects are painted.

M. De la Hire's argument in favour of the retina, Of 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 Priestley 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 sclerotis, 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 pulpy and softest; 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 so 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 insertion of the optic nerve.

Vision is distinguished into bright and obscure, distinct 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 it 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 distinctly, whether they be nearer or farther off, i.e., at different distances, 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... fore 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 proportionally 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 (a). 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 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 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. 5.), or crystalline humour \(e\), 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 \(gh\) 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 so, 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 \(gh\) 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 six inches; and angle of vision 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 8000th part of an inch; which he therefore calls a sensible point of the retina. On the other hand, Mr Courtryon 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. All lines can to a line of the same breadth with the circular spot will be seen unperceived at; because the quantity of impression from the line is much greater than from the spot; and a longer line is visible at a greater distance than a shorter one of the same breadth. He found by experience, that a silver wire could be seen when it subtended an angle of three seconds and an half; and that a silk thread could be seen when it subtended an angle of two seconds and a half.

This greater visibility of a line than of a spot, seems to arise only from the greater quantity of the impression; but without the limits of perfect vision, our author observes, that another cause concurs, whereby the difference of visibility between the spot and the line is rendered much more considerable. For the impression upon the retina made by the line is then not only much greater, but also much stronger than that of the spot; because the faint image, or penumbra, of any one point of the line, when the whole is placed beyond the limits of distinct vision, will fall within the faint image of the next point, and thereby much increase the light that comes from it.

In some cases our author found the cause of indistinct vision to be the unsteadiness of the eye; as our being able to see a single black line upon a white ground, or a single white line upon a black ground, and

(a) 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. and not a white line between two black ones on a white ground. In viewing either of the former objects, if the eye be imperceptibly moved, all the effect will be, that the object will be painted upon a different part of the retina; but, wherever it is painted, there will be but one picture, single and uncompounded with any other. But in viewing the other, if the eye fluctuate ever so little, the image of one or other of the black lines will be shifted to that part of the retina which was before possessed by the white line; and this must occasion such a dazzle in the eye, that the white line cannot be distinctly perceived, and distinguished from the black lines; which, by a continual fluctuation, will alternately occupy the space of the white line, whence must arise an appearance of one broad dark line, without any manifest separation.

By trying this experiment with two pins of known diameters, let in a window against the sky light, with a space between them equal in breadth to one of the pins, he found that the distance between the pins could hardly be distinguished when it subtended an angle of less than 40 seconds, though one of the pins alone could be distinguished when it subtended a much less angle. But though a space between two pins cannot be distinguished by the eye when it subtends an angle less than 40 seconds, it would be a mistake to think that the eye must necessarily commit an error of 40 seconds in estimating the distance between two pins when they are much farther from one another. For if the space between them subtend an angle of one minute, and each of the pins subtend an angle of four seconds, which is greater than the least angle the eye can distinguish, it is manifest that the eye may judge of the place of each pin within two seconds at the most; and consequently the error committed in taking the angle between them cannot at the most exceed four seconds, provided the instrument be sufficiently exact. And yet, says he, upon the like mistake was founded the principal objection of Dr Hooke against the accuracy of the celestial observations of Hevelius.

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, tho' 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 we do not see everything double? two eyes. It was the opinion of Sir Isaac Newton and others, that objects appear single because the two optic nerves unite before they reach the brain. But Dr Porterfield shews, 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 disjointed.

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 shews 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 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 have been just. In this he seems to have recourse to Refraction, the power of habit, tho' 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 Cheelden; 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 Cheelden, 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 thro' a multiplying glass.

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 one only.

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 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 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 fella 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 Of 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. Aepinus, who observed, that, when he was looking through a hole made in a plate of metal, about the tenth part of a line in diameter, with his left eye, both the hole itself appeared larger, and also the field of view seen thro' 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 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 \( \frac{3}{2} \) 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, Moucheronbroek, and Du Tour, concerning the place to which we refer an object viewed by one or both eyes. But the subject is not of much consequence. Any person may presently satisfy himself with respect to every thing belonging to this circumstance, either by experiment, holding his finger before his eyes, and looking at it and an object beyond it; or by figures, in which lines representing the optic axes may be made to cross one another at different distances from the eye.

§ 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; 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 somewhere 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 inferred, 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 when 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.

Prop. 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. 2.), and \( A b A c \) be two rays proceeding from thence, these rays passing out of a denser into a rarer medium, will be refracted from their respective perpendiculars \( b d; c e \), 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 \( A b; A c \), 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... 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. 3.) 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 EK 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 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 atmosphere higher than they need to do if they could come in a right line from the object: consequently the object will 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 towards the centre, that is, downwards as before, those which enter the eye must necessarily proceed as from some point above the object; wherefore the object will appear above its proper place.

From hence it is, that the sun, moon, and stars, appear 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 higher 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 rarer 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 evident from the sun and moon, which appear of 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 terminated by plane and parallel surfaces, appears nearer, brighter, and larger, than with the naked eye.

For instance, let A B (fig. 4.) 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, first, 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 refracted 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 figure, that it must be nearer the eye than that point; and because the same is true of all other pencils flowing from the object AB, the whole will be seen in the situation f g, nearer to the eye than the line A B.

As the rays RK, RL, would not have entered the eye, but have passed by it in the directions Kr, Lt, had they not been refracted in passing through the medium, the object appears brighter.

The rays Ah, Bi, will be refracted at h and i into the less converging lines k, l, and at the other surface into kM, lM, parallel to A b and B i produced; so that the extremities of the object will appear in the lines M k, M l produced, viz. in f and g, and under as large an angle fM g, as the angle A qB under which an eye at q would have seen it had there been no medium interposed to refract the rays; and therefore it appears larger to the eye at GH, being seen through the interposed 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 f g, because it is terminated by the lines M f and M g, 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.

Further, it is apparent from the figure, that the effect of a medium of this form depends wholly upon its thickness; for the distance between the lines R r and e c, and consequently the distance between the points e and R, depends upon the length of the line K a: Again, the distance between the lines AM and f M, depends on the length of the line b k; but both K u and k h depend on the distance between the surfaces CE and DF, and therefore the effect of this medium depends 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. 5.) be the object, CD the lens, and EF the eye. From A and B, the extremities of the object, draw the lines A Y r, B X r, crossing each other in the pupil of the eye; the angle A r B comprehended between these lines, is the angle under which the object would be seen with the naked eye. But by the interposition of a lens of this form, whose property it is to render converging rays more so, 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 extremities 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 AO and BP be such. These after refraction entering the eye at r, the extremities of the object will be seen in the lines r Q, r T, produced, and under the optic angle Q r T, which is larger than A r B, and therefore the apparent magnitude of the object will be increased.

3. Let GHI be a pencil of rays flowing from the point G; as it is the property of this lens to render diverging rays less diverging, parallel, or converging, it is evident, that some of those rays, which would proceed on to M and N, 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 object, 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 in 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. 6.) 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 e, where Ac, Bc, rays proceeding from the extreme points of the object, make not a much larger angle A c B, 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 the eye would receive if it were placed nearer to a or b, the object upon these 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 ab, 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 vivific 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 be situated farther from it on the other side than the place where the rays of the several pencils are collected into their respective foci, the object appears inverted, and pendulous in the air, between the eye and the lens.

To explain this, let AB (fig. 6.) 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 thro' 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 asserts. 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 can... not be collected together upon the retina, but fall up- on it as if they were the axes of so many distinct pen- cils coming through every point of the lens; where- fore 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 ap- plied 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. 7.) be the object, CD the pu- pil 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 diver- ging 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; where- fore the object will appear in the situation agb, less and nearer than without the lens. Farther, as the rays which proceed from G are rendered more diver- ging, some of them will be made to pass by the pu- pil 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 thro' a polygonous glass, that is, such as is terminated by several plain surfaces, is multiplied thereby.

For instance, let A (fig. 8.) be an object, and BC a polygonous 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 thro' 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 ap- pear also in its proper situation A.

Sect. II. 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 bo- dies 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 re- flected 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 po- sitions, scarce any rays will pass through both sides of the medium. A very considerable quantity is also un- accountably lost or extinguished at each reflecting sur- face; 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 ac- counted 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 suppo- sition 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 rea- sons.

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 an- other, there ought to be no rasures or unevennesses in that surface, the reflecting surface large enough to bear a sensible proportion to the magnitude of a ray of light; be- cause 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 rasures and scratches, which, tho' 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 per- fect 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 be- hind the body: And that it is not reflected by impin- ging 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 re- flected at the first surface of a body, by a repulsive force equally diffused over it; and at the second, by an attractive force.

I. If If there be a repulsive force diffused over the 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 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.

This argument concludes equally against a repulsive force uniformly diffused over the first surface of a body, and reflecting light there; because some bodies reflect the violet and transmit the red, others reflect the red and transmit the violet, at their first surface; which cannot possibly be upon this supposition, the rays of whichever of these colours we suppose to be the strongest.

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 forces 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 this 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 subtilance 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 subtilance. 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 conspires 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 first surface of a body, none is reflected there; but all that happens to it there is, that every ray that is not in a fit of easy transmission is there put into one, so that when they come at the other side (for this elastic subtilance, 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 thicknesses of the body, the intervals of the fits being different in rays of a different kind. This very well accounts 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. The same will happen if the rays be reflected from a metallic speculum; but the light will not be coloured; which shews, that the colours arise from that light which is reflected from the back-side, and that in the following manner: Besides that light which is regularly reflected from the farther surface of the glass, there is some reflected irregularly, which, passing from the back surface under the different obliquities, does as it were pass through glasses of different thicknesses, and therefore is in part reflected back again when it comes to the first surface, and is in part transmitted through it, the transmitted light, when received upon the white paper, exhibiting the rings of colours above-mentioned.

As to the light which is supposed to be reflected at the first surface, his opinion seems to be, That it is not there reflected, but that it really enters the surface, and is reflected from the back-side of the first series of particles that lie therein: so that, according as these particles are larger or smaller, the rays of light which at their entrance into them (for they are transparent, whether the body they compose be so or not) are thereby put into fits of easy transmission, at their emergence at the other side are some in a fit of easy transmission, others in a fit of easy reflection, according as the interval of their fits are large or small. So that the particles of a body may be of such a size that they shall reflect the red and transmit the violet, or that they may reflect the violet and transmit the red; or, in general, that the strongest and most forcible rays may be transmitted while the weaker are reflected; or the weaker may be transmitted while the stronger are reflected.

§ 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 plain 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 plain and spherical surfaces may be deduced.

I. Rays of light reflected from a plain 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 CD 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 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, everything 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 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. Part II.

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, everything 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 shewn 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 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 as 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, equal, all of it will pass into the second medium.

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

Prop. I. Of the reflection of rays from a plain surface.

"When rays fall upon a plain 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 so 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 from 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 plain 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 plain 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 IHF, 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 CAF 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 FH 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 E, 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 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 \(X, Y, \&c.\); 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 focus in the point \(F\), all the rays reflected from the convex surface would have proceeded as from the point \(F\), and those reflected from the concave surface would have fallen upon it, however distant their incident ones \(AB, DH\), might have been from each other. For in the parabola, all lines drawn parallel to the axis make 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 diverge, the distance of the focus of the reflected rays from the surface is to the distance of the radiant point from the same (or, if they converge, to that of the imaginary focus of the incident rays), as the distance of the focus of the reflected rays from the centre is to the distance of the radiant point (or imaginary focus 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 the distance of the imaginary focus of the incident rays from the same. Q.E.D.

**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. 5547.) 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 Elem.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. Q.E.D.

**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 \(RB, RD\), 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. Q.E.D.

The angles of incidence and reflection being equal, it is evident, that if, in any case, the reflected ray be made the incident one, the incident will become the reflected one; and therefore the four following cases may be considered respectively as the converse of the four foregoing; for in each of them the incident rays are supposed to coincide with the reflected ones in the other. Or they may be demonstrated independently of them as follows.

**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: Of Reflection.

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 the second,) 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. Q.E.D.

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 CDF and CDR will be so too; consequently the vertex of the triangle FDR is bisected by the line CD; therefore, 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. Q.E.D.

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. Q.E.D.

The two remaining cases may be considered as the converse of those under Prop. II. (p. 5548.), 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 surface 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 proposition 2d), 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 a supposition that R is at an infinite distance from B; that is, that the reflected rays BR and DR be parallel. Q.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. Q.E.D.

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 thro' the several points F, f, f, &c. The same is applicable in all the other cases.

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 ellipsis, 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 ellipsis 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 laid down in the third proposition, without making use of the quantity sought; we shall here give an instance whereby the method of doing it in all others 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 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 points, a method was there invented 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 case 2. and case 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 hath been laid 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 shewn 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 thro' 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, as 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 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 shewn 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, 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 thereat 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 with regard 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 line 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 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 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 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 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 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.

The 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 Of 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 reflecting surface; join the points AR and BR, and let AR represent a ray flowing from A; this will be reflected into RB: for C being the middle point between A and B, the angles ARC and CRB are equal; and a ray from B will likewise be reflected to A; and therefore the position of the image will be inverted with respect to that of the object.

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 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. 5543.) and therefore needs not be repeated here.

But as to what relates to the appearance of the object when the eye is placed nearer to the surface than the image, that was not there fully inquired into. That point shall therefore 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 w n; and so of the rest. Now if the same 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 OJ 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 assuming 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.

Dem. 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 H a K, which is equal to its vertical one Ma I, 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 H x L, which is larger than the angle H a K, 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 shewn 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 shewn 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 waysoever 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,

Further, 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 the 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, 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 thro' 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 flopped 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 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 coloured, 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 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 more 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, 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 thro' the prism upon the hole F, and turning the prism to and fro about its axis to make the image p t 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 p t 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 sines 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 beheld 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 having thus made the discovery, he contrived the following experiment to prove it at sight.

"In the middle of two thin boards, DE, de, 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 Plate 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, de, 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 de, and the rest being intercepted by the board, might paint upon it the coloured spectrum of the sun. And close behind this board I fixed another prism abc, to refract the light which came through the hole g. Then I returned speedily to the first prism ABC, and by turning it slowly to and fro about its axis, I caused the image which fell upon the second board de, to move up and down upon that board, that all its parts might pass successively through the hole in that board, and fall upon the prism behind it. And in the mean time I noted the places M, N, on the opposite wall, to which that light after its refraction in the second prism did pass; and by the difference of the places at M and N, I found that the light, which being most refracted in the first prism ABC, did go to the blue end of the image, was again more refracted by the second prism abc, than the light which went to the red end of that image. For when the lower part of the light which fell upon the second board de, was cast through the hole g, it went to a lower place M on the wall; and when the higher part of that light was cast through the same hole g, it went to a higher place, N, on the wall; and when any intermediate part of the light was cast through that hole, it went to some place in the wall between M and N. The unchanged position of the holes in the boards made the incidence of the rays upon the second prism to be the same in all cases. And yet in that common incidence some of the rays were more refracted and others less: and those were more refracted in this prism, which by a greater refraction in the first prism were more turned out of their way; and therefore, for their constancy of being more refracted, are deservedly called more refrangible."

Our author shews also, by experiments made with a convex glass, 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 reflectibility, and those rays are more reflexible than others which are more refrangible. A prism, ABC, 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..." angles ACB, and was refracted by it to G and H, began to be reflected into the line MN by its base BC, 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 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 homogeneal and similar; and that whose rays are some more refrangible than others, I call compound heterogeneal and dissimilar. The former light I call homogeneal, not because I would affirm it so 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, 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 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 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 found 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 fenorium 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 fenorium; and in the fenorium they are sensations of those motions under the forms of colours.

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 represent the circle CCXI, which all the most refrangible rays, propagated from fig. 13, 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 proportionably diminished. Let the circles AG, BH, CI, &c. remain as before; and let ag, bb, 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 p't, composed of the less circles, the three less circles ag, bb, 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 mixture mixture of the rays, we are to diminish the diameters of the circles. Now these would be diminished if the sun's diameter, to which they answer, could be made less than it is, or (which comes to the same purpose) if without doors, at 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, no not beyond the window. And therefore, instead of that hole, I used the hole in the window-shut as follows.

"In the sun's light let into my darkened chamber through a small round hole in my window-shut, at about 10 or 12 feet from the window, I placed a lens MN, by which the image of the hole F might be distinctly cast upon a sheet of white paper placed at I. Then immediately after the lens I placed a prism ABC, by which the trajeected 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 10th or 20th 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 might pass through the hole in the paper. This transmitted part of the light I refracted with a prism placed behind the paper; and, letting 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 shews that this light is refracted regularly without any dilatation of the rays, and is an ocular demonstration of the mathematical proposition mentioned above.

"In the homogeneous light I placed a paper circle of a quarter of an inch in diameter; and in the sun's unrefracted, heterogeneous, white light, I placed another paper circle of the same bigness; and going from these papers to the distance of some feet, I viewed both circles through a prism. The circle illuminated by the sun's heterogeneous light appeared very oblong, as in the second experiment, the length being many times greater than the breadth. But the other circle illuminated with homogeneous light appeared circular, and distinctly defined, as when it is viewed by the naked eye; which proves the whole proposition mentioned 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, rice, gold, silver, copper, glass, blue flowers, violets, bubbles of water tinged with various colours, colours, peacocks feathers, the tincture of lignum ne- phriticum, and such like, in red homogeneal light ap- peared totally red, in blue light totally blue, in green light totally green, and so of other colours. In the homogeneal light of any colour they all appeared to- tally 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 homogeneal 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 refrac- tions, 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 5th experiment, did in the progress from its end p, on which the most refrangible rays fell, unto its end z, on which the least refrangible rays fell, appear tin- ged with this series of colours; violet, indigo, blue, green, yellow, orange, red, together with all their in- termediate degrees in a continual succession perpetu- ally varying; so that there appeared as many degrees of colours as there were sorts of rays differing in re-

frangibility. And since these colours could not be changed by refractions nor by reflections, it follows, that all homogeneal light has its proper colour an- swering to its degree of refrangibility.

"Every homogeneal ray considered apart is refrac- ted, according to one and the same rule; so that its fine of incidence is to its fine of refraction in a given ratio: that is, every different coloured ray has a dif- ferent ratio belonging to it. This our author has proved by experiment, and by other experiments has determined by what numbers those given ratios are ex- pressed. 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 suc- ceed one another in the same line AC, and AD 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 77\(\frac{1}{2}\), of all the degrees of orange from 77\(\frac{1}{2}\) to 77\(\frac{3}{4}\), of yellow from 77\(\frac{3}{4}\) to 77\(\frac{5}{4}\), of green from 77\(\frac{5}{4}\) to 77\(\frac{7}{4}\), of blue from 77\(\frac{7}{4}\) to 77\(\frac{9}{4}\), of indigo from 77\(\frac{9}{4}\) to 77\(\frac{11}{4}\), and of violet from 77\(\frac{11}{4}\) to 78.

PART III.

END OF THE SEVENTH VOLUME.

Directions for placing the Plates in this Volume.

| Number of Plates | To face | Page | |-----------------|---------|------| | 192, or Plate CLXXV. | 211, or Plate CXCIV. | 4991 | | 193 | CLXXVI. | 4994 | | 194 | CLXXVII. | 4996 | | 195 | CLXXVIII. | 5037 | | 196 | CLXXIX. | 5038 | | 197 | CLXXX. | 5039 | | 198 | CLXXXI. | 5065 | | 199 | CLXXXII. | 5231 | | 200 | CLXXXIII. | 5253 | | 201 | CLXXXIV. | 5254 | | 202 | CLXXXV. | 5262 | | 203 | CLXXXVI. | 221 | | 204 | CLXXXVII. | 222 | | 205 | CLXXXVIII. | 223 | | 206 | CLXXXIX. | 224 | | 207 | CXC. | 225 | | 208 | CXCI. | 226 | | 209 | CXCII. | 227 | | 210 | CXCIII. | 228 |

N.B. ERRATA, OMISSIONS, &c. noticed and supplied in the APPENDIX at the end of the Work.