from ὀπτικός, to see, is that science which considers the nature, the composition, and the motion of light;—the changes which it suffers from the action of bodies;—the phenomena of vision, and the instruments in which light is the chief agent.

HISTORY.

Secr. I. Discoveries concerning the Refraction of Light.

THOUGH the ancients made few optical experiments, they nevertheless knew, that when light passed through media of different densities, it did not move in a straight line, but was bent or refracted out of its original direction. This was probably suggested to them by the appearance of a straight rod partly immersed in water; and accordingly we find many questions concerning this and other optical appearances in the works of Aristotle. Archimedes is said to have written a treatise on the appearance of a ring or circle under water, and therefore could not have been ignorant of the common phenomena of refraction. The ancients, however, were not only acquainted with these more ordinary appearances, but also with the production of colours by refraction. 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, 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 speculum, which, without having any colour of its own, assumes that of any other body. It appears, also, that the ancients were not ignorant of the magnifying power of glass globes filled with water, though they do not seem to have been acquainted with its cause; and the ancient engravers are supposed to have used a glass globe filled with magnifying water to magnify their figures. This indeed seems evident, from their lenticular and spherical gems of rock crystal which are still preserved, the effect of which, in magnifying at least, could scarcely have escaped the notice of those who had often occasion to handle them; if indeed, in the spherical or lenticular form, they were not solely intended for the purposes of burning. One of these, of the spherical kind, of about an inch and a half diameter, is preserved among the fossils presented by Dr Woodward to the university of Cambridge.

The first treatise of any consequence written on the subject of optics, was by the celebrated Ptolemy. The first treated treatise is now lost; but from the accounts of others, we find that he treated of astronomical refractions. The first astronomers were not aware that the intervals between stars appear less near the horizon than near the meridian; but it is evident that Ptolemy was aware of this circumstance, by the caution which he gives to allow something something for it, upon every recourse to ancient observations.

Ptolemy also advances a very sensible hypothesis to account for the greater apparent size of the sun and moon when seen near the horizon. The mind, he says, judges of the size of objects by means of a preconceived idea of their distance from us; and this distance is fancied to be greater when a number of objects intervene; which is the case when we see the heavenly bodies near the horizon. In his Almagest, however, he ascribes this appearance to a refraction of the rays by vapours, which actually enlarge the angle subtended by the luminaries.

The nature of refraction was afterwards considered by Alhazen an Arabian writer; insomuch that, having made experiments upon it at the common surface between air and water, air and glass, water and glass; and, being prepossessed with the ancient opinion of Alhazen's crystalline orbs in the regions above the atmosphere, he even suspected a refraction there also, and fancied he could prove it by astronomical observations. Hence this author concludes, that refraction 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 Vitellio, B. Waltherus, and by the excellent observations of Tycho Brahe. Alhazen observed, that refraction contracts the vertical diameters and distances of the heavenly bodies, and that it is the cause of the twinkling of the stars. But we do not find that either he, or his follower Vitellio, subjected it to mensuration. 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, when great attention was paid to the subject by Bernard Walther, Maestlin, and Tycho Brahe.

Alhazen supposed that the refraction of the atmosphere did not depend upon the vapours, but on the different transparency; by which, as Montucla conjectures, he meant the density of the gross air contiguous to the earth, and the ether or subtile air that lies beyond it. 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. He also observes, that the sky near the horizon is always imagined to be farther 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 Roger Bacon, 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 the useful invention of spectacles. 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.

Vitellio, a native of Poland, published a treatise of optics, containing all that was valuable in Alhazen. He observes, that light is always lost by refraction; 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. 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, thereby enabling it to make a stronger impression upon the eye. This writer also makes many attempts to ascertain the law of refraction. He likewise considers the foci of glass spheres, and the apparent size of objects seen through them; though upon these subjects his observations are inaccurate. It is sufficient indeed to show the state of knowledge, at that time, to observe, that both Vitellio, and his master Alhazen, account for objects appearing larger when seen under water, by the circular figure of its surface; since, being fluid, it conforms to the figure of the earth.

Contemporary with Vitellio was Roger Bacon, a man of extensive genius, who wrote upon almost every branch of science; yet in optics he does not seem to have made any considerable advances. Even some of the most absurd of the opinions of the ancients have had the sanction of his authority. He believed that visual rays proceed from the eye; because everything in nature is qualified to discharge its proper functions by its own powers, in the same manner as the sun and other celestial bodies. In his Specula Mathematica, he added some observations of little importance on the refraction of the light of the stars; the apparent size of objects; the enlargement of the sun and moon in the horizon. In his Opus Majus he demonstrates, what Alhazen had done before, that if a transparent body interposed between the eye and an object, be convex towards the eye, the object will appear magnified.

From this time, to that of the revival of learning in Europe, we have no treatise on optics. One of the first who distinguished himself in this way was Maurolycus, teacher of mathematics at Messina. In two works, entitled Theorema Lucis et Umbrae, and Diaphanorum Partes, &c., he demonstrates that the crystalline humour of the eye is a lens that collects the rays of light issuing from the object, 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.

While Maurolycus made such advances towards the discovery of the nature of vision, Baptista Porta of Naples invented the camera obscura, which throws still more light on the same subject. His house was resorted to by all the ingenious persons at Naples, whom he formed into an academy of secrets; each member being obliged to contribute something useful and not generally known. By this means he was furnished with materials for his Magia Naturae, which contains his account of the camera obscura, and which was published, as he informs us, when he was not quite 15 years old. He also gave History. gave the first hint of the magic lantern; which Kircher afterwards improved. His experiments with the camera obscura convinced him, that vision, as Aristotle supposed, is performed by the intromission of something into the eye, and not by visual rays proceeding from the eye, as had been formerly imagined by Empedocles; and he was the first who fully satisfied himself and others upon this subject. The resemblance indeed 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 on vision; and explains several cases in which we imagine things to be without the eye, when the appearances are occasioned by some affection of the organ itself, or some motion within it. He remarks 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 observes, that a round and flat surface plunged into water, will appear hollow as well as magnified to an eye above it; and he explains by a figure the manner in which this effect is produced.

The great problem concerning the measure of refractions was still unsolved. Alhazen and Vitellio, indeed, had attempted it; but failed, by trying to measure the angle instead of its sine. At last it was discovered by Snellius, professor of mathematics at Leyden. This philosopher, however, did not perfectly understand his own discovery, nor did he live to publish any account of it. It was afterwards explained by Professor Hortensius 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 assures us, were seen by Descartes. Before this time Kepler had published a New Table of Angles of Refraction, determined by his own experiments, for every degree of incidence. Kircher had done the same, and attempted a theory of refraction, on principles, which, if conducted with precision, would have led him to the law discovered by Snellius.

Descartes undertook to explain the cause of refraction by the resolution of forces. Hence 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, who asserted, contrary to the opinion of Descartes, that light suffers more resistance in water than air, and more in glass than in water; and maintained, that the resistance of different media with respect to light is in proportion to their densities. M. Leibnitz adopted the same general idea, upon the principle that nature accomplishes her ends by the shortest methods, and that light therefore ought to pass from one point to another, either by the shortest road, or that in which the least time is required.

At a meeting of the Royal Society, Aug. 31, 1664, it was found, with a new instrument prepared for that purpose, that the angle of incidence being 45 degrees, that of refraction is 30. About this time also we find the first mention of media not refracting the light in 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 difference of the angles of incidence to the sines of the angles of refraction was nearly the same as 4 to 3; and that, as spirit of wine occasions a greater refraction than common water, so oil of turpentine, which is lighter than spirit of wine, produces not only a greater refraction than common water, but a much greater than salt water. And at a meeting held November 9, the same year, Dr Hooke mentioned, that pure and clear salad oil produced a much greater refraction than any liquor which he had tried; the angle of refraction that answered to an angle of incidence of 30° being no less than 40° 30′; and the angle of refraction that answered to an angle of incidence of 20° being 29° 47′. M. de la Hire also made several experiments to ascertain the refractive power of oil, and found the sine of the angle of incidence to that of refraction 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 aqueous solutions of vitriol, saltpetre, and alum. 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. M. de la Hire also took a good deal of pains to determine whether 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 experiment made in 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 observes, that the refractive power of bodies is not proportioned to the density, at least not to the specific gravity, of the refracting medium. For the refractive power of glass to that of water is as 35 to 34, whereas its specific 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 direct proportion of their gravities.

The Royal Academy of Sciences at Paris endeavoured to repeat this 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, ordered Mr Hawksbee to make an instrument for the purpose, under the direction of Dr Halley, for the purpose of repeating the experiment. It consisted of a strong brass prism, two sides of which had sockets to receive two plane glasses, 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 turned 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 $64^\circ$; and the length of the telescope, having a fine hair in its focus, was about 10 feet. The event of this accurate experiment was as follows:—Having chosen a proper object, whose distance was 2588 feet, June 15. O.S. 1708, in the morning, the barometer being then at 29.73, and the thermometer at 62, 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 when the prism was full, the hair was observed to hide a mark 10½ inches below the former mark.

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. 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 $1'8''$, and the angle of incidence of the visual ray being $32$ degrees (because the angle of the glass planes was $64^\circ$), it follows from the known laws of refraction, that as the sine of $39^\circ$ is to that of $31^\circ59'26''$, differing from $32^\circ$ by $34''$, the half of $1'8''$; so is the sine of any other angle of 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 proportional to its density. And since the density of the atmosphere is as its weight directly, and its temperature 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.

The French academicians, being informed of the result of the above-mentioned experiment, employed M. De l'Isle the younger to repeat the former experiment with more care: He presently found, that the 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 of the Royal Society. The refraction was always proportional 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 regions above the atmosphere.

Dr Hooke first suggested the idea of making allowance for the effect of the refraction of light, in passing from the rarer to the 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 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, he considers very erroneous.

Dr Hooke ascribes the twinkling of the stars 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. That there is such an unequal distribution of the atmosphere, 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 steams 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 he first discovered this was by Viottelio's experiment already 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 refracted medium were exactly parallel to each other, no colours were produced. But of the true cause of those colours, he had not the least suspicion. This discovery was reserved for Sir Isaac Newton. Having procured a triangular glass prism to satisfy himself concerning the phenomena of colours; he was surprised at the oblong figure of the coloured spectrum, and the great disproportion betwixt its length and breadth; the former being about five times the measure of the latter. After various conjectures respecting the causes of these appearances, he suspected that the 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 in such a manner, that the light, passing through them both, might be refracted in opposite directions, and thus be returned by the latter into the same course from which it had been diverted by the former. In this manner he thought that the regular effects of the first prism would be augmented by the multiplicity of refractions. The event was, that the light, diffused by the first prism 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. He then hit upon what he calls the experimentum crucis, and found that light is not similar, or homogeneous; but that it consists of rays, some of which are more refrangible than others: so that, without any difference in their incidence on the same medium, some of them shall be more refracted than others; and therefore, that, according to their particular degrees of refrangibility, they will be transmitted through the prism to different parts of the opposite wall.

Since it appears from these experiments that different rays of light have different degrees of refrangibility, it follows, that the rules laid down by preceding philosophers concerning the refractive power of water, glass, &c., must be limited to the mean rays of the spectrum. Sir Isaac, however, proves, both geometrically and by experiment, that the sine of the incidence of every kind of light, considered apart, is to its sine of refraction in a given ratio.

The most important discovery concerning refraction since the time of Sir Isaac Newton is that of Mr Dollond, who found out a method of remedying the defects of refracting telescopes arising from the different refrangibility of light. Sir Isaac Newton imagined that the different rays were refracted in the same proportion by every medium, so that the refrangibility of the extreme rays might be determined if that of the mean ones were given. From this it followed, as Mr Dollond observes, that equal and contrary refractions must not only destroy each other, but that the divergency of the colours from one refraction would likewise be corrected by the other, and that there could be no possibility of producing any such thing as refraction without colour. Hence it was natural to infer, that all 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, philosophers despaired of bringing refracting telescopes to perfection. They therefore applied themselves chiefly to the improvement of the reflecting telescope; till 1747, when M. Euler, improving upon a hint of Sir Isaac Newton's, proposed to make object glasses of water and glass; hoping, that by their difference of refractive powers, the refractions would balance one another, and thereby prevent the dispersion of the rays that is occasioned by their difference of refrangibility. This memoir of M. Euler excited the attention of Mr Dollond. He went over all M. Euler's calculations, substituting for his hypothetical laws of refraction those which had been ascertained by Newton; and found, that it followed from Euler's own principles, that there could be no union of the foci of all kinds of colours, but in a lens infinitely large.

Euler did not mean to controvert the experiments of Newton: but 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 effected in the eye, which he considered an achromatic instrument. To this reasoning Mr Dollond made no reply, but by appealing to the experiments of Newton, and the circumspection with which it was known that he conducted all his inquiries.

This paper of Euler's was particularly noticed by M. Klingensierma of Sweden, who found that, from Newton's own principles, the result of his 8th experiment could not answer his description of it. Newton found, that when light passes out of air through several media, and thence goes out again into air, whether the refracting surfaces be parallel or inclined to one another, this 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, become tinged at its edges with colours. This he tried by refracting light with prisms of glass, placed within a prismatic vessel of water.

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

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

This paper of M. Klingensierma being communicated to Dollond, made him entertain doubts concerning Newton's report, and induced him to have recourse to experiment.

He therefore cemented together two plates of glass at their edges, so as to form a prismatic vessel, when stopped at the ends; 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 contrary to that of the water, in order that a ray of light, transmitted through both these refracting media, 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 and incident rays were parallel.

But according to the prevailing opinion, the object should have appeared of its natural colour; for if the difference of refrangibility had been equal in the two equal refractions, they would have rectified each other. This experiment, therefore, fully proved the fallacy of the received opinion, by showing the divergency of the light by the glass prism to be almost double of that by the water; for the image of the object was as much infected with the prismatic colours, as if it had been seen through a glass wedge only, whose refracting angle was near 30 degrees.

Mr Dollond was convinced 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 out colour; 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.

He, therefore, 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 quite free from any colours proceeding from the different refrangibility of the light; and as near as he could then measure, the refraction by the water was about \( \frac{1}{2} \) of that by the glass.

As these experiments proved, that different substances caused the light to diverge very differently in proportion to their general refractive power, Mr Dollond began to suspect that such a variety might possibly be found in different kinds of glass.

His next object, 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 whether the refraction and the divergency of the colours would vanish together.

From these experiments, which were not made till 1757, 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 kind, commonly called Venice glass, and the English crown glass, proved to be nearly alike in that respect; though, in general, the crown glass seemed to make light diverge less than the other. The common English plate glass made the light diverge more; and the white crystal, or English flint glass, most of all.

He then examined the particular qualities of every kind of glass that he could obtain, to fix upon two kinds in which the difference of their dispersive powers 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: so that any two wedges made in this proportion, and applied together, that they might refract in a contrary direction, would transmit the light without any dispersion of the rays. He found also, that the sine 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.

In order to apply these discoveries to the construction of telescopes, Mr Dollond considered, 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 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 the 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.

The difficulties which occurred in the application of this reasoning to practice, 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. And, thirdly, The difficulty of forming the four surfaces of the lenses exactly spherical. At length, however, after numerous trials, he was able to construct refracting telescopes, with such apertures and magnifying powers, under limited lengths, as far exceeded anything that had been produced before, representing objects with great distinctness, and in their natural colours.

As Mr Dollond did not explain the method by which he determined the curvatures of his lenses, the celebrated M. Clairaut, who had begun to investigate this subject, endeavoured to reduce it to a complete theory, from which rules might be deduced, for the benefit of the practical optician.

With this view, therefore, he endeavoured to ascertain the refractive power of different kinds of glass, and also their property of dispersing the rays of light. For this purpose he made use of two prisms, as Mr Dollond had done: but, instead of looking through them, he placed them in a dark room; and when the transmitted image of the sun was perfectly white, he concluded that the different refrangibility of the rays was corrected.

In order to ascertain more easily the true angles that prisms ought to have in order to destroy the effect of the difference of refrangibility, he constructed a prism 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 through them a beam of light.

In these experiments M. Clairaut was assisted by M. de Tournieres, and the results agreed with Mr Dollond's in general; but whereas Mr Dollond had made the dispersion of the rays in glass and in water to be as The subject of achromatic telescopes was also investigated by the illustrious D'Alembert. This excellent mathematician proposed a variety of new constructions, 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 these telescopes 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 does not seem to have occurred to any person before him. He even shows, that telescopes may be made to advantage, consisting of only one object glass, and an eye glass of a different refractive power. Some of his constructions have two or more eye glasses of different kinds of glass. This subject he considered at large in one of the volumes of his Opuscules Mathematiques. We have also three memoirs of M. D'Alembert upon this subject, among those of the French Academy; in the years 1764, 1765, and 1767.

The investigations of Clairaut and D'Alembert do not seem to have assisted the exertions of foreign artists. 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.

M. Euler who first gave occasion to this inquiry, having persuaded himself, both by reasoning and calculation, 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 achromatic telescopes, concluded that this extraordinary effect was partly owing to the crown glass not transmitting all the red light, which would have otherwise come to a different focus, and have distorted the image; but principally to his giving a just curvature to his glass, which he did not doubt would have produced the same effect if the lenses had all been made of the same kind of glass. At another time he imagined that the goodness of Mr Dollond's telescopes might be owing to the eye glass. If my theory, says he, be true, this disagreeable consequence follows, that Mr Dollond's object glasses cannot be exempt from the dispersion of colours: yet a regard to so respectable a testimony embarrasses me extremely, it being as difficult to question such express authority, as to abandon a theory which appears to me 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 to which others are subject.

Not doubting, however, but that Mr Dollond had made some improvement in the construction of telescopes, by the combination of glasses, he abandoned his former project, in which he had recourse to different media, 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, he could not help expressing his surprise that Mr Dollond should have been led to so important a discovery by reasoning in a manner quite contrary to the nature of things. At length, however, M. Euler was convinced of the reality and importance of Mr Dollond's discoveries; and frankly acknowledges, that perhaps he should never have been brought to assent to it, had he not been assured by his friend M. Clarraut that the experiments of the English optician might be depended upon. The experiments of M. Zeiber, however, gave him the most complete satisfaction with respect to this subject. This gentleman demonstrated, that it is the lead in the composition of glass which produces the variation in its dispersive power; 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.

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. Zeiber, dated Pittsburgh, 32nd of January 1764, in which he gives him a particular account of the success of his experiments on the composition of glass; and that, having correcting 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 | Ratio of the 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 : 1000 | | III. 1 : 1 | 1787 : 1000 | 3239 : 1000 | | IV. 1/2 : 1 | 1732 : 1000 | 2207 : 1000 | | V. 1/3 : 1 | 1724 : 1000 | 1800 : 1000 | | VI. 1/4 : 1 | 1664 : 1000 | 1354 : 1000 |

From this table it is evident, that a greater quantity of lead not only produces a greater dispersion of the rays, but also increases the mean refraction. The first of these kinds of glass, which contains three times as much minium as flint, will appear very extraordinary; since, hitherto, no transparent substance has been known, whose refractive power exceeded the ratio of two to one, and since the dispersion occasioned by this glass is almost five times as great as that of crown glass, which could scarcely be believed by those who entertained any doubt concerning the same property in flint glass, the effect of which is three times as great as crown glass.

Here, however, M. Euler announces to us another discovery of M. Zeiber, no less surprising than the former, and which disconcerted all his schemes for reconciling the above-mentioned phenomena. As the six kinds of glass mentioned in the preceding table were composed of nothing but minium and flint, M. Zeiber happened to think of mixing alkaline salts with them, in order to give the glass a consistence more proper for dioptric uses: This mixture, however, greatly diminishes... ed the mean refraction, almost without making any change in the dispersion. After many trials, he is said to have obtained a kind of glass, which 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.; though we have not heard that this kind of glass was ever used in the construction of telescopes.

Mr Dollond was not the only optician who had the merit of discovering the achromatic telescope, as this instrument appears to have been constructed by a private gentleman—Mr Chester More Hall. He observed that prisms of flint glass gave larger spectra than prisms of water, when the mean refraction was the same in both. He tried prisms of other glass, and found similar differences; and he applied this discovery to the same purposes as Mr Dollond. These facts came out in a process raised at the instance of Watkins optician, as also in a publication of Mr Ramsden. There is, however, no evidence that Dollond stole the idea from Mr Hall, or that they had not both claims to the discovery.

The best refracting telescopes, constructed on the principles of Mr Dollond, are still defective, on account of that colour which, by the aberration of the rays, they give to objects viewed through them, unless the object glass be of small diameter. This defect philosophers have endeavoured to remove by various contrivances, and Boscovich has, in his attempts for this purpose, displayed much ingenuity; but the philosopher whose exertions have been crowned with most success, and who has perhaps made the most important discovery in this science, is Dr Robert Blair professor of practical astronomy in the college of Edinburgh. By a judicious set of experiments, he has proved, that the quality of dispersing the rays in a greater degree than crown glass, is not confined to a few media, but is possessed by a great variety of fluids, and by some of these in a most extraordinary degree. He has shown, that though the greater refrangibility of the violet rays than of the red rays, when light passes from any medium whatever into a vacuum, may be considered as a law of nature; yet in the passage of light from one medium into another, it depends entirely on the qualities of the media which of these rays shall be the most refrangible, or whether there shall be any difference in their refrangibility. In order to correct the aberration arising from difference of refrangibility among the rays of light, he instituted a set of experiments, by which he detected a very singular and important quality in the muriatic acid. In all the dispersive media hitherto examined, the green rays, which are the mean refrangible in crown glass, were found among the less refrangible; but in the muriatic acid, these same rays were found to make a part of the more refrangible. This discovery led to complete success in removing the great defect of optical instruments, viz. that dissipation or aberration of the rays which arises from their unequal refrangibility, and has hitherto rendered it impossible to converge all of them to one point either by single or opposite refractions. A fluid, in which the particles of marine acid and metallic particles hold a due proportion, at the same time that it separates the extreme rays of the spectrum much more than crown glass, refracts all the orders of the rays in the same proportion that glass does: and hence rays of all colours, made to diverge by the refraction of the glass, may either be rendered parallel by a subsequent refraction in the confine of the glass and this fluid; or, by weakening the refractive density of the fluid, the refraction which takes place in the confine of it and glass may be rendered as regular as reflection, without the least colour whatever. The doctor has a telescope, not exceeding 15 inches in length, with a compound object glass of this kind, which equals in all respects, if it does not surpass, the best of Dollond's 42 inches long. See Phil. Trans. Edin. vol. iii.

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

Dr Nettleton observed a remarkable variety in the refractive power of the atmosphere, which demonstrates how little we can depend upon the calculated heights of mountains, when the observations are made with an instrument, and when the refractive power of the air is to be taken into the account. Being desirous to learn, by observation, how far the mercury would descend in the barometer at any given elevation, he proposed to measure 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 obtain no certain result. 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, that about 19 feet or more were required to make the mercury fall 1/10th of an inch; but afterwards, repeating the experiment, 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.

M. Euler considered the refractive power of the atmosphere, as affected by different degrees of heat and elasticity; in which he shows that its refractive power, to a considerable distance from the zenith, is sufficiently near the proportion of the tangent of that distance, and that the law of refraction follows the direct ratio of the 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.

As the density of the atmosphere varies with its altitude, and as the irregular curvature of the earth resolves causes a constant change in the inclination of the strata through which any ray of light passes to the eye, the refraction cannot be obtained from the density of the atmosphere, and the angular direction of the refracted ray. By comparing astronomical with meteorological observations, however, the celebrated M. La Place has given given a complete solution of this very important problem.

The phenomena known by the names of mirage, looming, and fata morgana, have been traced to irregularities of refractions arising from accidental changes in the temperature of the atmosphere. From the rarefaction of the air near the surface of water, buildings, or the earth itself, a distant object seen through this rarefied air sometimes appears depressed instead of raised by refraction; at other times it appears both elevated and depressed, so that the object seems double, and sometimes triple, one of the images being in an inverted position. This subject is much indebted to the researches of the ingenious Dr Wollaston, who has imitated these natural phenomena by viewing objects through the rarefied air contiguous to a red-hot poker, or through a saline or saccharine solution with water and spirit of wine floating upon its surface. This branch of optics has also been well illustrated by Mr Vince and Mr Huddart.

Sect. II. Discoveries concerning the Reflection of Light.

The followers of Plato were acquainted with the equality between the angles of incidence and reflection; and it is probable that they discovered this, by observing a ray of the sun reflected from standing water, or some other polished body; or from attending to the images of objects reflected by such surfaces. If philosophers paid any attention to this phenomenon, they could not but perceive, 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 observations upon these angles, the most rude and imperfect, could not fail to convince them of their equality, and that the incident and reflected rays were in the same plane.

Aristotle was sensible that it is the reflection of light from the atmosphere which prevents total darkness after the sun sets, and in places where he does not shine in the daytime. He was also of opinion, that rainbows, halos, and mock suns, were 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 two facts, the rectilineal progress of light, and the equality of the angles of incidence and reflection. The treatise of optics ascribed to Euclid is employed in determining the apparent size and figure of objects, from the angle which they subtend at the eye, and the apparent place of the image of an object reflected from a polished mirror. This place he fixes at the point where the reflected ray meets a perpendicular to the mirror drawn through the object. But this work is so imperfect and inaccurate, that it does not seem to be the production of Euclid.

It appears from Pliny and Lactantius, that burning glasses were known to the ancients. In one of the plays of Aristophanes, indeed, a person is introduced who proposes to destroy his adversary's papers by means of this instrument; and there is reason to believe that the Romans had a method of lighting their sacred fire by means of a concave speculum. It seems indeed to have been known A.C. 433, that there is an increase of heat in the place where the rays of light meet, after reflection from a concave mirror. The burning power of concave mirrors is noticed by the author of the work ascribed to Euclid. If we give any credit to what some ancient historians are said to have written concerning the exploits of Archimedes, we shall be induced to think that he constructed 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, which has not descended to our times.

B. Porta supposes that the burning mirrors of the ancients were parabolic and made of metal. It follows from the properties of this curve, that all the rays which fall upon it, parallel to its axis, will meet in the same point at the focus. Consequently, if the vertex of the parabola be cut off, as in fig. 1, it will make a convenient burning mirror. In some drawings of this instrument the 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, and that with a similar mirror Archimedes burnt the Roman fleet; using 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.

The nature of reflection was, however, very far from being understood. Even Lord Bacon, who made much greater advances in physics than his predecessors, 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.

Vitellio had endeavoured to show 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 from his description of the apparatus, it will be seen that the eye was to be directed towards the speculum placed within a room, while the object and the spectator were without it. But as no such effect can be produced by a convex mirror, Vitellio must have been under some deception with respect to his experiment.

B. Porta says, that this effect may be produced by a plain mirror only; and also by the combination of a plain and a concave mirror.

Kircher also speaks of the possibility of exhibiting these pendulous images, and supposes that they are reflected from the dense air: But 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.

It was Kepler who first discovered, that the apparent places of objects seen by reflecting mirrors depended on Kepler. Mr Boyle made some curious observations concerning the reflecting powers of differently coloured substances. In order to show that snow shines by a borrowed light and not by a native light, he placed a quantity of it in a room, from which all foreign light was excluded, and found that it was completely invisible. To try whether white bodies reflect more light than others, he held a sheet of white paper in a sunbeam 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.

To show that white bodies reflect the rays outwards, he adds, that common burning glasses require a long time to burn or discolour white paper; that the image of the sun was not so well defined upon white paper as upon black; that 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; and 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, ground into the form of a large concave speculum, and found that the image of the sun reflected from it was far from offending or dazzling his eyes, as it would have done from another speculum; and though this was large, he could not for a long time set a piece of wood on fire with it; though a far less speculum, of the same form, and of a more reflecting substance, would presently have made it flame.

To satisfy himself still farther with respect to this subject, he took a tile; and having made one half of its surface white and the other black, he exposed it to the summer sun. Having let it lie there some time, he found, that while the whitened part remained cool, the black part was very hot. 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 not so hot as the black part.

A remarkable property of lignum nephriticum (a species of guilandina) was first observed by Kircher. Mr Boyle has described this lignum nephriticum as a whitish kind of wood, which was brought from Mexico, 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 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 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.

Mr Boyle first distinctly noted the two very different colours which this remarkable tincture exhibits by transmitted and reflected light. If 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 influence 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 lively 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 shone upon it, he observed, that if he turned his back upon the sun, the shadow of any body 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.

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 frequent infusion 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.

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 sunbeams, 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 mixtures of light and shade.

It was not only in this tincture of lignum nephriticum that Mr Boyle perceived the difference between reflected and transmitted light. He observed it even in gold, though no person explained the cause of these appearances 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 more recent observations, 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.

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

This subject was more carefully investigated by Dr Hooke, who 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 he produced the famous coloured bubble of soap and water of which such use was afterwards made by Sir Isaac Newton, but which Dr Hooke and his contemporaries seem to have overlooked in Mr Boyle's treatise on colours, though it was published nine years before. It is no wonder that so curious an appearance excited the attention of that inquisitive body, and that they should desire him to bring an account of it in writing at their next meeting.

By the help of a small glass pipe, there were blown several small bubbles, out of a mixture of soap and water. At first, they appeared white and clear; but, 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 series was very bright. After these colours had passed through 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 were seen several holes, which by degrees grew very large, several of them running into one another.

Dr Hooke was the first who observed the beautiful colours that appear in thin plates of Muscovy glass. With a microscope 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 were disposed as in the outer rainbow. Some of them also were much brighter than others, and some of them very much broader. He also observed, that if there was a part where the colours were very broad, and conspicuous to the naked eye, they might be made, by pressing the part 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 split into plates of one-eighth or one-sixth of an inch in diameter, each of which will appear through a microscope to be uniformly adorned with some one vivid colour, and that these plates will be found upon examination to be of the same thickness throughout.

A phenomenon similar to this was noticed by Lord Brereton, who at a meeting of the Royal Society in 1666, produced some pieces of glass taken out of a church window, both on the north and on the south side of it; they were all eaten in by the air, but 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. It cannot be doubted, but that in all these cases, the glass is divided into thin plates, which exhibit colours, upon the same principle with those which Dr Hooke observed in the bubble of soap and water, and in the thin plate of glass, which we shall find more fully explained by Sir Isaac Newton.

The inquiries of M. Bouguer concerning the reflection of light are worthy of particular notice. They are fully detailed in his *Traité d'Optique*, a posthumous work published by La Caille in 1760.

In order to compare different degrees of light, he always contrived to place the radiant bodies or other bodies illuminated by them, in such a manner 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 two lights that came to his eye. After this, he had nothing more to do than to measure the distances EP and DP, and then the intensity of the lights was as EP² to DP².

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. 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.

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 and 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 which the light coming through the hole P, was reflected to R upon the screen GH, where it was compared with another beam of light that fell upon S, through the hole Q; which he made so much less than P, as that the spaces S and R were equally illuminated; and by the proportion that the apertures P and Q bore to each other, he calculated what quantity of light was lost by the reflection at O. It was necessary, he observes, that the two beams of light 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 of the same altitude, 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.

Before we proceed to detail the other experiments of Bouguer, we shall notice some which were made previous to them by Buffon on the diminution of light by reflection, and the transmission of it to considerable distances through the air.

By receiving the light of the sun in a dark room, and comparing it with the same light of the sun reflected by a mirror, he found that at small distances, as four or five feet, about one half was lost by reflection.

When the distances were 100, 200, and 300 feet, he could hardly perceive that it lost any of its intensity by being transmitted through such a space of air.

He afterwards made the same experiments with candles, in the following manner: He placed himself opposite to a looking glass, with a book in his hand, in a dark room; 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 could just distinguish the letters of the book, which was then 24 feet from the candle. He then received the light of the candle, reflected by the looking glass, upon his book, carefully excluding all the light that was reflected from anything else; and he found that the distance of the book from the candle, including the distance from the book to the looking glass (which was only half a foot) was in all 15 feet. He repeated the experiment several times, with nearly the same result; and therefore concluded, that the quantity of direct is to that of reflected light as 576 to 225; so that the light of five candles reflected from a plain 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, though curious, are inferior to those of M. Bouguer, both in extent and accuracy.

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

In the case of unpolished bodies, he found that a piece of white plaster, placed at an angle of 75°, with the incident rays, reflected 4/7 part of the light that is received from a candle nine inches from it. White paper, in the same circumstances, reflected in the same proportion; but at the distance of three inches, they both reflected 150 parts out of 1000.

Proceeding to make further 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 excepts the place where the shadow of the globe falls: because this is no more than a single point, with respect to the immensity of the spherical surface which receives the 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 large 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 eminences which cover them have not the same effect as the small polished hemispheres above mentioned.

He begins with observing, that of the light reflected from mercury, one fourth 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.

With regard to the quantities of light reflected at Great different angles of incidence, M. Bouguer found in general, 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 angle of incidence always more or less so in every thing that he tried. He found the greatest inequality in black marble, which, though not perfectly polished, yet with an angle of 3° 35' of incidence, it reflected almost as well as quicksilver. Of 1000 rays which it received, it returned 600: but when the angle of incidence was 14°, it reflected only 156; 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. M. Bouguer sometimes suspected, that, when the angles of incidence were very very small, the reflection from water was even greater than from quicksilver; though he rather thought that it was scarcely so great. In very small angles, he says, that water reflects nearly $\frac{1}{4}$ of the direct light.

The light reflected from a lake is sometimes $\frac{1}{3}$ or $\frac{1}{9}$, or even 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 altitude 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.

The light reflected from water at great angles of incidence is extremely small. M. Bouguer was assured, that, when the light was perpendicular, it reflected no more than the 37th part that quicksilver does in the same circumstances; for it did not appear that water reflects more than the 65th, or rather the 55th, part of perpendicular light. When the angle of incidence was $50^\circ$, the light reflected from the surface of water was about the 32d part of that which mercury reflected; and as the reflection from water increases as the angle of incidence diminishes, it was twice as strong in proportion at $30^\circ$; for it was then the 16th part of the quantity reflected from mercury.

In order to procure a common standard by which to measure the proportion of light reflected from various fluid substances, he selected water as the most commodious; and partly by observation and calculation he drew up the following table of the quantity of light reflected from its surface at different angles of incidence:

| Angles of incidence | Rays reflected of 1000 | |---------------------|------------------------| | $\frac{1}{2}$ | 721 | | 1 | 692 | | $\frac{3}{4}$ | 669 | | 2 | 639 | | $\frac{5}{4}$ | 614 | | 5 | 501 | | $\frac{7}{4}$ | 409 | | 10 | 333 | | $\frac{12}{4}$ | 271 | | 15 | 211 |

In the same manner, he constructed the following table containing the quantity of light reflected from the looking-glass not quicksilvered:

| Angles of incidence | Rays reflected of 1000 | |---------------------|------------------------| | $\frac{1}{2}$ | 584 | | 1 | 543 | | $\frac{3}{4}$ | 474 | | 2 | 412 | | $\frac{5}{4}$ | 356 | | 5 | 299 | | $\frac{7}{4}$ | 222 | | 10 | 157 |

When water floats upon mercury there will be two images of any object 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 65th 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 frequently observed, that there is a remarkable strong reflection into water, with respect to rays issuing from the water; and persons under water have seen images of things in the air in a manner peculiarly distinct and beautiful. In order to account for these facts, M. Bouguer observes 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. 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 nearly as transparent as when the light falls upon it from without.

This property belonging to the surfaces of transparent bodies, of absorbing the rays of light, is truly remarkable, and, as there is reason to believe, had not been noticed by any person before Bouguer.

That all the light is reflected at certain angles of incidence from air into denser substances, had frequently been noticed, especially in glass prisms; so that Newton made use of one of them, instead of a mirror, in the construction of his reflecting 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, one-fourth or one-third of the rays being extinguished, and two-thirds or three-fourths reflected. This property retains its full force as far as an angle of $49^\circ 49'$, (the proportion of the sines of the refraction being 31 and 20); but if the angle of incidence be increased but one degree, the quantity of light reflected inwards suddenly decreases, and a great part of the rays escape out of the glass, so that the surface becomes suddenly transparent.

All transparent bodies have the same property, with this difference, that the angle of incidence at which the strong reflection ceases, and at which the light which is not reflected is extinguished, is greater in some than in others. In water this angle is about $41^\circ 32'$; and in every medium it depends so much on the invariable proportion. 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 M. Bouguer proceeded to measure the quantity of light reflected by these internal surfaces at great angles of incidence, he had to struggle with many difficulties; but by 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.

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

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

Supposing the asperities of opaque bodies to consist of very small planes, it appears from these observations, that there are fewer of them in those bodies which reflect the light at small angles of incidence than at greater. 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 planes that compose those surfaces, and that are inclined to the general surface at the angles above mentioned, supposing that the whole surface contains 1000 of them that are parallel to itself, so as to reflect the light perpendicularly, when the luminous body is situated at right angles with respect to it. His conclusions reduced to a table, corresponding to the preceding, are as follow:

| Inclinations of the small surfaces with respect to the large one. | Silver | Plaster | Paper | |---------------------------------------------------------------|--------|---------|-------| | 0 | 1000 | 1000 | 1000 | | 15 | 777 | 736 | 937 | | 30 | 554 | 554 | 545 | | 45 | 333 | 374 | 338 | | 60 | 161 | 176 | 166 | | 75 | 53 | 50 | 52 |

These variations in the number of little planes, he expresses in the form of a curve; and afterwards shows, geometrically, what would be the effect if the bodies were enlightened in one direction, and viewed in another. Upon this subject he has several curious theorems and problems; but for these we must refer to the work itself.

Since the planets are more luminous at their edges than at their centres, he concludes, 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.

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

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

Having considered the surfaces of bodies as consisting of planes only, he observes that each small surface, separately taken, is extremely irregular, some of them really concave, and others convex; but, in reducing them to a middle state, they are to be regarded 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 effects, the rays always issue from an actual or imaginary focus, and after reflection always diverge from another.

The experiments of Lambert, related in his *Photometria*, have laid open to us many curious observations concerning the natural history of light. He was the first who determined that a radiating surface emits its light with nearly the same intensity in all directions, so that that every portion of it appears equally bright to an observer placed in any direction.

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 reflection 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 \( \frac{1}{100000} \)th of an inch, which space light describes in \( \frac{1}{1000000000000} \)th of a second. With so rapid a motion, therefore, may the internal parts of bodies be agitated by the influence of light, as to perform \( 125,000,000,000,000 \) 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; whence 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 gentleman observes, that the greatest quantity of rays, though crowded into the smallest space, will not of themselves produce any heat. 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, in an instant, be intensely heated.

The easiest way to be satisfied of this truth experimentally is, to hold a hair, or a piece of down, immediately above the focus of a lens or spectulum, 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 on account of its refraction, 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. See Priestley on Vision.

The attempts of the Abbé Nollet to fire inflammable substances by the concentration of the solar rays, have a near relation to the present subject. He attempted to fire liquid substances, but 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 suet.

He both threw the focus of these mirrors upon the substances themselves, and also upon the fumes that rose from them; but the only effect was, that the liquor boiled, and was dispersed in vapour or very small drops. 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 the rags thus prepared were longer in burning than those that were dry.

M. Beaune, who assisted M. Nollet in some of these experiments, observed further, that the same substances which were easily fired by the flame of burning bottles, could not be set on fire by the contact of the hottest bodies that did not actually flame. Neither either nor spirit of wine could be fired with a hot coal, or even red-hot iron, unless they were of a white heat.

By the help of optical principles, and especially by observations on the reflection of light, Mr Melville demonstrated that bodies which seem to touch one another are not always in actual contact. Upon examining the volatility and lustre of drops of rain that lie on the leaves of colocynth, and some other vegetables, 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 found also, 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, 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 a repulsive force. 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, the under surface becomes rough likewise, and thus by reflecting the light copiously in different directions, assumes the brilliant hue of unpolished silver.

It being thus proved by an optical argument, that the drop is not really in contact with the leaf, 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 singular miscellaneous observations, collated by Baron Alexander Funk, visiting some silver mines in Sweden, observed, that, in a clear day, it was as dark as pitch below 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 the depth of 156 fathoms. 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 is 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 observation was that of the ingenious Mr Grey. He 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, 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.

This method of accounting for the inverted image of the pin is evidently erroneous; for the same effect is produced, when the small aperture is formed of two semi-apertures at different distances from the eye, or when a small opening is made in the pigment on a piece of smoked glass. We have found indeed that the same phenomenon will appear, if, instead of looking at a hole in a piece of paper, we view a small luminous point so that it is expanded by indistinct vision into a circular image of light. The pin always increases in magnitude in proportion to its distance from the luminous point.

Sect. III. Discoveries concerning the Inflection of Light.

This property of light was not discovered till about the middle of the 17th 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 make his observations public till six years after Grimaldi.

Dr Hooke having darkened his room, admitted a beam of the sun's light through a very small hole in a brass plate. This beam spreading itself, formed a cone, the vertex of which was in the hole, and the base was on a paper, so placed as to receive it at some distance. In the 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 he ascribed to a property of light, that 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 that the circles intersected each other, he observed that there was not only a darker ring, encompassing the lighter circle, but a manifest dark line, or circle, as in fig. 6, which appeared even where the limb of the one interfered with that of the other.

In the light thus admitted, 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. In order to show that this light was not produced by reflection, 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 he concluded that they were occasioned by some new property of light.

He diversified this experiment, by placing the razor so as to divide the cone of light into two parts, and placing the paper so 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. This radiation always struck perpendicularly from the line of shadow, and, like the tail of a comet, extended more than 10 times the breadth of the remaining part of the circle. 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 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 rarefied, or gradually diverged into a quadrant; that this deflection is made towards the superficies of the opaque body perpendicularly; that those parts of the diverged radiations which are deflected by the greatest angle from the straight or direct radiations are the faintest, and those that are deflected by the least angles are the strongest; that rays cutting each other in one common aperture do not make the angles at the vertex equal; that colours may be made without refraction; that the diameter of the sun cannot be truly taken with common sights; that the same rays of light, falling upon the same point of an object, will turn into all sorts of colours, by the various inclinations of the object; and that 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 give an account of the discoveries of Father Grimaldi. Having introduced a ray of light, through a very small hole, AE, fig. 8, into a darkened room, he observed that the light was diffused in the form of a cone, the base of which was CD; and that if any opaque body, FE, was placed in this cone of light, at a considerable distance from the hole, and the shadow received upon a piece of white paper, the boundaries of it were not confined within GH, or the penumbra II, occasioned by the light proceeding from different parts of the aperture, and of the disk of the sun, but extended to MN: At this he was very much surprised, as he found that it was broader than it ought to 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 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 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 largest 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 colour 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 thus rendered either more intense or mixed.

Within the shadow itself, Grimaldi 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 it was necessary to have strong light, and to make the opaque body long and moderately broad. 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; though the light grew fainter in the same proportion.

The numbers of these streaks increased with the breadth of the plate. They were at least two, and sometimes four, if a thicker plate were made use of. But, with the same plate, 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, like those on the outside of the shadow, 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 they increased in number with the breadth of the plate. If the plate was very thin, the coloured streaks bent round from the opposite sides, and met one another as at B.

In order to obtain a more satisfactory proof, that rays of light really bend, in passing by the edges of bo- The third kind of radiation is horizontal, and is caused by the inflection of the light in passing between the eye-lashes.

Sir Isaac Newton's experiments of Grimaldi and Hooke were repeated and extended by Sir Isaac Newton, and were in some measure explained by that distinguished philosopher.

He made in a piece of lead a small hole the 42d part of an inch in diameter. Through this hole he let into his dark 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, that the hair acts upon the rays 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 hence it comes to pass that the shadow of the hair is much broader in proportion to the distance of the paper from the hair when it is nearer than when it is at a greater distance.

By wetting a polished plate of glass, and laying 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, he found that the shadow at the same distance was as big as before, so that this breadth of shadow must proceed from some other cause than the refraction of the air.

The shadows of all bodies placed in this light were bordered with three parallel fringes of coloured light, of which that which was nearest to the shadow was the broadest and most luminous, while that which was farthest from it was the narrowest, and so faint as to be scarcely visible. It was difficult to distinguish these colours, unless when the light fell very obliquely upon some 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.

Measuring these fringes and their intervals with the greatest accuracy, he found the former to be in the progression of the numbers $1, \sqrt{\frac{1}{3}}, \sqrt{\frac{2}{3}}, \sqrt{\frac{3}{3}}$, and their intervals to be in the same progression with them, that is, the fringes and their intervals together to be nearly in continual progression of the numbers $1, \sqrt{\frac{1}{3}}, \sqrt{\frac{2}{3}}, \sqrt{\frac{3}{3}}$.

Having made the aperture $\frac{1}{4}$ of an inch in diameter, and admitted the light as formerly, Sir Isaac placed, at the distance of two or three feet from the hole, a sheet of pasteboard, black on both sides; and in the middle of it he made a hole about $\frac{1}{4}$ of an inch square, and behind the hole he fastened to the pasteboard the blade of a sharp knife, to intercept some part of the light which passed through the hole. The planes of the pasteboard and blade of the knife were parallel to each other, 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 fall on a white paper, 2 or 3 feet beyond the knife, and there he saw two streams of faint light shoot out both ways from the beam of light into the shadow. But because the sun's direct light, by its brightness upon the paper, obscured these faint streams, so that he could scarcely 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 similar to 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 $\frac{1}{4}$ of an inch, and gradually decreased 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.

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 of an inch, the stream divided in the middle, and left a shadow between the two parts. This shadow was so dark, that all the light which passed between the knives seemed to be bent to the one hand or the other; and as the knives still approached each other, the shadow grew broader and the streams shorter next to it, till, upon the contact of the knives, all the light vanished.

Hence Sir Isaac concluded, 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; distance; and this distance, when the shadow began to appear between the streams, was about the 800th 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 stream vanished last which were farthest from the direct line.

In the experiment of one knife only, the coloured fringes did not appear; but, on account of the breadth of the hole in the window, became so broad as to run into one another, and, by joining, to make one continual light in the beginning of the streams; but in the last experiment, as the knives approached one another, a little before the shadow appeared between the two streams, the fringes began to appear on the inner ends of the streams, on either side of the direct light; three on one side, made by the edge of one knife, and three on the other side, made by the edge of the other knife. They were the most distinct when the knives were placed at the greatest distance from the hole in the window, and became still more distinct by making the hole less; so that he could sometimes see a faint trace of a fourth fringe beyond the three above mentioned: and as the knives approached one another the fringes grew more distinct and larger, till they vanished; the outermost vanishing first, and the innermost last. After they were all vanished, and the line of light 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 Newton concluded, that the light of the first fringe passed by the edge of the knife at a distance greater than the 800th 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 straight, and fixed their points into a board, so that their edges might contain a rectilinear angle. The distance of the edges of the knives from one another, at four inches from the angular point, was the 8th of an inch; so that the angle contained by their edges was about $1^\circ 54'$. The knives being thus fixed, he placed them in a beam of the sun's light let into his darkened chamber, through a hole the 42d of an inch wide, at the distance of 10 or 13 feet from the hole; and he let the light which passed between their edges fall very obliquely on a smooth white ruler, at the distance of $\frac{1}{9}$ inch, or an inch, from the knives; and there he saw the fringes made by the two edges of the knives run along the edges of the shadows of the knives, in lines parallel to those edges, without growing sensibly broader, till they met in angles equal to the angle contained by the edges of the knives; and where they met and joined, they ended, without crossing one another. But if the ruler was held at a much greater distance from the knives, the fringes, where they were farther from the place of their meeting, were a little narrower, and they became something broader as they approached nearer to one another, and after they met they crossed one another, and then became much broader than before.

From these observations he concluded, that the distances at which the light composing the fringes passed by the knives were not increased, or altered by the approach; 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 $\frac{1}{9}$ of an inch from the knives, the dark line between the first and second fringes 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 $\frac{1}{9}$ of an inch from the end of the light which passed between the knives, where their edges met; so that the distance of the edges of the knives, at the meeting of the dark lines, was the 160th 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 $\frac{1}{9}$ of an inch, the dark lines above mentioned met at a greater distance than $\frac{1}{9}$ 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 of an inch. For at another time, when the two knives were 8 feet 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 noted.

| Distance of the paper from the knives in inches | Distance between the edges of the knives in thousandth parts of an inch | |-----------------------------------------------|--------------------------------------------------| | 1$\frac{1}{9}$ | 0.012 | | 3$\frac{1}{9}$ | 0.020 | | 8$\frac{1}{9}$ | 0.034 | | 32 | 0.057 | | 96 | 0.081 | | 131 | 0.087 |

From these observations he concluded, that the light which forms 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, of the following dimensions. Let CA, CB, (fig. 2.) represent lines drawn upon the paper, parallel to the edges of the knives; and between which all the light would fall if it suffered no inflection. DE is a right line drawn through C, making the angles ACD, BCE, equal to one another, and terminating all the light which falls upon the paper, from the point where the edges of the knives meet. Then e i s, f k t, and g l v, will be three hyperbolic 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 x i p, y k q, and z l r, will be three other hyperbolic 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, which are similar, and equal to the former, 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 e i s, and x i p, till the meeting and crossing of the fringes; and then those lines cross the fringes in the form of dark lines terminating the first luminous fringes on the inside, and distinguishing them from another light, which begins to appear at i, and illuminates all the triangular space i p D E, 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.

Before the small hole in the window Newton placed a prism, 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, were bordered with fringes of the colour of the light in which they were held; and he found that those made in the red light were the largest, those made in the violet the least, and those made in the green of a middle bigness. The fringes with which the shadow of a man's hair were surrounded, 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 $\frac{1}{37.5}$ of an inch, and in the full violet $\frac{1}{45}$. 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 $\frac{1}{37.5}$, and in the violet $\frac{1}{45}$ of an inch; and these distances of the fringes held the same proportion at all distances from the hair, without any sensible variation.

From these observations it was evident, that the rays which formed 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 was held in the white beam of light, and cast a shadow bordered with three coloured fringes, those colours arose not from any new modifications impressed upon the rays of light by the hair, but only from the various inflections by which 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 who first made any experiments similar to Maraldi's observations chiefly respect the inflection of light towards other bodies, whereby their shadows are partially illuminated.

He exposed in the light of the sun a cylinder of wood three feet long, and $6\frac{1}{2}$ lines in diameter, when its shadow was everywhere equally black and well defined, even at the distance of 23 inches from it. At a greater distance the shadow appeared of two different densities; for its two extremities, 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 CCCLXXVI. 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 splendour diminished with the distance.

He repeated these experiments with three other cylinders of different dimensions; and from all of them 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, Maraldi supposes that 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 were 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 the shadows 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 repeating the experiments of Grimaldi and Newton, he 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 fringes 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.

A bristle, at the distance of nine feet from the hole, 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.

A plate, two inches long, and about half a line broad, 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, received at the distance of two feet and a half, was divided into four 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 CCLXXVI. fig. 4.

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

A second plate, 2 inches long and a line broad, being placed 14 feet from the hole, its shadow was received perpendicularly very near the plate, and was found to be 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 Fig. 6. 17 feet the black streaks were broader, more distinct, and more separated from the streaks that were less dark. At 42 feet, 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. 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. With plates from ¼ line to 2 lines broad, he could not observe any of the streaks of light, though the shadows were in some cases 56 feet from them.

The extraordinary size of the shadows of 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.

Maraldi having made the preceding experiments upon single long substances, placed 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.

He also placed in the rays of the sun a bristle and a plate of iron a line thick, so that they crossed one another obliquely; and when their shadows were received at the same distance, the light and dark streaks of the shadow of the bristle 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 bristle 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 bristle were visible upon it, as in fig. 8. To explain this appearance, he supposed that the rays of the sun glided a little along the bristle, so as to enlighten part of that which was behind the plate. But this seems to be an arbitrary and improbable supposition.

M. Maraldi also placed small globes in the solar light, admitted through a small aperture, and compared their shadows with those of the long substances, as he had done in the day light, and the appearances were still similar. It was 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 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 observe a difference of shades in those of the globes, taken at the same distance, though they were 2½ lines in diameter.

In order to explain the colours at the edges of these shadows, he threw some of the shadows upon others. 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.

He placed two plates of iron, each three or four lines broad, at a very small distance: 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 parallel streaks of a lively purple, separated by other black streaks; but between them there were other streaks, both of a very faint green, and also of a pale yellow.

The subject of inflection was next investigated by M. Mairan: but he only endeavoured to explain the facts which were known, by the hypothesis of an atmosphere surrounding all bodies; and consequently making two reflections and refractions of the light that falls upon them, one at the surface of the atmosphere, and the other at that of the body. This atmosphere he supposed to be of a variable density and refractive power, like the atmosphere.

M. Du Tour thought the variable atmosphere superfluous, and attempted to account for all the phenomena by an atmosphere of uniform density, and of a less refractive power than the air surrounding all bodies.

Only three fringes had been observed by preceding authors, but M. Du Tour was accidentally led to observe a greater number of them, and adopted from Grimaldi the following ingenious method of making them all appear very distinct.

He took a circular board 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 3 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 4 or 5 lines, in order to introduce a ray of the sun's light; and in the centre of the board C, he fixed a perpendicular pin about ¼ of a line in diameter.

This hoop being so placed, that a ray of light entering the chamber, through a vertical cleft of 2½ lines 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 m A n, 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, 1 or 1½ lines in breadth, which were generally bordered on both sides by a streak of orange colour.

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

In order to give some idea of his hypothesis, M. Du Tour shows that the ray a b, fig. 10, after being refracted at b, reflected at r and u, and again refracted at s and t, will be divided into its proper colours; the least refrangible or the red rays issuing at x, and the most refrangible or violet at y. 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.

The coloured streaks nearest the shadow of the pin, he supposes to be 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 r u, fig. 11. In this case, the red or least refrangible rays will issue at r, and the violet at u.

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

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

Sometimes M. Du Tour 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 accidental irregularity in the surface of the pin.

The white streaks mixed with the coloured ones he ascribes to small cavities in the surface of the pin; for they also changed their places when the pin was turned upon its axis.

He also found, that bodies of various kinds, and of different sizes, always produced fringes of the same dimensions.

Exposing two pieces of paper in the beam of light, so that part of it passed between two planes formed by them, them, M. du Tour observed, that the edges of this light were bordered with two orange streaks. To account for them, he supposes, that the more refrangible of the rays which enter at \( b \) are so refracted, that they do not reach the surface of the body at \( R \); so that the red and orange light may be reflected from thence in the direction \( dM \), where the orange 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 fringes at the borders of the white streaks, in the experiment of the hoop. He supposes, that the blue rays, which are not reflected at \( R \), pass on to \( I_3 \); and that these rays form the blue tinge observable in the shadows of some bodies. This, however, is mere trifling.

We may here make a general observation, applicable to all the attempts of philosophers to explain these phenomena by atmospheres. These attempts gave no explanation whatever of the physical cause of the phenomena. A phenomenon is some individual fact or event in nature. We are said to explain it, when we point out the general fact in which it is comprehended, and show the manner in which it is so comprehended, or the particular modification of the general fact. Philosophy resembles natural history, having for its subject the events of nature; and its investigations are nothing but the classification of these events, or the arrangement of them under the general facts of which they are individual instances. In the present instance there is no general fact referred to. The atmosphere is a mere gratuitous supposition; and all that is done is to show a resemblance between the phenomena of inflection of light to what would be the phenomena were bodies surrounded with such atmospheres; and even in this point of view, the discussions of Marian and Du Tour are extremely deficient. They have been satisfied with very vague resemblances to a fact observed in one single instance, namely, the refraction of light through the atmosphere of this globe.

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

But these analogical discussions are eminently deficient in another respect: They are held out as mechanical explanations of the changes of motion observed in rays of light. When it shall be shewn, that these are precisely such as are observed in refracting atmospheres, nothing is done towards deciding the original question; for the action of refracting atmospheres presents it in all its difficulties, and we must still ask how do these atmospheres produce this effect? No advance whatever is gained in science by thrusting in this hypothetical atmosphere; and Newton did wisely in attaching himself to the simple fact; and he thus gives us another step in science, by showing us a fact unknown before, viz. that the action of bodies on light is not confined to transparent bodies. He adds another general fact to our former stock, that light as well as other matter is acted on at a distance; and thus he made a very important deduction, that reflection, refraction, and inflection, are probably brought about by the same forces.

M. Le Cat has well explained a phenomenon of vision depending upon the inflection of light, which shows, sometimes that, in some cases objects by this means appear magnified. Looking at a distant steeple, when a wire, of a less diameter than the pupil of his eye, was held near to it, and drawing it several times betwixt his eye and that object, he found, that, every time the wire passed before his pupil, the steeple seemed to change its place, and some hills beyond the steeple appeared to have the same motion, just as if a lens had been drawn betwixt his eye and them. He found also, that there was a position of the wire 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 distinct and magnified. He then 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; and passing the wire before his eyes, 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 always, with the same result, the object being by this means magnified, and nearly doubled.

This phenomenon he explains by fig. 14, in which B represents the eye, A the steeple, and C a section of the wire. The black lines express the cone of light by which the natural image of the steeple A is formed, and which is much narrower than the diameter of the wire C; but the dotted lines include not only that cone of light, stopped and turned out of its course by the wire, but also more distant rays inflected by the wire, and thereby thrown more converging into the pupil; just as would have been the effect of the interposition of a lens between the eye and the object.

**Sect. IV. Discoveries concerning Vision.**

Maurolycus was the first who demonstrated that the crystalline humour of the eye is a lens which collects the light issuing from external objects, and converges them upon the retina. He did not, however, seem to be aware that an image of every visible object was thus formed upon the retina, though this seems hardly to have been a step beyond the discovery he had made. Montucia conjectures, that he was prevented from mentioning this part of the discovery by the difficulty of accounting for the upright appearance of objects. This discovery was made by Kepler; but he, too, was much puzzled with the inversion of the image upon the reti- 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. This is the true explanation of the difficulty, and is exactly the same as that which was lately given by Dr Reid.

These discoveries concerning vision were 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, he saw their images distinctly painted upon the retina. He did the same with the human eye, and exhibited this experiment at Rome in 1625.

Scheiner 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 does 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 traces the progress of the rays of light through all the humours; and after discussing every possible hypothesis concerning the seat of vision, he demonstrates that it is in the retina, and shows that this was the opinion of Alhazen, Vitellio, Kepler, and all the most eminent philosophers. He advances many reasons for this hypothesis; answers many objections to it; and, by a variety of arguments, refutes the opinion that the seat of vision is in the crystalline lens.

The subject of vision occupied the attention of Descartes. He explains the 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 induing of the size and distance of an object, by feeling 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 remarks, 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, in the same way as we imagine one stick to be two, when it is placed between two contiguous fingers laid across one another. 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 different distances by a change in the curvature of the crystalline, which he supposed to be a muscle, the tendons of it being the ciliary processes. 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 observes, appear larger than others, and also the parts contiguous to those on which the rays actually impinge; and for the same reason, if the objects be small, and placed at a great distance, they will always appear round, the figure of the angles disappearing.

The celebrated Dr Berkeley, bishop of Cloyne, published, in 1709, An Essay towards a New Theory of Vision, in which he solves many difficulties. He does not admit that it is by means of those lines and angles, theory of which are useful in explaining the theory of optics, that vision, different distances are estimated by the sense of sight; neither does he think that the mere direction of the optic axes, or the greater or less divergency of the rays of light, are sufficient for this purpose. "I appeal (says he) to experience, whether any one computes distance by the bigness of the angle made by the meeting of the two optic axes; or whether he ever thinks of the greater or less divergency of the rays which arrive from any point to his pupil: Nay, whether it be not perfectly impossible for him to perceive, by sense, the various angles wherewith the rays according to their greater or lesser divergency fell upon his eye." That there is a necessary connexion between these various angles, &c. and different degrees of distance, and that this connexion is known to every person skilled in optics, he readily acknowledges; but "in vain (he observes) shall mathematicians tell me, that I perceive certain lines and angles, which introduce into my mind the various notions of distance, so long as I am conscious of no such thing." He maintains that distance, magnitude, and even figure, are the objects of immediate perception only by the sense of touch; and that when we judge of them by sight, it is from different sensations felt in the eye, which experience has taught us to be the consequence of viewing objects of greater or less magnitude, of different figures, and at different distances. These sensations, with the respective distances, figures, and magnitudes by which they are occasioned, become so closely associated in the mind long before the period of distinct recollection, that the presence of the one instantly suggests the other; and we attribute to the sense of sight those notions which are acquired by the sense of touch, and of which certain visual sensations are merely the signs or symbols, just as words are the symbols of ideas. Upon these principles he accounts for single and erect vision. Subsequent writers have made considerable discoveries in the theory of vision; and among them there is hardly any one to whom this branch of science is so much indebted as to Dr Reid, and Dr Wells, whose reasonings we shall afterwards have occasion to detail.

Sect. V. Of Optical Instruments.

Glass globes, and specula, seem to have been the only optical instruments known to the ancients. Alhazen gave the first hint of the invention of spectacles. From close examination of Roger Bacon, it is not improbable that some monks gradually hit upon the construction of spectacles; to which Bacon's lesser segment was a nearer approach than Alhazen's larger one.

It is certain that spectacles were well known in the 13th century, and not long before. It is said that Alexander Spina, a native of Pisa, who died in 1313, 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. It is also inscribed on the tomb of Salvius Armatus, a nobleman of Florence, who died 1377, that he was the inventor of spectacles. Though both convex and concave lenses were sufficiently common, yet no attempt was made to combine them into a telescope till the end of the 16th century. Descartes considers James Metius as the first constructor of the telescope; and says, that as he was amusing himself with mirrors and burning glasses, he 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 Lippersheim, a spectacle-maker at Middleburgh, or rather by his children; who 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 entitled De vero telescopii inventore, gives this honour to Zacharius Joannides, i.e. Jansen, another spectacle-maker at the same place, who made the first telescope in 1590.

This ingenious mechanic had no sooner found the arrangement of glasses that magnified distant objects, than he enclosed them in a tube, and ran with his instrument to Prince Maurice; who, immediately conceiving that it might be useful in his wars, desired the author to keep it a secret. But this was found 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 Sirturus. Some person in Holland being very early supplied by him with a telescope, he passed with many for the inventor; but both Metius above mentioned, and Cornelius Drebell of Alcmaar, in Holland, applied to the inventor himself in 1620; as also did Galileo, and many others. The first telescope made by Jansen did not exceed 15 or 16 inches in length; but Sirturus, who says that he had seen it, and made use of it, thought it the best that he had ever examined.

Jansen directing his telescope to celestial objects, 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, Johannes Zacharias, observed 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 globular. Jupiter appeared round, and rather spherical; and sometimes he perceived two, sometimes three, and at other times even four small stars, a little above or below him; and, as far as he could observe, they performed revolutions round him.

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, 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.

About April or May, in 1609, it was reported at Venice, where Galileo (who was professor of mathematics in the university of Padua) then happened to be, that a Dutchman had presented to Count Maurice of Nassau, a certain optical instrument, by means of which, distant objects appeared as if they were near; but no farther account of the discovery had reached that place, verily, though this was near 20 years after the first discovery of the telescope. Struck, however, with this account, Galileo 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 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, where, from several eminences, he showed to some of the principal senators of that republic a variety of distant objects, to their very great astonishment. When he had made farther improvements in the instrument, he 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 it a written paper, in which he explained the structure and wonderful uses that might be made of the instrument both by land and sea. In return for so noble an entertainment, the republic, on the 25th of August, in the same year, more than tripled his salary as professor.

Galileo having amused himself for some time with the view of terrestrial objects, at length directed his tube towards the heavens; and found, that the surface of the moon was diversified with hills and valleys, like the earth. He found that the milky way and nebulæ 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 the ancients; and examining Jupiter, he found him attended by four stars, which, at certain periods, performed revolutions round him.

This discovery he made in January 1610, new style; and continuing his observations the whole of February following, he published, in the beginning of March, an account of all his discoveries, in his Nuncius Sidereus, printed at Venice.

The extraordinary discoveries contained in the Nuncius Sidereus, which was immediately reprinted both in Germany and France, were the cause of much debate among the philosophers of that time; many of whom could not give any credit to Galileo's account, while others endeavoured to decry his discoveries as nothing more than mere illusions.

In the beginning of July, 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 tergeminam.

Whilst he was still at Padua, he observed some spots on the face of the sun; but 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, as well as 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 among whom we find the name of Father Paul. This delay, however, was the cause of this discovery being contested with him by the famous Scheiner, who likewise made the same observation in October 1611, and we suppose had anticipated Galileo in the publication of it.

In November following Galileo 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, he went to Rome, where he gratified the cardinals, and all the principal nobility, with a view of the new wonders which he had discovered in the heavens.

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.

The honour of explaining the rationale of the telescope is due to the celebrated Kepler. He made several discoveries relating to the nature of vision; and not only explained the theory 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 converging and diverging the rays of a pencil of light. 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. He did not, however, investigate any rule for the foci of lenses unequally convex. He only says, in general, that they will fall somewhere in the middle, between the foci belonging to the two different degrees of convexity. We owe this investigation to Cavalieri, who laid down the following rule: As the sum of both the diameters is to one of them, so is the other to the distance of the focus.

The principal effects of telescopes depend upon these simple principles, 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 converged so as to form an image of the object; because they issue again from those points where there is no real substance, in certain directions, in the same manner as they did from the corresponding points in the objects themselves.

In fact, therefore, all that is effected by a telescope is, first, to make such an image of a distant object, by means of a lens or mirror; and then to give the eye some assistance for viewing that image as near as possible: so that the angle which it shall subtend at the eye, may be very large, compared with the angle which the object itself would subtend in the same situation. This is done by means of an eye-glass, which so refracts the pencils of rays, that they may afterwards be brought to their several foci by the 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 be looked at through a small hole in a thin plate or piece of paper, held close to the eye, it may be viewed very near to the eye, and, at the same distance, the apparent magnitude of the object will be the same in both cases. For if the hole be so small as to admit but a single ray from every point of the object, these rays will fall upon the retina in as many other points, and make a distinct image. They are only pencils 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 formed by the foci of the pencils without the eye, yet if, by the help of a concave eyeglass, the pencils of rays shall enter the pupil, just as they would have done from any place without the eye, the visual angle will be the same as if an image had actually been formed in that place. Objects will not appear inverted through this telescope, because the pencils which form the images of them, only cross one another once, viz at the object glass, as in natural vision they do in the pupil of the eye.

Such is the telescope that was first discovered and used by philosophers. The great inconvenience attending it is, that the field of view is exceedingly small, more difficult than ever since the pencils of rays enter the eye very much diverging from one another, but few of them can be than other intercepted by the pupil. This inconvenience increases with the magnifying power of the telescope; so that it is a matter of surprise how, with such an instrument, Galileo and others could have made such discoveries. No other telescope, however, than this, was so much as thought of for many years after the discovery. Descartes, who wrote 30 years after, mentions no others as actually constructed.

It is to the celebrated Kepler that we are indebted for the construction of what we now call the astronomical telescope. 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 theory into practice. Montucla conjectures, that the reason why he did not make trial of this 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 which he already possessed. He must also have foreseen, that the length 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.

The first person who actually made an instrument of Kepler's construction was Father Scheiner, who has given a description of it in his Rosa Ursina, published in 1635. If, says he, you insert two similar lenses 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 eyeglasses, 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. This construction, however, answered the end very imperfectly; and Father Rheita presently after discovered a better construction, using three eyeglasses instead of two.

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

Huygens was particularly eminent for his systematic knowledge of optics, and is the author of the chief improvements which have been made on all the dioptrical instruments till the discovery of the achromatic telescope. He was well acquainted with the theory of aberration arising from the spherical figure of the glasses, and has shown several ingenious methods of diminishing them by proper constructions of the eye-pieces. He first pointed out the advantages of two eye glasses in the astronomical telescope and double microscope, and gave rules for this construction, which both enlarges the field and shortens the instrument. Mr Dollond adapted his construction to the terrestrial telescope of De Rheita; and his five eyeglasses are nothing but the Huygenian eye-piece doubled. This construction has been too hastily given up by the artists of the present day for another, also of Mr Dollond's, of four glasses.

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

The first who distinguished themselves in grinding telescopic glasses were two Italians, Eustachio Divini at Rome, and Campani at Bologna, whose fame was much superior to that of Divini, or that of any other person of his time; though Divini himself pretended, that, in all the trials that were made with their glasses, and Campani's, of a greater focal length, performed better than those of Campani, and that his rival was not willing to try them 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 Louis XIV. and were of 86, 100, and 136 Paris feet in focal length.

Campani sold his lenses for a great price, and took every possible method to keep his art of making them secret. His laboratory was inaccessible, till after his death; when it was purchased by Pope Benedict XIV. who presented it to the academy called the Institute, 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 basons 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 them, and his great skill and address as a workman. It was also the general opinion that he owed much of his reputation to the secrecy and air of mystery which he affected; and that he made a great number of object-glasses which he rejected, showing only those that were very good. He made few lenses of a very great focal distance; and having the misfortune to break one of 141 feet in two pieces, he took incredible pains to join the two parts together, which he did at length so effectually, that it was used as if it had been entire; but it is not probable that he would have taken so much pains about it, if, as he pretended, he could very easily have made another as good.

Sir Paul Neille, Dr Hooke says, made telescopes of 36 feet, pretty good, and one of 50, but not of proportional goodness. Afterwards Mr Reive, and then Mr Cox, who were the most celebrated in England as grinders of optic glasses, made some good instruments of 50 and 60 feet focal length, and Mr Cox made one of 100.

These, and all other telescopes, were far exceeded by Extramdi's object glass of 600 feet focus made by M. Auzout, but he was never able to manage it. Hartsoeker is 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 enclosed 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 on this contrivance of use with Hartsoeker's. He placed the object-glass at the top of a long pole, have previously enclosed 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 silken 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 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 at pleasure, a stage that supported his object-glass.

M. de la Hire made some improvements in this method of managing the object-glass, by 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 the apparatus minutely, but shall only give a drawing of M. Huygen's pole, with a short explanation. 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 \(o\). When it was very dark, M. Huygens was obliged to make his object-glass visible by a lantern, \(y\), so constructed as to throw up to it the rays of light in a parallel direction.

Before leaving this subject, it must be observed, that M. Aurzout, 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 the subduplicate ratio of their lengths; and upon this supposition he drew up a table of the apertures of object-glasses of a great variety of focal lengths, from 4 inches to 40 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.

But all these improvements were diminished in value by the discovery of the 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, "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."

According to Dr Pringle, Mersennus was the man who entertained the first thought of a reflector. He certainly proposed a telescope with specula to the celebrated Descartes many years before Gregory's invention, though indeed in a manner so very unsatisfactory, that Descartes was so far from approving the proposal, that he endeavoured to convince Mersennus of its fallacy. Dr Smith, it appears, had never perused the two letters of Descartes to Mersennus which relate to that subject.

Gregory, a young man of uncommon genius, was led to the invention, in trying 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 incident rays, and to reflect them towards a small speculum of the same metal; this again was to return the image to an eye-glass placed behind the great speculum, which for that purpose was to be perforated in its centre. This construction he published in 1663, in his Optica Promota. But as Gregory, by his own account, was endowed with no mechanical dexterity, nor could find any workman capable of constructing his instrument, he was obliged to give up the pursuit: and probably, had not some new discoveries been made in light and colours, a reflecting telescope would never more have been thought of.

At an early period of life, Newton 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. Whilst he was thus employed, three years after Gregory's publication, he happened to examine the colours, formed by a prism, and having by 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 those which 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 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 parabolical 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 was towards the end of 1668, or in the beginning of the following year, when Newton being 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 the concave portion of a sphere; 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 substance that would be of a proper hardness, have the fewest pores, and receive the smoothest polish; a difficulty 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 similar irregularities in a refracting one. In another letter, written soon after, he informs the secretary, "that he was very sensible that metal reflects less light than glass transmits; but as he had found some metallic substances 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. Huygens, one of the greatest geniuses of the age, and a distinguished improver of the refracting telescope, 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 over 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 focal length, and consequently produce a much greater magnifying power 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 side of the glass: 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 substance: That the main business would be to find a substance for this speculum that would bear as good a polish as glass. Lastly, He believed that Mr Newton had not omitted to consider 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 exactness." Huygens was not satisfied with thus expressing to the society his high approbation of the invention; but drew up a favourable account of the new telescope, which he published in the Journal des Savans for 1672, by which channel it was soon known over Europe.

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 Cassegrain abroad as a rival to Newton's, no reflector was heard of for nearly half a century after. But when that period was elapsed, a reflecting telescope of the Newtonian form was at last produced by Mr Hadley, the inventor of the reflecting quadrant. The two telescopes which Newton had made were but six inches long; they were held in the hand for viewing objects, and in power were compared to a six feet refractor; whereas Hadley's was about 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 the manner of making the specula, we have, in the Philosophical 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 inventor.

The same celebrated artist, after finishing two telescopes of the Newtonian construction, accomplished a third of the Gregorian form; but, it would seem, less successfully. 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.

Mr James Short, as early as the year 1734, had signalized himself at Edinburgh by the excellence of his telescopes. 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 was well acquainted with the theory of optics. It was supposed that he had fallen upon a method of giving the parabolic figure to his great speculum; a point of perfection that Gregory and Newton had despaired of attaining; and that Hadley had never, as far as we know, attempted. Mr Short indeed affirmed, that 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, died with that ingenious artist. Mr Mudge, however, has lately realized the expectation of Sir Isaac Newton, who, above 100 years ago, presaged that the public would one day possess a parabolic speculum, not accomplished by mathematical rules, but by mechanical devices.

This was a 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 dexterous makers of every kind of mathematical instruments) so partial to his own abilities as not to acknowledge, that Mr Mudge has opened to them all some new and important lights, and has greatly improved the art of making reflecting telescopes."

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

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

The greatest improvement in refracting telescopes hitherto made public is that of Mr Dollond, of which an account has already been given in a preceding section, in which 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, 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 be 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 proportional 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 one 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.

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. 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 to be made to converge to a point on the retina; yet, if the glasses exceed a certain length, the colours may be spread too wide to be capable of being admitted through the pupil or aperture of the eye; which is the reason, that in long telescopes, constructed in the common manner, with three eye-glasses, the field is always very much contracted.

These considerations first set Mr Dollond on contriving how to enlarge the field, by increasing the number of eye-glasses without affecting 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 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 injuring 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 carried into foreign countries, 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.

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 a piece of machinery, by 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. As it is 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 most of the stars of the first and second magnitude have been seen even at mid-day, when the sun was shining bright; as This microscope was evidently a compound one, or rather something betwixt a telescope and a microscope; 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. 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 microscope; and this is clearly given, by the evidence of Borellus above mentioned, to Z Jansen, or his son.

The invention of compound microscopes is claimed by the same Fontana who arrogated to himself the discovery of telescopes; and though he did not publish any account of this invention till the year 1646 (notwithstanding he pretended to have made the discovery in 1618), Montucla, from not attending 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 title 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 very large, 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 about this time that Hartsoeker improved simple microscopes, by using small globules of glass, made socket 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 animaculæ in semine masculino, which gave rise to a new system of generation. A microscope of this kind, consisting of a globule of \( \frac{1}{2} \) th 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.

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 fixed on the point of a needle, which could be placed at any distance from the lens. If the objects were solid, he fastened them with glue; and if they were fluid, or required to be spread upon glass, he placed them on a small piece of Muscovy taffeta, or thin glass; which he afterwards glued to his needle. He had, however, a different apparatus for viewing the circulation of the blood, which he could attach to the same microscopes. M. Leeuwenhoek bequeathed the greatest part of his microscopes to the Royal Society. They were placed in a small Indian cabinet, in the drawers of which were 13 little boxes, each of which contained two microscopes, neatly fitted up in silver.

The glass of all these lenses is exceedingly clear, but none of them magnifies so much as those globules which are frequently used in other microscopes. Mr Folkes, who examined them, thought that they showed objects with much greater distinctness, a circumstance 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 experience in using them.

Mr Baker, who also examined these microscopes, and reported concerning them to the Royal Society, found that the greatest magnifier enlarged the diameter of an object about 165 times, but that all the rest fell much short of that power. He therefore concluded that M. Leeuwenhoek must have had other microscopes of much greater magnifying power for many of his discoveries.

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

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

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 several slips, not exceeding ¼ 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 hottest part of the flame, he had two globules, which he could increase or diminish at pleasure. If they were held a long time in the flame, they would have spots on them, so that he drew them out immediately after they became round. He broke off the stem 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 the same thread of very fine muslin appeared three or four times bigger than it did in the largest of Mr Wilson'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 small drops of water taken up with the 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 Grey, 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 beyond his 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 Montucia 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 3½ times more than they would have been in the usual way.

Temporary microscopes of a different kind have been constructed by Dr Brewster. They were composed of scopes of turpentine varnish, which was formed into a plane-convex lens, by laying a drop of it upon a piece of plain glass: the under surface of the glass was then smoked, and the black pigment removed immediately below the fluid lens. These lens lasted for a long time, and shewed objects distinctly, even when combined into a compound microscope. See Appendix to Ferguson's Lectures, vol. ii. and Microscope, p. 19.

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

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

Dr Smith invented a double reflecting microscope, of which a theoretical and practical account is given in reflecting his remarks at the end of the second volume of his System of Optics. As it is constructed on principles different from all others, and in the opinion of some, superior to all others, the reader will not be displeased with the following practical description.

A section of this microscope is shown in fig. 2, where ABC and abc are two specula, the former concave, and the latter convex, inclosed within the tube DEFG. The speculum ABC is perforated, and the object to be viewed. History. viewed is so placed between the centre and principal focus of that speculum, that the rays flowing from it to ABC are reflected towards an image p q. But before that image is formed, they are intercepted by the convex speculum a b c, and thence reflected through the hole BC in the vertex of the concave to a second image π, to be viewed through an eye-glass l. The object may either be situated between the two specula, or, which is perhaps better, between the principal focus and vertex c of the convex speculum a b c, a small hole being made in its vertex for the transmission of the incident rays. When the microscope is used, let the object be included between two little round plates of Muscovy-glass, fixed in a hole of an oblong brass plate u m, intended to slide close to the back side of the convex speculum: which must therefore be ground flat on that side, and so thin that the object may come precisely to its computed distance from the vertex of the speculum. The slider must be kept tight to the back of the metal by a gentle spring. The distance of the object being thus determined, distinct vision to different eyes, and through different eye-glasses, must be procured by a gentle motion of the little tubes that contain these glasses. These tubes must be made in the usual form of those that belong to Sir Isaac Newton's reflecting telescope, having a small hole in the middle of each plate, at the ends of the tube, situated exactly in each focus of the glass: The use of these holes and plates is to limit the visible area, and prevent any straggling rays from entering the eye. To the tube of the eye-glass is fastened the arm g, on which the adjusting screw turns. A similar arm u is attached to the fixed tube X, in which the neck of the screw turns; and by turning the button y, the eye-tube is moved farther from or nearer to the object, by which means different sorts of eyes obtain distinct vision.

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

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

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

M. Lieberkuhn, in 1738 or 1739, made two capital improvements in microscopes, by the invention of the solar microscope, and the microscope for opaque objects, and that when he was in England in the winter of 1739, he showed an apparatus for each of these purposes, made by himself, to several gentlemen of the Royal Society, as well as to some opticians.

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 specula and magnifiers of different powers, was brought to perfection by Mr Cuff in Fleet-street. M. Lieberkuhn made considerable improvements in his solar microscope, particularly in adapting it to the view of opaque objects; but in what manner this was effected, M. Æpinus, 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. Æpinus invites those who came into the possession of M. Lieberkuhn's apparatus to publish an account of this instrument; but it does not appear that his method was ever published.

This improvement of M. Lieberkuhn's induced M. Æpinus himself to attend to the subject; and he thus 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 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, scope and that nothing is necessary but a large concave mirror, magic lantern perforated as for a telescope; and the light should be turned so situated, that none of it may pass directly through the perforation, so as to fall on the images of the objects upon the screen. He proposes to have four different machines, for objects of different sizes; the first for those of six feet long, the second for those of one foot, the third for those of two inches, and the fourth for those of two lines; but it is needless to be particular in the description. PART I. THEORY OF OPTICS.

The science of optics is commonly divided into three parts, Dioptics, which treats of the laws of refraction, and the phenomena depending upon them; Catoptrics, which treats of the laws of reflection, and the phenomena connected with them; and, lastly, Chromatics, which treats of the phenomena of colour. But this division is of no use in a treatise of Optics, as most of the phenomena depend both on refraction and reflection, colour itself not excepted. For this reason, though we have given detached articles under the words Dioptics, Catoptrics, and Chromatics; we have reserved for this place the explanation of the laws of reflection and refraction, by which all optical phenomena may be explained.

CHAP. I. On Light.

Under the article Light we have given some account of the controversies concerning its nature. The opinions of philosophers may, in general, be arranged under these two: 1. That light is produced by the undulations of an elastic fluid, nearly in the same manner as sound is produced by the undulations of the air. This opinion was first offered to the public by Des Cartes, and afterwards by Mr Huygens. It was revived by Euler, and has lately found an able and ingenious defender in Dr Thomas Young.—2d, That the phenomena of vision are produced by the motion and action of matter emitted from the shining body with immense velocity, moving uniformly in straight lines, and acted on by other bodies so as to be reflected, refracted, or inflected, in various ways, by means of forces which act on it in the same manner as on other inert matter. Sir Isaac Newton has ably shown the dissimilarity between the phenomena of vision and the legitimate consequences of the undulations of an elastic fluid. All M. Euler's ingenious and laborious discussions have not removed Newton's objections in the smallest degree. Sir Isaac adopts the vulgar opinion, therefore, because the difficulties attending this opinion are not inconsistent with the established principles of mechanics, and are merely difficulties of conception to limited faculties like ours. We need not despair of being able to decide, by experiment, which of these opinions is nearest to the truth; because there are phenomena where the result should be sensibly different in the two hypotheses. At present, we shall content ourselves with giving some account of the legitimate consequences of the vulgar opinion, as modified by Sir Isaac Newton, viz. that light consists of small particles emitted with very great velocity, and attracted or repelled by other bodies at very small distances.

Every visible body emits or reflects inconceivably fine small particles of matter from each point of its surface, each point which issue from it continually, not unlike sparks from a burning coal, in straight lines and in all directions. These small particles face... Theory.

Refraction. particles entering the eye, and striking upon the retina (an expansion of the optic nerve over the back part of the eye to receive their impulses), excite in our minds the idea of light. And according 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 this, 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 being transmitted 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. Besides, 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 some miles of it every way with luminous particles, before it has lost the least sensible part of its substance in consequence of this copious emission.

It is evident that these particles proceed from every point of the surface of a visible body, and in all directions, 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. 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 by which philosophers estimate their velocity, is by observations made on the eclipses of Jupiter's satellites; which eclipses appear to us 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. Hence it is concluded, that they require about seven minutes to pass over a space equal to the distance of the earth from the sun.

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.

Chap. II. On Refraction.

Light, when proceeding from a luminous body, is invariably found to proceed in straight lines, without the least deviation. But, if it happens to pass obliquely from one medium to another, it always leaves the direction it had before, and assumes a new one; and this change of course is called its refraction. After having taken this new direction, it then proceeds invariably in a straight line till it meets with a different medium, when it is again turned out of its course. It must be observed, however, that though by this means we may cause the rays of light to make any number of angles in their course, it is impossible 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 these cases, it describes a curve of the hyperbolic kind; but at all other times it proceeds in straight lines, or in what may be taken for straight lines, without any sensible error.

Sect. I. On the cause of Refraction, and the Law by which it is performed.

The phenomena of refraction are explained by an attractive power in the medium through which light passes, in the following manner. All bodies being endowed with an attractive force, which is extended to some distance beyond their surfaces; when a ray of light passes out of a rarer into a denser medium (if this latter medium has a greater attractive force than the former, as is commonly the case), the ray, just before its entrance, will begin to be attracted towards the denser medium; and this attraction will continue to act upon it, till some time after it has entered the medium; and therefore, if a ray approaches a denser medium in a direction perpendicular to its surface, its velocity will be continually accelerated during its passage through the space in which that attraction exerts itself; and therefore, after it has passed that space, it will move on, till it arrive 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. 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 which is described before. These positions may be illustrated in the following manner.

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

As 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 then, there will not be so many particles above it which will attract it upwards, as below, that will attract it downwards. So that after it has entered the surface at N, in the direction OQ, it will not proceed in that direction, but will continue to describe a curve, as NS; after which it will proceed straight on towards the opposite side of the medium, being attracted equally every way; and therefore will at last proceed in the direction XST, still nearer the perpendicular PR than before.

If we suppose ABZY 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.

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

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

The space through which the attraction of cohesion of the particles of matter is extended is so very small, that in considering the progress of a ray of light out of one medium into another, the curvature it describes in passing through the space of attraction is generally neglected; and its path is supposed to be bent or 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; and 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, by which refraction is always performed; which 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 sine of the angle of incidence and that of the angle of refraction, will always have a constant ratio to one another in that medium.

To illustrate this: Let us suppose ABCD (fig. 4) to represent a rarer, and ABEF a denser medium; let GH be a ray of light passing through the first and entering the second at H, and let HI be the refracted ray: then supposing the perpendicular PR drawn through the point H, on the centre H, and with any radius, describe the circle ABPR; and from G and I, where the incident and refracted rays cut the circle, let fall the lines GK and IL perpendicularly upon the line PR; the former of these will be the sine of the angle of incidence, the latter of refraction. Now if in this case the ray GH is so refracted at H, that GK is double or triple, &c. of IL, then, whatever other inclination the ray GH might have had, the sine of its angle of incidence would have been double or triple, &c. to that of its angle of refraction. For instance, had the ray passed in the line MH before refraction, it would have passed in some line as HN afterwards, so situated that MO should have been double or triple, &c. of NQ.

The following table contains the refractive densities of several bodies.

| Substance | Refractive Density | |-----------------|--------------------| | Diamond | 2.500 | | Flint glass | 1.385 | | Plate glass | 1.502 | | Crown glass | 1.525 | | Sulphuric acid | 1.435 | | Solution of potash | 1.390 | | Olive oil | 1.469 | | Alcohol | 1.370 | | Atmospheric air | 1.000276 | | Ice | 1.31 | | Water | 1.336 |

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

**Lemma I.**

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

Let two bodies be uniformly accelerated from a state of rest in the points Aa, along the spaces AB, ab, fig. 5. Theory.

Law of Refraction, spaces described in equal times; it is evident, from what has been said under the articles Gravity and Acceleration, that because these spaces are described with motions uniformly accelerated, AC and ac are respectively the halves of the spaces which would be uniformly described during the same time with the velocities acquired at C and c, and are therefore measures of these velocities. And as these velocities are uniformly acquired in equal times, they are measures of the accelerating forces. Therefore, AC : ac = F : f.

Also, from the nature of uniformly accelerated motion, the spaces are proportional to the squares of the acquired velocities. Therefore, (using the symbols \( \sqrt{C} \), \( \sqrt{c} \), etc.) to express the squares of the velocities at C, c, etc., we have

\[ \sqrt{B} : \sqrt{C} = AB : AC \\ \sqrt{C} : \sqrt{c} = AC : ac \\ \sqrt{c} : \sqrt{b} = ac : ab \]

Therefore, by equality of compound ratios

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

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

Q. E. D.

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

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

Lemma II.

If a particle of matter, moving with any velocity along the line AC, be impelled by an accelerating or retarding force, acting in the same or in the opposite direction, and if the intensity of the force in the different points B, F, H, C, etc., be as the ordinates BD, FG, etc., to the line DGE, the areas BFGD, BHKD, etc., will be as the changes made on the square of the velocity, at B, when the particle arrives at the points F, H, etc.

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

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

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

This proposition, which is the 39th of the 1st book of the Principia, is perhaps the most important in the whole science of mechanics, being the foundation of every application of mechanical theory to the explanation of natural phenomena. No traces of it are to be found in the writings of philosophers before the publication of Newton's Principia, though it is assumed by John Bernoulli and other foreign mathematicians, as an elementary truth, without any acknowledgment of their obligations to its author. It is usually expressed by the equation \( f \cdot s = uv \) and \( f \cdot s = u^2 \), i.e., the sum of the momentary actions is equal to the whole or finite increment of the square of the velocity.

Proposition.

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

Let ST, KR, represent two planes (parallel to, and equidistant from, the refracting surface XY) which bound the space in which the light, during its passage, is acted on by the refracting forces.

The intensity of the refracting forces being supposed equal at equal distances from the bounding planes, though anyhow different at different distances from them, may be represented by the ordinates T, a, n, q, p, r, c, R, etc., of the curve a b n p c, of which the form must be... be determined from observation, and may remain for ever unknown. The phenomena of inflected light show us that it is attracted by the refracting substance at some distances, and repelled at others.

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

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

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

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

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

This specific velocity is easily determined for any substance in which the refraction is observed, by drawing \(Li\) perpendicular to \(EL\), meeting in \(i\) the circle described with the radius \(EF\). For \(Ei\) being equal to \(EF\), will represent the velocity of the refracted light, and \(EL\) represent the velocity of the incident light, and \(Ei = EL^2 + Li^2\), and therefore \(Li^2\) is the augmentation of the square of the initial velocity, and \(Li\) is the specific velocity.

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

Cor. 1. When light is refracted towards the perpendicular to the refracting surface it is accelerated; and it is retarded when it is refracted from the perpendicular, retarded. In the first case, therefore, it must be considered as or retarded having been acted on by forces conspiring (in part at least) with its motion, and vice versa. Therefore, because we see that it is always refracted towards the perpendicular, when passing from a void into any transparent substance, we must conclude that it is, on the whole, attracted by that substance. We must draw the same conclusion from observing, that it is refracted from the perpendicular in its passage out of any transparent substance whatever into a void. It has been attracted backwards by that substance. This acceleration of light in refraction is contrary to the opinion of those philosophers who maintain that illumination is produced by the undulation of an elastic medium. Euler attempts to prove, by mechanical laws, that the velocities of the incident and refracted light, are proportional to the sines of incidence and refraction, while our principles make them in this ratio inversely. Boscovich proposed a fine experiment for deciding this question. The aberration of the fixed stars arises from the combination of the motion of light with the motion of the telescope by which it is observed. Therefore this aberration should be greater or less when observed by means of a telescope filled with water, according as light moves slower or swifter through water than through air. He was mistaken in the manner in which the conclusion should be drawn from the observation made in the form prescribed by him; and the experiment has not yet been made in a convincing manner; because no fluid has been found of sufficient transparency to admit of the necessary magnifying power. It is an experiment of the greatest importance to optical science.

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

If the direction of the light be still more oblique, so that its perpendicular velocity is less than the specific velocity, it will not reach the plane ST, but be reflected as soon as it has penetrated so far that the specific velocity of the part penetrated (estimated by the compounding part of the area of forces) is equal to its perpendicular velocity. Thus the ray fE will describe the path E d D a penetrating to b c, so that the corresponding area of forces a b c e is equal to the square of f k, its perpendicular velocity.

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

It must be acknowledged that it is a very curious circumstance, that a body which is perfectly transparent should cease to be so at a certain obliquity; that certain obliquity should not hinder light from passing through a void into a piece of glass; but that the same wholly reflective obliquity should prevent it from passing from the glass into a void. The finest experiment for illustrating the fact is, to take two pieces of mirror-glass, not silvered, and put them together with a piece of paper between them, forming a narrow margin all round to keep them apart. Plunge this apparatus into water. When it is held nearly parallel to the surface of the water, everything at the bottom of the vessel will be seen clearly through the glasses; but when they are turned so as to be inclined about 50 degrees, they will intercept the light as much as if they were plates of iron. It will be proper to soak the paper in varnish, to prevent water from getting between the glasses.

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

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

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

It should follow, that the refraction of a star which is in our meridian at six o'clock in the evening should be greater than that of a star which comes on the meridian at six in the morning; because we are moving away from the first, and approaching to the last. But the difference is but 1/360 of the whole, and cannot be observed with sufficient accuracy in any way yet practised. A form of observation has been proposed by Dr Blair, professor of practical astronomy in the university of Edinburgh, which promises a very sensible difference of refraction. It is also to be expected, that a difference will be observed in the refraction of the light from the east and western ends of Saturn's ring. Its diameter is about 26 times that of the earth, and it revolves in 10h. 32' so that the velocity of its edge is about 1/7000 of the velocity of the sun's light. If therefore the light be reflected from it according to the laws of perfect elasticity, or in the manner here explained, that which comes to us from the western extremity will move more slowly than that which comes from the eastern extremity in the proportion of 2500 to 2501. And if Saturn can be seen distinctly after a refraction of 30° through a prism, the diameter of the ring will be increased one half in one position of the telescope, and will be as much diminished by turning the telescope half round its axis; and an intermediate position will exhibit the ring of a distorted shape. This experiment is one of the most interesting to optical science, as its result will be a severe touchstone of the theories which have been attempted for explaining the phenomena on mechanical principles.

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

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

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

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

All these consequences are agreeable to experience. The brilliant reflection from a dew-drop ceases when it touches the leaf on which it rests: The brilliancy of a diamond is greatly damaged by moisture getting behind it: The opacity of the combined mirror plates, mentioned in Cor. 2, is removed by letting water get between them: A piece of glass is distinctly or clearly seen in air, more faintly when immersed in water, still more faintly amidst olives, and it is hardly perceived in spirits of turpentine. These phenomena are incompatible with the notion that reflection is occasioned by impact on solid matter, whether of the transparent body, or of any other or other fancied fluid behind it; and their perfect coincidence with the legitimate consequences of the assumed principles, is a strong argument in favour of the truth of those principles.

It is worth while to mention here a fact taken notice of by Mr Beguelin, and proposed as a great difficulty in Newton's theory of refraction. In order to get the greatest possible refraction, and the simplest measure of the refracting power at the anterior surface of any transparent... transparent substance, Sir Isaac Newton enjoins us to employ a ray of light falling on the surface quam obliquisse. But Mr. Beguelin found, that when the obliquity of incidence in glass was about $89^\circ 50'$, no light was refracted, but that it was wholly reflected. He also observed, that when he gradually increased the obliquity of incidence on the superior surface of the glass, the light which emerged last of all did not skim along the surface, making an angle of $90^\circ$ with the perpendicular, as it should do by the Newtonian theory, but made an angle of more than ten minutes with the posterior surface. Also, when he began with very great obliquities, so that all the light was reflected back into the glass, and gradually diminished the obliquity of incidence, the first ray of light which emerged did not skim along the surface, but was raised about $10$ or $15$ minutes.

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

Cor. 6. Those philosophers who maintain the theory of undulation, are under the necessity of connecting the dispersive powers of bodies with their mean refractive powers. M. Euler has attempted to deduce a necessary difference in the velocity of the rays of different colours from the different frequency of the undulations, which he assigns as the cause of their different colorific powers. His reasoning on this subject is of the most delicate nature, and unintelligible to such as are not completely master of the infinitesimal calculus of partial differences, and is unsatisfactory to such as are able to go through its intricacies. It is contradicted by fact. He says, that musical sounds which differ greatly in acuteness are propagated through the air with different velocities: but one of the smallest bells in the chimes of St Giles's church in Edinburgh was struck against the rim of the very deep-toned bell on which the hours are struck. When the sound was listened to by a nice observer at the distance of more than two miles, no interval whatever could be observed. A similar experiment was exhibited to M. Euler himself, by means of a curious instrument used at St Petersburg, and which may be heard at three or four miles distance. But the experiment with the bells is unexceptionable, as the two sounds were produced in the very same instant. This connection between the refrangibility in general and the velocity must be admitted, in its full extent, in every attempt to explain refraction by undulation; and Euler was forced by it to adopt a certain consequence which made a necessary connection between the mean refraction and the dispersion of heterogeneous rays. Confident of his analysis, he gave a deaf ear to all that was told him of Mr Dollond's improvements on telescopes, and asserted, that they could not be such as were related; for an increase of mean refraction must always be accompanied with a determined increase of dispersion. Newton had said the same thing, being misled by a limited view of his own principles; but the dispersion assigned by him was different from that assigned by Euler. The dispute between Euler and Dollond was confined to the decision of this question only; and when some glasses made by a German chemist at Petersburg convinced Euler that his determination was erroneous, he did not give up the principle which had forced him to this determination of the dispersion, but immediately introduced a new theory of the achromatic telescopes of Dollond; a theory which took the artists out of the track marked out by mathematicians, and in which they had made considerable advances, and led them into another path, proposing maxims of construction hitherto untried, and inconsistent with real improvements which they had already made. The leading principle in this theory is to arrange the different ultimate images of leads art, a point which arise either from the errors of a spherical lens, figure or different refrangibility, in a straight line passing through the centre of the eye. The theory itself is specious; and it requires great mathematical skill to accomplish this point, and hardly less to decide on the propriety of the construction which it recommends. It is therefore but little known. But that it is a false theory, is evident from one simple consideration. In the most indistinct vision arising from the worst construction, this rectilineal arrangement of the images obtains completely in that pencil which is situated in the axis, and yet the vision is indistinct. But, what is to our present purpose, this new theory is purely mathematical, suiting any observed dispersive power, and has no connection with the physical theory of undulations, or indeed with any mechanical principles whatever. But, by admitting any dispersive power, whatever may be the mean refraction, all the physical doctrines in his Nova Theoria Lucis et Colorum are overlooked, and therefore never once mentioned, although the effects of M. Zeiller's glass are taken notice of as inconsistent with that mechanical proposition of Newton's which occasioned the whole dispute between Euler and Dollond.

They are indeed inconsistent with the universality of Law of that proposition. Newton advances it in his Optics merely as a mathematical proposition highly probable, but says that it will be corrected if he shall find it false.

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

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

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

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

The law of variation in the joint action of many particles adjoining to the surface of a refracting medium, is extremely different from that of a single particle; but variations in when this last is known, the other may be found out. We shall illustrate this matter by a very simple case.

Let DE be the surface of a medium, and let us suppose that the action of a particle of the medium on a particle from that of light extends to the distance ED, and that it is proportional to the ordinates ED, EF, FG, GH, &c., of the line A h C g f D; that is, that the action of the particle E of the medium on a particle of light in F, is to its action on a particle in H as Ff to HH, and that it is attracted at F but repelled at H, as expressed by the situation of the ordinates with respect to the abscissa. In the line AE produced to B, make EB, EC, ED, EF, &c. respectively equal to EA, EH, EC, EG, EF, &c.

It is evident that a particle of the medium at B will exert no action on the particle of light in E, and that the particles of the medium in A, B, C, D, will exert on it actions proportional to H h, G g, F f, ED. Therefore, supposing the matter of the medium continuous, the whole action exerted by the row of particles EB will be represented by the area A h CDE; and the action of the particles between B and D will be represented by the area A h C f F, and that of the particles between E and D by the area F/DE.

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

But let us suppose the particles of light to be within the medium, as at φ, and make φo = AE. It is again evident that it is acted on by the particles of the medium between φ and d with a force represented by the area A h CDE, and in the opposite direction by the particles in E φ with a force represented by the area F/DE. This balances an equal quantity of action, and there remains an action expressed by the area A h C f F. Therefore, if an equal and similar line to A h CDE be described on the abscissa EB, the action of the medium on a particle of light in φ will be represented by the area φ f x h B, lying beyond it.

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

Refraction of the whole row of particles. In the same manner may a curve be constructed, whose ordinates express the united action of the whole medium.

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

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

Lemma.

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

Let RVF, or rVf (fig. 10.) be the progress of the rays refracted at V (the angle r VR being considered in its nascent or evanescent state), and VC perpendicular to the refracting surface VA. From C draw CD, CB perpendicular to the incident and refracted rays RV, VF, cutting r V, Vf in A and B, and let C d, C b be perpendicular to r V, Vf.

Because the sines of incidence and refraction are in a constant ratio, their simultaneous variations are in the same constant ratio. Now the angle RV r is to the angle FV f in the ratio of \(\frac{B\beta}{BV}\) to \(\frac{D\delta}{DV}\); that is, of \(\frac{BC}{BV}\) to \(\frac{DC}{DV}\); that is, of \(\frac{\sin \text{ incid.}}{\cos \text{ incid.}}\) to \(\frac{\sin \text{ refr.}}{\cos \text{ refr.}}\); that is, of \(\tan \text{ incid.} : \tan \text{ refr.}\).

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

Problem.

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

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

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

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

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

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

"Retaining the former construction for the ray PF,"

(b) We refer to Graveshande, because we consider it of importance to make such a work as ours serve as a general index to science and literature. At the same time we take the liberty to observe, that the focus in question is virtually determined by the construction which we have given: for the points P, F of the line PF are determined, and therefore its position is also determined. The same is true of the position of p f, and therefore the intersection φ of the two lines is likewise determined. Refraction by Spherical Surfaces.

Draw PS perpendicular to PV, and RP perpendicular to RP, and make PR : PS = VR : VF. On PR describe the semicircle r RP, and on PS the semicircle S φ P, cutting the refracted ray PF in φ; draw p r, p S, p φ."

It follows from the lemma, that if φ be the focus of refracted rays, the variation P φ p of the angle of refraction is to the corresponding variation PR p of the angle of incidence as the tangent of the angle of incidence VRF to the tangent of the angle of incidence VRP. Now P p may be considered as coinciding with the arch of the semicircles. Therefore the angles PR p, P r p are equal, as also the angles P φ p, P S p. But PS p is to P r p as P r to PS; that is, as VR to VF; that is, as the cotangent of the angle of incidence to the cotangent of the angle of refraction; that is, as the tangent of the angle of refraction to the tangent of the angle of incidence. Therefore the point φ is the focus.

Sect. III. Of Refraction by Spherical Surfaces.

Problem.

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

Let PV π (figs. 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,) be a spherical surface whose centre is C, and let the incident light diverge from or converge to R. Draw the ray RC through the centre, cutting the surface in the point V, which we shall denominate the vertex, while RC is called the axis. This ray passes on without refraction, because it coincides with the perpendicular to the surface. Let RP be another incident ray, which is refracted at P; draw the radius PC. In RP make RE to RP as the sine of incidence m to the sine of refraction n; and about the centre R, with the distance RE, describe the circle EK, cutting PC in K; draw RK and RF parallel to it, cutting the axis in F. PF is the refracted ray, and F is the focus.

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

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

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

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

Cor. 2. If the incident ray RP is parallel to the axis RC, we have PO : CO as the sine of incidence to the sine of refraction. For the triangles RPK, PCO are similar, and \( PO : CO = RK : RP = m : n \).

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

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

Cor. 4. By construction we have RK : RP = \( \frac{m}{n} \) by similarity of triangles \( \frac{PF}{PR} = \frac{CF}{CR} \); therefore \( m \times PR \times CF = n \times CR \times PF \); therefore \( m \times PR : n \times CR = PF : CF \); and \( m \times PR : n \times CR : m \times PR = PF : CF : PF \); ultimately \( m \times VR : n \times CR : m \times VR = VC : VF \).

This is a very general optical theorem, and affords an easy method for computing the focal distance of refracted rays. For this purpose let VR, the distance of the radiant point, be expressed by the symbol r, the distance of the focus of refracted rays by the symbol f, and the radius of the spherical surface by a; we have

\[ \frac{mr}{m-a} : \frac{mr}{a-f}, \quad \text{and} \]

\[ \frac{f}{m-a} = \frac{m-a}{m-r+a}. \]

In its application due attention must be paid to the qualities of r and a, whether they be positive or negative, according to the conditions of last corollary.

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

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

Cor. 6. We have also Rq : RP = RV : RF, and ultimately for central rays RQ : RV = RV : RF, and RF = \( \frac{RV^2}{RQ} \). This proposition is true in lenses and mirrors, but not in single refracting surfaces.

Cor. 7. Also Rq : RC = RP : RF, and ultimately RQ : RV = RC : RF, and RF = \( \frac{RV \times RC}{RQ} \). N.B. These four points Q, V, C, F, either lie all one way from P, or two of them forward and two backward.

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

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

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

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

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

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

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

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

Sect. IV. On Lenses.

Lenses for optical purposes may be ground into nine different shapes. Lenses cut into five of those shapes, together with their axes, are described in vol. vi. page 33. (See Dioptrics). The other four are:

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

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

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

4. A concavo-convex glass, or meniscus, as C, which is seldom made use of in optical instruments.

A ray of light Gh falling perpendicularly on a plane glass EF, will pass through the glass in the same direction hi, and go out of it into the air in the same straight line iu.

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

Lemma.

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

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

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

Prof. I.

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

Let E be the centre of the lens, and r the centres of Fig. 7. to its surfaces, R r its axis, g EG a line parallel to the incident rays upon the surface B, whose centre is R. Parallel to g E draw a semidiameter BR, in which produced let V be the focus of the rays after their first refraction at the surface B, and joining r let it cut g E produced in G, and G will be the focus of the rays that lastly emerge from the lens.

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

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

Corol. 2. The last proportion gives the following rule for finding the focal distance of any thin lens. As R r, the interval between the centres of the surfaces, Theory.

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

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

The focus may also be found by this rule: \( QF : FE = QE : Eg \), for \( QG : GE = QA : Ag \).

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

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

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

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

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

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

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

Let \( A, B, C \), be the lenses, and \( RA \) a ray incident on the first of them. Let \( a, b, c \), be their foci for parallel rays coming in the opposite direction; draw the perpendicular \( ad \), cutting the incident ray in \( d \), and draw \( da \) through the centre of the lens: \( AB \) parallel to \( de \). Of Vision, to \( da \) will be the ray refracted by the first lens. Through the focus of the second lens draw the perpendicular \( ae \), cutting \( AB \) in \( e \); and draw \( eb \) through the centre of the second lens. \( BD \) parallel to \( eb \) will be the next refracted ray. Through the focus \( x \) of the third lens draw the perpendicular \( xf \), cutting \( BD \) in \( f \), and draw \( fc \) through the centre of the third lens. \( CE \) parallel to \( fc \), will be the refracted ray; and so on.

**Sect. V. On 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 external objects; it will be easy to understand how the rays are refracted by the humours of the eye, and are thereby collected into innumerable points on the retina, on which they form the images of the objects from which they flow. For the different humours of the eye, and particularly the crystalline, are to be considered as a convex glass; and the rays in passing through them as affected in the same manner in the one as in the other. A description of the coats and humours, &c. has been given in Anatomy; but it will be proper to repeat as much of the description as will be sufficient for our present purpose.

The eye is nearly globular, and consists of three coats and three humours. The part \( DHHG \) of the 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, \( mn \). The iris is composed of two sets of muscular fibres; the one of a circular form, which contracts the hole in the middle called the pupil, when the light would otherwise be too strong for the eye; and the other of radical fibres, tending everywhere from the circumference of the iris towards the middle of the pupil; which fibres, by their contraction, dilate and enlarge the pupil when the light is weak, in order to let in a greater quantity of it. 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 thrown the images of all visible objects.

Under the cornea is a fine transparent fluid like water, thence 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 refracting 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 of the consistence of hard jelly, and is to the specific gravity of water as 11 to 10. It is enclosed in a fine transparent membrane, called the capsule of the crystalline lens, from which proceed radial fibres \( oo \), called the ciliary ligaments, 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 of visco with the white of an egg, and very little exceeds the specific gravity and refractive power of water.

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

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, or the same thing may be better accomplished by paring the sclerotic coat so thin that it becomes a little transparent, and retains the vitreous humour.

Since the image is inverted, many have wondered why the object appears upright. But we are to consider, are seen:

1. That inverted is only a relative term: and, 2. That upright there is a very great difference between the real object and the 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 and upward; and know very well that we cannot feel the upper end by moving our hand downward. In the same manner 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. 9, 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 will be invisible. On this account, the optic nerve is wisely placed, not in the middle of the bottom of the eye, but towards the side next the nose; so that what appear ever part of the image falls upon the optic nerve of one eye, may not fall upon the optic nerve of the other, cause the optic nerve of the eye \( D \), but not of the eye \( E \); and the point of light... The nearer that any object is to the eye, the larger is the angle under which it is seen, and the magnitude of which it appears. Thus to the eye D, the object ABC is seen under the angle APC; and its image cba is very large upon the retina: but to the eye E, at a double distance, the same object is seen under the angle ApC, which is equal only to half the angle APC, as is evident by the figure. The image cba is likewise twice as large in the eye D, as the other image cba is in the eye E. In both these representations, a part of the image falls on the optic nerve, and the object in the corresponding parts 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, and thus forming the image of the object upon it, 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 part of the image which falls upon the middle of the optic nerve is lost, and consequently the corresponding part of the object is rendered invisible, is plain by experiment. For if a person fixes three patches, A, B, C, (fig. 2.) upon a white wall, at the height of the eye, and at the distance of about a foot from each other, and places himself before them, shutting the right eye, and directing the left towards the patch C, he will see the patches A and C, but the middle patch B will disappear. Or, if he shuts his left eye, and directs the right towards A, he will see both A and C, but B will disappear; and if he directs his eye towards B, he will see both B and A, but not C. For whatever patch is directly opposite to the optic nerve N, vanishes. This requires a little practice; after which he will find it easy to direct his eye so as to lose the sight of whatever patch he pleases.

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

M. Marriotte observes, that this improvement on his Of Vision experiment, by M. Picard, is ingenious, but difficult to execute, since the eyes must be considerably strained in looking at any object so near 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 stand opposite to them, at the distance of 12 or 13 feet, and hold your thumb before your eyes, at the distance of about eight inches, so that it may conceal from the right eye the paper that is to the left hand, and from the left eye the paper to the right hand. Then, if you look at your thumb steadily with both eyes, you will lose sight of both the papers; the eyes being so disposed, that each of them receives the image of one of the papers upon the base of the optic nerve, while the other is intercepted by the thumb.

M. Le Cat pursued this curious experiment a little farther than M. Marriotte. 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 following is the manner in which this curious experiment is now generally made. 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), when 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 is not surprising that M. Marriotte was led, by this remarkable observation, to suspect that the retina was the seat of vision. 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. He also thought that the dark colour of the choroid coat was intended to make it more susceptible of the impression of light. M. Pecquet, in answer to M. Marriotte's observation concerning the transparency of the retina, says, that it is very imperfectly so, resembling only oiled paper, or the horn that is used for lanterns; and besides, that its whiteness demonstrates it to be sufficiently opaque for stopping the rays of light, as much as is necessary for the purpose of vision; whereas, if vision be performed by means of those rays which are transmitted through such a substance as the retina, it must be very indistinct. The retina resembles very much the thin white film which intervenes between the white of an egg and its shell.

As to the blackness of the choroides, which M. Marriotte thought to be necessary for the purpose of vision, M. Pecquet observes, that it is not the same in all eyes, and that there are very different shades of it among the individuals of mankind, as also among birds, and some other animals, whose choroides is generally black; and that in the eyes of lions, camels, bears, oxen, 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. Marriotte replies, that there must be a great difference betwixt the state of that substance in living and dead subjects; and as a further 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.

M. Marriotte observes, in opposition to Pecquet's remark concerning the blood-vessels of the retina, 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 occupies no more than 1/8th part of it. Besides, if this were the cause of this want of vision, it would vanish gradually, and the space to which it is confined would not be so exactly terminated as it appears to be.

We must add, that M. Pecquet also observed, that notwithstanding the insensibility of the retina at the insertion of the optic nerve when the light is only moderate; yet luminous objects, such as a bright candle placed at the distance of four or five paces, do not absolutely disappear, in the same circumstances in which a white paper would; for this strong light may be perceived though the picture fall on the base of the nerve.

Dr Priestley, however, found that a large candle made no impression on that part of his eye, though 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.

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 choroid coat 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 was revived by an 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; whence he concluded, that the contraction of the iris is not produced by the action of the light. 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, in opposition to this argument of M. Mery, endeavours to show that fewer rays enter the eye under water, and that in those circumstances it is not so liable to be affected by them. Besides, it is obvious, 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 everything 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 the side of M. Marriotte in this controversy, it being peculiarly agreeable to his general hypothesis, viz. that the pia mater, of which the choroides is a production, and not the nerves themselves, is the proper instrument of sensation. He thought that the change which takes place in the eyes of old people (the choroides growing less black with age) favoured his hypothesis, as they do not see with the same distinctness as young persons. M. Le Cat supposed that the retina answers a purpose similar to that of the scarf-skin, covering the papilla pyramidales, which are the immediate organs 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 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 forms what Theory.

Of Vision, is called the choroides, or retina. He also showed that the sclerotic coat 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 contrary to the opinions of the celebrated Winslow, which he had advanced in his Traité de Sens.

To these arguments in favour of the choroides, we may add the following given by 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 interior surface of the retina, and the images of objects to be formed by direct rays, a considerable degree of confusion could not but arise from the light reflected by the choroides, in those animals in which it is white, or coloured. On the other hand, it would be impossible that vision should be performed at this place by light reflected from the choroides, because in many animals it is perfectly black; and yet such animals see even more distinctly than others.

If the seat of vision be at the farther surface of the retina, and if vision be performed by direct rays, a white choroid coat 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, perhaps, 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 on which it falls, 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, besides, peculiarly favourable to the opinion of Mariotte, that we can then see a sufficient reason for the diversity of its colour in different animals, according as they are circumstanced with respect to vision. In all terrestrial animals, which use 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 inclined to blue; and is therefore, probably, better adapted to see by the fainter coloured light, which chiefly prevails in the night; and we would add, is on the same account more liable to be strongly impressed by the colours to which they are chiefly exposed.

On the other hand, the choroides of birds in general, especially eagles, hawks, and other birds of prey, is black; by which means they are able to see with the greatest distinctness, but only in bright day light. The owl, however, seeking her food by night, has the choroides white like that of a cat. 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, and which makes both the retina and the choroides equally necessary to vision, and supposes it to be performed by the impression of light on the choroides communicated to the retina; Mr Michell observes, that the perceptions can hardly be supposed to be so acute, when the nerves 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 nerves, which are 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 Bernouilli, in the following manner. He placed a piece of money, O, 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 follow: 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. In order, therefore, that this space in which there is no vision may be as small as possible, it is necessary that the nerve should enter the eye perpendicularly, and that both this end, and also its entering the eye at a distance from its axis, are gained by the particular manner in which the two optic nerves. Of Vision. Nerves unite and become separate again, by crossing one another.

In support of one of the observations of Mr Michell, 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 day time. 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. Dr Priestley 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.

Dr Porterfield observes, that the reason why there is no vision at the entrance of the optic nerve into the eye, may be its want of that softness and delicacy which it has when it is expanded upon the choroides; and that, in those animals in which that nerve is inserted in the axis of the eye, it is observed to be equally delicate, and therefore probably equally sensible, in that place as in any other part of the retina. In general, the nerves, when embraced by their coats, have but little sensibility in comparison of what they are ended with when they are divested of them, and unfolded in a soft and pulpy substance.

Haller observes, that the choroides cannot be universally the seat of vision, because, 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, from the analogy of the senses, is much strengthened by considering that the retina is a large nervous apparatus, immediately exposed to the impression of light; whereas the choroides receives but a slender supply of nerves, in common with the sclerotica, the conjunctiva, and the eyelids, and that its nerves are much less exposed to the light than the naked fibres of the optic nerve.

That the optic nerve is of principal use in vision, is farther probable from several phenomena attending some of the diseases of the eye. 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, it was much harder, and cineritious. Morgagni 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.

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

The manner in which persons recover from an amaurosis, favours the supposition of the seat of vision being in the retina: since those parts which are the most distant from the insertion of the nerve, recover their sensibility the soonest, being in those places the most pulpy and soft; 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 in the axis of the eye, as when it was situated out of the axis. 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 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 to vision. 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 indistinct.—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 indistinct rays is collected into a focus exactly upon the retina; indistinct, 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 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 fibre of the iris, in order to take in a greater or smaller number of rays. 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.

In order that the rays be collected into points exactly upon the retina, that is, in order that objects may appear distinct, whether they be nearer or farther off, i.e., whether the rays proceeding from them diverge more or less The nature of this change has been a subject of great dispute among philosophers. While some have maintained, that the eye accommodates itself to different distances, by the muscular power of the ciliary ligament, which makes the crystalline lens approach to, or recede from, the retina; others are of opinion, that the form of the crystalline is altered by the ciliary ligament, or by the muscular power of the laminae of which it is composed. M. de la Hire supposes, that the eye is adapted to various distances by the contraction and dilatation of the pupil; and Dr Monro imagines, that its effect is produced by the pressure of the orbicular muscles upon the upper and under parts of the cornea, or by the action of the recti muscles, which elongate the axis of the eye, by pressing chiefly upon the sides of the eyeballs.—This subject has lately been accurately examined by Mr Ramsden, and Mr Everard Home, who found, that the adjustment of the eye is effected by three changes in the organ: 1. By an increase of curvature in the cornea, occasioned by the action of the recti muscles, which produces \( \frac{1}{2} \) of the effect. 2. By an elongation of the eyeball; and, 3. By a motion of the crystalline lens.

In those eyes where the cornea is very protuberant, and the rays of light suffer a 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, since the nearer an object is to the eye, the greater is its image, these people can see much smaller objects than others, as they see much nearer ones with the same distinctness; and their sight continues good longer than that of other people, because the cornea, as they grow old, becomes less protuberant, from the 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, if the rays diverge too much before they enter the eye, they cannot be brought to a focus when 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 \( abce \) (fig. 4), or crystalline humour \( cd \), 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 forms the image exactly 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 upon it; so that 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, and makes the rays unite at the retina, and form a distinct image of the object.

Such eyes as are of a proper convexity, cannot see any object distinctly at less distance than six inches; and there are numberless objects too small to be seen at that distance, because they cannot appear under any sensible angle.—Concerning the least angle under which any object is visible, there was a debate between Dr Hooke and Hevelius. The former asserted that no object could well be seen if it subtended an angle less than one minute; and, if the object be round as a black circular spot upon a white ground, or a white circle upon a black ground, it follows, from an experiment made by Dr Smith, that this is near the truth; and from this he calculates, that the diameter of the picture of such least visible point upon the retina is the 800th part of an inch; which he therefore calls a sensible point of the retina. On the other hand, Mr Courtivron found, by experiment, 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 he observes, that in order to our perceiving any impression 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 seconds, is not perceptible. Also a line of the same breadth with the circular spot will be visible at such a distance as the spot is not to be perceived at: because the quantity of impressions from the line is much greater than that 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 a 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, Dr Jurin 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 hole 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 Dr Jurin 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 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 so shifted to that part of the retina which was before possessed by Of Vision by the white line; and this must occasion such a dazzling 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 diameter, set 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 45 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 45 seconds, it does not follow that the eye must necessarily commit an error of 45 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. 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 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 perceptible will be much greater in proportion than the distance at which a space of equal breadth between two such objects ceases to be perceptible. 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 thus render it less perceptible; but the penumbra will add to the breadth of the single object, and will thereby make it more perceptible, 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 equal in breadth. In all these cases it made no difference whether the objects were placed in the shade or in the light of the sun; but when the degrees of light were small, their differences had a considerable effect, though by no means in proportion to the differences of the light. For if an object was illuminated to such a degree as to be just visible at the distance of nine feet, it would be visible at the distance of four feet, though the light was diminished above 160 times. It appeared in the course of these experiments, that common daylight is, at a medium, equal to that of 25 candles placed at the distance of one foot from the object.

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

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

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

This principle, however, has been thought sufficient to account for this appearance. Originally, every object, making two pictures, 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 has been thought verified by Mr Cheselden; 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 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 Forster, recited by Dr Smith and others; and thinks that the case of the young man crouched by Cheselden, who saw singly with both eyes immediately upon receiving his sight, is nearly decisive in proof of his supposition. He also found that three young gentlemen, whom he endeavoured to cure of squinting, saw objects singly, as soon as ever they were brought to direct the centres of both their eyes to the same object, though they had never been used to do so from their infancy; and he observes, that there are cases, in which, notwithstanding the fullest conviction of an object being single, no practice of looking at it will ever make it appear so, as when it is seen through a multiplying glass.

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

"The visible place of an object being composed of its visible distance and visible direction, to show how it may appear the same to both eyes, it will be necessary (says he*) to explain in what manner the distance and direction, which are perceived by one eye, may coincide with those which are perceived by the other."

With respect to visible distance, the author's opinion seems not to differ from that which we have stated elsewhere (see Metaphysics, No 49, 50); and therefore we have to attend only to what he says of visible direction.

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

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

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

He then proves by experiments, that objects situated in the common axis did not appear to be in that line, but in the axis of the eye by which they are not seen: that is, an object situated in the common axis appears to the right eye in the axis of the left, and vice versa. His next proposition, proved likewise by experiments, is, that "objects, situated in any line drawn through the mutual intersection of the optic axes to the visual base, do not appear to be in that line, but in another drawn through the same intersection, to a point in the visual base distant half this base from the similar extremity of the former line towards the left, if the objects be..." From these propositions he thus accounts for single vision by both eyes. "If the question be concerning an object at the concourse of the optic axes, it is seen single, because its two similar appearances, in regard to size, shape, and colour, are seen by both eyes in one and the same direction, or if you will, in two directions, which coincide with each other through the whole of their extent. It therefore matters not whether the distance be truly or falsely estimated; whether the object be thought to touch our eyes, or to be infinitely remote. And hence we have a reason, which no other theory or visible direction affords, why objects appeared single to the young gentleman mentioned by Mr Cheselden, immediately after his being couched, and before he could have learned to judge of distance by sight.

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

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

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 that the book was advanced considerably farther 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 determine the excess of this brightness; 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; and after a number of trials, 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 1/3 part.

He then proceeded to inquire, whether an object seen with both eyes appears larger than when seen with one; 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, however ingeniously it may be supported, that we see more distinctly with two eyes than with one; and that when a round object is near us, we see more of the surface in one case than in the other.

With respect to single vision with two eyes, Dr Hartley observes, that it deserves particular attention, that the optic nerves of men, and such other animals as look the same way with both eyes, unite in the cella turcica in a ganglion, or little brain, as one may call it, peculiar to themselves; and that the associations 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, Dr Smith 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 equidistant when seen separately by each eye; but when they are seen with both eyes, they appear much larger, and more distant also; and the objects seen through them 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 Theory.

Of Vision. 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. Epinus, 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 through it was more extended, whenever he shut his right eye; and both these effects were more remarkable when that eye was covered with his hand. He found considerable difficulty in measuring this augmentation of the 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 1\(\frac{3}{4}\) 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 cause of which he did not pretend to assign; but he observes, that it is wisely appointed by 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 eyelids not being impervious to the light. But the augmentation 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.

A great deal has been written by Gassendi, Le Clerc, Muschenbroek, and Du Tour, concerning the place to which we refer an object viewed by one or both eyes. But the most satisfactory account of this matter that we have met with, will be found in Dr Wells's Essay above quoted.

Sect. VI. Of the Appearance of Objects seen through Media of different Forms.

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

1. That as each point of an object, when viewed by the naked eye, appears in its proper place, and as that place is always to be found in the line in which the axis of a pencil of rays flowing from it enters the eye, or else in the line which Dr Wells calls the common axis; we hence acquire a habit of considering 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 through 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 asserted, that this is the precise distance; because the brightness, distinctness, and apparent magnitude of the object, on which its apparent distance in some measure depends, will also suffer an alteration by the refraction of the rays in passing through that medium.

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

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

5. And that, in some cases, the apparent brightness, distinctness, and magnitude of an object, are the only means by which 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 suppose it to be, unless where the contrary is expressed), appears nearer to the surface of the medium than it is.

Thus, if A (fig. 5.) be a point of an object placed within the medium BCDF, 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 direction 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 will appear, to an eye at H, nearer to the surface BC than it is.

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 this likewise it happens, that an object at the bottom of a vessel may be seen when the vessel is filled with water, though it be so placed with respect to the eye, that it cannot be seen when the vessel is empty. To explain this, let ABCD (fig. 6.) represent a vessel, and Fig. 6. let E be an object lying at the bottom of it. This object, when the vessel is empty, will not be seen by an In like manner, an object situated in the horizon appears above its true place, on account of the refraction of the rays which proceed from it in their passage through the atmosphere. 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 must appear above its proper place. Secondly, If the object be placed within the atmosphere, the case is still the same; for the rays which flow from it must continually enter a denser medium whose centre is below the eye; and therefore being refracted towards the centre, that is, downwards as before, those which enter the eye must necessarily proceed as from some point above the object; whence the object will appear above its proper place.

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.

Besides, 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: on which account the lower parts of them are apparently more elevated than the rest. 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 that they would do if the rays suffered no refraction, while their vertical ones are thus shortened.

**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 AB (fig. 7.) be the object, CDEF the medium, and GH the pupil of an eye, which is here drawn large to prevent confusion in the figure.—And, 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 c: 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 A B, the whole will be seen in the situation f g, nearer to the eye than the line AB.

2. As the rays RK, RL would not have entered the eye, but have passed by it in the directions KR, LR, had they not been refracted in passing through the medium, the object appears brighter. 3. The rays Ah, Bi, will be refracted at h and i into the less converging lines hk, il, and at the other surface into kM, lM, parallel to Ah and Bi produced; so that the extremities of the object will appear in the lines MK, ML produced, viz. in f and g, and under as large an angle fMG, as the angle AgB under which an eye at q would have seen it had there been no medium interposed to refract the rays: and therefore it appears larger to the eye GHI, 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 MF and MG, upon which account the object is made to appear less; and therefore the apparent magnitude of an object is not much augmented by being seen through a medium of this form.

Farther, it is apparent from the figure, that the effect of a medium of this form depends wholly upon its thickness; for the distance between the lines Kr and ec, and consequently the distance between the points e and R, depends upon the length of the line Ka:—Again, The distance between the lines AM and fM depends on the length of the line hk; but both Ka and kh 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. 8.) be the object, CD the lens, and EF the eye. 1. From A and B, the extreme points of the object, draw the lines AY, BX, crossing each other in the pupil of the eye; the angle AEB containing 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 concure 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; Theory.

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 rQ, rT, produced, and under the optic angle QrT, which is larger than A r B, and therefore the apparent magnitude of the object will be increased.—2. Let GHI be a pencil of rays flowing from the point G; as it is the property of this lens to render diverging rays less diverging, parallel, or converging, it is evident that some of those rays, which would proceed on to F and E, and miss the eye were they to suffer no refraction in passing through the lens, will now enter it; by which means the object will appear brighter. 3. The apparent distance of the object will vary according to the situation of it with respect to the focus of parallel rays of the lens. 1. Then, let us suppose the object placed so much nearer the lens than its focus of parallel rays, that the refracted rays KE and LF, though rendered less diverging by passing through it, may yet have a considerable degree of divergency, so that we may be able to form a judgment of the distance of the object thereby. In this case, the object ought to appear where EK, FL produced back concur; which, because they diverge less than the rays GH, GI, will be beyond G, that is, at a greater distance from the lens than the object is. But because both the brightness and magnitude of the object will at the same time be augmented, prejudice will not permit us to reckon 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 of by the direction of the rays KE and LF, yet on account of its brightness and magnitude 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, the rays flowing from it, 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 there will form an image of the object AB; and after crossing each other in the several points of it, as expressed in the figure, will pass on diverging as from a real object. Now if an eye be situated at c, where A c, B c, rays proceeding from the extreme points of the object, make not a much larger angle A c B, than they would do if no lens were interposed, and the rays belonging to the same pencil do not converge so much as those which the eye would receive if it were placed nearer to a or b, the object upon these accounts appearing very little larger or brighter than with the naked eye, is seen nearly in its proper place: but if the eye recede a little way towards a b, the object then appearing both brighter and larger seems to approach the lens: which is an evident proof of what has been so often asserted, viz. that we judge of the distance of an object in some measure by its brightness and magnitude; for the rays converge the more the farther the eye recedes from the lens; and therefore if we judged of the distance of the object by the direction of the rays which flow from it, we ought in this case to conceive it at a greater distance, than when the rays were parallel, or diverged at their entrance into the eye.

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

They imagined, that since 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 splendour.

Perhaps it may proceed from our judging of the distance of an object in some measure by its magnitude, that the 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 proper foci, the object appears inverted, and pendulous in the air, between the eye and the lens.

To explain this, let A B represent the object, C D the lens; and let the rays on the pencil ACD be collected circum- in a, and those of BCD in b, forming there an inverted image of the object AB, and let the eye be placed in F: it is apparent from the figure, that some of the refracted rays which pass through each point of the image appears 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 indistinct, viz. if at d, the rays which proceed from the same point of the object converge so very much, and if at e, they diverge so much, that they cannot be collected together upon the retina, but fall upon it as if they were the axes of so many distinct pencils coming through every point of the lens; wherefore little more than one single point of the object is seen at a time, and that appears all over the lens; whence nothing but indistinctness arises.

If the lens be so large that both eyes may be applied to it, as in h and k, the object will appear double; for it is evident from the figure, that the rays which enter the eye at h from either extremity of the object A or B, do not proceed as from the same point with that 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.

Forms

PROP. V.

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

Thus, let AB (fig. 10.) be the object, CD the pupil of an eye, and EF the lens. Now, as it is the property of a lens of this form to render diverging rays more so, and converging ones less so, the diverging rays GI, GL proceeding from the point G, will be made to diverge more, and so to enter the eye as from some nearer point g; and the rays AH, BI, which converge, will be made to converge less, and to enter the eye as from the points a and b; wherefore the objects will appear in the situation ag h, less and nearer than without the lens.

Further, As the rays which proceed from G are rendered more diverging, some of them will pass by the pupil of the eye, which otherwise would have entered it, and therefore each point of the object will appear less bright.

PROP. VI.

An object seen through a polygonal glass, that is, one which is terminated by several plain surfaces, is multiplied thereby.

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

SECT. VII. On 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 reflected from it; and it is by this reflected light that all bodies which have no light of their own become visible to us. Of that part of the ray which enters, another part is also reflected from the second surface, or that which is farthest from the luminous body. When this part arrives again at the first surface, part of it is reflected back from that surface; and thus it continues to be reflected between the two surfaces, and to pass backwards and forwards within the substance of the medium, till some part is totally extinguished and lost. Besides this inconsiderable quantity, however, which is lost in this manner, the second surface often reflects much more than the first; so that, in certain positions, scarcely any rays will pass through both sides of the medium. A very considerable quantity is also unaccountably lost at each reflecting surface; so that no body, however transparent, can transmit all the rays which fall upon it; neither, though it be ever so well fitted for reflection, will it reflect them all.

On the Cause of Reflection.

The reflection of light is not so easily accounted for as refraction. This last property may be accounted for in a satisfactory manner, by the supposition of an attractive power diffused throughout the medium, and extending a very little way beyond it; but with regard to the reflection of light, there seems to be no satisfactory hypothesis hitherto invented. Of the principal opinions on this subject Mr Rowning has given us the following account.

I. It was the opinion of philosophers, before Sir Isaac Newton discovered the contrary, that light is reflected not by impinging upon the solid parts of bodies. But that by this is not the case is evident from the following reasons, the solid parts of bodies cannot be regularly reflected, there should be no asperities or unevenness in the reflecting surface large enough face to bear a sensible proportion to the magnitude of a ray of light; because if the surface abound with these, the incident rays would be irregularly scattered rather 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 only to grind off the larger protuberances of the metal with the rough and sharp particles of emery, which must of necessity leave behind them an infinity of asperities and scratches, which, though inconsiderable with regard to the former roughnesses, and too minute to be discerned by us, must nevertheless bear a large proportion to, if not vastly exceed, the magnitude of the particles of light.

Secondly, It is not reflected at the second surface by nor at impinging against any solid particles. That it is not reflected by impinging upon the solid particles which constitute this second surface, is sufficiently obvious from the foregoing argument; the second surfaces of bodies being as incapable of a perfect polish as the first; and it is farther confirmed from this, viz. that the quantity of light reflected differs according to the different density of the medium behind the body. It is likewise not reflected by impinging upon the particles which constitute the surface of the medium behind it, because the strongest reflection at the second surface of a body, is when there is a vacuum behind it.

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

1. If there be a repulsive force diffused over the surface of bodies that repels the rays of light, 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.

2. As to the reflection at the second surface by the force supposed; Now the experiments of M. Bouguer show that bodies differ in their powers of thus separating light by reflection and refraction. It is not therefore a general property of light to be partly reflected and partly refracted, but a distinctive property of different bodies; and since we see that they possess it in different degrees, we are authorized to conclude that some bodies may want it altogether. We may therefore expect some success, by considering how bodies are affected by light, as well as how light is affected by bodies. Now, in all the phenomena of the material world we find bodies connected by mutual forces. We know no case where a body A tends towards a body B, or, in common language, is attracted by it, without, at the same time, the body B tending towards A. This is observed in the phenomena of magnetism, electricity, gravitation, corpuscular attraction, impulse, &c. We should therefore conclude from analogy, that as bodies change the motion of light, light also changes the motion of bodies; and that the particles near the surface are put into vibration by the passage of light through among them.

Suppose a parcel of cork balls all hanging as pendulums in a symmetrical order, and that an electrified ball passes through the midst of them; it is very easy ed. to show that it may proceed through this assemblage in various directions with a situated motion, and without touching any of them, and that its ultimate direction will have a certain inclination to its primary direction, depending on the outline of the assemblage, just as is observed in the motion of light; and, in the mean time, the cork balls will be variously agitated. Just so must it happen to the particles of a transparent body, if we suppose that they act on the particles of light by mutual attractions and repulsions.

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

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

Thus the objection stated in No. 164, is obviated, because the reflection and refraction are not here conceived as simultaneous, but as successive.

III. Some have supposed, that, by the action of light upon the surface of bodies, their parts are put into an undulatory motion; and that where the surface of it is subsiding light is transmitted, and in those places where it is rising light is reflected.

But to overlook the objections which we have just made to this theory of undulation, we have only to observe, that, were it admitted, it does not seem to advance us a step 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 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, from which it proceeds probably by some very subtle and elastic substance diffused through the universe, and that in the following manner. As bodies falling into water, or passing through the air, produce undulations in each, so the rays of light may excite vibrations in this elastic substance. The quickness of these 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 sounding 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 further supposes, that when light falls upon the surface of a body, if it be not in a fit of easy transmission, every ray is there put into one, so that when they come at the other side (for this elastic substance, pervading the pores of bodies, is capable of the same vibrations within the body as without it), the rays of one colour shall be in a fit of easy transmission, and those of another in a fit of easy reflection, according to the thickness of the body, the intervals of the fits being different in rays of a different kind. This seems to account for the different colours of the bubble and thin plate of air and water; and likewise for the reflection of light at the second surface of a thicker body; for the light thence reflected is also observed to be coloured, and to form rings according to the different thickness of the body, when not intermixed and confounded with other light, as will appear from the following experiment. If a piece of glass be ground concave on one side and convex on the other, both its concavity and convexity having one common centre; and if a ray of light be made to pass through a small hole in a piece of paper held in that common centre; and be permitted to fall on the glass; besides those rays which are regularly reflected back to the hole again, there will be others reflected to the paper, and form coloured rings surrounding the hole, not unlike those occasioned by the reflection of light from thin plates.

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

Of the Laws of Reflection.

The fundamental law of the reflection of light, is that the angle of reflection is always 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 impulse in bodies perfectly elastic. The axiom therefore holds good in every case of reflection, whether it be from plane or spherical surfaces; and hence the seven following propositions relating to the reflection of light from plane and spherical surfaces may be deduced.

I. Rays of light reflected from a plane surface have the same degree of inclination to one another that their respective incident ones have.—For the angle of reflection of each ray being equal to that of its respective incident one, it is evident, that each reflected ray will have the same degree of inclination to that portion of the surface from which it is reflected that its incident one has; but it is here supposed, that all those portions of surface from which the rays are reflected, are situated in the same plane; 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 face 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 a concave surface, are made Theory.

Laws of made to converge more.—For, every thing remaining as above, let GF, HB, be the incident rays. Now, because these rays have greater 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 concourse N farther from the point C than M, that to which the parallel rays AF and EB converged to in the foregoing case; and their precise degree of convergency will be greater than that wherein they converged before reflection.

IV. Diverging rays falling upon a concave 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, to some point on the contrary side of the centre, but situated farther from it than the radiant point. 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 radiant point; and if they diverge from the centre, they will be reflected thither again.

1. Let them diverge in the lines MF, MB, proceeding from the radiant point 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 shown to be reflected into them in the foregoing proposition; but the degree of their divergency will be less than their divergency before reflection.

3. Let them diverge 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 therefore proceed 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 rays diverge from Y, they will, after reflection, converge to X; those which were the incident rays in the former case being the reflected ones in this.

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 convex surfaces are rendered diverging.—For, let AB, GD, EF, be three parallel rays falling upon the convex surface BF, whose centre 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, 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 will return in the lines BK and FI, 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 convex surfaces are rendered more diverging.—For, things remaining as above, let GB, GF, be the incident rays. These having greater angles of incidence than the parallel ones AB and EF in the preceding case, their angles of reflection will also be greater; 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 of their divergency will exceed their divergency before reflection.

VII. Converging rays reflected from convex surfaces 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, 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 to which they converged; 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 seem to proceed from it.

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 FC, in which the rays GB, GF proceeded that were shown to be reflected into them by the last proposition: but the degree of their convergency will exceed their convergency 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. If the incident rays converge towards the centre, they coincide with their respective perpendiculars; and will therefore proceed after reflection as from that 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 this surface, is that which requires that the refracted angle (supposing the ray to pass out there) should be equal to or greater than... Of the reflection of rays from a plane surface.

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

This proposition admits of two cases.

Case 1. Of diverging rays.

Let AB, AC be two diverging rays incident on the plane 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, so 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 point 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 also equal: but the angle BCG is equal to its vertical angle FCE: therefore in the triangle ABC and GBC the angles at C are equal, the side BC common, and the right angles at B are equal; therefore AB = BG: 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.

Case 2. Of converging rays.

This is the converse of the former case. For supposing FC and AB to be two converging incident rays, CA and BA will be the reflected ones (the angles of incidence in the former case being now the angles of reflection, and vice versa), having the point A for their focus; but this 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.

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

The case of parallel rays incident on a plane surface is included in this proposition: for in that case we are to suppose the radiant point infinitely distant 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 it.

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.

Let AB, DH, 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 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 IHE, being the angles of incidence and reflection, will be equal. But HCF = DHI, the lines AC and DH being parallel; and CHF = IHE, wherefore the triangle CFH is isosceles, and consequently CF = FH: but supposing BH to vanish, FH = FB; and therefore upon this supposition FC = FB; that is, the focus of the reflected rays is the middle point between the centre of convexity and the surface.

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

Let AB, DH, 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 = FHC, being the angles of incidence and reflection; but HCF = DHC its alternate angle, and therefore the triangle CFH is isosceles. Wherefore CF = FH: but if we suppose BH to vanish, FB = FH, and therefore... Theory.

Laws of therefore CF = FB; that is, the focal distance of the reflected rays is the middle point between the centre and the surface.

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 greater the triangle CFH will become; and consequently, since it is always isosceles, 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 ray 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.

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

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

Had the surface BH in fig. 4 or 5 been formed by the revolution of a parabola about its axis having its 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.

Let RB, RD, represent two diverging rays flowing Fig. 7. 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 RHD = EDH the angle of reflection, and RHD = RDH the angle of incidence; therefore the triangle DRH is isosceles, and DR = RH.

Now the lines FD and RH being parallel, the triangles FDC and RHC are similar, or the sides are cut proportionally, and therefore FD : RH or RD = CF : CR; but BD vanishing, FD and RD differ not from FB and RB; wherefore FB : RB = CF : 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 point from it.

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

Let KD and CB be the converging incident rays, having their imaginary focus in the point R, which was the radiant point 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, the angles of reflection will be equal also; so that F will be the focus of the reflected rays: but it was there demonstrated, that FB : RB = CF : CR; that is, the distance of the focus from the surface is to the distance (in this case) of the imaginary focus of the incident rays, as the distance of the focus from the centre is to the distance of the imaginary focus of the incident rays from the same.

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

Let B represent a convex surface whose centre is C, Fig. 5, 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. Through D draw the perpendicular CD, and produce it to H; then will KDH = HDE, being the angles of incidence and reflection, and consequently RDC = CDF too. Therefore the triangle RDF is bisected by the line DC: wherefore (3 El. 6.) FD and DR, or BD vanishing, FB : BR = FC : 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 centre.

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

Let RB, RD, be the diverging rays incident upon Fig. 5. the concave surface BD, having their radiant point in R, the imaginary focus of the incident rays in the preceding case. Then as KD was in that case reflected into DE, RD will now be reflected into DF. But we had FB : RB = CF : 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 centre.

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 preceding; for in each of them the incident rays are supposed to coincide with the reflected ones in the other. Or they may be thus demonstrated independently of them.

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

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

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

Let FD and BF be two rays diverging from the point F, which was the imaginary focus of the incident rays in the preceding case. Then as ED was in that case reflected into DR, FD will be reflected into DK (for the reason mentioned in case 2.), so that the reflected ray will proceed as from the point R; but it was demonstrated in case 5, that RB : FB = CR : CF; that is, the distance of the focus from the surface is to that of the radiant from the surface, as the distance of the former from the centre is to that of the latter from the centre.

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

Let AB, ED 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; see Prop. vii. Through D draw the perpendicular CD, and produce it to H. Then will EDH = HDK, being the angles of incidence and reflection; but CDF = CDR, being their verticals; consequently the angle FDR is bisected by the line CD; wherefore RD : DF, or (3 Elem. 6.) BD vanishing, RB : BF = RC : 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 centre.

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

Let FB, FD be the incident rays radiating from F, the imaginary focus of the incident rays in the 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 case 7, that RB : FB = RC : CF; that is, the distance of the focus of the reflected rays from the surface is to the distance of the radiant from the surface, as the distance of the focus of the reflected rays from the centre is to the distance of the radiant from the centre.

The two remaining cases may be considered as the converse of those under Prop. ii. (p. 234.), because the incident rays in these are the reflected ones in them; or they may be demonstrated in the same manner with the preceding, as follows.

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

Let ED, RB represent two converging rays incident on the convex surface BD, and tending towards F, which we shall 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 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 : FB = RC : CF; but F in this case being supposed to be the focus of parallel rays, it is the middle point between C and R (by Prop. ii.), and therefore FB = FC, consequently RB = RC; which can only be upon the supposition that R is at an infinite distance from B; that is, that the reflected rays BR and DR be parallel.

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

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

It may here be observed that in the case of diverging rays falling upon a convex surface, 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 point R remains the same. For it is evident from the curvature of a circle, that the point D 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 appear if we draw several incident rays with their respective reflected ones, in such a manner that the angles of reflection may be 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 and converging rays incident upon a spherical surface. This is the reason, that, when rays are considered as reflected from a spherical surface, Theory.

From this 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 will not proceed after reflection as from any point in the line RB produced, but as from a curve line passing through the several points F, f, f, &c.

Had the curve BD been a hyperbola, 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 been an ellipse having its foci in F and R, the one of these being made the radiant (or imaginary focus of incident rays), the other would have been the focus of reflected ones, and vice versa. For both in the hyperbola and ellipse, lines drawn from each of their foci through any point make equal angles with the tangent to that point. Therefore, if the incident rays proceed to or from one of their foci, the reflected ones will all proceed as from or to the other focus. Therefore, 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 of the surface, and the focus of the reflected ones on the other; when they are both to fall on the same side, it must be formed by the revolution of an ellipse about its longer axis. However, as spherical surfaces are more easily formed, than those which are generated by the revolution of any of the conic sections about their axes, the latter are very rarely used.

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 example whereby the method of doing it in all others will readily appear.

Problem.

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

Call \( x \) the focal distance sought; 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. iii. we have the following proportion, \( x : 20 = 5 - x : 25 \); and multiplying extremes together and means together, we have \( 25x = 100 - 20x \), or \( x = \frac{100}{45} \).

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

Because it was observed in the preceding section, that different incident rays, though tending to or from one point, would after refraction proceed to or from different points, a method was there given 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 case 2. and of Bodies case 10.) But the method of determining the distinct point to or from which any 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.

Sect. VIII. Of the Appearance of Bodies seen by Light reflected from Plane and Spherical Surfaces.

Whatever has been said concerning the appearance of bodies seen through lenses, by refracted light, 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 a right line that passes through its correspondent point in the object, and is perpendicular to the reflecting surface.

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

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, do not proceed after reflection as from any point in that perpendicular, but as from other points situated in a certain curve, on which account these rays are neglected, as making an indistinct 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.

1. 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 before it, of the same magnitude, and directly opposite to it.

To explain this, let AB represent an object seen by surfaces, reflection from the plane surface SV; and let the rays Fig. 10. 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... Appearance 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 from the surface that the point itself is before it, and directly opposite to it: consequently, since the like may be shown of any other point B, 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, perpendicular to the plain surface, are parallel to each other, the image will also be of the same magnitude with the object.

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 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 from it, 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 any other B, 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 concursing 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 principal focus, the image falls on the opposite side of the surface, is more distant from it, and larger than the object.

Thus let AB be the object, SV the reflecting surface, F the principal focus, and C its centre. Through A and B, the extremities of the object, draw the lines CE, CR, which will be perpendicular to the surface; and let the rays AR, AG, be incident upon such points of it that they shall be reflected into an eye at H. Now, because the radiant points A and B are nearer the surface than the principal focus F, the reflected rays will diverge, and therefore proceed as from some points on the opposite side of the surface; which points, by the observation laid down at the beginning of this section, will be in the perpendiculars AE, BR, produced, suppose in I and M: but they will diverge in a less degree than their incident ones; 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 principal focus, 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 appearance, the reasons given in the foregoing case, being large and of Bodin distinct, we do not reckon it much farther from the surface than the image.

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

Thus let AB be the object, SV the reflecting surface, F its principal focus, and C its centre. Through A and B, 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 principal focus, 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; whence it is evident, that the image IM is larger than the object AB, that it falls on the contrary side of the centre, and that their positions are inverted with respect to each other.

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

This proposition is the converse of the preceding; 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 requires little explication; for the middle of the object being in the centre, rays flowing from it will fall perpendicularly upon the surface, and therefore necessarily return thither again; so that the middle of the image will be coincident with the middle of the object. But that the image should be inverted is perhaps not so clear. To explain this, let AB be the object, having its middle point C in the centre of the reflecting surface from 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 angle ACR = CRB; 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 ACR and BRC will not be equal, and part of the image only will therefore fall upon the object. 6. If in any of the three last cases, in each of which the image is formed on the same side of the reflecting surface with the object, the eye be situated farther from the surface than the place where the image falls, the rays of each pencil, crossing each other in the several points of the image, will enter the eye as from a real object situated there; so that the image will appear pendulous in the air between the eye and the reflecting surface, and in the position wherein it is formed, viz. inverted with respect to the object, in the same manner that an image formed by refracted light appears to an eye placed beyond it; which was fully explained under Prop. iv. and therefore needs not be repeated.

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

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

To explain this, let AB 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, Ka; 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 K a and L b, 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 m n; and so of the rest. Now, if the same pupil be removed into the situation ef, the reflected rays E e and G f will then enter the eye, and therefore one extremity of the object will appear to cover the space XY; and because the rays O f and L e will also enter it in their progress towards M, the point B, from which they came, will appear to cover ZV; the object therefore will appear larger and more confused than before. 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. 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) 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 minute of the image of the same object) and the image subtend equal angles at the eye.

Here we must suppose the pupil of the eye to be a point only, because the magnitude of it causes a small change alteration in the apparent magnitude of the object. Let face, 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 Ha Ka = Ma I, the angle 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 Gf O = If M, which the image subtends at the same place, and therefore the apparent object and image of it subtend equal angles at the eye.

Now if we suppose the pupil to have any sensible magnitude ab; then the object seen by the eye in that situation will appear under the angle HXL, which is larger than the angle Ha 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 shown at the end of this section); therefore, when the eye is nearer to the image than that point, the image will subtend a larger angle at it than the object does; and consequently, since the image and apparent object subtend equal angles at the eye, the apparent object must necessarily be seen under a larger angle than the object itself, wherever the eye be placed, between the surface and the image.

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

Though the extremities of the object AB and the centre C (fig. 17, 18, 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 appearance of Bo. FA = FB, and FG is perpendicular to the reflecting surfaces seen by face; and therefore the representation of the point A Reflection 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. Hence the proposition is obvious.

If it happens that the lines will not cross which way soever 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 the 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 this 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 = AGB, the angles which the image subtends at G the vertex of the reflecting surface, and 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.

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.

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 through transparent media, whose surfaces are properly disposed, so they may also by reflecting surfaces.

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

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

Sect. IX. Of the apparent Place, Distance, Magnitude, and Motion of Objects.

It had in general been taken for granted, that the place to which the eye refers any visible object seen by reflection or refraction, is that in which the visual rays meet a perpendicular from the object upon place, &c. of objects seen by the reflecting or refracting plane. But this method of judging of the place of objects was called in question by Dr Barrow, who contended that the arguments brought in favour of the opinion were not conclusive. These arguments are, that the images of objects appeared straight in a plane mirror, but curved rows thrown in a convex or concave one: that a straight thread, when partly immersed perpendicularly in water, does not appear crooked as when it is obliquely plunged in place of the fluid; but that which is within the water seems objects to be a continuation of that which is without. With respect to the reflected image, however, of a perpendicular right line from a convex to a concave mirror, he says, that it is not easy for the eye to distinguish the curve that it really makes; and that if the appearance of a perpendicular thread, part of which is immersed in water, be closely attended to, it will not favour the common hypothesis. If the thread is of any shining metal, as silver, and viewed obliquely, the image of the part immersed will appear to detach itself sensibly from that part which is without the water, so that it cannot be true that every object appears to be in the same place where the refracted ray meets the perpendicular; and the same observation, he thinks, may be extended to the case of reflection. According to Dr Barrow, we refer every point of an object to the place from which the pencils of light, that give us the image of it, issue, or from which they would have issued if no reflecting or refracting substance intervened. Pursuing this principle, he proceeds to investigate the place in which the rays issuing from each of the points of an object, and which reach the eye after one reflection or refraction, meet; and he found, that if the refracting surface was plane, and the refraction was made from a denser medium into a rarer, those rays would always meet in a place between the eye and a perpendicular to the point of incidence. If a convex mirror be used, the case will be the same; but if the mirror be plane, the rays will meet in the perpendicular, and beyond it if it be concave. He also determined, according to these principles, what form the image of a right line will take, when it is presented in different manners to a spherical mirror, or when it is seen through a refracting medium.

Though Dr Barrow reckoned the maxim which he endeavoured to establish, concerning the supposed place of visible objects, highly probable, he has the candour to mention an objection to it, of which he was not able to give a satisfactory solution. It is this. Let an object be placed beyond the focus of a convex lens; and if the eye be close to the lens, it will appear confused, but very near to its true place. If the eye be a little withdrawn, the confusion will increase, and the object will seem to come nearer; and when the eye is near the focus, the confusion will be exceedingly great, and the object will seem to be close to the eye. But in this experiment the eye receives no rays but those that are converging; and the point from which they issue is so far from being nearer than the object, that it is beyond it; notwithstanding which, the object is conceived to be much nearer than it is, though no very distinct idea can be formed of its precise distance. It may be observed, that in reality, the rays falling upon the eye in this case There are few persons, M. de la Hire remarks, who have both their eyes exactly equal, not only with respect to the limits of distinct vision, but also with regard to the colour with which objects appear tinged when they are viewed by them, especially if one of the eyes has been exposed to the impression of a strong light. To compare them together in this respect, he directs us to take two thin cards, and to make in each of them a round hole of a third or a fourth of a line in diameter, and, applying one of them to each of the eyes, to look through the holes on a white paper, equally illuminated, when a circle of the paper will appear to each of the eyes, and, placing the cards properly, these two circles may be made to touch one another, and thereby the appearance of the same object to each of the eyes may be compared to the greatest advantage. To make this experiment with exactness, it is necessary, he says, that the eyes be kept shut some time before the cards be applied to them.

By the following calculation, M. de la Hire gives us an idea of the extreme sensibility of the optic nerves. One may see very easily, at the distance of 4000 toises, the sail of a windmill; 6 feet in diameter; and the eye being supposed to be an inch in diameter, the picture of this sail, at the bottom of the eye, will be $\frac{1}{350}$ of an inch, which is less than the 666th part of a line, and is about the 666th part of a common hair, or the 8th part of a single thread of silk. So small, therefore, must one of the fibres of the optic nerve be, which, he says, is almost inconceivable, since each of these fibres is a tube that contains spirits.

The person who particularly noticed Dr Barrow's Berkeley's hypothesis was the ingenious Dr Berkeley, bishop of Cloyne, who distinguished himself so much by the objections which he started to the reality of a material world, and by his opposition to the Newtonian doctrine of fluxions. In his Essay towards a new Theory of Vision, he observes, that the circle formed upon the retina, by the rays which do not come to a focus, produce the same confusion in the eye, whether they cross one another before they reach the retina, or tend to do it afterwards; and therefore that the judgment concerning distance will be the same in both the cases, without any regard to the place from which the rays originally issued; so that in this case, as, by receding from the lens, the confusion, which always accompanies the nearness of an object, increases, the mind will judge that the object comes nearer.

But, says Dr Smith, if this be true, the object ought always to appear at a less distance from the eye than that at which objects are seen distinctly, which is not the case: and to explain this appearance, as well as every other in which a judgment is formed concerning distance, he maintains, that we judge of it chiefly if not only by the apparent magnitude of objects, so that, since the image grows larger as we recede from the lens through which it is viewed, we conceive the object to come nearer. He also endeavours to show, that in all cases in which glasses are used, we judge of distance by the same rule; from which he concludes, that the apparent distance of an object seen in a glass is to its apparent distance seen by the naked eye, as the apparent magnitude in the naked eye is to its apparent magnitude in the glass.

But that we do not judge of distance merely by the angle... angle under which objects are seen, is an observation place, &c., as old as Alhazen, who mentions several instances, in which, though the angles under which objects appear be different, the magnitudes are universally and instantaneously deemed not to be so. Mr Robins clearly shows the hypothesis of Dr Smith to be contrary to fact in the most common and simple cases. In microscopes, he says, it is impossible that the eye should judge the object to be nearer than the distance at which it has viewed the object itself, in proportion to the degree of magnifying. For when the microscope magnifies much, this rule would place the image at a distance, of which the sight cannot possibly form any opinion, as being an interval from the eye at which no object can be seen. In general, he says, he believes, that whoever looks at an object through a convex glass, and then at the object itself without the glass, will find it to appear nearer in the latter case, though it be magnified in the glass; and in the same trial with the concave glass, though by the glass the object be diminished, it will appear nearer through the glass than without it.

But the following experiment is the most convincing proof that the apparent distance of the image is not determined by its apparent magnitude. If a double convex glass be held upright before some luminous object, as a candle, there will be seen two images, one erect, and the other inverted. The first is made simply by reflection from the nearest surface, the second by reflection from the farther surface, the rays undergoing a refraction from the first surface both before and after the reflection. If this glass has not too short a focal distance when it is held near the object, the inverted image will appear larger than the other, and also nearer; but if the glass be carried off from the object, though the eye remain as near to it as before, the inverted image will diminish so much faster than the other, that, at length, it will appear very much less than it, but still nearer.

Here, says Mr Robins, two images of the same object are seen under one view, and their apparent distances, when immediately compared, seem to have no necessary connexion with the apparent magnitude. He also shows how this experiment may be made still more convincing, by sticking a piece of paper on the middle of the lens, and viewing it through a short tube.

M. Bouguer adopts the general maxim of Dr Barrow, in supposing that we refer objects to the place from which the pencils of rays seemingly converge at their entrance into the pupil. But when rays issue from below the surface of a vessel of water, or any other refracting medium, he finds that there are always two different places of this seeming convergence; one of them of the rays that issue from it in the same vertical circle, and therefore fall with different degrees of obliquity upon the surface of the refracting medium; and another, of those that fall upon the surface with the same degree of obliquity, entering the eye laterally with respect to one another. Sometimes, he says, one of these images is attended to by the mind, and sometimes the other, and different images may be observed by different persons. An object immersed in water affords an example, he says, of this duplicity of images.

If BA b be part of the surface of water, and the object be at O, there will be two images of it in two different places; one at G, on the caustic by refraction, and the other at E, in the perpendicular AO, which is as much a caustic as the other line. The former image place, &c., is visible by the rays ODM, O d m, which are one higher than the other, in their progress to the eye; whereas the image at E is made by the rays ODM, O e f, which enter the eye laterally. This, says he, may serve to explain the difficulty of Father Tacquet, Barrow, Smith, and many other authors.

G. W. Kraft has ably supported the opinion of Dr Barrow, that the place of any point, seen by reflection from the surface of any medium, is that in which rays issuing from it, infinitely near to one another, would meet; and considering the case of a distant object, viewed in a concave mirror, by an eye very near to it, when the image, according to Euclid and other writers, would be between the eye and the object, and the rule of Dr Barrow cannot be applied; he says that in this case the speculum may be considered as a plane, the effect being the same, only the image is more obscure.

Dr Porterfield gives a distinct view of the natural methods of judging concerning the distance of objects.

The conformation of the eye, he observes, can be of no use to us with respect to objects placed without the limits of distinct vision. As the object, however, does then appear more or less confused, according as it is more or less removed from those limits, this confusion assists the mind in judging of the distance of the object; it being always estimated so much the nearer, or the farther off, as the confusion is greater. But this confusion hath its limits; for when an object is placed at a certain distance from the eye, to which the breadth of the pupil bears no sensible proportion, the rays of light that come from a point in the object, and pass the pupil, are so little diverging, that they may be considered as parallel. For a picture on the retina will not be sensibly more confused, though the object be removed to a much greater distance.

The most general, and frequently the most certain means of judging of the distance of objects is, he says, by the angle made by the optic axis. For our two eyes are like two different stations, by the assistance of which distances are taken; and this is the reason why those persons who are blind of one eye, so frequently miss their marks in pouring liquor into a glass, snuffing a candle, and such other actions as require that the distance be exactly distinguished. To be convinced of the utility of this method of judging of the distance of objects, he directs us to suspend a ring in a thread, so that its side may be towards us, and the hole in it to the right and left hand; and taking a small rod, crooked at the end, retire from the ring two or three paces and having with one hand covered one of our eyes, to endeavour with the other to pass the crooked end of the rod through the ring. This, says he, appears very easy; and yet, upon trial, perhaps once in 100 times we shall not succeed, especially if we move the rod a little quickly.

The use of this second method of judging of distances Dechales limited to 120 feet; beyond which, he says, we are not sensible of any difference in the angle of the optic axis.

A third method of judging of the distance of objects, consists in their apparent magnitudes, on which so much stress was laid by Dr Smith. From this change in the magnitude Theory.

Hence we may see why we are so frequently deceived in our estimates of distance, by any extraordinary magnitudes of objects seen at the end of it; as, in travelling towards a large city, or a castle, or a cathedral church, or a mountain larger than common, we fancy them to be nearer than they really are. This also is the reason why animals, and little objects, seen in valleys, contiguous to large mountains, appear exceedingly small. For we think the mountain nearer to us than if it were smaller; and we should not be surprised at the smallness of the neighbouring animals, if we thought them farther off. For the same reason, we think them exceedingly small, when they are placed upon the top of a mountain, or a large building; which appear nearer to us than they really are, on account of their extraordinary size.

Dr Jurin accounts for our imagining objects, when seen from a high building, to be smaller than they are, and smaller than we fancy them to be when we view them at the same distance on level ground. It is, says he, because we have no distinct idea of distance in that direction, and therefore judge of things by their pictures upon the eye only; but custom will enable us to judge rightly even in this case.

Let a boy, says he, who has never been upon any high building, go to the top of a lofty spire, and look down into the street; the objects seen there, as men and horses, will appear so small as greatly to surprise him. But 10 or 20 years after, if in the mean time he has used himself now and then to look down from that and other great heights, he will no longer find the same objects to appear so small. And if he were to view the same objects from such heights as frequently as he sees them upon the same level with himself in the streets, he supposes that they would appear to him just of the same magnitude from the top of the spire, as they do from a window one story high. For this reason it is, that statues placed upon very high buildings ought to be made of a larger size than those which are seen at a nearer distance; because all persons, except architects, are apt to imagine the height of such buildings to be much less than it really is.

The fourth method by which Dr Porterfield says that we judge of the distance of objects, is the force with which their colour strikes upon our eyes. For if we be assured that two objects are of a similar and like colour, and that one appears more bright and lively than the other, we judge that the brighter object is the nearer of the two.

The fifth method consists in the different appearance of the small parts of objects. When these parts appear distinct, we judge that the object is near; but when they appear confused, or when they do not appear at all, we reckon the object to be at a greater distance. For the image of any object, or part of an object, diminishes as its distance increases.

The sixth and last method by which we judge of the distance of objects is, that the eye does not represent to our mind one object alone, but at the same time all those that are placed betwixt us and the principal object, whose distance we are considering; and the more this distance is divided into separate and distinct parts, the greater it appears to be. For this reason, distances upon uneven surfaces appear less than upon a plane; for the inequalities of the surfaces, such as hills, and holes, and rivers, that lie low and out of sight, either do not appear, or hinder the parts that lie behind them from appearing; and so the whole apparent distance is diminished by the parts that do not appear in it. This is the reason that the banks of a river appear contiguous to a distant eye, when the river is low and not seen.

Dr Porterfield very well explains several fallacies in vision which depend upon our mistaking the distances of objects. Of this kind, he says, is the appearance of parallel lines, and long vistas consisting of parallel rows of trees; for they seem to converge more and more as they are farther extended from the eye. The reason of this, he says, is because the apparent magnitudes of their perpendicular intervals are perpetually diminishing, while, at the same time, we mistake their distance. Hence we may see why, when two parallel rows of trees stand upon an ascent, whereby the more remote parts appear farther off than they really are, because the line that measures the length of the vistas now appears under a greater angle than when it was horizontal, the trees, in such a case, will seem to converge less, and sometimes, instead of converging, they will be thought to diverge.

For the same reason that a long vista appears to converge more and more the farther it is extended from the eye, the remoter parts of a horizontal walk or a long floor will appear to ascend gradually; and objects placed upon it, the more remote they are the higher they will appear, till the last be seen on a level with the eye; whereas the ceiling of a long gallery appears to descend towards a horizontal line, drawn from the eye of the spectator. For this reason, also, the surface of the sea, seen from an eminence, seems to rise higher and higher the farther we look; and the upper parts of high buildings seem to stoop, or incline forwards over the eye below, because they seem to approach towards a vertical line proceeding from the spectator's eye; so that statues on the top of such buildings, in order to appear upright, must recline, or bend backwards.

Dr Porterfield also shows the reason why a windmill, seen from a great distance, is sometimes imagined to move the contrary way from what it really does, by our taking the nearer end of the sail for the more remote. The uncertainty we sometimes find in the course of the motion of a branch of lighted candles, turned round at a distance, is owing, he says, to the same cause; as also our sometimes mistaking a convex for a concave surface, more especially in viewing seals and impressions with a convex glass or a double microscope; and lastly, that, upon coming in a dark night into a street, in which there is but one row of lamps, we often mistake the side of the street they are on.

Far more light was thrown upon this curious subject by M. Bouguer.

The proper method of drawing the appearance of two rows of trees that shall appear parallel to the eye, place, &c. is a problem which has exercised the ingenuity of several philosophers and mathematicians. That the apparent magnitude of objects decreases with the angle under which they are seen, has always been acknowledged. It is also acknowledged, that it is only by custom and experience that we learn to form a judgment both of magnitudes and distances. But in the application of these maxims to the above-mentioned problem, all persons, before M. Bouguer, made use of the real distance instead of the apparent one; by which only the mind can form its judgment. And it is manifest, that, if any circumstances contribute to make the distance appear otherwise than it is in reality, the apparent magnitude of the object will be affected by it; for the same reason, that, if the magnitude be misapprehended, the idea of the distance will vary.

For want of attending to this distinction, Tacquet pretended to demonstrate, that nothing can give the idea of two parallel lines (rows of trees for instance) to an eye situated at one of their extremities, but two hyperbolical curves, turned the contrary way; and M. Varignon maintained, that in order to make a vista appear of the same width, it must be made narrow, instead of wider, as it recedes from the eye.

M. Bouguer observes, that very great distances, and those that are considerably less than they, make nearly the same impression upon the eye. We, therefore, always imagine great distances to be less than they are; and for this reason the ground plan of a long vista always appears to rise. The visual rays come in a determinate direction; but as we imagine that they terminate sooner than they do, we necessarily conceive that the place from which they issue is elevated. Every large plane, therefore, as A B, viewed by an eye at O, will seem to lie in such a direction as A b; and consequently lines, in order to appear truly parallel on the plane A B, must be drawn so as that they would appear parallel on the plane A d, and be from thence projected to the plane A B.

To determine the inclination of the apparent ground-plan A d to the true ground plan A B, our ingenious author directs us to draw upon a piece of level ground two straight lines of a sufficient length (for which purpose lines fastened to small sticks are very convenient), making an angle of 3 or 4 degrees with one another. Then a person, placing himself within the angle, with his back towards the angular point, must walk backwards and forwards till he can fancy the lines to be parallel. In this situation, a line drawn from the point of the angle through the place of his eye, will contain the same angle with the true ground-plan which this does with the apparent one.

M. Bouguer then shows other more geometrical methods of determining this inclination; and says; that by these means he has often found it to be 4 or 5 degrees, though sometimes only 2 or 2½ degrees. The determination of this angle, he observes, is variable; depending upon the manner in which the ground is illuminated and the intensity of the light. The colour of the soil is also not without its influence, as well as the particular conformation of the eye, by which it is more or less affected by the same degree of light, and also the part of the eye on which the object is painted. When, by a slight motion of his head, he contrived, that certain parts of the soil, the image of which fell towards the bottom of his eye, should fall towards the top of the retina, he always thought that this apparent inclination became a little greater.

But what is very remarkable, is, that if he look towards a rising ground, the difference between the apparent ground-plan and the true one will be much more considerable, so that they will sometimes make an angle of 25 or 30 degrees. Of this he had made frequent observations. Mountains, he says, begin to be inaccessible when their sides make an angle from 35 or 37 degrees with the horizon, as then it is not possible to climb them but by means of stones or shrubs, to serve as steps to fix the feet on. In these cases, both he and his companions always agreed that the apparent inclination of the side of the mountain was 60 or 70 degrees.

These deceptions are represented in fig. 3, in which, when the ground-plan A M, or A N, is much inclined, the apparent ground-plan A m, or A n, makes a very large angle with it. On the contrary, if the ground dips below the level, the inclination of the apparent to the true ground-plan diminishes, till, at a certain degree of the slope, it becomes nothing at all; the two plans A P and A p being the same, so that parallel lines drawn upon them would always appear so. If the inclination below the horizon is carried beyond the situation A P, the error will increase; and what is very remarkable, it will be on the contrary side; the apparent plan A r being always below the true plan A R, so that if a person would draw upon the plan A R lines that shall appear parallel to the eye, they must be drawn converging, and not diverging, as is usual on the level ground; because they must be the projections of two lines imagined to be parallel, on the plan A r, which is more inclined to the horizon than A R.

These remarks, he observes, are applicable to different planes exposed to the eye at the same time. For if B H, fig. 4, be the front of a building, at the distance of A B from the eye, it will be reduced in appearance to the distance A b; and the front of the building will be b h, rather inclined towards the spectator, unless the distance be inconsiderable.

After making a great number of observations upon this subject, our author concludes, that when a man stands upon a level plane, it does not seem to rise sensibly but at some distance from him. The apparent plane, therefore, has a curvature in it, at that distance, the form of which is not very easy to determine; so that a man standing upon a level plane, of infinite extent, will imagine that he stands in the centre of a basin. This is also, in some measure, the case with a person standing upon the level of the sea.

He concludes with observing, that there is no difficulty in drawing lines according to these rules, so as to have any given effect upon the eye, except when some parts of the prospect are very near the spectator, and others very distant from him, because, in this case, regard must be had to the conical or conoidal figure of a surface. A right line passing at a small distance from the observer, and below the level of his eye, in that case almost always appears sensibly curved at a certain distance from the eye; and almost all figures in this case are subject to some complicated optical alteration to which the rules of perspective have not as yet been extended. If a circle be drawn near our feet, and within Theory.

Apparent that part of the ground which appears level to us, it place, &c. will always appear to be a circle, and at a very considerable distance it will appear an ellipse; but between these two situations, it will not appear to be either the one or the other, but will be like one of those ovals of Descartes, which is more curved on one of its sides than the other.

On these principles a parterre, which appears distorted when it is seen in a low situation, appears perfectly regular when it is viewed from a balcony or any other eminence. Still, however, the apparent irregularity takes place at a greater distance, while the part that is near the spectator is exempt from it. If AB, fig. 5, be the ground-plan, and A a be a perpendicular, under the eye, the higher it is situated, at O, to the greater distance will T, the place at which the plane begins to have an apparent ascent along T b, be removed.

All the varieties that can occur with respect to the visible motion of objects, are thus succinctly summed up by Dr Porterfield under eleven heads.

1. An object moving very swiftly is not seen, unless it be very luminous. Thus a cannon ball is not seen if it is viewed transversely: but if it be viewed according to the line it describes, it may be seen, because its picture continues long on the same place of the retina; which, therefore, receives a more sensible impression from the object.

2. A live coal swung briskly round in a circle appears a continued circle of fire, because the impressions made on the retina by light, being of a vibrating, and consequently of a lasting nature, do not presently perish, but continue till the coal performs its whole circuit, and returns again to its former place.

3. If two objects, unequally distant from the eye, move with equal velocity, the more remote one will appear the slower; or, if their celerities be proportional to their distances, they will appear equally swift.

4. If two objects, unequally distant from the eye, move with unequal velocities in the same direction, their apparent velocities are in a ratio compounded of the direct ratio of their true velocities, and the reciprocal one of their distances from the eye.

5. A visible object moving with any velocity appears to be at rest, if the space described in the interval of one second be imperceptible at the distance of the eye. Hence it is that a near object moving very slowly, as the index of a clock, or a remote one very swiftly, as a planet, seems to be at rest.

6. An object moving with any degree of velocity will appear at rest, if the space it runs over in a second of time be to its distance from the eye as 1 to 1400.

7. The eye proceeding straight from one place to another, a lateral object, not too far off, whether on the right or left, will seem to move the contrary way.

8. The eye proceeding straight from one place to another, and being sensible of its motion, distant objects will seem to move the same way, and with the same velocity. Thus, to a person running eastwards, the moon on his right hand appears to move the same way, and with equal swiftness; for on account of its distance, its image continues fixed upon the same place of the retina, from whence we imagine that the object moves along with the eye.

9. If the eye and the object move both the same way, only the eye much swifter than the object, the last will appear to go backwards.

10. If two or more objects move with the same velocity, and a third remain at rest, the moveable ones will appear fixed, and the quiescent one in motion the contrary way. Thus when the clouds move very swiftly, their parts seem to preserve their situation, and the moon to move the contrary way.

11. If the eye be moved with great velocity, lateral objects at rest appear to move the contrary way. Thus to a person sitting in a coach, and riding briskly through a wood, the trees seem to retire the contrary way; and to people in a ship, &c. the shores seem to recede.

At the conclusion of these observations, Dr Porterfield endeavours to explain another phenomenon of mo-field's action, which, though common and well known, had not been explained in a satisfactory manner. It is this: If a person turns swiftly round, without changing his place, move all objects about will seem to move round in a circle giddy partly the contrary way; and this deception continues not soon when ly while the person himself moves round, but, which is more surprising, it also continues for some time after he ceases to move, when the eye, as well as the object, is at absolute rest.

The reason why objects appear to move round the contrary way, when the eye turns round, is not so difficult to explain: for though, properly speaking, motion is not seen, as not being in itself the immediate object of sight; yet by the sight we easily know when the image changes its place on the retina, and thence conclude that either the object, the eye, or both, are moved. But by the sight alone we can never determine how far this motion belongs to the object, how far to the eye, or how far to both. If we imagine the eye at rest, we ascribe the whole motion to the object, though it be truly at rest. If we imagine the object at rest, we ascribe the whole motion to the eye, though it belongs entirely to the object; and when the eye is in motion, though we are sensible of its motion, yet, if we do not imagine that it moves so swiftly as it really does, we ascribe only a part of the motion to the eye, and the rest of it we ascribe to the object, though it be actually at rest. This last, he says, is what happens in the present case, when the eye turns round; for though we are sensible of the motion of the eye, yet we do not apprehend that it moves so fast as it really does; and therefore the bodies about appear to move the contrary way, as is agreeable to experience.

But the great difficulty still remains, viz. Why, after the eye ceases to move, objects should, for some time, still appear to continue in motion, though their pictures on the retina be really at rest, and do not at all change their place. This, he imagined, proceeds from a mistake we are in with respect to the eye, which, though it be absolutely at rest, we nevertheless conceive as moving the contrary way to that in which it moved before; from which mistake, with respect to the motion of the eye, the objects at rest will appear to move the same way which the eye is imagined to move; and, consequently, will seem to continue their motion for some time after the eye is at rest.

This is ingenious, but perhaps not just. An ac- for this count of this matter, which seems to us more satisfac-phenomenon- Apparent Tory, has been lately given to the public by Dr Wells, place, &c. "Some of the older writers upon optics (says this ingenious philosopher) imagined the visive spirits to be contained in the head, as water is in a vessel; which, therefore, when once put in motion by the rotation of our bodies, must continue in it for some time after this has ceased; and to this real circular movement of the visive spirits, while the body is at rest, they attributed the apparent motions of objects in giddiness. Dechales saw the weakness of this hypothesis; and conjectured, that the phenomenon might be owing to a real movement of the eyes; but produced no fact in proof of his opinion. Dr Porterfield, on the contrary, supposed the difficulty of explaining it to consist in showing, why objects at rest appear in motion to an eye which is also at rest. The solution he offered of this representation of the phenomenon, is not only extremely ingenious, but is, I believe, the only probable one which can be given. It does not apply, however, to the fact which truly exists; for I shall immediately show, that the eye is not at rest as he imagined. The last author I know of who has touched upon this subject is Dr Darwin. His words are, 'When any one turns round rapidly on one foot till he becomes dizzy, and falls upon the ground, the spectra of the ambient objects continue to present themselves in rotation, or appear to librate, and he seems to behold them for some time in motion.' I do not indeed pretend to understand his opinion fully; but this much seems clear, that if such an apparent motion of the surrounding objects depends in any way upon their spectra, or the illusive representations of those objects, occasioned by their former impressions upon the retinas, no similar motion would be observed, were we to turn ourselves round with our eyes shut, and not to open them till we became dizzy; for in this case, as the surrounding objects could not send their pictures to the retinas, there would consequently be no spectra to present themselves afterward in rotation. But whoever will make the experiment, will find, that objects about him appear to be equally in motion, when he has become dizzy by turning himself round, whether this has been done with his eyes open or shut. I shall now venture to propose my own opinion upon this subject.

Upon what data we judge visible objects to be in motion or at rest.

"If the eye be at rest, we judge an object to be in motion when its picture falls in succeeding times upon different parts of the retina; and if the eye be in motion, we judge an object to be at rest, as long as the change in the place of its picture upon the retina holds a certain correspondence with the change of the eye's position. Let us now suppose the eye to be in motion, while, from some disorder in the system of sensation, we are either without those feelings which indicate the various positions of the eye, or are not able to attend to them. It is evident, that in such a state of things an object at rest must appear to be in motion, since it sends in succeeding times its picture to different parts of the retina. And this seems to be what happens in giddiness. I was first led to think so from observing, that, during a slight fit of giddiness I was accidentally seized with, a coloured spot, occasioned by looking steadily at a luminous body, and upon which I happened at that moment to be making an experiment, was moved in a manner altogether independent of the positions I conceived my eyes to possess. To determine this point, I again produced the spot, by looking some time at the flame of a candle: then turning myself round till I became giddy, I suddenly discontinued this motion, and directed place, &c., my eyes to the middle of a sheet of a paper, fixed upon the wall of my chamber. The spot now appeared upon the paper, but only for a moment; for it immediately after seemed to move to one side, and the paper to the other, notwithstanding I conceived the position of my eyes to be in the mean while unchanged. To go on with the experiment, when the paper and spot had proceeded to a certain distance from each other, they suddenly came together again; and this separation and conjunction were alternately repeated a number of times, the limits of the separation gradually becoming less, till at length the paper and spot both appeared to be at rest, and the latter to be projected upon the middle of the former. I found also, upon repeating and varying the experiment a little, and when I had turned myself from left to right, the paper moved from right to left, and the spot consequently the contrary way; but that when I had turned from right to left, the paper would then move from left to right. These were the appearances observed while I stood erect. When I inclined, however, my head in such a manner as to bring the side of my face parallel to the horizon, the spot and paper would then move from each other, one upward and the other downward. But all these phenomena demonstrate, that there was a real motion in my eyes at the time I imagined them to be at rest; for the apparent situation of the spot, with respect to the paper, could not possibly have been altered, without a real change of the position of those organs. To have the same thing proved in another way, I desired a person to turn quickly round, till he became very giddy; then to stop himself, and look steadfastly at me. He did so, and I could plainly see, that although he thought his eyes were fixed, they were in reality moving in their sockets, first toward one side and then toward the other."

M. Le Cat well explains a remarkable deception, by a remarkable deception, which a person shall imagine an object to be on the opposite side of a board, when it is not so, and also inverted and magnified. It is illustrated by fig. 6, in which M. le Cat, D represents the eye, and CB a large black board, pierced with a small hole. E is a large white board, placed beyond it, and strongly illuminated; and a pin, or other small object, held betwixt the eye and the first board. In these circumstances, the pin shall be imagined to be at F, on the other side of the board, where it will appear inverted and magnified; because what is in fact perceived, is the shadow of the pin upon the retina; and the light that is stopped by the upper part of the pin coming from the lower part of the enlightened board, and that which is stopped by the lower part coming from the upper part of the board, the shadow must necessarily be inverted with respect to the object. This is nothing more than Mr Grey's experiment, in which he saw an inverted image of the pin, and which we have already noticed.

There is a curious phenomena relating to vision, which some persons have ascribed to the inflection of light, but which Mr Melville explains in a very different and very simple manner.

When any opaque body is held at the distance of three or four inches from the eye, so that a part of some more distant luminous object, such as the window, or the plained flame of a candle, may be seen by rays passing near its Mr Melv Theory.

Apparent edge, if another opaque body, nearer to the eye, be brought across from the opposite side, the edge of the first body will seem to swell outwards, and meet the latter; and in doing so will intercept a portion of the luminous object that was seen before.

This appearance he explains in the following manner:

Let AB represent the luminous object to which the sight is directed, CD the more distant opaque body, GH the nearer, and EF the diameter of the pupil. Join ED, FD, EG, FG, and produce them till they meet AB in K, N, M, and L. It is plain that the parts AN, MB, of the luminous object cannot be seen. But taking any point a between N and K, and drawing a D d, since the portion d F of the pupil is filled with light flowing from that point, it must be visible. Any point b, between a and K, must fill f F, a greater portion of the pupil, and therefore must appear brighter. Again, Any point c between b and K, must appear brighter than b, because it fills a greater portion g F with light. The point K itself, and every other point in the space KL, must appear very luminous, since they send entire pencils of rays EKF, ELF, to the eye; and the visible brightness of every point from L towards M, must decrease gradually, as from K to N, that is, the spaces KN, LM, will appear as dim shadowy borders, or fringes, adjacent to the edges of the opaque bodies.

When the edge G is brought to touch the right line KF, the penumbras unite; and as soon as it reaches NDF, the above phenomenon begins; for it cannot pass that right line without meeting some line a D d, drawn from a point between N and K, and, by intercepting all the rays that fall upon the pupil, render it invisible. In advancing gradually to the line KDE, it will meet other lines b D f, c D g, &c., and therefore render the points b, c, &c., from N to K, successively invisible; and therefore the edge of the fixed opaque body CD must seem to swell outwards, and cover the whole space NK; while GH, by its motion, covers MK. When GH is placed at a greater distance from the eye, CD continuing fixed, the space OP to be passed over in order to intercept NK is less; and therefore, with an equal motion of GH, the apparent swelling of CD must be quicker; which is found true by experience.

If ML represent a luminous object, and REFQ any plane exposed to its light, the space FQ will be entirely shaded from the rays, and the space FE will be occupied by a penumbra, gradually darker, from E to F. Let now GH continue fixed, and CD move parallel to the plane EF; and as soon as it passes the line LF, it is evident that the shadow QF will seem to swell outwards; and when CD reaches ME, so as to cover with its shadow the space RE, QF, by its extension, will cover FE. This is found to hold true likewise by experiment.

Sect. X. On Aberration of Figure or Sphericity.

The great practical use of the science of optics is to aid human sight; but it has been repeatedly observed during the progress of this article, that in constructing dioptrical instruments for this purpose, great difficulties arise from the aberration of light. It has been shown how to determine the concourse of any refracted ray PF' with the ray RVCF', which passes through the centre C, and therefore falls perpendicularly on the spherical surface at the vertex V, and suffers no refraction. This is the conjugate focus to R for the two rays RP, RV, and for another ray flowing from R and falling on the surface at an equal distance on the opposite side to P. In short, it is the conjugate focus for all the rays flowing from R, and falling on the spherical surface in the circumference of a circle described by the revolution of the point P round the axis RVCF'; that is, of all the rays which occupy the conical surface described by the revolution of RP, and the refracted rays occupy the conical surface produced by the revolution of PF'.

But no other rays flowing from R are collected at F'; for it appeared in the demonstration of that proposition, that rays incident at a greater distance from the axis RC were collected at a point between C and F'; and then the rays which are incident on the whole arch PC, or the spherical surface generated by its revolution round RC, although they all cross the axis RC, are diffused over a certain portion of it, by what has been called the aberration of figure. It is called also (but improperly) the aberration from the geometrical focus, by which is meant the focus of an infinitely slender pencil of rays, of which the middle ray (or axis of the pencil) occupies the lens RC, and suffers no refraction. But there is no such focus. But if we make mRV = nRC; mRV = VC : VF, the point F is called the geometrical focus, and is the remotest limit from C of all the foci (equally geometrical) of rays flowing from R. The other limit is easily determined by constructing the problem for the extreme point of the given arch.

It is evident from the construction, that while the point of incidence P is near to V, the line CK increases but very little, and therefore CF diminishes little, and the refracted rays are but little diffused from F'; and therefore they are much denser in its vicinity than any other point of the axis. It will soon be evident that they are incomparably denser. It is on this account that the point F has been called the conjugate focus to R, and the geometrical focus, and the diffusion has been called aberration. A geometrical point R is thus represented by a very small circle at F', and F has drawn the chief attention. And as, in the performance of optical instruments, it is necessary that this extended representation of a mathematical point R be very small, that it may not sensibly interfere with the representations of the points adjacent to R, and thus cause indistinct vision, a limit is thus set to the extent of the refracting surface which must be employed to produce this representation. But this evidently diminishes the quantity of light, and renders the vision obscure though distinct. Artists have therefore endeavoured to execute refracting surfaces of forms not spherical, which collect accurately to one point the light issuing from another, and the mathematicians have furnished them with forms having this property: but their attempts have been fruitless. Spherical surfaces are the only ones which can be executed with accuracy. All are done by grinding the refracting substance in a mould of proper materials. When this is spherical, the two work themselves, with moderate attention, into an exact sphere; because if any part is more prominent than another, it is ground away, and the whole gets of necessity one curvature. And it is astonishing to what degree of accuracy this is done. An error of the millionth part of an inch would totally destroy stroy the figure of a mirror of an inch focal distance, so as to make it useless for the coarsest instrument. Therefore all attempts to make other figures are given up. Indeed other reasons make them worse than spherical, even when accurately executed. They would not collect to accurate focuses the rays of oblique pencils.

It is evident from these observations, that the theory of aberrations is absolutely necessary for the successful construction of optical instruments; and it must be acceptable to the reader to have a short account of it in this place. Enough shall be said here to show the general nature and effects of it in optical instruments, and in some of the more curious phenomena of nature. Under the article Telescope the subject will be resumed, in such a manner as to enable the reader who possesses a very moderate share of mathematical knowledge, not only to understand how aberrations are increased and diminished, but also how, by a proper employment of contrary aberrations, their hurtful effects may be almost entirely removed in all important cases. And the manner in which the subject shall be treated in the present general sketch, will have the advantage of pointing out at the same time the maxims of construction of the greatest part of optical instruments, which generally produce their effects by means of pencils of rays which are either out of the axis altogether, or are oblique to it; cases which are seldom considered in elementary treatises of optics.

Let PV be a spherical surface of a refracting substance (glass for instance), of which C is the centre, and let an indefinitely slender pencil of rays AP a p be incident on it, in a direction parallel to a ray CV passing through the centre. It is required to determine the focus f of this pencil.

Let AP be refracted into PF. Draw CI, CR the sines of incidence and refraction, and CP the radius. Draw RB perpendicular to CP, and Bf parallel to AP or CV. I say, first, f is the focus of the indefinitely slender pencil, or, more accurately speaking, f is the remotest limit from P of the concourse of rays with PF refracted by points lying without the arch VP, or the nearest limit for rays incident between V and P.

Draw the radius C p c', the line p f; and draw p g parallel to P f, and P o perpendicular to P f. It is evident, that if f be the focus, c' p f is the angle of refraction corresponding to the angle of incidence a p C, as C' p f is the angle corresponding to APC. Also PC p is the increment of the angle of incidence, and the angle c' p g is equal to the sum of the angle C' p f and C' C c, and the angle g f f is equal to the angle p f P. Therefore c' p f = C' p f + P, C p + P f p. Therefore PC p + F f p is the corresponding increment of the angle of refraction. Also, because RP o = CP p (being right angles) the angles p P o = RPC, and P o : P f = PR : PC.

Therefore by a preceding Lemma in this article, we have PC p + P f p : PC p = tan. ref. : tan. incid. = T, R : T, I; and P f p : PC p = T, R - T, I : T, I,

= diff. : T, I; but P f p : PC p = P o : P p = PR : PC

= PR : P f = DR : DB (because DB is parallel to Bf by construction) = tan. CPR - tan. CPI : tan. CPI. Now CPI is the angle of incidence; and therefore CPR is the angle properly corresponding to it as an angle of refraction, and the point f is properly determined.

Hence the following rule. As the difference of the tangents of incidence and refraction is to the tangent of incidence, so is the radius of the surface multiplied by the cosine of refraction to the distance of the focus of an infinitely slender pencil of parallel incident rays.

N.B. We here consider the cosine of refraction as a number. This was first done by the celebrated Euler, and is one of the greatest improvements in mathematics which this century can boast of. The sines, tangents, secants, &c. are considered as fractional numbers, of which the radius is unity. Thus CP × sin. 30°, is the same thing with \( \frac{1}{2} \) CP, or \( \frac{CP}{2} \). And in like manner, CB, drawn perpendicular to the axis × sin. 19° 28' 16" 32" is the same thing with \( \frac{1}{3} \) of CB. Also \( \frac{CB}{\cos. 60°} \) is the same thing with twice CB, &c.

In this manner, BE = BC × sin. BCE, and also BE = CE × tan. BCE, and CB = CE × sec. BCE, &c. &c. This manner of considering the lines which occur in geometrical constructions is of immense use in all parts of mixed mathematics; and nowhere more remarkably than in optics, the most beautiful example of them. Of this an important instance shall now be given.

Cor. 1. The distance f G of this lateral focus from the axis CV (that is, from the line drawn through the centre parallel to the incident light) is proportional to the cube of the semi-aperture PH of the spherical surface.

For f G = BE. Now BE = CB × sin. BCE, = CB × sin. CPA; and CB = RC × cos. RCB, = RC × sin. CPR, and RC = CP × sin. CPR: Therefore BE = PC × sin. CPR × sin. PCA = PC × sin. refr. × sin. incid.

but sin. refr. = \( \frac{m}{n} \) sin. incid. Therefore finally, BE,

or f G = PC × \( \frac{m^3}{n^3} \) × sin. incid.: But PC, sin. incid. is evidently PH the semi-aperture; therefore the proposition is manifest.

Cor. 2. Now let this slender pencil of rays be incident at the vertex V. The focus will now be a point F in the axis, determined by making CV : CF = m : n. Let the incident pencil gradually recede from the axis CF, still, however, keeping parallel to it. The focus f will always be found in a curve line DC'F, so constituted that the ordinate G will be as the cube of the line PH, perpendicular to the axis intercepted between the axis and that point of the surface which is cut by a tangent to the curve in f.

All the refracted rays will be tangents to this curve, and the adjacent rays will cross each other in these lateral foci f; and will therefore be incomparably more dense along the curve than anywhere within its area. This is finely illustrated by receiving on white paper the light of the sun refracted through a globe or cylinder of glass filled with water. If the paper is held parallel to the axis of the cylinder, and close to it, the illuminated part will be bounded by two very bright parallel lines, where it is cut by the curve; and these lines will gradually approach each other as the paper is withdrawn from the vessel, till they coalesce into one very bright bright line at F, or near it. If the paper be held with its end touching the vessel, and its plane nearly perpendicular to the axis, the whole progress of the curve will be distinctly seen.

As such globes were used for burning glasses, the point of greatest condensation (which is very near but not exactly in F) was called the focus. When these curves were observed by Mr Tchirnhaus, he called them caustics; and those formed by refraction he called dia-caustics, to distinguish them from the catacaustics formed by reflection.

It is somewhat surprising, that these curves have been so little studied since the time of Tchirnhaus. The doctrine of aberrations has indeed been considered in a manner independent on their properties. But whoever considers the progress of rays in the eye-piece of optical instruments, will see that the knowledge of the properties of dia-caustic curves determines directly, and almost accurately, the foci and images that are formed there. For, let the object-glass of a telescope or microscope be of any dimensions, the pencils incident on the eye-glasses are almost all of this evanescent bulk. These advantages will be shown in their proper places; and we proceed at present to extend our knowledge of aberrations in general, first considering the aberrations of parallel incident rays.

Abiding by the instance represented by the figure, it is evident that the caustic will touch the surface in a point \( \phi \), so situated that \( c \phi : \phi x = m : n \). The refracted ray \( \phi \) will touch the surface, and will cross the axis in \( \phi \), the nearest limit of diffusion along the axis. If the surface is of smaller extent, as PV, the caustic begins at \( f \), when the extreme refracted ray \( P f \) touches the caustic, and crosses the axis in \( F' \), and the opposite branch of the caustic in K. If there be drawn an ordinate KO \( k \) to the caustic, it is evident that the whole light incident on the surface PVII passes through the circle whose diameter is K \( k \), and that the circle is the smallest space which receives all the refracted light.

It is of great importance to consider the manner in which the light is distributed over the surface of this circle of smallest diffusion: for this is the representation of one point of the infinitely distant radiant object. Each point of a planet, for instance, is represented by this little circle; and as the circles representing the different adjacent points must interfere with each other, an indistinctness must arise similar to what is observed when we view an object through a pair of spectacles which do not fit the eye. The indistinctness must be in proportion to the number of points whose circles of diffusion interfere; that is, to the area of these circles, provided that the light is uniformly diffused over them: but if it be very rare at the circumference, the impression made by the circles belonging to the adjacent points must be less sensible. Accordingly, Sir Isaac Newton, supposing it incomparably rarer at the circumference than towards the centre, affirms that the indistinctness of telescopes, arising from the spherical figure of the object-glass, was some thousand times less than that arising from the unequal refrangibility of light; and, therefore, that the attempts to improve them by diminishing or removing this aberration were needless, while the indistinctness from unequal refrangibility remained. It is surprising, that a philosopher so eminent for sagacity and for mathematical knowledge should have made such a mistake, and unfortunate that the authority of his great name hindered others from examining the matter, trusting to his assertion that the light was so rare at the border of this circle. His mistake is surprising, because the very nature of a caustic should have showed him that the light was infinitely dense at the borders of the circle of smallest diffusion. The first person who detected this oversight of the British philosopher was the Abbé Boscovich, who, in a dissertation published at Vienna in 1767, showed, by a very beautiful analysis, that the distribution was extremely different from what Newton had asserted, and that the superior indistinctness arising from unequal refrangibility was incomparably less than he had said. We shall attempt to make this delicate and interesting matter conceived by those who have but small mathematical preparation.

Let the curve DVCI \( c v d \) be the caustic (magnified), EI its axis, I the focus of central rays, B the focus of extreme rays, and IB the line containing the foci of all the intermediate rays, and CO \( c \) the diameter of the circle of smallest diffusion.

It is plain, that from the centre O there can be drawn two rays OV, O \( v \), touching the caustic in V, \( v \). Therefore the point O will receive the ray EO, which passes through the vertex of the refracting surface, and all the rays which are incident on the circumference of a circle described on the refracting surface by the extremity of the ray OV, or O \( v \). The density of the light at O will therefore be indefinitely great.

From the point C there can be drawn two rays; one of them CX touching the caustic in C, and the other C, touching it at \( d \) on the opposite side. The rays which touch the caustic in the immediate vicinity of C\( y \), both in the arch CV and the arch CI will cut OC in points indefinitely near to each other; because their distance from each other in the line OC will be to their uniform distance on the refracting surface as the distance between their points of contact with the caustic to the distance of these points from the refracting surface. Here therefore at C the density of the light will also be indefinitely great.

From any point H, lying between O and C, may be drawn three rays. One of them LHT, P, touching the arch CD of the caustic in T, cutting the refracting surface in P, and the axis in L; another \( t \) HP, touching the arch CI of the caustic in \( t \). The third is \( H \tau \), touching the arch \( c d \) of the opposite branch of the caustic in \( \tau \).

It will greatly assist our conception of this subject, Fig. 1, if we consider a ray of light from the refracting surface as a thread attached at J of this figure, or at F of fig. 1, and gradually unlooped from the caustic DVCI on one side, and then lopped on the opposite branch \( I c v d \); and attend to the point of its intersection with the diameter \( c OC \) of the circle of smallest diffusion.

Therefore, 1. Let the ray be first supposed to pass through the refracting surface at F, the right hand extremity of the aperture. The thread is then folded up on the whole right hand branch ICVD of the caustic; and if the straight part of it FD be produced, it will cut the diameter of the circle of smallest diffusion in the opposite extremity \( c \). Or suppose a ruler in place of the thread, applied to the caustic at D and to the refracting surface at F, the part of it \( Dc \), which which is detached from the caustic, cuts CO in the point c. 2. Now suppose the ruler to revolve gradually, its extremity moving across the arch FA of the refracting surface while the edge is applied to the caustic; the point of contact with the caustic will shift gradually down the branch DV of the caustic, while its edge passes across the line eC; and when the point of contact arrives at V, the extremity will be at Y on the refracting surface, and the intersection of the edge will be at O. 3. Continuing the motion, the point of contact shifts from V to Z, the extremity from Y to Q', and the intersection from O to Q, so that $OQ' = \frac{OC}{2}$, as will presently appear. 4. After this, the point of contact will shift from Z to C, the extremity from Q' to X, half way from F to A, as will soon be shown, and the intersection from Q to C. 5. The point of contact will now shift from C down to I, the extremity will pass from X to A, and the intersection will go back from C to O. 6. The ruler must now be applied to the other branch of the caustic I c v d, and the point of contact will ascend from I to c, the extremity will pass from A to x, half way to f from A, and the intersection from O to c. 7. The point of contact will ascend from C to z, the extremity passes from x to q', and the intersection from C to q, O q' being $\frac{OC}{2}$. 8. While the contact of the ruler and caustic shifts from z to v, the extremity shifts from q' to y, and the intersection from q to O.

9. The contact rises from v to d, the extremity passes from y to f, and the intersection from O to C; and then the motion across the refracting surface is completed, the point of contact shifting down from D to I along the branch DVZCI, and then ascending along the other branch I c v d, while the intersection passes from c to C, back again from C to c, and then back again from c to C, where it ends, having thrice passed through every intermediate point of cC.

We may form a notion of the density of the light in any point H, by supposing the incident light of uniform density at the refracting surface, and attending to the constipation of the rays in the circle of smallest diffusion. Their vicinity may be estimated both in the direction of the radii OH, and in the direction of the circumference described by its extremity H, during its revolution round the axis; and the density must be conceived as proportional to the number of originally equidistant rays, which are collected into a spot of given area. These have been collected from a corresponding spot or area of the refracting surface; and as the number of rays is the same in both, the density at H will be to the density of the refracting surface, as the area occupied of the refracting surface, to the corresponding area at H.

The vicinity of the rays in the direction of the radius depends on the proportion between PT and TH. For the ray adjacent to PTH may be supposed to cross it at the point of contact T; and therefore the uniform distance between them at the surface of that medium is to the distance between the same rays at H as the distance of T from the refracting surface to its distance from H. Therefore the number of rays which occupy a tenth of an inch, for example, of the radius AP, is to the number which would occupy a tenth of an inch at H as TH to TP; and the radial density at P is to the radial density at H, also as TH to TP. In the next place, of the circumferential density at P is to that at H as the radius AP to the radius OH. For supposing the figure to turn round its axis AI, the point P of the refracting surface will describe a circumference whose radius is AP, and H will describe a circumference whose radius is OH; and the whole rays which pass through the first circumference pass also through the last, and therefore their circumferential densities will be in the inverse proportion of the spaces into which they are collected. Now the radius AP is to the radius OH as AL to OL; and circumferences have the same proportion with their radii. Therefore the circumferential density at P is to that at H as AL to OL inversely; and it was found that the radial density was as AN to ON inversely, being as TH to TP, which are very nearly in this ratio. Therefore the absolute density (or number of rays collected in a given space) at P will be to that at H, in the ratio compounded of these ratios; that is, in the ratio of ON × OL to AN × AL. But as NL bears but a very small ratio to AN or AL, AN × AL may be taken as equal to AO² without any sensible error. It never differs from it in telescopes 100th part, and is generally incomparably smaller. Therefore the density at H may be considered as proportional to ON × OL inversely. And it will afterwards appear that NS is $\frac{3}{2}$ OL. Therefore the density at H is inversely as ON × NS.

Now describe a circle on the diameter OS, and draw NT cutting the circumference Nφ = ON × NS, and the density at H is as Nφ inversely. This gives us a very easy estimation of the density, viz., draw a line from the point of contact of the ray which touches the part VC of the caustic, and the density is in the inverse subduplicate ratio of the part of this line intercepted between the axis and the circumference SφO. It will afterwards appear that the density corresponding to this ray is one half of the density corresponding to all the three: or a better expression will be had for the density at H by drawing Rβ perpendicular to Rφ, and βo perpendicular to φβ, making φR in φβ, then φo is as $\frac{1}{\phi N^2}$, or is proportional to the density, as is evident.

When H is at O, N is at S, and φo is infinite. As H moves from O, N descends, and φo diminishes, till H comes to Q, and T to x, and φ to ξ, and o to R. When H moves from Q towards C, T descends below x, φ again increases, till it is again infinite, when H is at C, T at C, and N at O.

Thus it appears, without any minute consideration, that the light has a density indefinitely great in the centre O; that the density decreases to a minimum in some intermediate point Q, and then increases again to infinity at the margin C. Hence it follows, that the indistinctness arising from the spherical figure of the refracting surfaces is incomparably greater than Newton supposed; and that the valuable discovery of Mr Dollond of achromatic lenses, must have failed of answering his fond expectations, if his very method of producing them had not, at the same time, enabled him to remove that other indistinctness by employing contrary aberrations. And now, since the discovery by Dr Blair of substances which disperse the different colours in the same proportions, but very different degrees, Theory.

Optics.

of glasses, has enabled us to employ much larger portions of the sphere than Mr Dollond could introduce into his object-glasses, it becomes absolutely necessary to study this matter completely, in order to discover and ascer- tain the amount of the errors which perhaps unavoid- ably remain.

This slight sketch of the most simple case of aber- ration, namely, when the incident rays are parallel, will serve to give a general notion of the subject; and the reader can now see how contrary aberrations may be employed in order to form an ultimate image which shall be as distinct as possible. For let it be proposed to converge parallel rays accurately to the focus F, by the refraction of spherical surfaces of which V is the vertex. Let PV be a convex lens of such a form, that rays flowing from F, and passing through it immediately round the vertex V, are collected to the conjugate focus R, while the extreme ray FP, inci- dent on the margin of the lens P, is converged to r, nearer to V, having the longitudinal aberration R r. Let p V be a plano-concave lens, of such sphericity that a ray A p, parallel to the axis CV, and incident on the point p, as far from its vertex V as P in the other lens is from its vertex, is dispersed from r, the distance V being equal to V, while the central rays are dispersed from P, as far from V as R is from V. It is evident, that if these lenses be joined as in fig. 4, a ray A' p', parallel to the common axis CV, will be collected at the distance VF equal to VF in the fig. 4, and that rays passing through both lenses in the neigh- bourhood of the axis will be collected at the same point F.

This compound lens is said to be without spheri- cal aberration; and it is true that the central and the extreme rays are collected in the same point F; but the rays which fall on the lens between the centre and margin are a little diffused from F, and it is not pos- sible to collect them all to one point. For in the rules for computing the aberration, quantities are neglected which do not preserve, in different apertures, the same ratio to the quantities retained. The diffusion is least when the aberration is corrected, not for the very ex- tremity, but for a certain intermediate point (varying with the aperture, and having no known ratio to it); and when this is done, the compound lens is in its state of greatest perfection, and the remaining aberration is quite insensible. See Telescope.

Sect. VI. On 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 nearly in the words of Sir Isaac Newton, who first discovered it; es- pecially as his account is more full and perspicuous than those of succeeding writers.

"In a dark chamber, at a round hole F, about one third of an inch broad, made in the shutter of a window, I placed a glass prism ABC, whereby the beam of the sun's light, SF, which came in at that hole, might be refracted upwards, toward the opposite wall of the cham- ber, and there form a coloured image of the sun, repre- sented at PT. The axis of the prism was, in this and the following experiments, perpendicular to the incident rays. About this axis I turned the prism slowly, and saw the refracted or coloured image of the sun, first to On the dif- descend, and then to ascend. Between the descent and ascent, when the image seemed stationary, I stopped the prism and fixed it in that posture.

"Then I let the refracted light fall perpendicularly upon a sheet of white paper, MN, placed at the oppo- site wall of the chamber, and observed the figure and dimensions of the solar image, PT, formed on the pa- per 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 indistinctly, the light there 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 8 inches; and ACB, the refracting angle of the prism, by which so great a length was made, was 64 degrees. With a less angle the length of the image was less, the breadt remaining the same. It is farther to be observed, that the rays went on in straight lines from the prism to the image, and therefore at their going out of the prism had all that inclination to one another from which the length of the image proceeded. This image PT was 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 in- termediate degrees in a continual succession perpetually varying."

Our author concludes from this and other experi- ments, "that the light of the sun consists of a mixture of sev- eral sorts of coloured rays, some of which at equal sorts incidences are more refracted than others, and therefore are called more refrangible. The red at T, being near- est to the place Y, where the rays of the sun would go refrangible, directly if the prism was taken away, is the least refrac- ted of all the range; and the orange, yellow, green, blue, indigo, and violet, are continually more and more re- fracted, as they are more and more diverted from the course of the direct light. For by mathematical rea- soning 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 be- tween 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. There- fore, since it is found by experience that this image is not round, but about five times longer than broad, it follows, that all the rays are not equally refracted. This conclusion is farther confirmed by the following experiments.

"In the sunbeam SF, which was propagated into the Fig. 2. room through the hole in the window-shutter EG, at the distance of some feet from the hole, I held the prism ABC in such a posture, that its axis might be perpendi- cular to that beam; then I looked through the prism upon the hole F, and turning the prism to and fro about its axis to make the image 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 E, I ob- served the length of its refracted image p t to be many times greater than its breadth; and that the most re- fracted part thereof appeared violet at p; the least re- fracted, at t; and the middle parts indigo, blue, green,

yellow. On the different refrangibility of light 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 unfolded the whole mystery of colours, we see our author was not only beholding to the experiments themselves, which many others had made before him, but also to his skill in geometry; which was absolutely necessary to determine what the figure of the refracted image ought to be upon the old principle of an equal refraction of all the rays: but having thus made the discovery, he contrived the following experiment to prove it at sight.

"In the middle of two thin boards, DE d e, I made a round hole in each, at G and g, a third part of an inch in diameter; and in the window-shut a much larger hole being made, at F, to let into my darkened chamber a large beam of the sun's light, I placed a prism, ABC, behind the shut in that beam, to refract it towards the opposite wall; and close behind this prism I fixed one of the boards DE, in such a manner that the middle of the refracted light might pass through the hole made in it at G, and the rest be intercepted by the board. Then at the distance of about 12 feet from the first board, I fixed the other board, d e, in such manner that the middle of the refracted light, which came through the hole in the first board, and fell upon the opposite wall, might pass through the hole g in this other board d e, and the rest being intercepted by the board, might paint upon it the coloured spectrum of the sun. And close behind this board I fixed another prism a b c, to refract the light which came through the hole g. Then I returned speedily to the first prism ABC, and by turning it slowly to and fro about its axis, I caused the image which fell upon the second board d e, to move up and down upon that board, that all its parts might pass successively through the hole on that board, and fall upon the prism behind it. And in the mean time I noted the places, M, N, on the opposite wall, to which that light after its refraction in the second prism did pass; and by the difference of the places at M and N, I found that the light, which being most refracted in the first prism ABC, did go to the blue end of the image, was again more refracted by the second prism a b c, than the light which went to the red end of that image. For when the lower part of the light which fell upon the second board d e, was cast through the hole g, it went to a lower place M on the wall; and when the higher part of that light was cast through the same hole g, it went to a higher place N on the wall; and when any intermediate part of the light was cast through that hole, it went to some place in the wall between M and N. The unchanged position of the hole 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."

Sir Isaac shows also, by experiments made with convex glass, that lights, reflected from natural bodies, which differ in colour, differ also in 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 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 thereby 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, above that of the red and yellow at t: and afterwards, when the rest of the light, which was green, yellow, and red, began to be totally reflected and vanished at G, the light of those colours at t, on the paper p t, received as great an increase as the violet and blue had received before. Which puts it past dispute, that those rays became first of all totally reflected at the base BC, which before at equal incidences with the rest upon the base BC had suffered the greatest refraction. I do not here take any 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 45 degrees. For the rays FM begin to be totally reflected when the angle CMF is about 50 degrees; and therefore they will then make a right angle of 90 degrees with AC.

"It appears also from experiments, that the beam of light MN, reflected by the base of the prism, being augmented first by the more refrangible rays and afterwards by the less refrangible, is composed of rays differently refrangible.

"The light whose rays are all alike refrangible, I call simple, homogeneous, and similar; and that whose rays are some more refrangible than others, I call compound, heterogeneous, and dissimilar. The former light I call homogeneous, not because I would affirm it so in all respects; The colours of homogeneous lights I call primary, homogeneous, and simple; and those of heterogeneous lights, heterogeneous and compound. For these are always compounded of homogeneous lights, as will appear in the following discourse.

The homogeneous 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, blue-making, violet-making; and so of the rest. And if at any time I speak of light and rays as coloured or endowed with colours, I would be understood to speak not philosophically and properly, but grossly, and according to such conceptions as vulgar people in seeing all these experiments would be apt to frame. For the rays, to speak properly, are not coloured. In them there is nothing else than a certain power and disposition to stir up a sensation of this or that colour. For as sound, in a bell or musical string or other sounding body, is nothing but a trembling motion, and in the air nothing but that motion propagated from the object, and in the sensorium it is a sense of that motion under the form of sound; so colours in the object are nothing but a disposition to reflect this or that sort of rays more copiously than the rest: in rays they are nothing but their dispositions to propagate this or that motion into the sensorium; and in the sensorium they are sensations of those motions under the forms of colours. See Chromatics.

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 which all the most refrangible rays, propagated from the whole disk of the sun, will illuminate and paint upon the opposite wall if they were alone; EL the circle, which all the least refrangible rays would in like manner illuminate if they were alone; BH, CI, DK, the circles which so many intermediate sorts would paint upon the wall, if they were singly propagated from the sun in successive order, the rest being intercepted; and conceive that there are other circles without number, which innumerable other intermediate sorts of rays would successively paint upon the wall, if the sun should successively emit every sort apart. And seeing the sun emits all these sorts at once, they must all together illuminate and paint innumerable equal circles; of all which, being according to their degrees of refrangibility placed in order in a continual series, that oblong spectrum PT is composed, which was described in the first experiment.

Now if these circles, whilst their centres keep their distances and positions, could be made less in diameter, their interfering one with another, and consequently the mixture of the heterogeneous rays, would be proportionally diminished. Let the circles AG, BH, CI, &c. remain as before; and let ag, bh, ci, &c. be so many less circles lying in a like continual series, between two parallel right lines ae and gl, with the same distance between their centres, and illuminated with the same sorts of rays; that is, the circle ag with the same sort by which the corresponding circle AG was illuminated; and the rest of the circles bh, ci, dk, el, respectively with the same sorts of rays by which the corresponding circles BH, CI, DK, EL, were illuminated. In the figure PT, composed of the great circles, three of those, AG, BH, CI, are so expanded into each other, that three sorts of rays, by which those circles are illuminated, together with innumerable other sorts of intermediate rays, are mixed at QR in the middle of the circle BH. And the like mixture happens throughout almost the whole length of the figure PT. But in the figure pt, composed of the less circles, the three less circles ag, bh, ci, which answer to those three greater, do not extend into one another; nor are there anywhere mingled so much as any two of the three sorts of rays by which those circles are illuminated, and which in the figure PT are all of them intermingled at QR. So then, if we would diminish the mixture of the rays, we are to diminish the diameters of the circles. Now these would be diminished if the sun's diameter, to which they answer, could be made less than it is, or (which comes to the same purpose), if without doors, at great distance from the prism towards the sun, some opaque body were placed with a round hole in the middle of it to intercept all the sun's light, except so much as coming from the middle of his body could pass through that hole to the prism. For so the circles AG, BH, and the rest, would not any longer answer to the whole disk of the sun, but only to that part of it which could be seen from the prism through that hole; that is, to the apparent magnitude of that hole viewed from the prism. But that these circles may answer more distinctly to that hole, a lens is to be placed by the prism to cast the image of the hole (that is, every one of the circles AG, BH, &c.) distinctly upon the paper at PT; after such a manner, as by a lens placed at a window the pictures of objects abroad are cast distinctly upon the paper within the room. If this be done, it will not be necessary to place that hole very far off, nor beyond the window. And therefore, instead of that hole, I used a 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, about 10 or 12 feet from the window, I placed a lens MN, fig. 6, by which the image of the hole F might be distinctly cast upon a sheet of white paper placed at I. Then immediately after the lens I placed a prism ABC, by which the projected light might be refracted either upwards or sidewise, 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 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 images pt became 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 On the dif: the breadth of the image p t, the circle of heterogeneous rays that compose it may be separated from each other as much as you please. Yet instead of the circular hole of Light F, it is better to substitute a 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 p t 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 part 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 the refracted light fall perpendicularly upon a white paper, two or three feet distant from the prism, I found that the spectrum formed on the paper by this light was not oblong, as when it is made in the first experiment, by refracting the sun's compound light, but was, so far as I could judge by my eye, perfectly circular, the length being nowhere greater than the breadth; which shows that this light is refracted regularly without any dilatation of the rays, and is an ocular demonstration of the mathematical proposition mentioned above.

"In the homogeneous light I placed a paper circle of a quarter of an inch in diameter: and in the sun's unrefracted, heterogeneous, white light, I placed another paper circle of the same bigness; and going from these papers to the distance of some feet, I viewed both circles through a prism. The circle illuminated by the sun's heterogeneous light appeared very oblong, as in the second experiment, the length being many times greater than the breadth. But the other circle, illuminated with homogeneous light appeared circular, and distinctly defined, as when it is viewed by the naked eye; which proves the whole proposition mentioned in the beginning of this article.

"In the homogeneous light I placed flies and such like minute objects, and viewing them through a prism I saw their parts as distinctly defined as if I had viewed them with the naked eye. The same objects placed in the sun's unrefracted heterogeneous light, which was white, I viewed also through a prism, and saw them most confusedly defined, so that I could not distinguish their smaller parts from one another. I placed also the letters of a small print one while in the homogeneous light, and then in the heterogeneous; and viewing them through a prism, they appeared in the latter case so confused and indistinct that I could not read them; but in the former, they appeared so distinct that I could read readily, and thought I saw them as distinct as when I viewed them with my naked eye: in both cases, I viewed the same objects through the same prism, at the same distance from me, and in the same situation. There was no difference but in the lights by which the objects were illuminated, and which in one case was simple, in the other compound; and therefore the distinct vision ferent 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 refractions. And as these colours were not changed by refractions, so neither were they by reflections. For all white, gray, red, yellow, green, blue, violet bodies, as paper, ashes, red lead, orpiment, indigo, bice, gold, silver, copper, grass, blue flowers, violets, bubbles of water tinged with various colours, peacock feathers, the tincture of lignum nephriticum, and such like, in red homogeneous light appeared totally red, in blue light totally blue, in green light totally green, and so of other colours. In the homogeneous light of any colour they all appeared totally of that same colour; with this only difference, that some of them reflected that light more strongly, others more faintly. I never yet found any body which by reflecting homogeneous light could sensibly change its colour.

"From all which it is manifest, that if the sun's light consisted of but one sort of rays, there would be but one colour in the world, nor would it be possible to produce any new colour by reflections and refractions; and by consequence, that the variety of colours depends upon the composition of light.

"The solar image p t, formed by the separated rays in the 6th experiment, did in the progress from its end p, on which the most refrangible rays fell, unto its end t, on which the least refrangible rays fell, appear tinged with this series of colours; violet, indigo, blue, green, yellow, orange, red, together with all their intermediate degrees in a continual succession perpetually varying; so that there appeared as many degrees of colours as there were sorts of rays differing in refrangibility. And since these colours could not be changed by refractions nor by reflections, it follows that all homogeneous light has its proper colour answering to its degree of refrangibility.

"Every homogeneous ray considered apart is refracted according to one and the same rule; so that its sine of incidence is to its sine of refraction in a given ratio: that is, every different coloured ray has a different ratio belonging to it. This our author has one and proved by experiment, and by other experiments has the same determined by what numbers those given ratios are expressed. For instance, if an heterogeneous white ray of the sun emerges out of glass into air; or, which is the same thing, if rays of all colours be supposed to succeed one another in the same line AC, and AD their common sine of incidence in glass be divided into 50 equal parts, then EF and GH, the sines 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 sines of refraction of all the degrees of red will have all intermediate degrees of magnitude from 77 to 77½, of all the degrees of orange from 77½ to 77¾, of yellow from 77¾ to 77½, of green from 77½ to 77¾, of blue from 77½ to 77¾, of indigo from 77¾ to 77½, and of violet from 77½ to 78." See Chromatics, Supplement. PART II. EXPLANATION OF OPTICAL PHENOMENA.

Sect. I. Of the Rainbow.

The observations of the ancients, and the philosophers of the middle ages, concerning the rainbow, were such as could not have escaped the notice of the most illiterate husbandmen; and their various hypotheses deserve no notice. It is a considerable time, even after the dawn of true philosophy, before we find any discovery of importance on this subject. Maurolycus was the first who pretended to have measured the diameters of the two rainbows with much exactness; and he found that of the inner bow to be $45^\circ$, and that of the outer bow $56^\circ$; from which Descartes takes occasion to observe, how little we can depend upon the observations of those who were not acquainted with the cause of the phenomena.

Clichotovius, who died in 1543, had maintained, that the second bow is the image of the first, which he thought was evident from the inverted order of the colours. For, said he, when we look into the water, all the images that we see reflected by it are inverted with respect to the objects themselves; the tops of the trees, for instance, that stand near the brink, appearing lower than the roots.

As the rainbow was opposite to the sun, it was natural to imagine, that its colours were produced by some kind of reflection of the rays of light from the drops of rain. No person seems to have thought of ascribing these colours to refraction, till one Fletcher of Breslaw, in a treatise published in 1571, endeavoured to account for them by means of a double refraction and one reflection. But he imagined that a ray of light, after entering a drop of rain, and suffering a refraction both at its entrance and exit, was afterwards reflected from another drop, before it reaches the eye of the spectator. He seems to have overlooked the reflection at the posterior surface of the drop, or to have imagined that all the bendings of the light within the drop would not make a sufficient curvature to bring the rays of the sun to the eye of the spectator. That he should think of two refractions, was the necessary consequence of his supposing that the ray entered the drop at all. This supposition, therefore, was all that he instituted to explain the phenomena. B. Porta supposed that the rainbow is produced by the refraction of light in the whole body of rain or vapour, but not in the separate drops.

It is to a man who had no pretensions to philosophy, that we are indebted for the true explanation. This was Antonio De Dominis, bishop of Spalatro, whose treatise De Radiis Visus et Lucis, was published by J. Bartolus in 1611. He first maintained, that the double refraction of Fletcher, with an intervening reflection, was sufficient to produce the colours of the bow, and also to bring the rays that formed them to the eye of the spectator, without any subsequent reflection. He distinctly describes the progress of a ray of light entering the upper part of the drop, where it suffers one refraction, and after being thereby thrown upon the back part of the inner surface, is thence reflected to the lower part of the drop; at which place undergoing a second refraction, it is thereby bent, so as to come directly to the eye. To verify this hypothesis, De Dominis proceeded in a very sensible and philosophical manner. He procured a small globe of solid glass, and viewing it when it was exposed to the rays of the sun, in the same manner in which he had supposed that the drops of rain were situated with respect to them, he actually observed the same colours which he had seen in the true rainbow, and in the same order.

Thus the circumstances in which the colours of the rainbow were formed, and the progress of a ray of light through a drop of water, were clearly understood; but philosophers were a long time at a loss when they endeavoured to assign reasons for all the particular colours, and for the order of them. Indeed nothing but the doctrine of the different refrangibility of the rays of light, could furnish a complete solution of this difficulty. De Dominis supposed that the red rays were those which had traversed the least space in the inside of a drop of water, and therefore retained more of their native force; and consequently, striking the eye more briskly, gave it a stronger sensation; that the green and blue colours were produced by those rays, the force of which had been, in some measure, obtunded in passing through a greater body of water; and that all the intermediate colours were composed (according to the hypothesis which generally prevailed at that time) of a mixture of these three primary ones. That the different colours were produced by some difference in the impulse of light upon the eye, was an opinion which had been adopted by many persons, who had ventured to depart from the authority of Aristotle.

Afterwards the same De Dominis observed, that all the rays of the same colour must leave the drop of water in a part similarly situated with respect to the eye, in order that each of the colours may appear in a circle, the centre of which is a point of the heavens, in a line drawn from the sun through the eye of the spectator. The red rays, he observed, must issue from the drop nearest to the bottom of it, in order that the circle of red may be the outermost, and the most elevated in the bow.

Though De Dominis conceived so justly the manner in which the inner rainbow is formed, he was far from having as just an idea of the cause of the exterior bow. This he endeavoured to explain in the very same manner as the interior, viz. by one reflection of the light within the drop, preceded and followed by a refraction; supposing only that the rays which formed the exterior bow were returned to the eye by a part of the drop lower than that which transmitted the red of the interior bow. He also supposed that the rays which formed one of the bows came from the upper limb of the sun, and those which formed the other from the lower limb, without considering that the bows ought thus to have been contiguous; or rather, that an indefinite number of bows would have had their colours all intermixed.

When Sir Isaac Newton discovered the different refrangibility of the rays of light, he immediately applied the discovery to the phenomena of the rainbow, taking Let \(a\) be a drop of water, and \(S\) a pencil of light; which, on its leaving the drop reaches the eye of the spectator. This ray, at its entrance into the drop, begins to be decomposed into its proper colours; and upon leaving the drop, after one reflection and a second refraction, it is farther decomposed into as many small differently-coloured pencils as there are primitive colours in the light. Three of them only are drawn in this figure, of which the blue is the most, and the red the least, refracted.

The theory of the different refrangibility of light enables us to assign a reason for the size of a bow of each particular colour. Newton, having found that the sines of refraction of the most refrangible and least refrangible rays, in passing from rain water into air, are in the ratio of 185 to 182, when the sine of incidence is 138, computed the size of the bow; and found, that if the sun was only a physical point, the breadth of the inner bow would be 2°; and if to this 30' were added for the apparent diameter of the sun, the whole breadth would be 2°. But as the outermost colours, especially the violet, are extremely faint, the breadth of the bow will not appear to exceed two degrees. He found by the same principles, that the breadth of the exterior bow, if it was everywhere equally vivid, would be 4° 20'. But in this case there is a greater deduction to be made, on account of the faintness of the light of the exterior bow; so that it will not appear to be more than 3 degrees broad.

The principal phenomena of the rainbow are explained on Sir Isaac Newton's principles in the following propositions.

**Prop. I.**

When the rays of the sun fall upon a drop of rain and enter into it, some of them, after one reflection and two refractions, may come to the eye of a spectator who has his back towards the sun, and his face towards the drop.

If XY be a drop of rain, and if the sun shine upon it in any lines \(s, f, s, d, s, a\), &c., most of the rays will enter into the drop; some of them only will be reflected from the first surface; those rays which are thence reflected do not come under our present consideration, because they are never refracted at all. The greatest part of the rays then enter the drop, and those passing on to the second surface, will most of them be transmitted through the drop. At the second surface or hinder part of the drop, at \(p, g\), some few rays will be reflected, whilst the rest are transmitted; those rays proceed in some such lines as \(r, n, q\); and coming out of the drop in the lines \(r, v, q, t\), may fall upon the eye of the spectator, who is placed anywhere in those lines, with his face towards the drop, and consequently with his back towards the sun, which is supposed to shine upon the drop in the lines \(s, f, s, d, s, a\), &c. These rays are twice refracted and once reflected; they are refracted when they pass out of the air into the drop; they are reflected from the second surface, and are refracted again when they pass out of the drop into the air.

**Def.** When rays of light reflected from a drop of rain come to the eye, those are called effectual which are able to excite a sensation.

**Prop. II.**

When rays of light come out of a drop of rain, they will not be effectual, unless they are parallel and contiguous.

There are but few rays that can come to the eye at all; for since the greatest part of those rays which enter the drop XY between X and \(a\), pass out of the drop through the hinder surface \(p, g\); only few are thence reflected, and come out through the nearer surface between \(a\) and Y. Now, such rays as emerge, or come out of the drop, between \(a\) and Y, will be ineffectual, unless they are parallel to one another, as \(r, v, q, t\); because such rays as come out diverging from one another will be so far asunder when they come to the eye, that all of them cannot enter the pupil; and the very few that can enter it will not be sufficient to excite any sensation. But even rays, which are parallel, as \(r, v, q, t\), will not be effectual, unless there are several of them contiguous or very near to one another. The two rays \(r, v\) and \(q, t\) alone will not be perceived, though both of them enter the eye; for so very few rays are not sufficient to excite a sensation.

**Prop. III.**

When rays of light come out of a drop of rain after one reflection, those will be effectual which are reflected from the same point, and which entered the drop near to one another.

Any rays, as \(s, b\) and \(c, d\), when they have passed out of the air into a drop of water, will be refracted towards the perpendiculars \(b, l, d, l\); and as the ray \(s, b\) falls farther from the axis \(a\) than the ray \(c, d\), \(s, b\) will be more refracted than \(c, d\); so that these rays, though parallel to one another at their incidence, may describe the lines \(b, c\) and \(d, e\) after refraction, and be reflected from the same point \(e\). Now all rays, which are thus reflected from the same point, when they have described the lines \(e, f, g\), and after reflection emerge at \(f\) and \(g\), will be so refracted, when they pass out of the drop into the air, as to describe the parallel lines \(h, i, j\). If these rays were to return from \(e\) in the lines \(c, b, e, d\), and were to emerge at \(b\) and \(d\), they would be refracted into the lines of their incidence \(b, x, d, c\). But if these rays, instead of being returned in the lines \(c, b, e, d\), are reflected from the same point \(e\) in the lines \(e, g, e, f\), the lines of reflection \(e, g\) and \(e, f\) will be inclined to one another and to the surface of the drop, just as much as the lines \(c, b\) and \(e, d\) are. First, \(e, b\) and \(e, g\) make the same angle with the surface of the drop; for the angle \(b, x\), which \(e, b\) makes with the surface of the drop, is the complement of incidence, and the angle \(e, r\), which \(e, g\) makes with the surface, is the complement of reflection; and these two are equal to one another. In the same manner it might be shown, that \(e, d\) and \(e, f\) make equal angles with the surface of the drop. Secondly, The angle \(b, c, d = f, g\); or the reflected rays \(e, g, e, f\), and the incident rays \(b, e, d, e\), are equally inclined to each other. For the angle of incidence \(b, c, d = f, g\), the angle of reflection, and the angle of incidence \(d, e, l = f, e, l\), the angle... angle of reflection; consequently, the difference between the angles of incidence is equal to the difference between the angles of reflection, or \( b e l - d e l = g e l - f e l \), or \( b e d = g e f \). Since therefore either the lines \( e g, e f \), or the lines \( e b, e d \), are equally inclined both to one an- other and to the surface of the drop; the rays will be refracted in the same manner, whether they return in the lines \( e b, e d \), or are reflected in the lines \( e g, e f \). But if they return in the lines \( e b, e d \), the refraction, when they emerge at \( b \) and \( d \), would make them parallel. Therefore, if they are reflected from one and the same point \( c \) in the lines \( e g, e f \), the refraction, when they emerge at \( g \) and \( f \), will likewise make them parallel.

But though such rays as are reflected from the same point in the hinder part of a drop of rain, are parallel to one another when they emerge, and so have one condition that is requisite towards making them effec- tual, yet there is another condition necessary; for rays that are effectual must be contiguous as well as paral- lel. And though rays, which enter the drop in different places, may be parallel when they emerge, those only will be contiguous which enter it nearly at the same place.

Let \( XY \) be a drop of rain, \( a g \) the axis or diameter of the drop, and \( s a \) a ray of light that enters the drop at \( a \). This ray \( s a \), being perpendicular to both the sur- faces, will pass through the drop in the line \( a g h \) without being refracted; but any collateral rays, such as those that fall about \( s b \), will be made to converge to the axis, and passing out at \( n \) will meet the axis at \( k \); Rays which fall farther from the axis than \( s b \), such as those which fall about \( s c \), will likewise be made to converge; but their focus will be nearer to the drop than \( h \). Suppose therefore \( i \) to be the focus of the rays that fall about \( s c \), any ray \( s c \), when it has described the line \( c o \) within the drop, and is tending to the focus \( i \), will pass out of the drop at the point \( o \). The rays that fall upon the drop about \( s d \), will converge to a focus still nearer than \( i \), as at \( k \). These rays therefore go out of the drop at \( p \). The rays, that fall about \( s c \), will con- verge to a focus nearer than \( k \), as suppose at \( l \); and the ray \( s c \), when it has described the line \( c o \) within the drop, and is tending to \( h \), will pass out at the point \( o \). The rays that fall still more remote from the axis will converge to a focus still nearer. Thus the ray \( s f \) will after refraction converge to a focus at \( m \), which is nearer than \( l \); and having described the line \( f n \) within the drop, it will pass out to the point \( n \). Now we may here observe, that as any rays \( s b \) or \( s c \), fall farther above the axis \( s a \), the points \( n \) or \( o \), where they pass out behind the drop, will be farther above \( g \); or that, as the incident ray rises from the axis \( s a \), the arc \( g n o \) increases, till we come to some ray \( s d \), which passes out of the drop at \( p \); and this is the highest point where any ray that falls upon the quadrant or quarter \( ax \) can pass out: for any rays \( f e \), or \( s f \), that fall higher than \( s d \), will not pass out on any point above \( p \), but at the points \( o \), or \( n \), which are below it. Consequently, though the arc \( g n o \) increases, whilst the distance of the incident ray from the axis \( s a \) increases, till we come to the ray \( s d \); yet afterwards, the higher the ray falls above the axis \( s a \), this arc \( p o n g \) will decrease.

We have hitherto spoken of the points on the pos- terior part of the drop, where the rays pass out of it; but this was for the sake of determining the points from which those rays are reflected, which do not pass out behind the drop. For, in explaining the rainbow, we have no further reason to consider those rays which go through the drop; since they can never come to the eye of a spectator placed anywhere in the lines \( r v \) or \( q t \) with his face towards the drop. Now, as there are many rays which pass out of the drop between \( g \) and \( p \), so some rays will be thence reflected: and consequently the several points between \( g \) and \( p \), which are the points where some of the rays pass out of the drop, are like- wise the points of reflection for the rest which do not pass out. Therefore in respect of those rays which are reflected, we may call \( g p \) the arc of reflection; and may say, that this arc of reflection increases, as the dis- tance of the incident ray from the axis \( s a \) increases, till we come to the ray \( s d \); the arc of reflection is \( g n \) for the ray \( s b \), it is \( g o \) for the ray \( s c \), and \( g p \) for the ray \( s d \). But after this, as the distance of the incident ray from the axis \( s a \) increases, the arc of reflection de- creases; for \( o g \) less than \( p g \) is the arc of reflection for the ray \( s c \), and \( p g \) is the arc of reflection for the ray \( s f \).

Hence it is obvious, that some ray, which falls above \( s d \), may be reflected from the same point with some other ray which falls below \( s d \). Thus, for instance, the ray \( s b \) will be reflected from the point \( n \), and the rays \( s f \) will be reflected from the same point; and conse- quently, when the reflected rays \( n r, n q \), are refracted as they pass out of the drop at \( r \) and \( q \), they will be par- allel. But since the intermediate rays, which enter the drop between \( s f \) and \( s b \), are not reflected from the same point \( n \), these two rays alone will be parallel to one another when they come out of the drop, and the intermediate rays will not be parallel to them. And consequently these rays \( r v, q t \), though they are parallel after they emerge at \( r \) and \( q \), will not be contiguous, and for that reason will not be effectual; the ray \( s d \) is reflected from \( p \), which has been shown to be the limit of the arc of reflection; such rays as fall just above \( s d \), and just below \( s d \), will be reflected from nearly the same point \( p \), as appears from what has been already shown. These rays therefore will be parallel, because they are reflected from the same point \( p \); and they will likewise be contiguous, because they all of them enter the drop at the same place very near to \( d \). Conse- quently, such rays as enter the drop at \( d \), and are re- flected from \( p \) the limit of the arc of reflection, will be effectual; since, when they emerge at the part of the drop between \( a \) and \( y \), they will be both parallel and contiguous.

If it can be shown that the rainbow is produced by the rays of the sun which are thus reflected from drops of rain as they fall while the sun shines upon them, this proposition may serve to show us, that this appearance is not produced by any rays that fall upon any part, and are reflected from any part of those drops: since this appearance cannot be produced by any rays but those which are effectual; and effectual rays must al- ways enter each drop at one certain place in the ante- rior part of it, and must likewise be reflected from one certain place in the posterior surface.

Prop. IV.

When rays that are effectual emerge from a drop of rain after one reflection and two refrac- tions, those which are most refrangible will,

K k at their immersion, make a less angle with the incident rays than those which are least refrangible; and by this means the rays of different colours will be separated from one another.

Let \( f h \) and \( g i \) be effectual violet rays emerging from the drop at \( f g \); and \( f n \), \( g p \), effectual red rays emerging from the same drop at the same place. Now, though all the violet rays are parallel to one another, because they are supposed effectual, and though all the red rays are likewise parallel to one another from the same reason; yet the violet rays will not be parallel to the red rays. These rays, as they have different degrees of refrangibility, will diverge from one another; any violet ray \( g i \), which emerges at \( g \), will diverge from any red ray \( g p \), which emerges at the same place. Now, both the violet ray \( g i \), and the red ray \( g p \), as they pass out of the drop of water into the air, will be refracted from the perpendicular \( k o \). But the violet ray is more refrangible than the red one; and for that reason \( g i \), or the refracted violet ray, will make a greater angle with the perpendicular than \( g p \), the refracted red ray; or the angle \( i g o \) will be greater than the angle \( p g o \).

Suppose the incident ray \( s b \) to be continued in the direction \( s k \), and the violet ray \( i g \) to be continued backwards in the direction \( ik \), till it meets the incident ray at \( k \). Suppose likewise the red ray \( g \) to be continued backwards in the same manner, till it meets the incident ray at \( w \). The angle \( i k s \) is that which the violet ray, or most refrangible ray at its immersion, makes with the incident ray; and the angle \( p w s \), is that which the red ray or least refrangible ray at its immersion, makes with the incident ray. The angle \( i k s \) is less than the angle \( p w s \). For, in the triangle, \( g w k \), \( g w s \), or \( p w s \), is the external angle at the base, and \( g k w \) or \( i k s \) is one of the internal opposite angles. (Eucl. B. I. Prop. xvi.). What has been shown to be true of the rays \( g i \) and \( g p \) might be shown in the same manner of the rays \( f h \) and \( f n \), or of any other rays that emerge respectively parallel to \( g i \) and \( g p \). But all the effectual violet rays are parallel to \( g i \), and all the effectual red rays are parallel to \( g p \). Therefore the effectual violet rays at their immersion make a less angle with the incident ones than the effectual red ones. For the same reason, in all the other sorts of rays, those which are most refrangible, at their immersion from a drop of rain after one reflection, will make a less angle with the incident rays, than those do which are less refrangible.

Otherwise: When the rays \( g i \) and \( g p \) emerge at the same point \( g \), as they both come out of the water into air, and consequently are refracted from the perpendicular, instead of going straight forwards in the line \( e g \) continued, they will both be turned round upon the point \( g \) from the perpendicular \( g o \). Now it is easy to conceive, that either of these lines might be turned in this manner upon the point \( g \) as upon a centre, till they became parallel to \( s b \) the incident ray. But if either of these lines or rays were refracted so much from \( g o \) as to become parallel to \( s b \), the ray thus refracted, would, after emersion, make no angle with \( s k \), because it would be parallel to it. Consequently that ray which is most turned round upon the point \( g \), or that ray which is most refrangible, will after emersion be nearest parallel to the incident ray, or will make the least angle with it. The same may be proved of all other rays emerging parallel to \( g i \) and \( g p \) respectively, or of all effectual rays; those which are most refrangible will after emersion make a less angle with the incident rays, than those do which are least refrangible.

But since the effectual rays of different colours make different angles with \( s k \) at their emersion, they will be separated from one another: so that if the eye were placed in the beam \( f g h i \), it would receive only rays of one colour from the drop \( x o g v \); and if it were placed in the beam \( f g n p \), it would receive only rays of some other colour.

The angle \( s w p \), which the least refrangible or red rays make with the incident ones when they emerge so as to be effectual, is found by calculation to be \( 42^\circ 2' \). And the angle \( s k i \), which the most refrangible rays make with the incident ones when they emerge so as to be effectual, is found to be \( 45^\circ 17' \). The rays which have the intermediate degrees of refrangibility, make with the incident ones intermediate angles between \( 42^\circ 2' \) and \( 45^\circ 17' \).

**Prop. V.**

If a line be supposed to be drawn from the centre of the sun through the eye of the spectator, the angle which any effectual ray, after two refractions and one reflection, makes with the incident ray, will be equal to the angle which it makes with that line.

Let the eye of the spectator be at \( i \), and let \( q t \) be the line supposed to be drawn from the centre of the sun through the eye of the spectator; the angle \( g i t \), which any effectual ray makes with this line, will be equal to the angle \( i k s \), which the same ray makes with the incident ray \( s b \) or \( s k \). If \( s b \) is a ray coming from the centre of the sun, then since \( q t \) is supposed to be drawn from the same point, these two lines, upon account of the remoteness of the point from whence they are drawn, may be looked upon as parallel to one another. But the right line \( k r \) crossing these two parallel lines will make the alternate angles equal. (Eucl. B. I. Prop. xxix.). Therefore \( k i t \) or \( g i t = s k i \).

**Prop. VI.**

When the sun shines upon the drops of rain as they are falling, the rays that come from those drops to the eye of a spectator, after one reflection and two refractions, produce the primary rainbow.

If the sun shines upon the rain as it falls, there are commonly seen two bows, as \( AFB \), \( CHD \); or if the bow seen cloud and rain does not reach over that whole side of the sky where the bows appear, then only a part of one or of both bows is seen in that place where the rain falls. Of these two bows, the innermost \( AFB \) is the more vivid of the two, and this is called the primary bow. The outer part \( TFY \) of the primary bow is red, the inner part \( VEX \) is violet; the intermediate parts, reckoning from the red to the violet, are orange, yellow, green, blue, and indigo. Suppose the spectator's eye to be at \( O \), and let \( LOP \) be an imaginary line drawn... drawn from the centre of the sun through the eye of the spectator: if a beam of light S coming from the sun fall upon any drop F; and the rays that emerge at F in the line FO, so as to be effectual, make an angle FOP of $42^\circ 2'$ with the line LP; then these effectual rays make an angle of $42^\circ 2'$ with the incident rays, by the preceding proposition, and consequently these rays will be red, so that the drop F will appear red. All the other rays, which emerge at F, and would be effectual if they fell upon the eye, are refracted more than the red ones, and consequently will pass above the eye. If a beam of light S fall upon the drop E, and the rays that emerge at E in the line EO, so as to be effectual, make an angle of $40^\circ 17'$ with the line LP; then these effectual rays make likewise an angle of $40^\circ 17'$ with the incident rays, and the drop E will appear of a violet colour. All the other rays, which emerge at E, and would be effectual if they came to the eye, are refracted less than the violet ones, and therefore pass below the eye. The intermediate drops between F and E will for the same reasons be of the intermediate colours.

Thus we have shown why a set of drops from F to E, as they are falling, should appear of the seven primary colours. It is not necessary that the several drops, which produce these colours, should all of them fall at exactly the same distance from the eye. The angle FOP, for example, is the same whether the distance of the drop from the eye is OF, or whether it is in any other part of the line OF something nearer to the eye. And whilst the angle FOP is the same, the angle made by the emerging and incident rays, and consequently the colour of the drop, will be the same. This is equally true of any other drop. So that though in the figure the drops F and E are represented as falling perpendicularly one under the other, yet this is not necessary in order to produce the bow.

But the coloured line FE, which we have already accounted for, is only the breadth of the bow. It still remains to be shewn, why not only the drop F should appear red, but why all the other drops from A to B in the arc ATFYB should appear of the same colour. Now it is evident, that wherever a drop of rain is placed, if the angle which the effectual rays make with the line LP is equal to the angle FOP, that is, if the angle which the effectual rays make with the incident rays is $42^\circ 2'$, any of those drops will be red, for the same reason that the drop F is of this colour.

If FOP were to turn round upon the line OP, so that one end of this line should always be at the eye, and the other be at P opposite to the sun; such a motion of this figure would be like that of a pair of compasses turning round upon one of the legs OP with the opening FOP. In this revolution the drop F would describe a circle, P would be the centre, and ATFYB would be an arc in this circle. Now since, in this motion of the line and drop OF, the angle made by FO with OP, that is, the angle FOP, continues the same; if the sun were to shine upon this drop as it revolves, the effectual rays would make the same angle with the incident rays, in whatever part of the arc ATFYB the drop was to be. Therefore, whether the drop be at A, or at T, or at Y, or at B, or wherever else it is in this whole arc, it would appear red, as it does at F.—The drops of rain, as they fall, are not indeed turned round in this manner: but then, as great numbers of them are falling at once in right lines from the cloud, whilst one drop is at F, there will be others at Y, at T, at B, at A, and in every other part of the arc ATFYB: and all these drops will be red for the same reason that the drop F would have been red, if it had been in the same place. Therefore, when the sun shines upon the rain as it falls, there will be a red arc ATFYB opposite to the sun. In the same manner, because the drop E is violet, we might prove that any other drop, which, whilst it is falling, is in any part of the arc AVEXB, will be violet; and consequently, at the same time that the red arc ATFYB appears, there will likewise be a violet arc AVEXB below or within it. FE is the distance between these two coloured arcs; and from what has been said, it follows, that the intermediate space between these two arcs will be filled up with arcs of the intermediate colours, orange, yellow, blue, green, and indigo. All these coloured arcs together make up the primary rainbow.

**Prop. VII.**

The primary rainbow is never a greater arc than a semicircle.

Since the line LOP is drawn from the sun through the eye of the spectator, and since P is the centre of the rainbow; it follows, that the centre of the rainbow is always opposite to the sun. The angle FOP is an angle of $40^\circ 2'$, as was observed, or F the highest part of the bow is $42^\circ 2'$ from P the centre of it. If the sun is more than $42^\circ 2'$ high, P the centre of the rainbow, which is opposite to the sun, will be more than $42^\circ 2'$ below the horizon; and consequently F the top of the bow, which is only $42^\circ 2'$ from P, will be below the horizon; that is, when the sun is more than $42^\circ 2'$ high, no primary rainbow will be seen. If the altitude of the sun be something less than $40^\circ 2'$, then P will be something less than $42^\circ 2'$ below the horizon; and consequently F, which is only $42^\circ 2'$ from P, will be just above the horizon; that is, a small part of the bow at this height of the sun will appear close to the ground opposite to the sun. If the sun be $22^\circ$ high, then P will be $22^\circ$ below the horizon; and F the top of the bow, being $42^\circ 2'$ from P, will be $22^\circ 2'$ above the horizon; therefore, at this height of the sun, the bow will be an arc of a circle whose centre is below the horizon; and consequently that arc of the circle which is above the horizon, or the bow, will be less than a semicircle. If the sun be in the horizon, then P, the centre of the bow, will be in the opposite part of the horizon; F, the top of the bow, will be $42^\circ 2'$ above the horizon; and the bow itself, because the horizon passes through the centre of it, will be a semicircle. More than a semicircle can never appear; because if the bow were more than a semicircle, P the centre of it must be above the horizon; but P is always opposite to the sun, therefore P cannot be above the horizon, unless the sun is below it; and when the sun is set, or is below the horizon, it cannot shine upon the drops of rain as they fall; and consequently, when the sun is below the horizon, no bow at all can be seen.

**Prop VIII.**

When the rays of the sun fall upon a drop of rain, some of them, after two reflections and two re- fractions, may come to the eye of a spectator who has his back towards the sun and his face towards the drop.

If HGW is a drop of rain, and parallel rays coming from the sun, as v, y w, fall upon the lower part of it, they will be refracted towards the perpendiculars v l, w l, as they enter into it, and will describe some such lines as v h, w i. At h and i great part of these rays will pass out of the drop; but some of them will be reflected from thence in the lines h f, i g. At f and g again, great part of the rays that were reflected thither will pass out of the drop. But these rays will not come to the eye of a spectator at o. Here, however, all the rays will not pass out; but some will be reflected from f and g, in some such lines as f d, g h; and these, when they emerge out of the drop of water into the air at b and d, will be refracted from the perpendiculars, and, describing the lines d t, b o, may come to the eye of the spectator who has his back towards the sun and his face towards the drop.

Prop. IX.

Those rays, which are parallel to one another after they have been once refracted and once reflected in a drop of rain, will be effectual when they emerge after two refractions and two reflections.

No rays can be effectual, unless they are contiguous and parallel. It appears from what was said, that when rays come out of a drop of rain contiguous to one another, either after one or after two reflections, they must enter the drop nearly at the same place. And if such rays as are contiguous are also parallel after the first reflection, they will emerge parallel, and therefore will be effectual. Let v r and y w be contiguous rays which come from the sun, and are parallel when they fall upon the lower part of the drop; suppose these rays to be refracted at v and w, and to be reflected at h and i; if they are parallel, as h f, i g, after this first reflection, then, after they are reflected a second time from f and g, and refracted a second time as they emerge at d and b, they will go out of the drop in the parallel lines d t and b o, and will therefore be effectual.

The rays v r, y w, are refracted towards the perpendiculars v l, w l, when they enter the drop, and will be made to converge. As these rays are very oblique, their focus will not be far from the surface v w. If this focus be at k, the rays, after they have passed the focus, will diverge from thence in the directions k h, k i; and if k i is the principal focal distance of the concave reflecting surface h i, the reflected rays h f, i g, will be parallel. These rays e f, i g, are reflected again from the concave surface f g, and will meet in a focus at e, so that g e will be the principal focal distance of this reflecting surface f g. And because h i and f g are parts of the same sphere, the principal focal distances g e and k i will be equal. When the rays have passed the focus e, they will thence diverge in the line e d, e b; and we are to show, that when they emerge at d and b, and are refracted there, they will become parallel.

Now if the rays v k, w k, when they have met at k, were to be turned back again in the directions k v, k w, and were to emerge at v and w, they would be refracted into the lines of their incidence, v x, w y, and therefore would be parallel. But since g e = i k, as has already been shown, the rays e d, e b, that diverge from c, fall in the same manner upon the drop at d and b, as the rays k v, k w, would fall upon it at v and w; and e d, e b, are just as much inclined to the refracting surface d b, as k v, k w would be to the surface v w. Hence it follows, that the rays e d, e b, emerging at d and b, will be refracted in the same manner, and will have the same direction in respect of one another, as k v, k w would have. But k v and k w would be parallel after refraction. Therefore the rays e d and e b will emerge in the lines d p, b o, parallel to one another, and consequently effectual.

Prof. X.

When rays that are effectual emerge from a drop of rain after two reflections and two refractions, those which are most refrangible will at their emersion make a greater angle with the incident rays than those do which are least refrangible; and by this means the rays of different colours will be separated from one another.

If rays of different colours, which are differently refrangible, emerge at any point b, these rays will not be all of them equally refracted from the perpendicular. Thus, if b o is a red ray, which is of all others the least refrangible, and b m is a violet ray, which is of all others the most refrangible; when these two rays emerge at b, the violet ray will be refracted more from the perpendicular b x than the red ray, and the refracted angle x b m will be greater than the refracted angle x b o. Hence it follows, that these two rays, after emersion, will diverge from one another. In like manner, the rays that emerge at d will diverge from one another; a red ray will emerge in the line d p, a violet ray in the line d t. So that though all the effectual red rays of the beam b d m t are parallel to one another, and all the effectual red rays of the beam b d o p are likewise parallel, yet the violet will not be parallel to the red beam. Thus the rays of different colours will be separated from one another.

This will appear farther, if we consider what the proposition affirms, That any violet or most refrangible ray will make a greater angle with the incident rays, than any red or least refrangible ray makes with the same incident rays. Thus if y w be an incident ray, b m a violet ray emerging from the point b, and b o a red ray emerging from the same point; the angle which the violet ray makes with the incident one is y r m, and that which the red ray makes with it is y s o. Now y r m is greater than y s o. For in the triangle b r s, the internal angle b r s is less than b s y the external angle at the base. (Eucl. B. I. Prop. xvi.). But y r m is the complement of b r s or of b r y to two right ones, and y s o is the complement of b s y to two right ones. Therefore, since b r y is less than b s y, the complement of b r y to two right angles will be greater than the complement of b s y to two right angles; or y r m will be greater than y s o.

Otherwise: Both the rays b o and b m, when they are refracted in passing out of the drop at b, are turned round upon the point b from the perpendicular b x. Now either of these lines b o or b m might be turned round. Of the round in this manner, till it made a right angle with \( y w \). Consequently, that ray which is most turned round upon \( b \), or which is most refracted, will make an angle with \( y w \), that will be nearer to a right one than that ray makes with it which is least turned round upon \( b \), or which is least refracted. Therefore that ray which is most refracted will make a greater angle with the incident ray than that which is least refracted.

But since the emerging rays, being differently refrangible, make different angles with the same incident ray \( y w \), the refraction which they suffer at emersion will separate them from one another.

The angle \( y r m \), which the most refrangible or violet rays make with the incident ones, is found by calculation to be \( 54^\circ 7' \); and the angle \( y s o \), which the least refrangible or red rays make with the incident ones, is found to be \( 59^\circ 57' \); the angles, which the rays of the intermediate colours, indigo, blue, green, yellow, and orange, make with the incident rays, are intermediate angles between \( 54^\circ 7' \) and \( 59^\circ 57' \).

**Prop. XI.**

If a line is supposed to be drawn from the centre of the sun through the eye of the spectator; the angle which, after two refractions and two reflections, any effectual ray makes with the incident ray, will be equal to the angle which it makes with that line.

If \( y w \) is an incident ray, \( b o \) an effectual ray, and \( q n \) a line drawn from the centre of the sun through \( o \) the eye of the spectator; the angle \( y s o \), which the effectual ray makes with the incident ray, is equal to \( s o n \) the angle which the same effectual ray makes with the line \( q n \). For \( y w \) and \( q n \), considered as drawn from the centre of the sun, are parallel; \( b o \) crosses them, and consequently makes the alternate angles \( y s o, s o n \), equal to one another. *Eucl. B. I. Prop. xxix.*

**Prop. XII.**

When the sun shines upon the drops of rain as they are falling, the rays, that come from these drops to the eye of a spectator, after two reflections and two refractions, produce the secondary rainbow.

The secondary rainbow is the outermost CHD. When the sun shines upon a drop of rain \( H \); and the rays \( HO \), which emerge at \( H \) so as to be effectual, make an angle \( HOP \) of \( 54^\circ 7' \) with \( LOF \) a line drawn from the sun through the eye of the spectator; the same effectual rays will make likewise an angle of \( 54^\circ 7' \) with the incident rays \( S \), and the rays which emerge at this angle are violet ones, by what was observed above. Therefore, if the spectator's eye is at \( O \), none but violet rays will enter it: for as all the other rays make a less angle with \( OP \), they will fall above the spectator's eye. In like manner, if the effectual rays that emerge from the drop \( G \) make an angle of \( 59^\circ 57' \) with the line \( OP \), they will likewise make the same angle with the incident rays \( S \); and consequently, from the drop \( G \) no rays will come to the spectator's eye at \( O \) but red ones; for all the other rays making a greater angle with the line \( OP \), will fall below the eye at \( O \). For the same reason, the rays emerging from the intermediate drops between \( H \) and \( G \), and coming to the spectator's eye at \( O \), will emerge at intermediate angles, and therefore will have the intermediate colours. Thus if there are seven drops from \( H \) to \( G \) inclusively, their colours will be violet, indigo, blue, green, yellow, orange, and red. This coloured line is the breadth of the secondary rainbow.

Now, if \( HOP \) were to turn round upon the line \( OP \), like a pair of compasses upon one of the legs \( OP \) with the opening \( HOP \), it is plain from the supposition, that, in such a revolution of the drop \( H \), the angle \( HOP \) would be the same, and consequently the emerging rays would make the same angle with the incident ones. But in such a revolution the drop would describe a circle of which \( P \) would be the centre, and \( CNHRD \) an arc. Consequently, since, when the drop is at \( N \), or at \( R \), or anywhere else in that arc, the emerging rays make the same angle with the incident ones as when the drop is at \( H \), the colour of the drop will be the same to an eye placed at \( O \), whether the drop is at \( N \), or at \( H \), or at \( R \), or anywhere else in that arc. Now, though the drop does not thus turn round as it falls, and does not pass through the several parts of this arc, yet, since there are drops of rain falling everywhere at the same time, when one drop is at \( H \), there will be another at \( R \), another at \( N \), and others in all parts of the arc; and these drops will all be violet-coloured, for the same reason that the drop \( H \) would have been of this colour if it had been in any of those places. In like manner, as the drop \( G \) is red when it is at \( G \), it would likewise be red in any part of the arc \( CWGQD \); and so will any other drop when, as it is falling, it comes to any part of that arc. Thus as the sun shines upon the rain, whilst it falls, there will be two arcs produced, a violet-coloured arc \( CNHRD \), and a red one \( CWGQD \); and for the same reasons the intermediate space between these two arcs will be filled up with arcs of the intermediate colours. All these arcs together make up the secondary rainbow.

**Prop. XIII.**

The colours of the secondary rainbow are fainter than those of the primary rainbow; and are arranged in the contrary order.

The primary rainbow is produced by such rays as have been only once reflected; the secondary rainbow colours are produced by such rays as have been twice reflected. But at every reflection some rays pass out of the drop of rain without being reflected; so that the fewer the rays are reflected, the fewer of them are left. Therefore, those of the primary bow, the colours of the secondary bow are produced by primary, fewer rays, and consequently will be fainter, than the colours of the primary bow.

In the primary bow, reckoning from the outside of it, the colours are arranged in this order; red, orange, yellow, green, blue, indigo, violet. In the secondary bow, reckoning from the outside, the colours are violet, indigo, blue, green, yellow, orange, red. So that the red, which is the outermost or highest colour in the primary bow, is the innermost or lowest colour in the secondary one.

Now the violet rays, when they emerge so as to be Concavity effectual after one reflection, make a less angle with of the Sky; the incident rays than the red ones; consequently the violet rays make a less angle with the lines OP than the red ones. But, in the primary rainbow, the rays are only once reflected, and the angle which the effec- tual rays make with OP is the distance of the colour- ed drop from P the centre of the bow. Therefore the violet drops, or violet arc, in the primary bow, will be nearer to the centre of the bow than the red drops or red arc; that is, the innermost colour in the primary bow will be violet, and the outermost colour will be red. And, for the same reason, through the whole primary bow, every colour will be nearer the centre P, as the rays of that colour are more refrangible.

But the violet rays, when they emerge so as to be effectual after two reflections, make a greater angle with the incident rays than the red ones; consequently the violet rays will make a greater angle with the line OP, than the red ones. But in the secondary rainbow the rays are twice reflected, and the angle which the effectual rays make with OP is the distance of the col- oured drop from P the centre of the bow. Therefore the violet drops or violet arc in the secondary bow will be farther from the centre of the bow, than the red drops or red arc; that is, the outermost colour in the secondary bow will be violet, and the innermost co- lour will be red. And, for the same reason, through the whole secondary bow, every colour will be farther from the centre P, as the rays of that colour are more refrangible.

Sect. II. Of Coronas, Parhelia, &c.

Under the articles CORONA and PARHELION, a pret- ty full account is given of the different hypotheses con- cerning these phenomena, and likewise of the method by which these hypotheses are supported, from the known laws of refraction and reflection. See also the article CHROMATICS in the Supplement.

Sect. III. Of the Concave Figure of the Sky.

The apparent concavity of the sky is only an optical deception founded on the incapacity of our organs of vi- sion to take in very large distances. Dr Smith has de- monstrated, that, if the surface of the earth were perfect- ly plane, the distance of the visible horizon from the eye would scarcely exceed the distance of 5000 times the height of the eye above the ground: beyond this dis- tance, all objects would appear in the visible horizon. For, let OP be the height of the eye above the line PA drawn upon the ground; and if an object AB=PO, be removed to a distance PA equal to 5000 times that height, it will hardly be visible by reason of the small- ness of the angle AOB. Consequently any distance AC, however great, beyond A, will be invisible. For since AC and BO are parallel, the ray CO will always cut AB in some point D between A and B; and there- fore the angle AOC, or AOD, will always be less than AOB, and therefore AD or AC will be invisible. Consequently all objects and clouds, as CE and FG, placed at all distances beyond A, if they be high enough to be visible, or to subtend a bigger angle at the eye than AOB, will appear at the horizon AB; because the di- stance AC is invisible.

Hence, if we suppose a long row of objects, or a long wall ABZY, built upon this plane, and its perpendicu- lar distance OA from the eye at O to be equal to or con- cavity greater than the distance Oa of the visible horizon, it of the Sky will not appear straight, but circular, as if it were built upon the circumference of the horizon a e g y: and if the wall be continued to an immense distance, its ex- treme parts YZ will appear in the horizon at y z, where it is cut by a line O y parallel to the wall. For, supposing a ray YO, the angle YO y will become insen- sibly small. Imagine this infinite plane OAY y, with the wall upon it, to be turned about the horizontal line O like the lid of a box, till it becomes perpendicu- lar to the other half of the horizontal plane LMy, and the wall parallel to it, like a vast ceiling overhead; and then the wall will appear like the concave figure of the clouds overhead. But though the wall in the horizon appear in the figure of a semicircle, yet the ceiling will not, but much flatter. Because the hori- zontal plane was a visible surface, which suggested the idea of the same distances quite round the eye: but in the vertical plane extended between the eye and the ceiling, there is nothing that affects the sense with an idea of its parts but the common line O y; consequently the apparent distances of the higher parts of the ceiling will be gradually diminished in ascending from that line. Now when the sky is overcast with clouds of equal gravities, they will all float in the air at equal heights above the earth, and consequently will compose a sur- face resembling a large ceiling, as flat as the visible surface of the earth. Its concavity therefore is only apparent: and when the heights of the clouds are un- equal, since their real shapes and magnitudes are all unknown, the eye can seldom distinguish the unequal distances of those clouds that appear in the same di- rections, unless when they are very near us, or are driven by contrary currents of the air. So that the visible shape of the whole surface remains alike in both cases. And when the sky is either partly overcast or partly free from clouds, it is matter of fact that we re- tain much the same idea of its concavity as when it was quite overcast.

The concavity of the heavens appears to the eye, Why the which is the only judge of an apparent figure, to be a concavity less portion of a spherical surface than a hemisphere of the sky Dr Smith says, that the centre of the concavity is than a he- much below the eye: and by taking a medium amongmishere, several observations he found the apparent distance of its parts at the horizon to be generally between three and four times greater than the apparent distance of its parts overhead. For let the arch ABCD repre- sent the apparent concavity of the sky, O the place of the eye, OA and OC the horizontal and vertical ap- parent distances, whose proportion is required. First observe when the sun or the moon, or any cloud or star, is in such a situation at B, that the apparent arches BA, BC, extended on each side of this object towards the horizon and zenith, seem equal to the eye; then taking the altitude of the object B with a quadrant, or a cross staff, or finding it by astronomy from the given time of observation, the angle AOB is known. Drawing therefore the line OB in the position thus determined, and taking in it any point B, in the ver- tical line CO produced downwards, find the centre E of a circle ABC, whose arches BA, BC, intercepted between B and the legs of the right angle AOC, shall be equal to each other; then will this arch ABCD re- Blue colour present the apparent figure of the sky. For by the eye of the sky, we estimate the distance between any two objects in the heavens by the quantity of sky that appears to lie between them; as upon earth we estimate it by the quantity of ground that lies between them. The centre E may be found geometrically by constructing a cubic equation, or as quickly and sufficiently exact by trying whether the chords BA, BC, of the arch ABC drawn by conjecture are equal, and by altering its radius BE till they are so. Now in making several observations upon the sun, and some others upon the moon and stars, they seemed to our author to bisect the vertical arch ABC at B, when their apparent altitudes or the angle AOB was about 23 degrees; which gives the proportion of OC to OA as 3 to 10 or as 1 to 3 nearly. When the altitude of the sun was 30°, the upper arch seemed always less than the under one; and, in our author's opinion, always greater when the sun was about 18 or 20 degrees high.

**SECT. IV. Of the Blue Colour of the Sky, and of Blue and Green Shadows.**

The opinions of ancient writers concerning the colour of the sky merit no notice. The first who gave any rational explanation was Fromondus. He supposed that the blueness of the sky proceeded from a mixture of the white light of the sun with the black space beyond the atmosphere, where there is neither refraction nor reflection. This opinion very generally prevailed, and was maintained by Otto Guerick and all his contemporaries, who asserted, that white and black may be mixed in such a manner as to make a blue. M. Bouguer had recourse to the vapours diffused through the atmosphere, to account for the reflection of the blue rays rather than any other. He seems, however, to suppose, that it arises from the constitution of the air itself, from which the fainter-coloured rays are incapable of making their way through any considerable tract of it. Hence he is of opinion, that the colour of the air is properly blue; to which opinion Dr Smith seems also to have inclined.

To this blue colour of the sky is owing the appearance of blue and green shadows in the morning and evening.—These were first observed by M. Buffon in 1742, when he noticed that the shadows of trees which fell upon a white wall were green. He was at that time standing upon an eminence, and the sun was setting in the cleft of a mountain, so that he appeared considerably lower than the horizon. The sky was clear, excepting in the west, which, through free from clouds, was lightly shaded with vapours, of a yellow colour, inclining to red. Then the sun itself was exceedingly red, and was apparently at least four times as large as he appears to be at mid-day. In these circumstances he saw very distinctly the shadows of the trees, which were 50 or 40 feet from the white wall, coloured with a light green inclining to blue. The shadow of an arbour, which was three feet from the wall, was exactly drawn upon it, and looked as if it had been newly painted with verdigrise. This appearance lasted near five minutes; after which it grew fainter, and vanished at the same time with the light of the sun.

The next morning at sunrise, he went to observe other shadows, upon another white wall; but instead of finding them green as before, he observed that they were of the colour of lively indigo. The sky was serene, except a slight covering of yellowish vapours in the east; and the sun rose behind a hill, so that it was elevated above his horizon. In these circumstances, the blue shadows were only visible three minutes; after which they appeared black, and in the evening of the same day he observed the green shadows exactly as before. On another day at sunset he observed that the shadows were not green, but of a beautiful sky-blue. He also observed that the sky was in a great measure free from vapours at that time, and that the sun set behind a rock, so that it disappeared before it came to his horizon. Afterwards, he often observed the shadows both at sunrise and sunset; but always perceived them to be blue, though with a great variety of shades.

The first person who attempted to explain this phenomenon was the Abbé Mazées. He observed that of these when an opaque body was illuminated by the moon and a candle at the same time, and the two shadows were cast upon the same white wall, that which was enlightened by the candle was reddish, and that which was enlightened by the moon was blue. He supposed, however, the change of colour to be occasioned by the diminution of the light; but M. Melville and M. Bouguer, both independent of one another, seem to have hit upon the true cause of this curious appearance, and which guer's explanation has already hinted at. The former of these gentlemen, in his attempts to explain the blue colour of the sky, observes, that since it is certain that no body assumes any particular colour, but because it reflects one sort of rays more abundantly than the rest; and since it cannot be supposed that the constituent parts of pure air are gross enough to separate any colours of themselves; we must conclude with Sir Isaac Newton, that the violet and blue making rays are reflected more copiously than the rest, by the finer vapours diffused through the atmosphere, whose parts are not big enough to give them the appearance of visible opaque clouds. And he shows that in proper circumstances, the bluish colour of the sky light may be actually seen on bodies illuminated by it, as, he says, it is objected should always happen upon this hypothesis. For that if, on a cloudless day, a sheet of white paper be exposed to the sun's beams, when any opaque body is placed upon it, the shadow which is illuminated by the sky only will appear remarkably bluish compared with the rest of the paper, which receives the sun's direct rays.

M. Bouguer, who has taken the most pains with this subject, observes, that as M. Buffon mentions the shadows appearing green only twice, and that at all other times they were blue, this is the colour which they regularly have, and that the blue was changed into green by some accidental circumstance. Green, he says, is only a composition of blue and yellow, so that this accidental change may have arisen from the mixture of some yellow rays in the blue shadow; and that perhaps the walls might have had that tinge, so that the blue is the only colour for which a general reason is required. This, he says, must be derived from the colour of pure air, which always appears blue, and which always reflects that colour upon all objects without distinction; but which is too faint to be perceived when our eyes are strongly affected by the light of the sun, reflected from other objects around us. To confirm this hypothesis, he adds some interesting observations of his own, in which this appearance is agreeably diversified. Being at the village of Boucholtz in July 1764, he observed the shadows projected on the white paper of his pocket-book when the sky was clear.

At half an hour past six in the evening, when the sun was about 4° high, he observed that the shadow of his finger was of a dark gray, while he held the paper opposite to the sun; but when he inclined it almost horizontally, the paper had a bluish cast, and the shadow upon it was of a beautiful bright blue.

When his eye was placed between the sun and the paper laid horizontally, it always appeared of a bluish cast; but when he held the paper thus inclined between his eye and the sun, he could distinguish, upon every little eminence occasioned by the inequality of the surface of the paper, the chief prismatic colours. This multitude of coloured points, red, yellow, green, and blue, almost effaced the natural colour of the objects.

At 6h 45' the shadows began to be blue, even when the rays of the sun fell perpendicularly. The colour was the most lively when the rays fell upon it at an angle of 45°; but with a less inclination of the paper, he could distinctly perceive, that the blue shadow had a border of a stronger blue on that side which looked towards the sky, and a red border on that side which was turned towards the earth. To see these borders, it was necessary to place the body that made the shadow very near the paper; and the nearer it was, the more sensible was the red border. At the distance of three inches, the whole shadow was blue. At every observation, after having held the paper towards the sky, he turned it towards the earth, which was covered with verdure; holding it in such a manner, that the sun might shine upon it while it received the shadows of various bodies; but in this position he could never perceive the shadow to be blue or green at any inclination with respect to the sun's rays.

At seven o'clock, the altitude of the sun being still about two degrees, the shadows were of a bright blue, even when the rays fell perpendicularly upon the paper, but were brightest when it was inclined 45°. At this time he was surprised to observe, that a large tract of sky was not favourable to the production of this blue colour, and that the shadow falling upon the paper placed horizontally was not coloured, or at least the blue was very faint. This singularity, he concluded, arose from the small difference between the light of that part of the paper which received the rays of the sun and that which was in the shade in this situation. In a situation precisely horizontal, the difference would vanish, and there could be no shadow. Thus too much or too little of the sun's light produced, but for different reasons, the same effect; for they both made the blue light reflected from the sky become insensible. This gentleman never saw any green shadows; but supposes that the cause of those seen by Buffon might be the mixture of yellow rays, reflected from the vapours, which he observes were of that colour.

These blue shadows, our author observes, are not confined to the times of the sunrise and sunsetting; on the 19th of July, when the sun has the greatest force, he observed them at three o'clock in the afternoon, but the sun at that time shone through a mist.

If the sky be clear the shadows begin to be blue, when, if they be projected horizontally, they are eight times as long as the height of the body that produces of the sky, them, that is, when the altitude of the sun's centre is 7° 8'. This observation, he says, was made in the beginning of August.

Besides these coloured shadows, which are produced by the interception of the direct rays of the sun, our author observed others similar to them at every hour of the day, in rooms into which the light of the sun was reflected from some white body, if any part of the clear sky could be seen from the place, and all unnecessary light was excluded as much as possible. He remarks, that the blue shadows may be seen at any hour of the day, even with the direct light of the sun; and that this colour will disappear in all those places of the shadow from which the blue sky cannot be seen.

All the observations that our author made upon the yellow or reddish borders of shadows above mentioned, led him to conclude, that they were occasioned by the interception of the sky light, whereby part of the shadow was illuminated either by the red rays reflected from the clouds when the sun is near the horizon, or from some terrestrial bodies in the neighbourhood. This conjecture is favoured by the necessity he was under of placing any body near the paper, in order to produce this bordered shadow, as he says it is easily demonstrated, that the interception of the sky light can only take place when the breadth of the opaque body is to its distance from the white ground on which the shadow falls, as twice the line of half the amplitude of the sky to its cosine.

At the conclusion of his observations on these blue shadows, he gives a short account of another kind of them, which he supposes to have the same origin. These shadows often saw early in the spring when reading by the light of a candle in the morning, and consequently with the twilight mixed with that of his candle. In these circumstances, the shadow made by intercepting the light of his candle, at the distance of about six feet, was of a beautiful and clear blue, which became deeper as the opaque body which made the shadow was brought nearer to the wall, and was exceedingly deep at the distance of a few inches only. But where the day light did not come, the shadows were all black without the least mixture of blue.

The explanations of the blue colour of the sky given by Newton and Bouguer are far from satisfactory, and we presume that the following method of accounting for that phenomenon affords the true explanation. The light which flows from any portion of the blue sky is obviously reflected light, which is thrown out into the atmosphere in all directions by the earth, and the clouds and vapours which surround it. The red or least refrangible rays of this light having a greater momentum than the blue or most refrangible rays, penetrate much farther into the atmosphere, and though a few may be reflected, yet almost all of them will be absorbed or lost before they can return to the earth's surface. On the contrary, the blue rays, having less momentum, are not capable of penetrating so far into a resisting medium, and are therefore reflected to the earth's surface, and give a blue colour to the expanse of the heavens. The blue colour of the sky is exactly the converse of the red colour which is perceived at great depths in the sea, and of the red hue of the morning and evening clouds. These phenomena Irradiations phenomena being produced by transmitted or refracted of the Sun's light; the red rays make their way through the medium light, &c., to the observer's eye, while the blue ones are reflected or absorbed.

Sect. V. Of the Irradiations of the Sun's Light appearing through the interstices of the Clouds.

This is an appearance which every one must have observed when the sky was pretty much overcast, and the clouds have many breaks or openings. At that time several large beams of light, something like the appearance of the light of the sun admitted into a smoky room, will be seen generally with a very considerable degree of divergency, as if the radiant point was situated at no great distance above the clouds. Dr Smith observes that this appearance is one of those which serve to demonstrate that very high and remote objects in the heavens do not appear to us in their real shapes and positions, but according to their perspective projections in the apparent concavity of the sky. He acquaints us, that though these beams are generally seen diverging, as represented in fig. 11, it is not always the case. He himself, in particular, once saw them converging towards a point diametrically opposite to the sun; for, as near as he could conjecture, the point to which they converged was situated as much below the horizon as the sun was then elevated above the opposite part of it. This part is represented by the line \( t D \), and the point below it in position to the sun is \( E_3 \) towards which all the beams \( v t, v t, \&c. \) appeared to converge.

Perceiving that the point of convergence was opposite to the sun, he suspected that this unusual phenomenon was but a case of the usual apparent divergence of the beams of the sun from his apparent place among the clouds, as represented in fig. 11; for though nothing is more common than for rays to diverge from a luminous body, yet the divergence of these beams in such large angles is not real, but apparent. Because it is impossible for the direct rays of the sun to cross one another at any point of the apparent concavity of the sky, in a greater angle than about half a degree. For the diameter of the earth being so very small, in comparison to the distance of the sun, as to subtend an angle at any point of his body of about 20 seconds; and the diameter of our visible horizon being extremely smaller than that of the earth; it is evident, that all the rays which fall upon the horizon from any given point of the sun, must be inclined to each other in the smallest angles imaginable: the greatest of them being as much smaller than that angle of 20" as the diameter of the visible horizon is smaller than that of the earth. All the rays that come to us from any given point of the sun may therefore be considered as parallel; as the rays \( e B g \) from the point \( e \), or \( f B h \) from the opposite point \( f \); and consequently the rays of these two pencils that come from opposite points of the sun's real diameter, and cross each other in the sun's apparent place \( B \) among the clouds, can form no greater an angle with each other than about half a degree; this angle of their intersection \( B f \) being the same as the sun would subtend to an eye placed among the clouds at \( B \), or (which is much the same) to an eye at \( O \) upon the ground. Because the sun's real distance \( OS \) is inconceivably greater than his apparent distance \( OB \). Therefore the rays of the sun, as \( B g, B h \), do really diverge from his apparent place \( B \) in no greater angles \( g B h \) than about half a degree. Nevertheless they appear to diverge from the place \( B \) in all possible angles, and even in opposite directions. Let us proceed then to an explanation of this apparent divergence, which is by no means self-evident; though at first sight we are apt to think it is, by not distinguishing the vast difference between the true and apparent distances of the sun.

Supposing all the rays of the sun to fall accurately parallel to each other upon the visible horizon, as they do very nearly, yet in both cases they must appear to diverge in all possible angles. Let us imagine the heavens to be partly overcast with a spacious stratum of broken clouds, \( v, v, v, \&c. \), parallel to the plane of the visible horizon, represented by the line \( AOD \); and when the sun's rays fall upon these clouds in the parallel lines \( s v, s v, \&c. \), let some of them pass through their interstices in the lines \( vt, vt, \&c. \), and fall upon the plane of the horizon at the places \( t, t, \&c. \). And since the rest of the incident rays \( sv, sv, \&c. \), are supposed to be intercepted from the place of the spectator at \( O \) by the cloud \( x \), and from the intervals between the transmitted rays \( vt, vt, \&c. \), by the clouds \( v, v, \&c. \), a small part of these latter rays \( vt, vt, \&c. \), when reflected every way from some certain kind of thin vapours floating in the air, may undoubtedly be sufficient to affect the eye with an appearance of lights and shades, in the form of bright beams in the places \( vt, vt, \&c. \), and of dark ones in the intervals between them; just as similar beams of light and shade appear in a room by reflections of the sun's rays from smoke or dust flying within it; the lights and shades being here occasioned by the transmission of the rays through some parts of the window, and by their interruption at other parts.

Now, if the apparent concavity of this stratum of clouds \( v, v, \) to the eye at \( O \), be represented by the arch \( ABCD \), and be cut in the point \( B \) by the line \( OB \), parallel to the beams \( vt \); it will be evident by the rules of perspective, that these long beams will not appear in their real places, but upon the concave \( AB CD \) diverging every way from the place \( B \), where the sun himself appears, or the cloud \( x \) that covers his body, as represented separately in full view in fig. 11.

And for the same reason, if the line \( BO \) be produced towards \( E_3 \) below the plane of the horizon \( AOD \), and the eye be directed towards the region of the sky directly above \( E_3 \), the lower ends of the same real beams \( vt, vt, \) will now appear upon the part \( DF \) of this concave; and will seem to converge towards the point \( E_3 \), situated just as much below the horizon as the opposite point \( B \) is above it: which is separately represented in full view in fig. 12.

For if the beams \( vt, vt, \) be supposed to be visible throughout their whole lengths, and the eye be directed in a plane perpendicular to them, here represented by the line \( OF \); they and their intervals will appear broadest in and about this plane, because these parts of them are nearest to the eye; and therefore their remoter parts and intervals will appear gradually narrower towards the opposite ends of the line \( BE \). As a farther illustration of this subject, we may conceive the spectator at \( O \) to be situated upon the top of so large a descent \( OHI \) towards a remote valley \( IK \), and the sun to be so very low, that the point \( E_3 \) opposite to him, may be seen above the horizon of this shady valley. In this case it is manifest, that the spectators at O of the Sun's would now see these beams converging so far as to meet each other at the point E in the sky itself.

This phenomenon is not seen in moonlight, probably because her light is too weak after reflections from any kind of vapours, to cause a sensible appearance of lights and shades so as to form these beams. And in the phenomenon of fig. 12, the converging sunbeams towards the point below the horizon were not quite so bright and strong as those usually are that diverge from him; and the sky beyond them appeared very black (several showers having passed that way), which certainly contributed to this appearance. Hence it is probable that the thinness and weakness of the reflected rays from the vapours opposite the sun, is the chief cause that this appearance is so very uncommon in comparison to that of diverging beams. For as the region of the sky round about the sun is always brighter than the opposite one, so the light of the diverging beams ought also to be brighter than that of the converging ones. For, though rays are reflected from rough unpolished bodies in all directions, yet more of them are reflected forwards obliquely, than are reflected more directly backwards. Besides, in the present case, the incident rays upon the opposite region to the sun, are more diminished by continual reflections from a longer tract of the atmosphere, than the incident rays upon the region next the sun.

The common phenomenon of diverging beams is more frequent in summer than in winter, and also when the sun is lower than when higher up; probably because the lower vapours are denser, and therefore more strongly reflective than the higher; because the lower sky light is not so bright as the upper; because the air is generally more quiet in the mornings and evenings than about noon-day; and lastly, because many sorts of vapours are more plentifully exhaled in summer than in winter, from many kinds of volatile vegetables; which vapours, when the air is cooled and condensed in the mornings and evenings, may become dense enough to reflect a sensible light.

Sect. VI. Of the Illumination of the Earth's Shadow in Lunar Eclipses.

The ancient philosophers, who knew nothing of the refractive power of the atmosphere, were much perplexed to account for the body of the moon being visible when totally eclipsed. At such times she generally appears of a dull red colour, like tarnished copper. This, they thought, was the moon's native light, by which she became visible when hid from the brighter light of the sun. Plutarch, indeed, attributes this appearance to the light of the fixed stars reflected to us by the moon; but this is too weak to produce the effect. The true cause of it is the scattered beams of the sun bent into the earth's shadow by refractions through the atmosphere in the following manner.

Let the body of the sun be represented by the circle ab, and that of the earth by cd; and let the lines ace and bdc touch them both, and meet in e beyond the earth; then the angular space ced will represent the conical figure of the earth's shadow, which would be totally dark, were none of them bent into it by the refraction of the atmosphere. The rays ah and bi, which touch its opposite sides, will proceed unrefracted, and meet each other at k. Then the two nearest rays to these that flow within them, from the same points a and b, being refracted inwards through the margin of the atmosphere, will cross each other at a point l, somewhat nearer to the earth than k; and in like manner, two opposite rays next within the two last will cross each other at a point m, somewhat nearer to the earth than l, having suffered greater refractions, by passing through longer and denser tracts of air lying somewhat nearer to the earth. The like approach of the successive intersections k, l, m, is to be understood of innumerable couples of rays, till you come to the intersection n of the two innermost; which we may suppose just to touch the earth at the points o and p. It is plain then, that the space bounded by these rays on, np, will be the only part of the earth's shadow wholly unenlightened. Let fmg be part of the moon's orbit when it is nearest the earth, at a time when the earth's dark shadow onp, is longest: in this case, the ratio of tm to ta is about 4 to 3; and consequently the moon, though centrally eclipsed at m, may yet be visible by means of the scattered rays, first transmitted to the moon by refraction through the atmosphere, and thence reflected to the earth.

For let the incident and emergent parts ag, rn, of Fig. 2, the ray ag or n, that just touches the earth at o, be produced till they meet at u, and let agu meet the axis st produced in w; and joining us and um, since the refractions of a horizontal ray passing from o to r, or from o to q, would be alike and equal, the external angle nuw is double the quantity of the usual refraction of a horizontal ray; and the angle aus is the apparent measure of the sun's semidiameter seen from the earth; and the angle ust is that of the earth's semidiameter tu seen from the sun (called his horizontal parallax); and lastly, the angle umt is that of the earth's semidiameter seen from the moon (called her horizontal parallax); because the elevation of the point u above the earth is too small to make a sensible error in the quantity of these angles; whose measures by astronomical tables are as follow:

Sun's least app. semidiam. = aus = 15° 30' Sun's horizontal parallax = ust = 10° 10' Their difference * is = tux = 15° 40' Excl. Twice horizontal refraction = nuw = 67° 30' Prop. Their sum + is = tnu = 83° 10' + Nud. Moon's greatest horiz. parallax = tmu = 62° 10'

Therefore (by a preceding prop.) we have tmu = tnu + (ang. tnu - ang. tmu = 83° - 10° = 62° - 10° = 4° 30' in round numbers; which was to be proved. It is easy to collect from the moon's greatest horizontal parallax of 62° - 10°, that her least distance tm is about 55½ semidiameters of the earth; and therefore the greatest length tn of the dark shadow, being three quarters of tm, is about 41½ semidiameters.

The difference of the last-mentioned angles tnu, tmu is mun = 21°, that is, about two-thirds of 31° - 42°, the angle which the whole diameter of the sun subtends at u. Whence it follows, that the middle point m of the moon centrally eclipsed, is illuminated by rays which come from two-thirds of every diameter of the sun's disk, and pass by one side of the earth; and also by rays that come from the opposite two-thirds of every. PART III. ON THE CONSTRUCTION OF OPTICAL INSTRUMENTS.

CHAP. I. Description of Optical Instruments.

Of the mechanism of optical instruments, particular accounts are given in this work under their respective names. These it would be improper to repeat; but as it belongs to the science of optics to explain, by the laws of refraction and reflection, the several phenomena which those instruments exhibit, we must here enumerate the instruments themselves, omitting entirely, or stating very briefly, such facts as are given at large in other places.

SECT. I. The Multiplying Glass.

The multiplying glass is made by grinding down the convex side \( h k \) of a plano-convex glass \( A B \), into several flat surfaces, as \( h b, b l d, d k \). An object \( C \) will not appear magnified when seen through this glass by the eye at \( H \); but it will appear multiplied into as many different objects as the glass contains plane surfaces. For, since rays will flow from the object \( C \) to all parts of the glass, and each plane surface will refract these rays to the eye, the same object will appear to the eye in the direction of the rays which enter it through each surface. Thus, a ray \( g i H \), falling perpendicularly on the middle surface, will go through the glass to the eye without suffering any refraction; and will therefore show the object in its true place at \( C \): whilst a ray \( a b \) flowing from the same object, and falling obliquely on the plane surface \( b h \), will be refracted in the direction \( b e \), by passing through the glass; and, upon leaving it, will go on to the eye in the direction \( e H \); which will make the same object \( C \) appear also at \( E \), in the direction of the ray \( H e \), produced in the right line \( H a n \).

And the ray \( c d \), flowing from the object \( C \), and falling obliquely on the plane surface \( d k \), will in the same way be refracted to the eye at \( H \); which will cause the same object to appear at \( D \), in the direction \( H f m \).

If the glass be turned round the line \( g i H \), as an axis, the object \( C \) will keep its place, because the surface \( b l d \) is not removed; but all the other objects will seem to go round \( C \), because the oblique planes, on which the rays \( a b c d \) fall, will turn round by the motion of the glass.

SECT. II. Mirrors.

It has been already observed, that there are three kinds of mirrors principally used in optical experiments (see CATOPTRICS, Sect. I.); the plane mirror, the spherical convex mirror, and the spherical concave mirror. Of these the plane mirror first claims our attention, as it is more common, and of greater antiquity, than the other two. We have shown that the image reflected by this mirror appears as far behind the surface as the object is before it; that the image will appear of the same size and in the same position with the object; that every plane mirror will reflect an image of twice its own length and breadth; and that in certain circumstances it will reflect several images of the same object. These phenomena we shall now explain by the laws of reflection.

Let \( A B \) be an object placed before the reflecting surface \( g h i \) of the plane mirror \( C D \); and let the eye be at \( o \). Let \( A h \) be a ray of light flowing from the top \( A \) of the object, and falling upon the mirror at \( h \), and \( h m \) be a perpendicular to the surface of the mirror at \( h \); the ray \( A h \) will be reflected from the mirror to the eye at \( o \), making an angle \( m h o \) equal to the angle \( L i 2 \). Optical In. A h m: then will the top of the image E appear to the eye in the direction of the reflected ray o h produced to E, where the right line A p E, from the top of the object, cuts the right line o h E, at E. Let B i be a ray of light issuing from the foot of the object at B to the mirror at i; and n i a perpendicular to the mirror from the point i, where the ray B i falls upon it; this ray will be reflected in the line r o, making an angle n r o equal to the angle B i n, with that perpendicular, and entering the eye at o; then will the foot F of the image appear in the direction of the reflected ray o i, produced to F, where the right line B F cuts the reflected ray produced to F. All the other rays that flow from the intermediate points of the object A B, and fall upon the mirror between h and i, will be reflected to the eye at o; and all the intermediate points of the image E F will appear to the eye in the direction of these reflected rays produced. But all the rays that proceed from the object and fall upon the mirror above h, will be reflected back above the eye at o; and all the rays that flow from the object, and fall upon the mirror below i, will be reflected back below the eye at o; so that none of the rays that fall above h, or below i, can be reflected to the eye at o; and the distance between h and i is equal to half the length of the object A B.

Hence it appears, that if a man sees his whole image in a plane looking-glass, the part of the glass that reflects his image must be just half as long and half as broad as himself, let him stand at any distance from it whatever; and that his image must appear just as far behind the glass as he is before it. Thus, the man A B viewing himself in the plane mirror C D, which is just half as long as himself, sees his whole image as at E F, behind the glass, exactly equal to his own size. For a ray A C proceeding from his eye at A, and falling perpendicularly upon the surface of the glass at C, is reflected back to his eye, in the same line C A; and the eye of his image will appear at E, in the same line produced to E, beyond the glass. And a ray B D, flowing from his foot, and falling obliquely on the glass at D, will be reflected as obliquely on the other side of the perpendicular a b D, in the direction D A; and the foot of his image will appear at F, in the direction of the reflected ray A D, produced to F, where it is cut by the right line B G F, drawn parallel to the right line A C E; just the same as if the glass were taken away, and the real man stood at F, equal in size to the man standing at B: For to his eye at A, the eye of the other man at E would be seen in the direction of the line A C E; and the foot of the man at F would be seen by the eye A, in the direction of the line A D F.

If the glass be brought nearer the man A B, suppose to c b, he will see his image at C D G: for the reflected ray C A (being perpendicular to the glass) will show the eye of the image at C; and the incident ray B b, being reflected in the line b A, will show the foot of his image at G; the angle of reflection a b A being always equal to the angle of incidence B b a; and so of all the intermediate rays from A to B. Hence, if the man A B advances towards the glass C D, his image will approach towards it; and if he recedes from the glass, his image will also recede from it.

If the object be placed before a common looking-glass, and viewed obliquely, three, four, or more images of it, will appear behind the glass.

To explain this, let A B C D represent the glass; and optical In. let E F be the axis of a pencil of rays flowing from E, striking a point in an object situated there. The rays of this pencil will in part be reflected at F, supposo into the line F G. What remains will (after refraction at F, which we do not consider here) pass on to H; from whence (on account of the quicksilver which is spread over the second surface of the glass) they will be strongly reflected to K, where part of them will emerge and enter an eye at L. By this means one representation of the point E will be formed in the line L K produced, suppose in M: Again, Another pencil, whose axis is E N, Why the first reflected at N, then at O, and afterwards at P, will form a second representation of the same point at Q: And, thirdly, Another pencil, whose axis is E R, after seen in successive reflections at the several points R, S, H, T V, will exhibit a third representation of the same point at X; and so on ad infinitum. The same being true of each point in the object, the whole will be represented in the like manner; but the representations will be faint, in proportion to the number of reflections which the rays suffer, and the length of their progress within the glass. We may add to these another representation of the same object in the line L O produced, made by such of the rays as fall upon O, and are thence reflected to the eye at L. This experiment may be tried by placing a candle before the glass as at F, and viewing it obliquely, as from L.

2. Of Concave Mirrors. The effects of these in magnifying and diminishing objects, have in general been already explained; but in order to understand the nature of reflecting telescopes, it will still be proper to subjoin the following particular description of the effects of concave mirrors.

When parallel rays, as d f a, C m b, e b c, fall upon a concave mirror A b B, they will be reflected back from that mirror, and meet in a point m, at half the distance of the surface of the mirror from C the centre of its concavity; for they will be reflected at as great an angle from a perpendicular to the surface of the mirror, as they fell upon it with regard to that perpendicular, but on the other side thereof. Thus, let C be the centre of concavity of the mirror A b B; and let the parallel rays d f a, C m b, and e b c, fall upon it at the points a, b, and c. Draw the lines C r a, C m b, and C h c, from the centre C to these points; and all these lines will be perpendicular to the surface of the mirror. Make the angle C a h = d a C, and draw the line a m h, which will be the direction of the ray d f a, after it is reflected from the point a of the mirror; so that the angle of incidence d a C = C a h, the angle of reflection; the rays making equal angles with the perpendicular C i a on its opposite sides.

Draw also the perpendicular C h c to the point c, where the ray e b c touches the mirror; and having made the angle C c i = C c e, draw the line c m i, which will be the course of the ray e b c, after it is reflected from the mirror. The ray C m b passing through the centre of concavity of the mirror, and falling upon it at b, is perpendicular to it; and is therefore reflected back from it in the same line b m C. All these reflected rays meet in the point m; and in that point the image of the body which emits the parallel rays d a, C b, and e c, will be formed; which point is distant from the mirror equal to half the radius b m C of its concavity. As the rays which proceed from any celestial object may be esteemed parallel, the image of that object will be formed at \( m \), when the reflecting surface of the concave mirror is turned directly to the object. Hence the focus \( m \) of parallel rays is not in the centre of the mirror's concavity, but half way between the mirror and that centre.

The rays which proceed from any remote terrestrial object are not strictly parallel, but come diverging to it, in separate pencils, from each point of the side of the object next the mirror; and therefore they will not be converged to a point at the distance of half the radius of the mirror's concavity from its reflecting surface, but into separate points at a little greater distance from the mirror. The nearer the object is to the mirror, the farther these points will be from it: and an inverted image of the object will be formed in them, which will seem to hang in the air, and will be seen by an eye placed beyond it (with regard to the mirror) in all respects similar to the object, and as distinct as the object itself.

Let \( A c B \) be the reflecting surface of a mirror, whose centre of concavity is at \( C \); and let the upright object \( DE \) be placed beyond the centre \( C \), and send out a canonical pencil of diverging rays from its upper extremity \( D \), to every point of the concave surface of the mirror \( A c B \). But to avoid confusion, we only draw three rays of that pencil, as \( DA, DC, DB \).

From the centre of concavity \( C \), draw the three right lines \( CA, CC, CB \), touching the mirror in the same points where the three rays touch it; and all these lines will be perpendicular to the surface of the mirror. Make \( CA = DAC \), and draw the right line \( AD \) for the course of the reflected ray \( DA \); make \( CC = DCC \), and draw the right line \( CD \) for the course of the reflected ray \( DC \); make also \( CB = DBC \), and draw the right line \( BD \) for the course of the reflected ray \( DB \).

All these reflected rays will meet in the point \( d \), where they will form the extremity \( d \) of the inverted image \( cd \) similar to the extremity \( D \) of the upright object \( DE \).

If the pencil of rays \( Ef, Eg, Eh \), be also continued to the mirror, and their angles of reflection from it be made equal to their angles of incidence upon it, as in the former pencil from \( D \), they will all meet at the point \( e \) by reflection, and form the extremity \( e \) of the image \( cd \), similar to the extremity \( E \) of the object \( DE \).

And as each intermediate point of the object, between \( D \) and \( E \), sends out a pencil of rays in like manner to every part of the mirror, the rays of each pencil will be reflected back from it, and meet in all the intermediate points between the extremities \( e \) and \( d \) of the image; and so the whole image will be formed in an inverted position not at \( i \), half the distance of the mirror from its centre of concavity \( C \), but at a greater distance between \( i \) and the object \( DE \).

This being well understood, the reader will easily understand how the image is formed by the large concave mirror of the reflecting telescope, when he comes to the description of that instrument.

When the object is more remote from the mirror than its centre of concavity \( C \), the image will be less than the object, and between the object and mirror: when the object is nearer than the centre of concavity, the image will be more remote and bigger than the object. Thus, if \( ED \) be the object, \( dc \) will be its image: For as the object recedes from the mirror, the image approaches nearer to it; and as the object approaches nearer to the mirror, the image recedes farther from it; on account of the lesser or greater divergency of the pencil of rays which proceed from the object: for the less they diverge, the sooner they are converged to points by reflection; and the more they diverge, the farther they proceed before they meet.

If the radius of the mirror's concavity, and the distance of the object after refraction, be known, the distance of the image from the mirror is found by this rule: Divide the product of the distance and radius by double the distance made less by the radius, and the quotient is the distance required.

If the object be in the centre of the mirror's concavity, the image and object will be coincident, and equal in bulk.

If a man place himself directly before a large concave mirror, but farther from it than its centre of concavity, he will see an inverted image of himself in the air, between him and the mirror, and of a less size than himself. If he holds out his hand towards the mirror, the hand of the image will come out towards his hand, and coincide with it, of an equal bulk, when his hand is in the centre of concavity; and he will imagine he may shake hands with his image. If he reaches his hand farther, the hand of the image will pass by his hand, and come between his hand and his body: and if he moves his hand towards either side, the hand of the image will move towards the other; so that whatever way the object moves, the image will move the contrary. All the while a bystander will see nothing of the image, because none of the reflected rays that form it enter his eyes.

**Sect. III. Camera Obscura.**

The camera obscura having already been fully described under the word **Dioptrics**, we shall at present only direct the reader's attention to an improvement which has lately been made upon this amusing instrument.

"The improvements (says Dr Brewster) which have been made upon the camera obscura since its first invention, regard chiefly its external form; and no attempts have been made to increase the brilliancy and distinctness of the image. When we compare the picture of external objects, which is formed in a dark chamber by the object-glass of a common refracting telescope, with that which is formed with an achromatic object-glass, we shall find the difference between their distinctness much less than we should have at first expected. Although the achromatic lens form an image of the minutest parts of the landscape, yet when this image is received on paper, these minute parts are obliterated by the small hairs and asperities on its surface, and the effect of the picture is very much impaired. In the Royal Observatory at Greenwich the image is received upon a large concave piece of stucco; but this substance does not seem to be more favourable for the reception of images than a paper ground. In order to obviate these imperfections, I tried a number of white substances of different degrees of smoothness, and several metallic surfaces with different degrees of polish, but did not succeed in finding any surface superior to paper. It happened, however, to receive the image on the silvered back of a looking-glass, and was surprised at the brilliancy and distinctness with which external objects were represented. The little spherical protuberances, how- Optical In-ever, which arise from the roughness of the tinfoil, have a tendency to detract from the precision of the image, and certainly injure it considerably when examined narrowly with the eye. In order to remove these small eminences, I ground the surface carefully with a bed of hones which I had used for working the plane specula of Newtonian telescopes. By this operation, which is exceedingly delicate, and may be performed without injuring the mirror, I obtained a surface finely adapted for the reception of images. The minute parts of the landscape, when received on this substance, are formed with so much precision, and the brilliance of the colouring is so uncommonly fine, as to equal, if not surpass the images formed in the air by means of concave specula. Notwithstanding the bluish colour of the metallic ground, white objects are represented in their true colour, and the verdure of the foliage is so rich and vivid, that the image seems to surpass in beauty even the object itself. On account of the metallic lustre of the surface, the distinctness of the image will always be greatest when the eye of the observer is placed in the direction of the reflected rays.

"The common portable camera obscura, which has already been described (see Dioptrics), is necessarily on a small scale, and has many disadvantages. These disadvantages are completely remedied in the camera obscura, invented by the Rev. Mr Thomson of Duddington, which is represented in figures 1. and 2. of Plate CCCLXXXIX. In fig. 1. A is a metallic or wooden ring, in which the four wooden bars AF, AI, AG, AH, move by means of joints at A, and are kept together by the cross pieces BC, DE, which move round B and D as centres, and fold up along BA and DA, when the instrument is not used. The surface FIHG, on which the image is received, consists of a piece of silk covered with paper. It is made to roll up at IH, which moves in a joint at I, so that the whole surface FIHG, when winded upon HI, can be folded upon the bar IA. By this means the instrument, which is covered with green silk covered with a black substance, may be put together and carried as an umbrella. It is shewn more fully in fig. 2. where A is the aperture for placing the lens, and BC a semicircular opening for viewing the image. A black veil may be fixed to the circumference of BC, and thrown over the head of the observer to prevent the admission of any extraneous light."

Sect. IV. Microscopes.

Under the article Microscope a full account has been given of the external construction of those instruments as they are now made by the most eminent artists.

It did not fall within the plan of that article to explain the way in which an enlarged picture of the object is formed upon the retina by means of the microscope, and the means of ascertaining its magnifying power; but we shall now direct the reader's attention to this interesting subject.

1. The Single Microscope, the simplest of all microscopes is nothing more than a small globule of glass, or a convex lens whose focal distance is extremely short. The magnifying power of this microscope is thus ascertained by Dr Smith, "A minute object pq, seen distinctly through a small glass AE by the eye put close to it, appears so much greater than it would to the naked eye, placed at the least distance qL from whence it appears sufficiently distinct, as this latter distance qL optical is it greater than the former qE. For having put your strument eye close to the glass EA, in order to see as much of the object as possible at one view, remove the object pq to and fro till it appear more distinctly, suppose at the distance E q. Then conceiving the glass AE to be removed, and a thin plate, with a pin-hole in it, to be put in its place, the object will appear distinct and as large as before, when seen through the glass, only not so bright. And in this latter case it appears so much greater than it does to the naked eye at the distance qL, either with a pin-hole or without it, as the angle pE q is greater than the angle pLq, or as the latter distance qL is greater than the former qE. Since the interposition of the glass has no other effect than to render the appearance distinct, by helping the eye to increase the refraction of the rays in each pencil, it is plain that the greater apparent magnitude is entirely owing to a nearer view than could be taken by the naked eye. As the human eye is so constructed, as, for reasons already assigned, to have distinct vision only when the rays which fall upon it are parallel or nearly so; it follows that if the eye be so perfect as to see distinctly by pencils of parallel rays falling upon it, the distance E q, of the object from the glass, is then the focal distance of the glass. Now, if the glass be a small round globule, of about \(\frac{1}{4}\) th of an inch diameter, its focal distance E q, being three quarters of its diameter, is \(\frac{3}{4}\) th of an inch; and if qL be eight inches, the distance at which we usually view minute objects, this globule will magnify in the proportion of \(8\) to \(\frac{3}{4}\), or of \(160\) to \(1\).

Mr Gray's Water Microscope is represented in Plate CCCLXXXIX, fig. 4. The drop of water taken up on the point of a pin is introduced into the small hole D, \(\frac{1}{4}\) of an inch in diameter, in the piece of brass DE, about \(\frac{1}{4}\) of an inch thick. The hole D is in the middle of a spherical cavity, about \(\frac{1}{4}\) of an inch in diameter, and a little deeper than half the thickness of the brass; on the opposite side of the brass is another spherical cavity, half as broad as the former, and so deep as to reduce the circumference of the small hole to a sharp edge. The water being placed in these cavities, will form a double convex lens with unequal convexities. The object, if it is solid, is fixed upon the point C of the supporter AB, and placed at its proper distance from the water lens by the screw FG. When the object is fluid, it is placed in the hole A, but in such a manner as not to be spherical; and this hole is brought opposite the fluid lens by moving the extremity G of the screw into the slit GH.

2. The Double or Compound Microscope, consists of an object-glass c d, and an eye-glass e f. The small object a b is placed at a little greater distance from the glass c d than its principal focus; so that the pencils of rays flowing from the different points of the object, and passing through the glass, may be made to converge, and unite in as many points between g and h, where the image of the object will be formed; which image is viewed by the eye through the eye-glass e f. For the eye-glass being so placed, that the image g h may be in its focus, and the eye much about the same distance on the other side, the rays of each pencil will be parallel after going out of the eye-glass, as at e and f, till they come to the eye at k, where they will begin to converge by the refractive power of the humours; and af- Optical instruments having crossed each other in the pupil, they will be collected into points on the retina, and form upon it the large inverted image AB.

By this combination of lenses, the aberration of the light from the figure of the glass, which in a globule of the kind above mentioned is very considerable, is in some measure corrected. This appeared so sensibly to be the case, even to former opticians, that they very soon began to make the addition of another lens. For, says Mr Martin, it is not only evident from the theory of this aberration, that the image of any point is rendered less confused by refraction through two lenses than by an equal refraction through one; but it also follows, from the same principle, that the same point has its image still less confused when formed by rays refracted through three lenses than by an equal refraction through two; and therefore a third lens added to the other will contribute to make the image more distinct, and consequently the instrument more complete. At the same time the field of view is amplified, and the use of the microscope rendered more agreeable, by the addition of the other lens. Thus also we may allow a somewhat larger aperture to the object lens, and thus increase the brightness of objects, and greatly heighten the pleasure of viewing them. For the same reason, Mr Martin has proposed a four-glass microscope, which answers the purposes of magnifying and of distinct vision still more perfectly.

The magnifying power of double microscopes is easily understood, thus: The glass L next the object PQ is very small, and very much convex, and consequently its focal distance LF is very short; the distance LQ of the small object PQ is but a little greater than LF: Greater it must be, that the rays flowing from the object may converge after passing through the glass, and crossing one another, form an image of the object; and it must be but a little greater, that the image p q may be at a great distance from the glass, and consequently may be much larger than the object itself. This picture p q being viewed through a convex glass AE, whose focal distance is q E, appears distinct as in a telescope. Now the object appears magnified for two reasons; first, because, if we viewed its picture p q with the naked eye, it would appear as much greater than the object, at the same distance, as it really is greater than the object, or as much as L q is greater than L Q; and secondly, because this picture appears magnified through the eye-glass as much as the least distance at which it can be seen distinctly with the naked eye, is greater than q E, the focal distance of the eye-glass. If this latter ratio be five to one, and the former ratio of L q to L Q be 20 to 1; then, upon both accounts, the object will appear 5 times 20, or 100 times greater than to the naked eye.

The section of a compound microscope with three lenses is represented in fig. 10. By the middle one GK the pencil of rays coming from the object-glass are refracted so as to tend to a focus at O; but being intercepted by the proper eye-glass DF, they are brought together at I, which is nearer to that lens than its proper focus at L; so that the angle DIF, under which the object now appears, is larger than DLF, under which it would have appeared without this additional glass; and consequently the object is more magnified in the same proportion. Dr Hooke informs us, that, in most of his observations, he made use of a double microscope with this broad middle glass when he wanted to see much of an object at one view, and taking it out when he would examine the small parts of an object more accurately; for the fewer refractions there are, the more bright and clear the object appears.

The following rule for finding the magnifying power of compound microscopes with three lenses, has been given by Dr Brewster in his Appendix to Ferguson's Lectures, vol. ii. p. 468. "Divide the difference between the distance of the two first lenses, or those next the object, and the focal distance of the second or amplifying compound glass, by the focal distance of the second glass, and the quotient will be a first number. Square the distance between the two first lenses, and divide it by the difference between that distance, and the focal distance of the second glass, and divide this quotient by the focal distance of the third glass, or that next the eye, and a second number will be obtained. Multiply together the first and second numbers, and the magnifying power of the object glass (as found by one of the following tables), and the product will be the magnifying power of the compound microscope."

Having in the historical part of this article given a short account of the construction of Dr Smith's double reflecting microscope, it may not be improper in this place to point out the method of ascertaining its magnifying power. This we shall do from the author himself, because his symbols, being general, are applicable to such microscopes of all dimensions.

Between the centre E and principal focus T of a concave speculum ABC, whose axis is EQTC, place an object PQ; and let the rays flowing from it be reflected from the speculum AB towards an image p q; but before they unite in it, let them be received by a convex speculum a b c, and thence be reflected, through a hole BC in the vertex of the concave, to a second image π x, to be viewed through an eye-glass l.

The object may be situated between the specula C, e; or, which is better, between the principal focus t and vertex c of the convex one, a small hole being made in its vertex for the incident rays to pass through.

In both cases we have TQ, TE, Tq, continual proportions in some given ratio, suppose of r to n; and also tg, tc, tx, continual proportions in some other given ratio, suppose of s to m. Then if d be the usual distance at which we view minute objects distinctly with the naked eye, and π l the focal distance of the least eye-glass, through which the object appears sufficiently bright and distinct, it will be magnified in the ratio of m n d to π l.

For the object PQ, and its first image p q, are terminated on one side by the common axis of the specula, and on the other by a line PE, drawn through the centre E of the concave ABC. Likewise the images p q and π x are terminated by the common axis and by the line ep π, drawn through the centre e of the convex a b c (Euclid, v. 12.). Hence, by the similar triangles π x e, p q e, and also p q E, PQE, we have π x : p q = e : e = m : 1; and p q : PQ = g E : QE = n : 1; and consequently π x : PQ = m n : 1 whence π x = m n × PQ. Now if l' be the focal distance of the eye-glass l, the points P, Q of the object, are seen through it by the rays of two pencils emerging parallel to the lines π x l. Optical In. \( \pi l x l \) respectively; that is, PQ appears under an angle equal to \( \frac{\pi}{l} \), which is as \( \frac{\pi}{l} = \frac{m n P Q}{x l} \); and to the naked eye at the distance d from PQ, it appears under an angle PoQ which is as \( \frac{P Q}{d} \), and therefore is magnified in the ratio of these angles, that is, of \( m n d \) to \( x l \).

Cor. Having the numbers \( m, n, d \), to find an eyeglass which shall cause the microscope to magnify M times in diameter, take \( x l = \frac{m n d}{M} \). For the apparent magnitude is to the true as \( M : 1 = m n d : x l \).

We shall conclude this part of our subject with the following easy method of ascertaining the magnifying power of such microscopes as are most in use.

The apparent magnitude of any object, as must appear from what has been already said, is measured by the angle under which it is seen; and this angle is greater or smaller according as the object is nearer to or farther from the eye; and of consequence the less the distance at which it can be viewed, the larger it will appear. The naked eye is unable to distinguish any object brought exceedingly near it; but by looking through a convex lens at an object placed in its focus, however near the focus of that lens be, an object may be distinctly seen; and the smaller the lens is, the nearer will be its focus, and in the same proportion the greater will be its magnifying power. From these principles it is easy to find the reason why the first or greatest magnifiers are so extremely minute; and also to calculate the magnifying power of any convex lens employed in a single microscope: For as the focal distance of the lens is to the distance at which we see objects distinctly with the naked eye, so is it to the magnifying power.

If the focal length of a convex lens, for instance, be one inch, and the distance at which we look at small objects eight inches, which is the common standard, an object may be seen through that lens at one inch distance from the eye, and will appear in its diameter eight times larger than it does to the naked eye; but as the object is magnified every way, in length as well as in breadth, we must square this diameter to know how much it really is enlarged; and we then find that its superficies is magnified 64 times.

Again, Suppose a convex lens whose focal distance is only one-tenth of an inch; as in eight inches, the common distance of distinct vision with the naked eye, there are 80 tenths, an object may be seen through this glass 80 times nearer than with the naked eye. It will, of consequence, appear 80 times longer, and as much broader, than it does to common sight; and is therefore magnified 6400 times. If a convex glass be so small that its focus is only \( \frac{1}{25} \)th of an inch distant, we find that eight inches contain 160 of these twentieth parts; and consequently the length and breadth of any object seen through such a lens will be magnified 160 times, and the whole surface 25,600 times. As it is easy to melt a drop or globule of a much smaller diameter than a lens can be ground, and as the focus of a globule is no farther off than one-fourth of its own diameter, it must therefore magnify to a prodigious degree. But this excessive magnifying power is much more than counterbalanced by its admitting so little light, want of distinctness, and showing such a small portion of the object to be examined; for which reason, these globules, though greatly valued some time ago, are now almost entirely rejected. According to Mr Folkes's description of the single microscopes of convex lenses which Leeuwenhoek left to the Royal Society, they were all exceedingly clear, and showed the object very bright and distinct; which Mr Folkes considered as owing to the great care this gentleman took in the choice of his glass, his exactness in giving it the true figure, and afterwards reserving only such for his use as upon trial he found to be most excellent. Their powers of magnifying are different, as different objects may require; and as on the one hand, being all ground glasses, none of them are so small, or consequently magnify to so great a degree, as some of the globules frequently used in other microscopes; yet the distinctness of these very much exceeds those which are commonly used.

In order to find the magnifying power of a single microscope, no more is necessary than to bring it to its true focus, the exact place of which will be known by an object's appearing perfectly distinct and sharp when placed there. Then, with a pair of small compasses, measure, as nearly as possible, the distance from the centre of the glass to the object which is viewed, and how many parts of an inch that distance is. When this is known, compute how many times those parts of an inch are contained in eight inches, and the result will give the number of times the diameter is magnified: squaring the diameter will give the superficies; and if the solid content is wanted, it will be shown by multiplying the superficies by the diameter.

The superficies of one side of an object only can be seen at one view; and to compute how much that is magnified, is most commonly sufficient; but sometimes it is satisfactory to know how many minute objects are contained in a larger; as suppose we desire to know how many animalcules are contained in the bulk of a grain of sand: and to answer this, the cube, as well as the surface, must be taken into the account.

For the satisfaction of those who are not much versant in these subjects, we shall here subjoin the following tables taken from the Appendix to Ferguson's Lectures.

The first column contains the focal length of the convex lens in hundredths of an inch. The second contains the number of times which such a lens will magnify the diameter of objects: The third shews the number of times that the surface is magnified; and the fourth the number of times that the cube of the object is magnified. A table of a similar kind, though upon a much smaller scale, has already been published; but the nearest distance at which the eye can see distinctly, is there supposed to be eight inches, which we are confident, from experience, is too large an estimate for the generality of eyes. Table I. is therefore computed upon the supposition that the distance alluded to is seven inches.

"When we consider however (says the editor of the work now quoted) that the eye examines very minute objects at a less distance than it does objects of a greater magnitude, we shall find that the magnifying power of lenses ought to be deduced from the distance at which the eye examines objects really microscopic. This circumstance has been overlooked by every writer on optics, and merits our attentive consideration. We have now before us two specimens of engraved characters..." Optical Instruments.

The one is so large that it can be easily read at the distance of ten inches; and the other is so exceedingly minute that it cannot be read at a greater distance than five inches. Now we maintain that, if these two kinds of engraving are seen through the same microscope, the one will be twice as much magnified as the other. This indeed is obvious; for as the magnifying power of a lens is equal to the distance at which the object is examined by the naked eye divided by the focal length of the lens, we shall have \( \frac{5}{x} \) for the number of times which the minute engraving is magnified, and \( \frac{10}{x} \) for the number of times that the large engraving is magnified, \( x \) being the focal length of the lens. It follows, therefore, that the number of times that any lens magnifies objects really microscopic should be determined, by making the distance at which they are examined by the naked eye about five inches.

Upon this principle we have computed Table II., which contains the magnifying power of convex lenses when employed to examine microscopic objects.

### Table I.

A New Table of the magnifying power of small convex lenses or single microscopes, the distance at which the eye sees distinctly being seven inches.

| Focal distance of the lens or microscope | Number of times that the diameter of an object is magnified | Number of times that the surface of an object is magnified | Number of times that the cube of an object is magnified | |-----------------------------------------|-------------------------------------------------------------|-----------------------------------------------------------|--------------------------------------------------------| | Inches and roodths of an inch | Dec. Times of a time | Times | Times | | 1 or 100 | 7.00 | 49 | 343 | | 1/2 or 75 | 9.33 | 87 | 810 | | 1/3 or 50 | 14.00 | 196 | 2744 | | 1/4 or 40 | 17.50 | 306 | 5360 | | 1/5 or 30 | 23.33 | 544 | 12698 | | 1/6 or 20 | 35.00 | 1225 | 42875 | | 1/7 or 19 | 36.84 | 1354 | 49836 | | 1/8 or 18 | 38.89 | 1513 | 58864 | | 1/9 or 17 | 41.18 | 1697 | 69935 | | 1/10 or 16 | 43.75 | 1910 | 83453 | | 1/11 or 15 | 46.66 | 2181 | 101848 | | 1/12 or 14 | 50.00 | 2500 | 125000 | | 1/13 or 13 | 53.85 | 2804 | 155721 | | 1/14 or 12 | 58.33 | 3399 | 198156 | | 1/15 or 11 | 63.67 | 4945 | 257259 | | 1/16 or 10 | 70.00 | 4900 | 343000 | | 1/17 or 9 | 77.58 | 6053 | 470911 | | 1/18 or 8 | 87.59 | 7656 | 669922 | | 1/19 or 7 | 100.00 | 10000 | 1000000 | | 1/20 or 6 | 116.66 | 13689 | 1601613 | | 1/21 or 5 | 140.00 | 19000 | 274400 | | 1/22 or 4 | 175.00 | 30625 | 5359375 | | 1/23 or 3 | 233.33 | 54289 | 12649337 | | 1/24 or 2 | 359.00 | 122500 | 42875000 | | 1/25 or 1 | 700.00 | 490000 | 3430000000 |

The greatest magnifier in Mr Leeuwenhoek's cabinet of microscopes, presented to the Royal Society, has its focus nearly at one-twentieth of an inch distance from its centre; and consequently magnifies the diameter of an object 166 times, and the superficies 25,600. But the greatest magnifier in Mr Wilson's single microscopes, as they are now made, has usually a focal length only of the 50th part of an inch; whereby it has a power of enlarging the diameter of an object 400, and its superficies 160,000 times.

The magnifying power of the solar microscope must be calculated in a different manner; for here the distance of the screen or sheet on which the image of the object is cast, divided by the focal length of the lens, gives its magnifying power. Suppose, for instance, the lens made use of has its focus at half an inch, and the screen is placed at the distance of five feet, the object from that will then appear magnified 20 times, and the superficies of others, 14,400 times; and, by putting the screen at a greater distance, you may magnify the object almost as much as you please: but the screen should be placed just at that distance where the object is seen most distinct and clear. With regard to the double reflecting microscope, Mr Baker observes, that the power of the object-lens is indeed greatly increased by the addition of two eyeglasses; but as no object-lens can be used with them so minute a diameter, or which magnifies itself near so much as those that can be used alone, the glasses of this microscope, upon the whole, magnify little or nothing more than those of Mr Wilson's single one; the chief advantage arising from a combination of lenses being the sight of a larger portion of the object.

Sect. V. Telescopes.

I. The Refracting Telescope.

1. The Astronomical Telescope.—From what has been said concerning the compound microscope, the nature of the common astronomical telescope will easily be understood: for it differs from the microscope only in this, that the object is placed at so great a distance from it, that the rays of the same pencil flowing from the object, may be considered as falling parallel upon the object-glass; and therefore the image made by that lens is considered as coincident with its focus of parallel rays.

This will appear very plain from fig. 4, in which AB is the object emitting the several pencils of rays A c d, B c d, &c., but supposed to be at so great a distance from the object-glass, c d, that the rays of the same pencil may be considered as parallel to each other; they are therefore supposed to be collected into their respective foci at the points m and p, situated at the focal distance of the object-glass c d. Here they form an image E, and crossing each other proceed diverging to the eye-glass h g; which being placed at its own focal distance from the points m and p, the rays of each pencil, after passing through that glass, will become parallel among themselves; but the pencils themselves will converge considerably with respect to one another, even so as to cross at e, very little farther from the glass g h than its focus; because, when they entered the glass, their axes were almost parallel, as coming through the object-glass at the point k, to whose distance the breadth of the eye-glass in a long telescope bears very small proportion. So that the place of the eye will be nearly at the focal distance of the eye-glass, and the rays of each respective pencil being parallel among themselves, and their axes crossing each other in a larger angle than they would do if the object were to be seen by the naked eye, vision will be distinct, and the object will appear magnified.

The magnifying power in this telescope is as the focal length of the object-glass to the focal length of the eye-glass.

In order to prove this, we may consider the angle A k B as that under which the object would be seen by the naked eye; for in considering the distance of the object, the length of the telescope may be omitted, as bearing no proportion to it. Now the angle under which the object is seen by means of the telescope is g e h, which is to the other A k B, or its equal g k h, as the distance from the centre of the object-glass to that of the eye-glass. The angle, therefore, which an object subtends to an eye assisted by a telescope of this kind, is to that under which it subtends to the naked eye, as the focal length of the object-glass to the focal length of the eye-glass.

It is evident from the figure, that the visible area, or space which can be seen at one view, when we look through this telescope, depends on the breadth of the eye-glass, and not of the object-glass; for if the eye-glass be too small to receive the rays g m, p h, the extremities of the object could not have been seen at all: a larger breadth of the object-glass conduces only to the rendering each point of the image more luminous, by receiving a larger pencil of rays from each point of the object.

It is in this telescope as in the compound microscope, objects where we see not the object itself, but only its image seen therethrough: now that image being inverted with respect to it, the object, because the axis of the pencils that flow from the object cross each other at k, objects seen through a telescope of this kind necessarily appear inverted.

This is a circumstance not at all regarded by astronomers; but for viewing objects upon the earth, it is convenient that the instrument should represent them in their natural posture; to which use the telescope with three eye-glasses, as represented fig. 13, is peculiarly adapted.

AB is the object sending out the several pencils A c d, B c d, &c., which passing through the object-glass c d, are collected into their respective foci in CD, where they form an inverted image. From this they proceed to the first eye-glass e f, whose focus being at refracting l, the rays of each pencil are rendered parallel among themselves, and their axes, which were nearly parallel before, are made to converge and cross each other: the second eye-glass g h, being so placed that its focus shall fall upon m, renders the axes of the pencils which diverge from thence parallel, and causes the rays of each, which were parallel among themselves, to meet again at its focus EF on the other side, where they form a second image inverted with respect to the former, but erect with respect to the object. Now this image being seen by the eye at a b through the eye-glass i k, affords a direct representation of the object, and under the same angle that the first image CD would have appeared, had the eye been placed at l, supposing the eye-glasses to be of equal convexity; and therefore the object is seen equally magnified in this as in the former telescope, that is, as the focal distance of the object-glass to that of any one of the glasses, and appears erect.

2. The Galilean Telescope with the concave eye-glass is constructed as follows.

AB is an object sending forth the pencils of rays g h i, k l m, &c., which, after passing through the object-glass c d, tend towards e F f (where we shall suppose the focus of it to be), in order to form an inverted image there as before; but in their way to it are made to pass through the concave glass n o, so placed that its focus may fall upon E, and consequently the rays of the several pencils which were converging towards those respective focal points e, E, f, will be rendered parallel, but the axes of those pencils crossing each other at P, and diverging from thence, will be rendered more diverging, as represented in the figure. Now these rays entering the pupil of an eye, will form a large and distinct image a b upon the retina, which will be inverted with Optical In. with respect to the object, because the axis of the pen- cil cross in F. The object of course will be seen erect, and the angle under which it will appear will be equal to that which the lines a F, b F, produced back through the eye-glass, form at F.

It is evident, that the less the pupil of the eye is, the less is the visible area seen through a telescope of this kind; for a less pupil would exclude such pencils as proceed from the extremities of the object AB, as is evident from the figure. This inconvenience renders this telescope unfit for many uses; and is only to be re- medied by the telescope with the convex eye-glasses, where the rays which form the extreme parts of the image are brought together in order to enter the pupil of the eye, as explained above.

It is apparent also, that the nearer the eye is placed to the eye-glass of this telescope, the larger is the area seen through it; for, being placed close to the glass, as in the figure, it admits rays that come from A and B, the extremities of the object, which it could not if it was placed farther off.

The degree of magnifying in this telescope is in the same proportion with that in the other, viz., as the fo- cal distance of the object glass is to the focal distance of the eye-glass.

For there is no other difference but this, viz., that as the extreme pencils in that telescope were made to con- verge and form the angle g e h or i n k (fig. 13.), these are now made to diverge and form the angle a F b (fig. 1.) which angles, if the concave glass in one has an equal refractive power with the convex one in the other, will be equal, and therefore each kind will exhi- bit the object magnified in the same degree.

There is a defect in all these kinds of telescopes, not to be remedied in a single lens by any means what- ever, which was thought only to arise from the spher- ical aberration of the object-glass. But it was dis- covered by Sir Isaac Newton, that the imperfection of this sort of telescope, so far as it arises from the spher- ical form of the glasses, bears little proportion to that which is owing to the different refrangibility of light. This diversity in the refraction of rays is about a 28th part of the whole; so that the object-glass of a tele- scope cannot collect the rays which flow from any one point in the object into less space than a circle whose diameter is about the 56th part of the breadth of the glass.

To show this, let AB represent a convex lens, and let CDF be a pencil of rays flowing from the point D; let H be the point at which the least refrangible rays are collected to a focus; and I, that where the most refrangible concur. Then, if IH be the 28th part of EH, IK will be a proportionable part of EC (the tri- angles HIK and HEC being similar): consequently IK will be the 28th part of FC. But MN will be the least space into which the rays will be collected, as ap- pears by their progress represented in the figure. Now MN is but about half of KL; and therefore it is about the 56th part of the breadth of that part of the glass through which the rays pass; which was to be shown.

Since therefore each point of the object will be re- presented in so large a space, and the centres of those spaces will be contiguous, because the points in the object the rays flow from are so; it is evident, that the image of an object made by such a glass must be a most confused representation, though it does not appear so when viewed through an eye-glass that magnifies in a moderate degree; consequently the degree of magni- fying in the eye-glass must not be too great with re- spect to that of the object-glass, lest the confusion be- come sensible.

Notwithstanding this imperfection, a dioptrical te- lescope may be made to magnify in any given degree, provided it be of sufficient length; for the greater the focal distance of the object-glass is, the less may be the proportion which the focal distance of the eye-glass may bear to that of the object-glass, without render- ing the image obscure. Thus, an object-glass, whose focal distance is about four feet, will admit of an eye- telescope glass whose focal distance shall be little more than an inch, and consequently will magnify almost 48 times; to their but an object-glass of 40 feet focus will admit of an length. eye-glass of only four inches focus; and will therefore magnify 120 times; and an object-glass of 100 feet focus will admit of an eye-glass of little more than six inches focus, and will therefore magnify almost 200 times.

The reason of this disproportion in their several de- grees of magnifying may be explained thus: Since the diameter of the spaces, into which rays flowing from the several points of an object are collected, are as the breadth of an object-glass, it is evident that the degree of confusedness in the image is as the breadth of that glass; for the degree of confusedness will only be as the diameters or breadth of those spaces, and not as the spaces themselves. Now the focal length of the eye- glass, that is, its power of magnifying, must be as that degree; for, if it exceeds it, it will render the confused- ness sensible; and therefore it must be as the breadth or diameter of the object-glass. The diameter of the object-glass, which is as the square root of its aperture or magnitude, must be as the square root of the power of magnifying in the telescope; for unless the aper- ture itself be as the power of magnifying, the image will want light: the square root of the power of mag- nifying will be as the square root of the focal distance of the object-glass; and therefore the focal distance of the eye-glass must be only as the square root of that of the object-glass. So that in making use of an object-glass of a longer focus, suppose, than one that is given, you are not obliged to apply an eye-glass of a proportionably longer focus than what would suit the given object-glass, but such a one only whose fo- cal distance shall be to the focal distance of that which will suit the given object-glass, as the square root of the focal length of the object-glass you make use of, is to the square root of the focal length of the given one. And this is the reason that longer telescopes are capable of magnifying in a greater degree than shorter ones, without rendering the object confused or coloured.

Upon these principles the following new table, taken from the appendix to Ferguson's Lectures, vol. ii., p. 471, second edition, has been computed. It is founded on a telescope of Huygens, mentioned in his Astroscopia Compendiaria, which had an object-glass 34 feet in fo- cal length, and which bore an eye-glass of 2½ inches fo- cal distance, and therefore magnified 163 times. The table for refracting telescopes, which has been given by preceding optical writers, was copied from Smith's Op- tics. Optical Injuries, as the production of the celebrated Huygens, while it was calculated only by the editors of his Dioptrics, from a telescope made by that celebrated optician; which, however, seems to have been inferior to that which is the foundation of the following table. The table is suited to Rhineland measure; but the second and third columns may be converted into English measure by dividing them by .7, the focal distances of the object-glasses being supposed English feet.

A New Table of the apertures, focal lengths, and magnifying power of refracting telescopes.

| Focal length of the object-glass | Sine or aperture of the eye-glass | Focal distance of the eye-glass | Magnifying Power | |---------------------------------|----------------------------------|-------------------------------|-----------------| | Feet | Inch. Dec. | Inch. Dec. | Times | | 1 | 0.65 | 0.50 | 28 | | 2 | 1.03 | 0.62 | 39 | | 3 | 1.30 | 0.75 | 48 | | 4 | 1.45 | 0.87 | 55 | | 5 | 1.61 | 1.00 | 60 | | 6 | 1.79 | 1.07 | 67 | | 7 | 1.96 | 1.15 | 73 | | 8 | 2.14 | 1.21 | 77 | | 9 | 2.20 | 1.30 | 83 | | 10 | 2.32 | 1.38 | 87 | | 13 | 2.63 | 1.58 | 99 | | 15 | 2.81 | 1.70 | 106 | | 20 | 3.31 | 1.95 | 123 | | 25 | 3.73 | 2.15 | 139 | | 30 | 4.01 | 2.40 | 150 | | 35 | 4.34 | 2.58 | 163 | | 40 | 4.64 | 2.76 | 174 | | 45 | 4.92 | 2.93 | 184 | | 50 | 5.20 | 3.08 | 195 | | 55 | 5.48 | 3.22 | 205 | | 60 | 5.71 | 3.36 | 214 | | 70 | 6.16 | 3.64 | 231 | | 80 | 6.58 | 3.90 | 246 | | 90 | 7.02 | 4.12 | 262 | | 100 | 7.39 | 4.35 | 276 | | 200 | 10.41 | 6.17 | 389 | | 300 | 12.89 | 7.52 | 479 | | 400 | 14.72 | 8.71 | 551 | | 500 | 16.52 | 9.71 | 618 |

Sect. VI. On Achromatic Telescopes.

The inconvenience of very long telescopes is so great, that different attempts have been made to remove it. Of these, the most successful have been by Dollond and Blair; and the general principles upon which these eminent opticians proceeded have been mentioned in the historical part of this article, and in the preceding section. A fuller account of Dr Blair's discovery will be seen in the Transactions of the Royal Society of Edinburgh; and of Dollond's, it may be sufficient to observe, in addition to what has been already said, that the object-glasses of his telescopes are composed of three distinct lenses, two convex and one concave; of which the concave one is placed in the middle, as is represented in fig. 3, where a and c show the two convex lenses, and b b the concave one, which is by the British artists placed in the middle. The two convex ones are made of London crown glass, and the middle one of white flint glass; and they are all ground to spheres of different radii, according to the refractive powers of the different kinds of glass and the intended focal distance of the object-glass of the telescope. According to Boscovich, the focal distance of the parallel rays for the concave lens is one-half, and for the convex glass one-third of the combined focus. When put together, they refract the rays in the following manner. Let a b, a b, Fig. 4, be two red rays of the sun's light falling parallel on the first convex lens c. Supposing there was no other lens present but that one, they would be converged into the lines b e, b e, and at last meet in the focus q. Let the lines g h, g h, represent two violet rays falling on the surface of the lens. These are also refracted, and will meet in a focus; but as they have a greater degree of refrangibility than the red rays, they must of consequence converge more by the same power of refraction in the glass, and meet sooner in a focus, suppose at r.—Let now the concave lens d d be placed in such a manner as to intercept all the rays before they come to their focus. Were this lens made of the same materials, and ground to the same radius with the convex one, it would have the same power to cause the rays diverge that the former had to make them converge. In this case, the red rays would become parallel, and move on in the line o o, o o: But the convex lens, being made of flint glass, and upon a shorter radius, has a greater refractive power, and therefore they diverge a little after they come out of it; and if no third lens was interposed, they would proceed diverging in the lines o p t, o p t; but, by the interposition of the third lens o v o, they are again made to converge, and meet in a focus somewhat more distant than the former, as at x. By the concave lens the violet rays are also refracted, and made to diverge; but having a greater degree of refrangibility, the same power of refraction makes them diverge somewhat more than the red ones; and thus, if no third lens was interposed, they would proceed in such lines as l m n, l m n. Now as the differently coloured rays fall upon the third lens with different degrees of divergence, it is plain, that the same power of refraction in that lens will operate upon them in such a manner as to bring them all together to a focus very nearly at the same point. The red rays, it is true, require the greatest power of refraction to bring them to a focus; but they fall upon the lens with the least degree of divergence. The violet rays, though they require the least power of refraction, yet have the greatest degree of divergence; and thus all meet together in the point x, or nearly so.

But, though we have hitherto supposed the refraction of the concave lens to be greater than that of the convex ones, it is easy to see how the errors occasioned by the first lens may be corrected by it, though it should have even a less power of refraction than the convex one. Thus, let a b, a b, be two rays of red light falling upon the convex lens c, and refracted into the focus q; let also g h, g h, be two violet rays converged into a focus at r; it is not necessary, in order to their convergence into a common focus at x, that the concave lens should make them diverge: it is sufficient if the glass Optical instruments have a power of dispersing the violet rays somewhat more than the red ones; and many kinds have this power of dispersing some kinds of rays, without a very great power of refraction. It is better, however, to have the object-glass composed of three lenses; because there is then another correction of the aberration by means of the third lens; and it might be impossible to find two lenses, the errors of which would exactly correct each other. It is also easy to see, that the effect may be the same whether the concave glass is a portion of the same sphere with the others or not; the effect depending upon a combination of certain circumstances, of which there is an infinite variety.

By means of this correction of the errors arising from the different refrangibility of the rays of light, it is possible to shorten refracting telescopes considerably, and yet leave them equal magnifying powers. The reason of this is, that the errors arising from the object-glass being removed, those which are occasioned by the eye-glass are inconsiderable: for the error is always in proportion to the length of the focus in any glass; and in very long telescopes it becomes exceedingly great, being no less than \( \frac{1}{4} \)th of the whole; but in glasses of a few inches focus it becomes trifling. Refracting telescopes, which go by the name of Dollond's, are therefore now constructed in the following manner. Let AB represent an object-glass composed of three lenses as above described, and converging the rays 1, 2, 3, 4, &c., to a very distant focus as at x. By means of the interposed lens CD, however, they are converged to one much nearer, as at y, where an image of the object is formed. The rays diverging from thence fall upon another lens EF, where the pencils are rendered parallel, and an eye placed near that lens would see the object magnified and very distinct. To increase the magnifying power still more, however, the pencils thus become parallel are made to fall upon another at GH; by which they are again made to converge to a distant focus: but, being intercepted by the lens IK, they are made to meet at the nearer one z; whence diverging to LM, they are again rendered parallel, and the eye at N sees the object very distinctly.

From an inspection of the figure it is evident, that Dollond's telescope thus constructed is two telescopes combined together; the first ending with the lens EF, and the second with LM. In the first we do not perceive the object itself, but the image of it formed at y; and in the second we perceive only the image of that image formed at z. Such telescopes are nevertheless exceedingly distinct, and represent objects so clearly as to be preferred, in viewing terrestrial things, even to reflectors. The latter indeed have greatly the advantage in their powers of magnifying, but they are much deficient in point of light. Much more light is lost by reflection than by refraction: and as in these telescopes the light must unavoidably suffer two reflections, a great deal of it is lost; nor is this loss counterbalanced by the greater aperture which these telescopes will bear, which enables them to receive a greater quantity of light than the refracting ones. The metals of reflecting telescopes also are very much subject to tarnish, and require much more dexterity to clean them than the glasses of refractors; which makes them more troublesome and expensive, though for making discoveries in the heavens they are undoubtedly the only proper instruments which have been hitherto constructed.

II. The Reflecting Telescope.

The inconveniences arising from the great length of Newtonian refracting telescopes, before the discovery of the achromatic telescope, are sufficiently obvious; and these, together with the difficulties occasioned by the different refrangibility of light, induced Sir Isaac Newton to turn his attention to the subject of reflection, and endeavour to realize the ideas of himself and others concerning the possibility of constructing telescopes upon that principle.—The instrument which he contrived is represented, fig. 7, where ABCD is a large tube, open at AD, closed at BC, and of a length at least equal to the distance of the focus from the metallic spherical concave speculum GH placed at the end BC. The rays EG, FH, &c., proceeding from a remote object PR, intersect one another somewhere before they enter the tube, so that EG, &c., are those that come from the lower part of the object, and FH, from its upper part: these rays after falling on the speculum GH, will be reflected so as to converge and meet in mn, where they will form a perfect image of the object.—But as this image cannot be seen by the spectator, they are intercepted by a small plane metallic speculum KK, intersecting the axis at an angle of 45°, by which the rays tending to mn will be reflected towards a hole LL in the side of the tube, and the image will be less distinct, because some of the rays which would otherwise fall on the concave speculum GH, are intercepted by the plane speculum; nevertheless it will appear in a considerable degree distinct, because the aperture AD of the tube, and the speculum GH, are large. In the lateral hole LL is fixed a convex lens, whose focus is at S q; and therefore this lens will refract the rays that proceed from any point of the image, so as at their exit they will be parallel, and those that proceed from the extreme points S q will converge after refraction, and form an angle at O, where the eye is placed; which will see the image S q, as if it were an object through the lens LL; consequently the object will appear enlarged, inverted, bright, and distinct. In LL lenses of different convexities may be placed, which by being moved nearer to the image or farther from it, would represent the object more or less magnified, provided that the surface of the speculum GH be of a perfectly spherical figure. If, in the room of one lens LL, three lenses be disposed in the same manner with the three eye-glasses of the refracting telescope, the object will appear erect, but less distinct than when it is observed with one lens.

On account of the position of the eye in this telescope, it is extremely difficult to direct the instrument towards any object. Huygens, therefore, first thought of adding to it a small refracting telescope, the axis of which is parallel to that of the reflector. This is called a finder or director. When the Newtonian telescope is large, and placed upon its lower end to view bodies in great altitudes, the common finder can be of no use, from the difficulty of getting the eye to the eye-piece. On this account Dr Brewster proposes (Appendix to Ferguson's Lectures, vol. ii. p. 478.) to bend the tube of the finder to a right angle, and place a plane mirror at the angular point, so as to throw the image above the upper part of the tube that Optical Instrumentation

In that the eye-piece of the finder may be as near as possible to the eye-piece of the telescope. The angular part where the plain mirror is to be fixed, should be placed as near as possible to the focal image, in order that only a small part of the finder may stand above the tube; and in this way the eye can be transferred with the greatest facility from the one eyepiece to the other.

The advantages of this construction will be understood from fig. 3. Plate CCCCLXXXIX., where TT is part of a Newtonian telescope, D the eye-piece, and ABC the finder. The image formed by the object-glass A is reflected upwards by the plain mirror B, placed at an angle of 45° with the axis of the tube, and the image is viewed with the eye-glass AC. Those who have been in the habit of using the Newtonian telescope with the common finder will be sensible of the convenience resulting from this contrivance.

In order to determine the magnifying power of this telescope, it is to be considered that the plane speculum KK is of no use in this respect. Let us then suppose, that one ray proceeding from the object coincides with the axis GLA of the lens and speculum; let bb be another ray proceeding from the lower extreme of the object, and passing through the focus I of the speculum KH: this will be reflected in the direction b'd', parallel to the axis GLA, and falling on the lens d'Ld, will be refracted to G; so that GL will be equal to Ll, and d'G=d'I. To the naked eye the object would appear under the angle I b i = b I A; but by means of the telescope it appears under the angle d'GL=d'Ll=Idi; and the angle Idi is to the angle I b i :: I b : I d; consequently the apparent magnitude by the telescope is to that by the naked eye as the distance of the focus of the speculum from the speculum, to the distance of the focus of the lens from the lens.

The following new table of the apertures and magnifying power of Newtonian telescopes is taken from the Appendix to Ferguson's Lectures, vol. ii. p. 480. It is founded on a Newtonian telescope constructed by Hadley, in which the focal length of the great speculum was three feet three inches, and the magnifying power 226. Its aperture varied from three and a half to four and a half inches according to the want of brightness in the objects to be examined. The first column contains the focal length of the great speculum in feet, and the second its linear aperture in inches, and hundredths of an inch. The third and fourth columns contain Sir Isaac Newton's numbers, by means of which the apertures of any kind of reflecting telescopes may be easily computed. The fifth column contains the focal length of the eye-glasses in thousandths of an inch, and the sixth contains the magnifying power of the instrument.

| Focal length of the concave speculum | Aperture of the concave speculum | Sir Isaac Newton's numbers | Focal length of the eye-glass | Magnifying power | |-------------------------------------|---------------------------------|--------------------------|-------------------------------|----------------| | Feet | Inch. Dec. | | Inch. Dec. | Times | | 1 | 1.34 | 100 | 0.107 | 56 | | 2 | 2.23 | 168 | 0.129 | 93 | | 3 | 3.79 | 283 | 0.152 | 158 | | 4 | 5.14 | 383 | 0.168 | 214 | | 5 | 6.36 | 476 | 0.181 | 265 | | 6 | 7.51 | 562 | 0.192 | 313 | | 7 | 8.64 | 645 | 0.200 | 360 | | 8 | 9.67 | 800 | 0.209 | 403 | | 9 | 10.44 | 946 | 0.218 | 445 | | 10 | 11.69 | 1084 | 0.222 | 487 | | 11 | 12.65 | 1151 | 0.228 | 527 | | 12 | 13.58 | 1245 | 0.233 | 566 | | 13 | 14.50 | 1345 | 0.238 | 604 | | 14 | 15.41 | 1445 | 0.243 | 642 | | 15 | 16.25 | 1545 | 0.248 | 677 | | 16 | 17.11 | 1645 | 0.252 | 713 | | 17 | 17.98 | 1745 | 0.256 | 749 | | 18 | 18.82 | 1845 | 0.260 | 784 | | 19 | 19.63 | 1945 | 0.264 | 818 | | 20 | 20.45 | 2045 | 0.268 | 852 | | 21 | 21.24 | 2145 | 0.271 | 885 | | 22 | 22.06 | 2245 | 0.274 | 919 | | 23 | 22.85 | 2345 | 0.277 | 952 | | 24 | 23.62 | 2445 | 0.280 | 984 |

Let TYYT be a brass tube, in which LldD is a Gregorian metallic concave speculum, perforated in the middle telescope X; and EF a less concave mirror, so fixed by the arm or strong wire RT, which is moveable by means of a long screw on the outside of the tube, as to be moved nearer to or farther from the larger speculum LldD, its axis being kept in the same line with that of the great one. Let AB represent a very remote object, from each part of which issue pencils of rays, e.g. cd, CD, from A the upper extreme of the object, and LL, i't, from the lower part B; the rays LL, CD, from the extremes crossing one another before they enter the tube. These rays falling upon the larger mirror LD, are reflected from it into the focus K H, where they form an inverted image of the object AB, as in the Newtonian telescope. From this image the rays, issuing as from an object, fall upon the small mirror EF, the centre of which is at e; so that after reflection they would meet in their foci at QQ, and there form an erect image. But since an eye at that place could see but a small part of an object, in order to bring rays from more distant parts of it into the pupil, they are intercepted by the plano-convex lens MN, by which means a smaller erect image is formed at PV, which is viewed from Optical Instruments.

Part III. OPTICS.

Optical instruments both make the rays of each pencil parallel and magnify the image PV. At the place of this image all the foreign rays are intercepted by the perforated partition ZZ. For the same reason the hole near the eye O is very narrow. When nearer objects are viewed by this telescope, the small speculum EF is removed to a greater distance from the larger LD, so that the second image may be always formed in PV; and this distance is to be adjusted (by means of the screw on the outside of the great tube) according to the form of the eye of the spectator. It is also necessary, that the axis of the telescope should pass through the middle of the speculum EF, and its centre, the centre of the speculum LL, and the middle of the hole X, the centres of the lenses MN, SS, and the hole near O. As the hole X in the speculum LL can reflect none of the rays issuing from the object, that part of the image which corresponds to the middle of the object must appear to the observer more dark and confused than the extreme parts of it. Besides, the speculum EF will also intercept many rays proceeding from the object; and therefore unless the aperture TT be large, the object must appear in some degree obscure.

In the best reflecting telescopes, the focus of the small mirror is never coincident with the focus of the great one, where the first image KH is formed, but a little beyond it (with respect to the eye), as at n; the consequence of which is, that the rays of the pencils will not be parallel after reflection from the small mirror, but converge so as to meet in points about QqQ, where they would form a larger upright image than PV, if the glass R was not in their way; and this image might be viewed by means of a single eyeglass properly placed between the image and the eye; but then the field of view would be less, and consequently not so pleasant; for which reason, the glass R is still retained, to enlarge the scope or area of the field.

To find the magnifying power of this telescope, multiply the focal distance of the great mirror by the distance of the small mirror from the image next the eye, and multiply the focal distance of the small mirror by the focal distance of the eye-glass; then divide the former product by the latter, and the quotient will express the magnifying power. For a table of the apertures and powers of Gregorian telescopes, see Appendix to Ferguson's Lectures, vol. ii. p. 472, 473.

One great advantage of the reflecting telescope is, that it will admit of an eye-glass of a much shorter focal distance than a refracting telescope; and consequently it will magnify so much the more: for the rays are not coloured by reflection from a concave mirror, if it be ground to a true figure, as they are by passing through a convex glass, let it be ground ever so true.

The nearer an object is to the telescope, the more its pencils of rays will diverge before they fall upon the great mirror, and therefore they will be the longer of meeting in points after reflection; so that the first image KH will be formed at a greater distance from the large mirror, when the object is near the telescope, than when it is very remote. But as this image must be formed farther from the small mirror than its principal focus n, this mirror must be always set at a greater distance from the larger one, in viewing near objects, than in viewing remote ones. And this is done by turning the screw on the outside of the tube, until the small mirror be so adjusted, that the object (or rather its image) appears perfect.

In looking through any telescope towards an object, we never see the object itself, but only that image of it which is formed next the eye in the telescope. For if a man holds his finger or a stick between his bare eye and an object, it will hide part (if not the whole) of the object from his view: But if he ties a stick across the mouth of a telescope before the object-glass, it will hide no part of the imaginary object he saw through the telescope before, unless it covers the whole mouth of the tube: for all the effect will be, to make the object appear dimmer, because it intercepts part of the rays. Whereas, if he puts only a piece of wire across the inside of the tube, between the eye-glass and his eye, it will hide part of the object which he thinks he sees; which proves, that he sees not the real object, but its image. This is also confirmed by means of the small mirror EF, in the reflecting telescope, which is made of opaque metal, and stands directly between the eye and the object towards which the telescope is turned; and will hide the whole object from the eye at O, if the two glasses ZZ and SS are taken out of the tube.

If the small mirror of the preceding instrument be Cassegrain-convex instead of concave, it is then called the Cassegrainian telescope. As the small mirror is in this case placed between the great speculum and its focus, a Cassegrainian telescope will be shorter than a Gregorian one of the same magnifying power by twice the real length of the small mirror. For a table of the apertures, &c. of this instrument, see Appendix to Ferguson's Lectures, vol. ii. p. 474, 475.

Sect. VII. On the Merits of different Microscopes and Telescopes.

The advantages arising from the use of microscopes and telescopes depend, in the first place, upon their property of magnifying the minute parts of objects, so that they can by that means be more distinctly viewed by the eye; and, secondly, upon their throwing more light into the pupil of the eye than what is done without them. The advantages arising from the magnifying power would be extremely limited, if they were not also accompanied by the latter: for if the same quantity of light is spread over a large portion of surface, it becomes proportionably diminished in force; and therefore the objects, though magnified, appear proportionably dim. Thus, though any magnifying glass should enlarge the diameter of the object 10 times, and consequently magnify the surface 100 times, yet if the focal distance of the glass was about eight inches (provided this was possible), and its diameter only about the size of the pupil of the eye, the object would appear 100 times more dim when we looked through the glass, than when we beheld it with our naked eyes; and this, even on a supposition that the glass transmitted all the light which fell upon it, which no glass can do. But if the focal distance of the glass was only four inches, though its diameter remained as before, the inconvenience would be vastly diminished, because the glass could then then be placed twice as near the object as before, and consequently would receive four times as many rays as in the former case, and therefore we would see it much brighter than before. Going on thus, still diminishing the focal distance of the glass, and keeping its diameter as large as possible, we will perceive the object more and more magnified, and at the same time very distinct and bright. It is evident, however, that with regard to optical instruments of the microscopic kind, we must sooner or later arrive at a limit which cannot be passed. This limit is formed by the following particulars. 1. The quantity of light lost in passing through the glass. 2. The diminution of the glass itself, by which it receives only a small quantity of rays. 3. The extreme shortness of the focal distance of great magnifiers, whereby the free access of the light to the object which we wish to view is impeded, and consequently the reflection of the light from it is weakened. 4. The aberrations of the rays, occasioned by their different refrangibility.

To understand this more fully, as well as to see how far these obstacles can be removed, let us suppose the lens made of such a dull kind of glass that it transmits only one half of the light which falls upon it. It is evident that such a glass, of four inches focal distance, and which magnifies the diameter of an object twice, still supposing its own breadth equal to that of the pupil of the eye, will show it four times magnified in surface; but only half as bright as if it was seen by the naked eye at the usual distance; for the light which falls upon the eye from the object at eight inches distance, and likewise the surface of the object in its natural size, being both represented by 1, the surface of the magnified object will be 4, and the light which makes that magnified object visible only 2: because though the glass receives four times as much light as the naked eye does at the usual distance of distinct vision, yet one half is lost in passing through the glass. The inconvenience in this respect can therefore be removed only as far as it is possible to increase the clearness of the glass, so that it shall transmit nearly all the rays which fall upon it; and how far this can be done, hath not yet been ascertained.

The second obstacle to the perfection of microscopic glasses is the small size of great magnifiers, by which, notwithstanding their near approach to the object, they receive a smaller quantity of rays than might be expected. Thus suppose, a glass of only \(\frac{1}{16}\)th of an inch focal distance; such a glass would increase the visible diameter 80 times, and the surface 6400 times. If the breadth of the glass could at the same time be preserved as great as that of the pupil of the eye, which we shall suppose \(\frac{1}{16}\)th of an inch, the object would appear magnified 6400 times, at the same time that every part of it would be as bright as it appears to the naked eye. But if we suppose that this magnifying glass is only \(\frac{1}{16}\)th of an inch in diameter, it will then only receive \(\frac{1}{4}\)th of the light which otherwise would have fallen upon it; and therefore, instead of communicating to the magnified object a quantity of illumination equal to 6400, it would communicate only one equal to 1600, and the magnified object would appear four times as dim as it does to the naked eye. This inconvenience, however, is still capable of being removed, not indeed by increasing the diameter of the lens, because this must be in proportion to its focal distance, but by throwing a greater quantity of light on the object. Thus, in the above-mentioned example, if four times the quantity of light which naturally falls upon it could be thrown upon the object, it is plain that the reflection from it would be four times as great as in the natural way; and consequently the magnified image, at the same time that it was as many times magnified as before, would be as bright as when seen by the naked eye. In transparent objects this can be done very effectually by a concave speculum, as in the reflecting microscope already described: but in opaque objects the case is somewhat more doubtful; neither do the contrivances for viewing these objects seem entirely to make up for the deficiencies of the light from the smallness of the lens and shortness of the focus.—When a microscopic lens magnifies the diameter of an object forty times, it hath then the utmost possible magnifying power, without diminishing the natural brightness of the object.

The third obstacle arises from the shortness of the focal distance in large magnifiers: but in transparent objects, where a sufficient quantity of light is thrown on the object from below, the inconvenience arises at last from straining the eye, which must be placed nearer the glass than it can well bear; and this entirely supersedes the use of magnifiers beyond a certain degree.

The fourth obstacle arises from the different refrangibility of the rays of light, and which frequently causes such a deviation from truth in the appearances of things that many people have imagined themselves to have made surprising discoveries, and have even published them to the world: when in fact they have been only as many optical deceptions, owing to the unequal refractions of the rays. For this there seems to be no remedy, except the introduction of achromatic glasses into microscopes as well as telescopes. How far this is practicable, hath not yet been tried; but when these glasses shall be introduced (if such introduction is practicable,) microscopes will then undoubtedly have received their ultimate degree of perfection.

With regard to telescopes, those of the refracting Dollond kind have evidently the advantage of all others, where the aperture is equal, and the aberrations of the rays refracted are corrected according to Mr Dollond's method; because the image is not only more perfect, but a much greater quantity of light is transmitted than what can be reflected from the best materials hitherto known. Unluckily, however, the imperfections of the glass set a limit to these telescopes, as has been already observed, so that they cannot be made above three feet and a half long. On the whole, therefore, the reflecting telescopes are preferable in this respect, that they may be made of dimensions greatly superior; by which means they can both magnify to a greater degree, and at the same time throw much more light into the eye.

With regard to the powers of telescopes, however, they are all of them exceedingly less than what we would be apt to imagine from the number of times which they magnify the object. Thus, when we hear of a telescope which magnifies 200 times, we are apt to imagine, that, on looking at any distant object through it, we should perceive it as distinctly as we would with our naked eye at the 200th part of the The best method of trying the goodness of any telescope is to observe how much farther off you are able to read with it than with the naked eye. But that all deception may be avoided, it is proper to choose something to be read where the imagination cannot give any assistance, such as a table of logarithms, or something which consists entirely of figures; and hence the truly useful power of the telescope is easily known.

In this way Mr Short's large telescope, which magnifies the diameter of objects 1200 times, is yet unable to afford sufficient light for reading at more than 200 times the distance at which we can read with our naked eye.

With regard to the form of reflecting telescopes, it is now pretty generally agreed, that when the Gregorian ones are well constructed, they have the advantage of those of the Newtonian form. One advantage evident at first sight is, that with the Gregorian telescope an object is perceived by looking directly through it, and consequently is found with much greater ease than in the Newtonian telescope, where we must look into the side. The unavoidable imperfection of the specula common to both, also gives the Gregorian an advantage over the Newtonian form. Notwithstanding the utmost care and labour of the workmen, it is found impossible to give the metals either a perfectly spherical or a perfectly parabolical form. Hence arises some indistinctness of the image formed by the great speculum; which is frequently corrected by the little one, provided they are properly matched. But if this is not done, the error will be made much worse; and hence many of the Gregorian telescopes are far inferior to the Newtonian ones; namely, when the specula have not been properly adapted to each other. There is no method by which the workman can know the specula which will fit one another without a trial; and therefore it is necessary to have many specula ready made of each sort, that in fitting up a telescope those may be chosen which best suit each other.

The brightness of any object seen through a telescope, in comparison with its brightness when seen by the naked eye, may in all cases be easily found by the following formula. Let \( n \) represent the natural distance at which an object can be distinctly seen; and let \( d \) represent its distance from the object-glass of the instrument. Let \( m \) be the magnifying power of the instrument; that is, let the visual angle subtended at the eye by the object when at the distance \( n \), and viewed without the instrument, be to the visual angle produced by the instrument as \( r \) to \( m \). Let \( a \) be the diameter of the object-glass, and \( p \) that of the pupil. Let the instrument be so constructed, that no parts of the pencils are intercepted for want of sufficient apertures of the intermediate glasses. Lastly, let the light lost in reflection or refraction be neglected.

The brightness of vision through the instrument will be expressed by the fraction

\[ \frac{a^4}{n^3} \]

the brightness of natural vision being 1. But although this fraction may exceed unity, the vision through the instrument will not be brighter than natural vision. For, when this is the case, the pupil does not receive all the light transmitted through the instrument.

In microscopes, \( n \) is the nearest limits of distinct vision, nearly seven inches. But a difference in this circumstance, arising from a difference in the eye, makes no change in the formula, because \( m \) changes in the same proportion with \( n \).

In telescopes \( n \) and \( d \) may be reckoned equal, and the formula becomes

\[ \frac{a^4}{m^3} \]

A view of the history and construction of the telescope is given in the article Achromatic Glasses, in the Supplement.

**Sect. VIII. Apparatus for Measuring the Intensity of Light.**

That some luminous bodies give a stronger, and others a weaker light, and that some reflect more light than others, was always known; but no person, before M. Bouguer hit upon a tolerable method of ascertaining the proportion that two or more lights bear to one another. The methods he most commonly used were the following.

He took two pieces of wood or pasteboard EC and CD, in which he made two equal holes P and Q, over which he drew pieces of oiled or white paper. Upon these holes he contrived that the light of the different bodies he was comparing should fall; while he placed a third piece of pasteboard FG, so as to prevent the two lights from mixing with one another. Then placing himself sometimes on one side, and sometimes on the other, but generally on the opposite side of this instrument, with respect to the light, he altered their position till the papers in the two holes appeared to be equally enlightened. This being done, he computed the proportion of their light by the squares of the distances at which the luminous bodies were placed from the objects. If, for instance, the distances were as three and nine, he concluded that the lights they gave were as nine and eighty-one. Where any light was very faint, he sometimes made use of lenses, in order to condense it; and he enclosed them in tubes or not as his particular application of them required.

To measure the intensity of light proceeding from the heavenly bodies, or reflected from any part of the sky, he contrived an instrument which resembles a kind of portable camera obscura. He had two tubes, of which the inner was black, fastened at their lower extremities by a hinge C. At the bottom of these tubes were two holes, R and S, three or four lines in diameter, covered with two pieces of fine white paper. The two other extremities had each of them a circular aperture, an inch in diameter; and one of the tubes consisted of two, one of them sliding into the other, which produced the same effect as varying the aperture at the end. When this instrument is used, the observer has his head, and the end of the instrument C, so covered, that no light can fall upon his eye, besides that which comes through the two holes S and R, while an assistant manages the instrument, and draws out or shortens the tube DE, as the observer directs. When the two holes appear equally illuminated, the intensity of the lights is judged to be inversely as the squares of the tubes. In using this instrument, it is necessary that the object should subtend an angle larger than the aperture A or D, seen from the other end of the tube; for, otherwise, the lengthening of the tube has no effect. To avoid, in this case, making the instrument of an inconvenient length, or making the aperture D too narrow, he has recourse to another expedient. He constructs an instrument, represented (fig. 6.), consisting of two object-glasses, AE and DF, exactly equal, fixed in the ends of two tubes six or seven feet, or, in some cases, 10 or 12 feet long, and having their foci at the other ends. At the bottoms of these tubes B, are two holes, three or four lines in diameter, covered with a piece of white paper; and this instrument is used exactly like the former.

If the two objects to be observed by this instrument be not equally luminous, the light that issues from them must be reduced to an equality, by diminishing the aperture of one of the object-glasses; and then the remaining surface of the two glasses will give the proportion of their lights. But for this purpose, the central parts of the glass must be covered in the same proportion with the parts near the circumference, leaving the aperture such as is represented (fig. 7.), because the middle part of the glass is thicker and less transparent than the rest.

If all the objects to be observed lie nearly in the same direction, Bouguer remarks, that these two long tubes may be reduced into one, the two object-glasses being placed close together, and one eye-glass sufficing for them both. The instrument will then be the same with that of which he published an account in 1748, and which he called a heliometer, or astrometer.

It is not, however, the absolute quantity, but only the intensity of the light, that is measured by these two instruments, or the number of rays, in proportion to the surface of the luminous body; and it is of great importance that these two things be distinguished. The intensity of light may be very great, when the quantity, and its power of illuminating other bodies, may be very small, on account of the smallness of its surface; or the contrary may be the case, when the surface is large.

Having explained these methods which M. Bouguer took to measure the different proportions of light, we shall subjoin a few examples of his application of them.

It is observable, that when a person stands in a place where there is a strong light, he cannot distinguish objects that are placed in the shade; nor can he see anything upon going immediately into a place where there is very little light. It is plain, therefore, that the action of a strong light upon the eye, and also the impression which it leaves upon it, makes it insensible to the effect of a weaker light. M. Bouguer had the curiosity to endeavour to ascertain the proportion between the intensities of the two lights in this case; and by throwing the light of two equal candles upon a board, he found that the shadow made by intercepting the light of one of them, could not be perceived by his eye, upon the place enlightened by the other, at little more than eight times the distance; from whence he concluded, that when one light is eight times eight, or 64 times less than another, its presence or absence will not be perceived. He allows, however, that the effect may be different on different eyes; and supposes that the boundaries in this case, with respect to different persons, may lie between 60 and 80.

Applying the two tubes of his instrument, mentioned above, to measure the intensity of the light reflected from different parts of the sky; he found that when the sun was 25 degrees high, the light was four times stronger at the distance of eight or nine degrees from his body, than it was at 31 or 32 degrees. But what struck him the most was to find, that when the sun is 15 or 20 degrees high, the light decreases on the same parallel to the horizon to 110 or 120 degrees, and then increases again to the place exactly opposite to the sun.

The light of the sun, our author observes, is too strong, and that of the stars too weak, to determine the variation of their light at different altitudes; but as, in both cases, it must be in the same proportion with the diminution of the light of the moon in the same circumstances, he made his observations on that luminary, and found, that its light at 19° 16', is to its light at 66° 11', as 1681 to 2500; that is, the one is nearly two thirds of the other. He chose those particular altitudes, because they are those of the sun at the two solstices at Croisiac, where he then resided. When one limb of the moon touched the horizon of the sea, its light was 2000 times less than at the altitude of 66° 11'. But this proportion he acknowledges must be subject to many variations, the atmosphere near the earth varying so much in its density. From this observation he concludes, that at a medium light is diminished in the proportion of about 2500 to 1681, in traversing 7469 toises of dense air.

M. Bouguer also applied his instrument to the different parts of the sun's disk, and found that the centre is considerably more luminous than the extremities of parts of it. As near as he could make the observation, it disks of the sun was more luminous than a part of the disk ½ this of the semidiameter from it, in the proportion of 35 to 28; which, as he observes, is more than in the proportion of the sines of the angles of obliquity. On the other hand, he observes, that both the primary and secondary planets are more luminous at their edges than near their centres.

The comparison of the light of the sun and moon is a subject that has frequently exercised the thoughts of philosophers; but we find nothing but random conjectures, before Bouguer applied his accurate measures in this case. In general, the light of the moon is imagined to bear a much greater proportion to that of the sun than it really does: and not only are the imaginations of the vulgar, but those of philosophers also, imposed upon with respect to it. It was a great surprise to M. de la Hire to find that he could not, by the help of any burning mirror, collect the beams of the moon in a sufficient quantity to produce the least sensible heat. Other philosophers have since made the like attempts with mirrors of greater power, though without any greater success; but this will not surprise us, when we see the result of M. Bouguer's observations on this subject.

In order to solve this curious problem concerning the comparison of the light of the sun and moon, he compared each of them to that of a candle in a dark room, one in the day-time, and the other in the night following, the moon... ing, when the moon was at her mean distance from the earth; and, after many trials, he concluded that the light of the sun is about 300,000 times greater than that of the moon; which is such a disproportion, that, at he observes, it can be no wonder that philosophers have had so little success in their attempts to collect the light of the moon with burning glasses. For the largest of them will not increase the light 1000 times; which will still leave the light of the moon, in the focus of the mirror, 300 times less than the intensity of the common light of the sun.

To this account of the proportion of light which we actually receive from the moon, it cannot be displeasing to the reader, if we compare it with the quantity which would have been transmitted to us from that opaque body, if it reflected all the light it receives. Dr Smith thought that he had proved, from two different considerations, that the light of the full moon would be to our day-light as 1 to about 90,900, if no rays were lost at the moon.

In the first place, he supposes that the moon enlightened by the sun, is as luminous as the clouds are at a medium. He therefore supposed the light of the sun to be equal to that of a whole hemisphere of clouds, or as many moons as would cover the surface of the heavens. But on this Dr Priestley observes, that it is true, the light of the sun shining perpendicularly upon any surface would be equal to the light reflected from the whole hemisphere, if every part reflected all the light that fell upon it; but the light that would in fact be received from the whole hemisphere (part of it being received obliquely) would be only one-half as much as would be received from the whole hemisphere, if every part of it shone directly upon the surface to be illuminated.

In his Remarks, par. 97, Dr Smith demonstrates his method of calculation in the following manner.

Let the little circle cfdg represent the moon's body half enlightened by the sun, and the great circle ab, a spherical shell concentric to the moon, and touching the earth; ab, any diameter of that shell perpendicular to a great circle of the moon's body, represented by its diameter cd; e the place of the shell receiving full moon light from the bright hemisphere fgd. Now, because the surface of the moon is rough like that of the earth, we may allow that the sun's rays, incident upon any small part of it, with any obliquity, are reflected from it every way alike, as if they were emitted. And, therefore, if the segment df shone alone, the points ae, would be equally illuminated by it; and likewise if the remaining bright segment dg shone alone, the points be would be equally illuminated by it. Consequently, if the light at the point a was increased by the light at b, it would become equal to the full moon light at e. And conceiving the same transfer to be made from every point of the hemispherical surface habik to their opposite points in the hemisphere kaceh, the former hemisphere would be left quite-dark, and the latter would be uniformly illuminated with full moon light; arising from a quantity of the sun's light, which immediately before its incidence on the moon, would uniformly illuminate a circular plane equal to a great circle of her body, called her disk. Therefore the quantities of light being the same upon both surfaces, the density of the sun's incident light is to the density of full moon light, as that hemispherical surface hck is to the said disk; that is, as any other hemispherical surface whose centre is at the eye, to that part of it which the moon's disk appears to possess very nearly, because it subtends but a small angle at the eye: that is, as radius of the hemisphere to the versed sine of the moon's apparent semidiameter, or as 10,000,000 to 11662 or as 90,400 to 1; taking the moon's mean horizontal diameter to be 16' 7".

Strictly speaking, this rule compares moon light at the earth with day light at the moon; the medium of which, at her quadratures, is the same as our day-light; but is less at her full in the duplicate ratio of 365 to 366, or thereabout, that is, of the sun's distances from the earth and full moon; and therefore full moon light would be to our day light as about 1 to 90,900, if no rays were lost at the moon.

Secondly, I say that full moon light is to any other moon light as the whole disk of the moon to the part that appears enlightened, considered upon a plane surface. For now let the earth be at b, and let dl be perpendicular to fg, and gm to cd: then it is plain, that gl is equal to dm; and that gl is equal to a perpendicular section of the sun's rays incident upon the arch dgl which at b appears equal to dm; the eye being unable to distinguish the unequal distances of its parts. In like manner, conceiving the moon's surface to consist of innumerable physical circles parallel to cfdg, as represented at A, the same reason holds for every one of these circles as for cfdg. It follows then, that the bright part of the surface visible at b, when reduced to a flat as represented at B, by the crescent pdgm, will be equal and similar to a perpendicular section of all the rays incident on that part, represented at C by the crescent pgqlp. Now the whole disk being in proportion to this crescent as the quantities of light incident upon them; and the light falling upon every rough particle, being equally rarefied in diverging to the eye at b, considered as equidistant from them all; it follows, that full moon light is to this moon light as the whole disk pdq to the crescent pdgm.

Therefore by compounding this ratio with that in the former remark, day-light is to moon-light as the surface of an hemisphere whose centre is at the eye, to the part of that surface which appears to be possessed by the enlightened part of the moon.

Mr Michell made his computation in a much more simple and easy manner, and in which there is much less danger of falling into any mistake. Considering the distance of the moon from the sun, and that the density of the light must decrease in the proportion of the square of that distance, he calculated the density of the sun's light, at that distance, in proportion to its density at the surface of the sun; and in this manner he found, that if the moon reflected all the light it receives from the sun, it would only be the 45,000th part of the light we receive from the greater luminary. Admitting, therefore, that moon-light is only a 300,000th part of the light of the sun, Mr Michell concludes, that it reflects no more than between the 6th and 7th part of what falls upon it.

Could Rumford, has constructed a photometer, in which the shadows, instead of being thrown upon a photometer paper spread out upon the wainscot, or side of the room, are projected upon the inside of the back part. Apparatus of a wooden box \( \frac{7}{8} \) inches wide, \( \frac{10}{8} \) inches long, and \( \frac{3}{4} \) inches deep, in the clear. The light is admitted into it through two horizontal tubes in the front, placed so as to form an angle of \( 60^\circ \); their axes meeting at the centre of the field of the instrument. In the middle of the front of the box, between these two tubes, is an opening through which is viewed the field of the photometer (see fig. 52). This field is formed of a piece of white paper, which is not fastened immediately upon the inside of the back of the box, but is pasted upon a small pane of very fine ground glass; and this glass, thus covered, is let down into a groove, made to receive it, in the back of the box. The whole inside of the box, except the field of the instrument, is painted of a deep black dead colour.

To the under part of the box is fitted a ball and socket, by which it is attached to a stand which supports it; and the top or lid of it is fitted with hinges, in order that the box may be laid quite open, as often as it is necessary to alter any part of the machinery it contains.

The count had found it very inconvenient to compare two shadows projected by the same cylinder, as these were either necessarily too far from each other to be compared with certainty, or, when they were nearer, were in part hid from the eye by the cylinder. To remedy this inconvenience, he now makes use of two cylinders, which are placed perpendicularly in the bottom of the box just described, in a line parallel to the back part of it, distant from this back \( \frac{2}{15} \) inches, and from each other 3 inches, measuring from the centres of the cylinders; when the two lights made use of in the experiment are properly placed, these two cylinders project four shadows upon the white paper upon the inside of the back part of the box, or the field of the instrument; two of which shadows are in contact, precisely in the middle of that field, and it is these two alone that are to be attended to. To prevent the attention being distracted by the presence of unnecessary objects, the two outside shadows are made to disappear; which is done by rendering the field of the instrument so narrow, that they fall without it, upon a blackened surface, upon which they are not visible. If the cylinders be each \( \frac{4}{15} \) of an inch in diameter, and \( \frac{2}{15} \) inches in height, it will be quite sufficient that the field be \( \frac{2}{15} \) inches wide; and as an unnecessary height of the field is not only useless, but disadvantageous, as a large surface of white paper not covered by the shadows produces too strong a glare of light, the field ought not to be more than \( \frac{1}{15} \) of an inch higher than the tops of the cylinders. That its dimensions, however, may be occasionally augmented, the covered glass should be made \( \frac{5}{8} \) inches long, and as wide as the box is deep, viz. \( \frac{3}{4} \) inches; since the field of the instrument can be reduced to its proper size by a screen of black pasteboard, interposed before the anterior surface of this covered glass, and resting immediately upon it. A hole in this pasteboard, in the form of an oblong square, \( \frac{1}{15} \) inch wide, and two inches high, determines the dimensions, and forms the boundaries of the field. This screen should be large enough to cover the whole inside of the back of the box, and it may be fixed in its place by means of grooves in the sides of the box, into which it may be made to enter. The position of the opening above mentioned is determined by the height of the cylinders; the top of it being \( \frac{1}{15} \) of an inch higher than the tops of the cylinders; and as the height of it is only two inches, while the height of the cylinders is \( \frac{2}{15} \) inches, it is evident that the shadows of the lower parts of the cylinders do not enter the field. No inconvenience arises from that circumstance; on the contrary, several advantages are derived from that arrangement.

That the lights may be placed with facility and precision, a fine black line is drawn through the middle of the field, from the top to the bottom of it, and another (horizontal) line at right angles to it, at the height of the top of the cylinders. When the tops of the shadows touch this last mentioned line, the lights are at a proper height; and farther, when the two shadows are in contact with each other in the middle of the field, the lights are then in their proper directions.

We have said that the cylinders, by which the shadows are projected, are placed perpendicularly in the bottom of the box; but as the diameters of the shadows of these cylinders vary in some degree, in proportion as the lights are broader or narrower, and as they are brought nearer to or removed farther from the photometer, in order to be able in all cases to bring these shadows to be of the same diameter, which is very advantageous, in order to judge with greater facility and certainty when they are of the same density, the count renders the cylinders moveable about their axes, and adds to each a vertical wing \( \frac{1}{15} \) of an inch wide, \( \frac{1}{15} \) of an inch thick, and of equal height with the cylinder itself, and firmly fixed to it from the top to the bottom. This wing commonly lies in the middle of the shadow of the cylinder, and as long as it remains in that situation it has no effect whatever; but when it is necessary that the diameter of one of the shadows be increased, the corresponding cylinder is moved about its axis, till the wing just described, emerging out of the shadow, and intercepting a portion of light, brings the shadow projected upon the field of the instrument to be of the width or diameter required. In this operation it is always necessary to turn the cylinder outwards, or in such a manner that the augmentation of the width of the shadow may take place on that side of it which is opposite to the shadow corresponding to the other light. The necessity for that precaution will appear evident to any one who has a just idea of the instrument in question, and of the manner of making use of it. They are turned likewise without opening the box, by taking hold of the ends of their axes, which project below its bottom.

As it is absolutely necessary that the cylinders should constantly remain precisely perpendicular to the bottom of the box, or parallel to each other, it will be best to construct them of brass; and, instead of fixing them immediately to the bottom of the box (which, being of wood, may warp), to fix them to a strong thick piece of well-hammered plate brass; which plate of brass may be afterwards fastened to the bottom of the box by means of one strong screw. In this manner two of the count's best instruments are constructed; and, in order to secure the cylinders still more firmly in their vertical positions, they are furnished with broad flat rings, or projections, where they rest upon the brass plate; which rings are \( \frac{1}{15} \) of an inch thick, and equal in diameter to the projection of the wing of the cylinder, to the bottom of which they afford a firm support. These cylinders are likewise forcibly pushed, or rather pulled, against against the brass plate upon which they rest, by means of compressed spiral springs placed between the under side of that plate and the lower ends of the cylinders.

Of whatever material the cylinders be constructed, and whatever be their forms or dimensions, it is absolutely necessary that they, as well as every other part of the photometer, except the field, should be well painted of a deep black dead colour.

In order to move the lights to and from the photometer with greater ease and precision, the observer should provide two long and narrow, but very strong and steady, tables; in the middle of each of which there is a straight groove, in which a sliding carriage, upon which the light is placed, is drawn along by means of a cord which is fastened to it before and behind, and which, passing over pulleys at each end of the table, goes round a cylinder; which cylinder is furnished with a winch, and is so placed, near the end of the table adjoining the photometer, that the observer can turn it about, without taking his eye from the field of the instrument.

Many advantages are derived from this arrangement: First, the observer can move the lights as he finds necessary, without the help of an assistant, and even without removing his eye from the shadows; secondly, each light is always precisely in the line of direction in which it ought to be, in order that the shadows may be in contact in the middle of the vertical plane of the photometer; and, thirdly, the sliding motion of the lights being perfectly soft and gentle, that motion produces little or no effect upon the lights themselves, either to increase or diminish their brilliancy.

These tables must be placed at an angle of 60 degrees from each other, and in such a situation, with respect to the photometer, that lines drawn through their middles, in the direction of their lengths, meet in a point exactly under the middle of the vertical plane or field of the photometer, and from that point the distances of the lights are measured; the sides of the tables being divided into English inches, and a vernier, showing tenths of inches, being fixed to each of the sliding carriages upon which the lights are placed, and which are so contrived that they may be raised or lowered at pleasure; so that the lights may be always in a horizontal line with the tops of the cylinders of the photometer.

In order that the two long and narrow tables or platforms, just described, may remain immovable in their proper positions, they are both firmly fixed to the stand which supports the photometer; and, in order that the motion of the carriages which carry the lights may be as soft and gentle as possible, they are made to slide upon parallel brass wires, 9 inches asunder, about \( \frac{1}{2} \) of an inch in diameter, and well polished, which are stretched out upon the tables from one end to the other.

The structure of the apparatus will be clearly understood by a bare inspection of Plate CCCLXXXIX.

Fig. 5. is a plan of the inside of the box, and the adjoining parts of the photometer. Fig. 6. Plan of the two tables belonging to the photometer. Fig. 7. The box of the photometer on its stand. Fig. 8. Elevation of the photometer, with one of the tables and carriages.

Having sufficiently explained all the essential parts of this photometer, it remains for us to give some account of the precautions necessary to be observed in using it. And, first, with respect to the distance at which lights, whose intensities are to be compared, should be placed from the field of the instrument, the ingenious and accurate inventor found, that when the weakest of the lights in question is about as strong as a common wax candle, that light may most advantageously be placed from 30 to 36 inches from the centre of the field; and when it is weaker or stronger, proportionally nearer or farther off. When the lights are too near, the shadows will not be well defined; and when they are too far off, they will be too weak.

It will greatly facilitate the calculations necessary in drawing conclusions from experiments of this kind, if some steady light, of a proper degree of strength for that purpose, be assumed as a standard by which all others may be compared. Our author found a good Argand's lamp much preferable for this purpose to any other lamp or candle whatever. As it appears, he says, from a number of experiments, that the quantity of light emitted by a lamp, which burns in the same manner with a clear flame, and without smoke, is in all cases as the quantity of oil consumed, there is much reason to suppose, that, if the Argand's lamp be so adjusted as always to consume a given quantity of oil in a given time, it may then be depended on as a just standard of light.

In order to abridge the calculation necessary in these inquiries, it will always be advantageous to place the standard-lamp at the distance of 100 inches from the photometer, and to assume the intensity of its light at its source equal to unity; in this case (calling this standard light A, the intensity of the light at its source \( x = 1 \), and the distance of the lamp from the field of the photometer \( m = 100 \)), the intensity of the illumination at the field of the photometer \( \left( \frac{x}{m^2} \right) \) will be expressed by the fraction \( \frac{1}{100} \); and the relative intensity of any other light which is compared with it, may be found by the following proportion: Calling this light B, putting \( y \) = its intensity at its source, and \( n \) = its distance from the field of the photometer, expressed in English inches, as it is \( \frac{y}{n^2} = \frac{x}{m^2} \), or, instead of \( \frac{x}{m^2} \), writing its value \( \frac{1}{10000} \), it will be \( \frac{y}{n^2} = \frac{1}{10000} \); and consequently \( y \) is to 1 as \( n^2 \) is to 10000; or the intensity of the light B at its source, is to the intensity of the standard light A at its source, as the square of the distance of the light B from the middle of the field of the instrument, expressed in inches, is to 10000; and hence it is \( y = \frac{n^2}{10000} \).

Or, if the light of the sun, or that of the moon, be compared with the light of a given lamp or candle C, the result of such comparison may be best expressed in words, by saying, that the light of the celestial luminary in question, at the surface of the earth, or, which is the same thing, at the field of the photometer, is equal to the light of the given lamp or candle, at the distance found by the experiment; or, putting \( a \) = the intensity of the light of this lamp C at its source, and \( p \) = its distance, Apparatus distance, in inches, from the field, when the shadows corresponding to this light, and that corresponding to the celestial luminary in question, are found to be of equal densities, and putting \( x = \frac{a}{p^2} \); or the real value of \( a \) being determined by a particular experiment, made expressly for that purpose with the standard lamp, that value may be written instead of it. When the standard lamp itself is made use of, instead of lamp C, then the value of \( A \) will be 1.

The count's first attempts with his photometer were to determine how far it might be possible to ascertain, by direct experiments, the certainty of the assumed law of the diminution of the intensity of the light emitted by luminous bodies; namely, that the intensity of the light is everywhere as the squares of the distances from the luminous body inversely. As it is obvious that this law can hold good only when the light is propagated through perfectly transparent spaces, so that its intensity is weakened merely by the divergency of its rays, he instituted a set of experiments to ascertain the transparency of the air and other mediums.

With this view, two equal wax candles, well trimmed, and which were found, by a previous experiment, to burn with exactly the same degree of brightness, were placed together, on one side, before the photometer, and their united light was counterbalanced by the light of an Argand's lamp, well trimmed, and burning very equally, placed on the other side over against them. The lamp was placed at the distance of 100 inches from the field of the photometer, and it was found that the two burning candles (which were placed as near together as possible, without their flames affecting each other by the currents of air they produced) were just able to counterbalance the light of the lamp at the field of the photometer, when they were placed at the distance of 60.8 inches from that field. One of the candles being now taken away and extinguished, the other was brought nearer to the field of the instrument, till its light was found to be just able, singly, to counterbalance the light of the lamp; and this was found to happen when it had arrived at the distance of 43.4 inches. In this experiment, as the candles burnt with equal brightness, it is evident that the intensities of their united and single lights were as 2 to 1, and in that proportion ought, according to the assumed theory, the squares of the distances, 60.8 and 43.4, to be; and, in fact, \( 60.8^2 = 3696.64 \) is to \( 43.4^2 = 1883.56 \) as 2 is to 1 very nearly.

Again, in another experiment, the distances were,

| With two candles | 54 inches | Square = 2916 | |-----------------|-----------|--------------| | With one candle | 38.6 | = 1489.96 |

Upon another trial,

| With two candles | 54.6 inches | Square = 2981.16 | |------------------|-------------|------------------| | With one candle | 39.7 | = 1576.09 |

And, in the fourth experiment,

| With two candles | 58.4 inches | Square = 3410.56 | |------------------|-------------|------------------| | With one candle | 42.2 | = 1780.84 |

And, taking the mean of the results of these four experiments,

Squares of the distances

| With two candles | With one candle | |------------------|-----------------| | N° 1. 3696.64 | 1883.56 | | N° 2. 2916 | 1489.96 | | N° 3. 2981.16 | 1576.09 | | N° 4. 3410.56 | 1780.84 |

Means 3251.09 and 1682.61

which again are very nearly as 2 to 1.

With regard to these experiments, it may be observed, that were the resistance of the air to light, or the diminution of the light from the imperfect transparency of air, sensible within the limits of the inconsiderable distances at which the candles were placed from the photometer, in that case the distance of the two equal lights united ought to be, to the distance of one of them single, in a ratio less than that of the square root of 2 to the square root of 1. For if the intensity of a light emitted by a luminous body, in a space void of all resistance, be diminished in the proportion of the squares of the distances, it must of necessity be diminished in a still higher ratio when the light passes through a resisting medium, or one which is not perfectly transparent; and from the difference of those ratios, namely, that of the squares of the distances, and that other higher ratio found by the experiment, the resistance of the medium might be ascertained. This he took much pains to do with respect to air, but did not succeed; the transparency of air being so great, that the diminution which light suffers in passing through a few inches, or even through several feet of it, is not sensible.

Having found, upon repeated trials, that the light of a lamp, properly trimmed, is incomparably more equal than that of a candle, whose wick, continually growing longer, renders its light extremely fluctuating, he substituted lamps to candles in these experiments, and made such other variations in the manner of conducting them as he thought bid fair to lead to a discovery of the resistance of the air to light, were it possible to render that resistance sensible within the confined limits of his machinery. But the results of them, so far from affording means for ascertaining the resistance of the air to light, do not even indicate any resistance at all; on the contrary, it might almost be inferred, from some of them, that the intensity of the light emitted by a luminous body in air is diminished in a ratio less than that of the squares of the distances; but as such a conclusion would involve an evident absurdity, namely, that light moving in air, its absolute quantity, instead of being diminished, actually goes on to increase, that conclusion can by no means be admitted.

Why not? Theories must give place to facts; and if this fact can be fairly ascertained, instead of rejecting the conclusion, we ought certainly to rectify our notions of light, the nature of which we believe no man fully comprehends. Who can take it upon him to say, that the substance of light is not latent in the atmosphere, as heat or caloric is now acknowledged to be latent, and that the agency of the former is not called forth by the passage of a ray through a portion of air, as the agency of the latter is known to be excited by The ingenious author's experiments all conspired to show that the resistance of the air to light is too inconsiderable to be perceptible, and that the assumed law of the diminution of the intensity of light may be depended upon with safety. He admits, however, that means may be found for rendering the air's resistance to light apparent; and he seems to have thought of the very means which occurred for this purpose to M. de Saussure.

That eminent philosopher, wishing to ascertain the transparency of the atmosphere, by measuring the distances at which determined objects cease to be visible, perceived at once that his end would be attained, if he should find objects of which the disappearance might be accurately determined. Accordingly, after many trials, he found that the moment of disappearance can be observed with much greater accuracy when a black object is placed on a white ground, than when a white object is placed on a black ground; that the accuracy was still greater when the observation was made in the sun than in the shade; and that even a still greater degree of accuracy was obtained, when the white space surrounding a black circle, was itself surrounded by a circle or ground of a dark colour. This last circumstance was particularly remarkable, and an observation quite new.

If a circle totally black, of about two lines in diameter, be fastened on the middle of a large sheet of paper or pasteboard, and if this paper or pasteboard be placed in such a manner as to be exposed fully to the light of the sun, if you then approach it at the distance of three or four feet, and afterwards gradually recede from it, keeping your eye constantly directed towards the black circle, it will appear always to decrease in size the farther you retire from it, and at the distance of 33 or 34 feet will have the appearance of a point. If you continue still to recede, you will see it again enlarge itself; and it will seem to form a kind of cloud, the darkness of which decreases more and more according as the circumference becomes enlarged. The cloud will appear still to increase in size the farther you remove from it; but at length it will totally disappear. The moment of the disappearance, however, cannot be accurately ascertained; and the more experiments were repeated, the more were the results different.

M. de Saussure, having reflected for a long time on the means of remedying this inconvenience, saw clearly, that as long as this cloud took place, no accuracy could be obtained; and he discovered that it appeared in consequence of the contrast formed by the white parts which were at the greatest distance from the black circle. He thence concluded, that if the ground was left white near this circle, and the parts of the pasteboard at the greatest distance from it were covered with a dark colour, the cloud would no longer be visible, or at least almost totally disappear.

This conjecture was confirmed by experiment. M. de Saussure left a white space around the black circle equal in breadth to its diameter, by placing a circle of black paper a line in diameter, on the middle of a white circle three lines in diameter, so that the black circle was only surrounded by a white ring a line in breadth. The whole was pasted upon a green ground. A green colour was chosen, because it was dark enough to make the cloud disappear, and the easiest to be procured.

The black circle surrounded in this manner with white on a green ground, disappeared at a much less distance than when it was on a white ground of a large size.

If a perfectly black circle, a line in diameter, be pasted on the middle of a white ground exposed to the open light, it may be observed at the distance of from 44 to 45 feet; but if this circle be surrounded by a white ring a line in breadth, while the rest of the ground is green, all sight of it is lost at the distance of only 15 feet.

According to these principles M. de Saussure delineated several black circles, the diameters of which increased in a geometrical progression, the exponent of which was \( \frac{1}{2} \). His smallest circle was \( \frac{1}{2} \) or \( \frac{1}{2} \) of a line in diameter; the second, \( \frac{1}{3} \); the third, \( \frac{1}{4} \); and so on to the sixteenth, which was 87.527, or about 7 inches 3½ lines. Each of these circles was surrounded by a white ring, the breadth of which was equal to the diameter of the circle, and the whole was pasted on a green ground.

M. de Saussure, for his experiments, selected a straight road or plain of about 1200 or 1500 feet in circumference, which towards the north was bounded by trees or an ascent. Those who repeat them, however, must pay attention to the following remarks: When a person retires backwards, keeping his eye constantly fixed on the pasteboard, the eye becomes fatigued, and soon ceases to perceive the circle; as soon therefore as it ceases to be distinguishable, you must suffer your eyes to rest; not, however, by shutting them, for they would when again opened be dazzled by the light, but by turning them gradually to some less illuminated object in the horizon. When you have done this for about half a minute, and again directed your eyes to the pasteboard, the circle will be again visible, and you must continue to recede till it disappear once more. You must then let your eyes rest a second time in order to look at the circle again, and continue in this manner till the circle becomes actually invisible.

If you wish to find an accurate expression for the want of transparency, you must employ a number of circles, the diameters of which increase according to a certain progression; and a comparison of the distances at which they disappear will give the law according to which the transparency of the atmosphere decreases at different distances. If you wish to compare the transparency of the atmosphere on two days, or in two different places, two circles will be sufficient for the experiment.

According to these principles, M. de Saussure caused to be prepared a piece of white linen cloth eight feet square. In the middle of this square he sewed a perfect circle, two feet in diameter, of beautiful black wool; around this circle he left a white ring two feet in breadth, and the rest of the square was covered with pale green. In the like manner, and of the same materials, he prepared another square; which was, however, equal to only \( \frac{1}{2} \) of the size of the former, so that each side of it was 8 inches; the black circle in the middle was two inches in diameter, and the white space around the circle was 2 inches also.

If two squares of this kind be suspended vertically and parallel to each other, so that they may be both illuminated in an equal degree by the sun; and if the atmosphere, at the moment when the experiment is made, be perfectly transparent, the circle of the large square, which is twelve times the size of the other, must be seen at twelve times the distance. In M. de Saussure's experiments the small circle disappeared at the distance of 314 feet, and the large one at the distance of 3588 feet, whereas it should have disappeared at the distance of 3768. The atmosphere, therefore, was not perfectly transparent. This arose from the thin vapours which at that time were floating in it. M. de Saussure calls his instrument a diaphanometer; but it serves one of the purposes of a photometer.

From a number of experiments made with the photometer, Count Rumford found, that, by passing through a pane of fine, clear, well polished glass, such as is commonly made use of in the construction of looking-glasses, light loses .1973 or its whole quantity, i.e. of the quantity which impinged on the glass; that when light is made to pass through two panes of such glass standing parallel, but not touching each other, the loss is .3184 of the whole; and that in passing through a very thin, clear, colourless pane of window-glass, the loss is only .1263. Hence he infers, that this apparatus might be very usefully employed by the optician, to determine the degree of transparency of glass, and direct his choice in the provision of that important article of his trade. The loss of light when reflected from the very best plain glass mirror, the author ascertained, by five experiments, to be ¼d of the whole which fell upon the mirror.

An ingenious photometer has also been invented by Professor Leslie, and fully described in his celebrated work on Heat, to which we must refer the reader for a complete description of this instrument. It measures the calorific effect of heat, and is founded upon this principle, "that if a body be exposed to the sun's rays, it will, in every possible case, be found to indicate a measure of heat exactly proportioned to the quantity of light which it has absorbed." See Essay on Heat, p. 103.

**CHAP. II. On the method of forming the Lenses and Specula, of Refracting and Reflecting Telescopes.**

**Sect. I. On the Method of grinding and polishing Lenses.**

Having fixed upon the proper aperture and focal distance of the lens, take a piece of sheet copper, and strike a fine arch upon its surface, with a radius equal to half that distance, if it is to be plano-convex, and let the length of this arch be a little greater than the given aperture. Remove with a file that part of the copper which is without the circular arch, and a convex gage will be formed. Strike another arch with the same radius, and having removed that part of the copper which is within it, a concave gage, will be obtained. Prepare two circular plates of brass, about ¼ of an inch thick, and half an inch greater in diameter than the breadth of the lens, and solder them upon a cylinder of lead of the same diameter, and about an inch high. These tools are then to be fixed upon a turning lathe, and one of them turned into a portion of a concave sphere, so as to suit the convex gage; and the other into a portion of a convex sphere, so as to answer the concave gage. After the surfaces of the brass plates are turned as accurately as possible, they must be ground upon one another, alternately, with flour emery; and when the two surfaces exactly coincide, the grinding tools will be ready for use.

Procure a piece of glass whose dispersive power is as small as possible, if the lens is not for achromatic instruments, and whose surfaces are parallel; and by means of a pair of large scissors or pincers, cut it into a circular shape, so that its diameter may be a little greater than the required aperture of the lens. When the roughness is removed from its edges by a common grindstone (A), it is to be fixed with black pitch to a wooden handle of a smaller diameter than the glass, and about an inch high, so that the centre of the handle may exactly coincide with the centre of the glass.

The glass being thus prepared, it is then to be ground Mode of with fine emery upon the concave tool, if it is to be grinding, convex, and upon the convex tool, if it is to be concave. To avoid circumlocution, we shall suppose that the lens is to be convex. The concave tool, therefore, which is to be used, must be firmly fixed to a table or bench, and the glass wrought upon it with circular strokes, so that its centre may never go beyond the edges of the tool. For every 6 circular strokes, the glass should receive 2 or 3 cross ones along the diameter of the tool, and in different directions. When the glass has received its proper shape, and touches the tool in every point of its surface, which may be easily known by inspection, the emery is to be washed away, and finer kinds (B) successively substituted in its room, till by the same alternation of circular and transverse strokes, all the scratches and asperities are removed from its surface. After the finest emery has been used, the roughness which remains may be taken away, and a slight polish superinduced by grinding the glass with pounded pumice-stone in the same manner as before. While the operation of grinding is going on, the convex tool should, at the end of every five

(A) When the focal distance of the lens is to be short, the surface of the piece of glass should be ground upon a common grindstone, so as to suit the gage as nearly as possible; and the plates of brass, before they are soldered on the lead, should be hammered as truly as they can be done into their proper form. By this means much labour will be saved both in turning and grinding.

(B) Emery of different degrees of fineness may be made in the following manner. Take five or six clean vessels, and having filled one of them with water, put into it a considerable quantity of flour emery. Stir it well with a piece of wood, and after standing for 5 seconds pour the water into the second vessel. After it has stood about 12 seconds, pour it out of this into a third vessel, and so on with the rest; and at the bottom of each vessel will be found emery of different degrees of fineness, the coarsest being in the first vessel, and the finest in the last. In this manner the small lenses of simple and compound microscopes, the eye-glasses and the object-glasses of telescopes, are to be ground. In grinding concave lenses, Mr Imison* employs leaden wheels with the same radius as the curvature of the lens, and with their circumferences of the same convexity as the lens is to Arts, part be concave. These spherical zones are fixed upon a turning lathe, and the lens, which is held steadily in the hand, is ground upon them with emery, while they are revolving on the spindle of the lathe. In the same way convex lenses may be ground and polished, by fixing the concave tool upon the lathe; but these methods, however simple and expeditious they may be, should never be adopted for forming the lenses of optical instruments, where an accurate spherical figure is indispensable. It is by the hand alone that we can perform with accuracy those circular and transverse strokes, the proper union of which is essential to the production of a spherical surface. Appendix to Ferguson's Lectures, vol. ii. p. 452.

Sect. II. On the Method of Casting, Grinding, and Polishing the Specula of Reflecting Telescopes.

The metals of reflecting telescopes are generally composed of 32 parts of copper, and 15 of gun tin, with the addition of two parts of arsenic, to render the composition more white and compact. The Reverend Mr Edwards found, from a variety of experiments, that if one part of brass, and one of silver, be added to the preceding composition, and only one part of arsenic used, a most excellent metal will be obtained, which is the whitest, hardest, and most reflective, that he ever met with. The superiority of this composition, indeed, has been completely evinced by the excellence of Mr Edwards' telescopes, which excel other reflectors in brightness and distinctness, and show objects in their natural colours. But as metals of this composition are extremely difficult to cast, as well as to grind and polish, it will be better for those who are inexperienced in the art, to employ the composition first mentioned.

After the flasks of sand (D) are prepared, and a mould made for the metal by means of a wooden or metallic casting the pattern, so that its face may be downwards, and a few small holes left in the sand at its back, for the free egress of the included air—melt the copper in a crucible by itself, and when it is reduced to a fluid state, fuse the tin in a separate crucible, and mix it with the melted copper, by stirring them together with a wooden spatula. The proper quantity of powdered arsenic, wrapped up in a piece of paper, is then to be added, the operator retaining his breath till its noxious fumes are completely dissipated; and when the scoria is removed from the fluid mass, it is to be poured out as quickly as possible into the flasks. As soon as the metal is become solid, remove it from the sand into some hot ashes or coals, for the purpose of annealing.

(c) As colcothar of vitriol is obtained by the decomposition of martial vitriol, it sometimes retains a portion of this salt. When this portion of martial vitriol is decomposed by dissolution in water, the yellow ochre which results penetrates the glass, forms an incrustation upon its surface, and gives it a dull and yellowish tinge, which is communicated to the image which it forms.

(d) The finest sand which we have met with in this country, is to be found at Roxburgh castle, in the neighbourhood of Kelso. Method of annealing it, and let it remain among them till they are completely cold. The ingate is then to be taken from the metal by means of a file; and the surface of the speculum must be ground upon a common grindstone, till all the imperfections and asperities are taken away. When Mr Edwards' composition is employed, the copper and tin should be melted according to the preceding directions, and, when mixed together, should be poured into cold water, which will separate the mass into a number of small particles. These small pieces of metal are then to be collected and put into the crucible, along with the silver and brass, after they have been melted together in a separate crucible; the proper quantity of arsenic is to be added, and a little powdered rosin thrown into the fluid metal before it is poured into the flasks.

When the metal is cast, and prepared by the common grindstone for receiving its proper figure, the gages and grinding tools are to be formed in the same manner as for convex lenses, with this difference only, that the radius of the gages must always be double the focal length of the speculum. In addition to the convex and concave brass tools, which should be only a little broader than the metal itself, a convex elliptical tool of lead and tin should also be formed with the same radius, so that its transverse may be to its conjugate diameter as 10 to 9, the latter being exactly equal to the diameter of the metal. On this tool the speculum is to be ground with flour emery, in the same manner as lenses, with circular and cross strokes alternately, till its surface is freed from every imperfection, and ground to a spherical figure. It is then to be wrought with great circumspection, on the convex brass tool, with emery of different degrees of fineness, the concave tool being sometimes ground upon the convex one, to keep them all of the same radius; and when every scratch and appearance of roughness is removed from its surface, it will be fit for receiving the final polish. Before the speculum is brought to the polisher, it has been the practice to smooth it on a bed of hones, or a convex tool made of common blue hones. This additional tool, indeed, is absolutely necessary, when silver and brass enter into the composition of the metal, in order to remove that roughness which will always remain after the finest emery has been used; but when these metals are not ingredients in the speculum, there is no occasion for the bed of hones. Without the intervention of this tool I have finished several specula, and given them as exquisite a lustre as they could possibly have received. Mr Edwards does not use any brass tools in his process, but transfers the metal from the elliptical leaden tool to the bed of hones. By this means the operation is simplified, but we doubt much if it is, in the least degree, improved. As a bed of hones is more apt to change its form than a tool of brass, it is certainly of great consequence that the speculum should have as true a figure as possible before it is brought to the hones; and we are persuaded, from experience, that this figure may be better communicated on a brass tool, which can always be kept at the same curvature by its corresponding tool, than on an elliptical block of lead. We are certain, however, that when the speculum is required to be of a determinate focal length, this length will be obtained more precisely with the brass tools than without them. But Mr Edwards has observed, that these tools are not only unnecessary, but 'really detrimental.' That Mr Edwards found them unnecessary, we cannot doubt, from the excellence of the specula which he formed without their assistance; but it seems inconceivable how the brass tools can be in the least degree detrimental. If the mirror is ground upon 20 different tools before it is brought to the bed of hones, it will receive from the last of these tools a certain figure, which it would have received even if it had not been ground on any of the rest; and it cannot be questioned, that a metal wrought upon a pair of brass tools, is equally, if not more, fit for the bed of hones, than if it had been ground merely on a tool of lead.

When the metal is ready for polishing, the elliptical leaden tool is to be covered with black pitch, about the metal, one-twentieth of an inch thick, and the polisher formed in the same way as in the case of lenses, either with the concave brass tool, or with the metal itself. The colcothar of vitriol should then be triturated between two surfaces of glass, and a considerable quantity of it applied at first to the surface of the polisher. The speculum is then to be wrought in the usual way upon the polishing tool till it has received a brilliant lustre, taking care to use no more of the colcothar, if it can be avoided, and only a small quantity of it, if it should be found necessary. When the metal moves stiffly on the polisher, and the colcothar assumes a dark muddy hue, the polish advances with great rapidity. The tool will then grow warm, and would probably stick to the speculum, if its motion were discontinued for a moment. At this stage of the process, therefore, we must proceed with great caution, breathing continually on the polisher till the friction is so great as to retard the motion of the speculum. When this happens, the metal is to be slipped off the tool at one side, cleaned with soft leather, and placed in a tube for the purpose of trying its performance; and if the polishing has been conducted with care, it will be found to have a true parabolic figure.

Appendix to Ferguson's Lectures vol. ii. p. 457.

See the articles ACHROMATIC GLASSES, CHROMATICS, and OPTICS, in the Supplement.

INDEX.

A.

ABERRATION, theory of, No 199. Evils of—remedy, 200. Light distributed by, over the smallest circle of diffusion, 201. Contrary aberrations correct each other, 203. Adams's method of making globules for large magnifiers, 98.

Aerial speculums mentioned by Mr Grey, No 46. Aerial images formed by concave mirrors, 241. Aethers, supposed, do not solve the phenomena of inflection, 60. Air, refractive power of, 13, 14. Strongly reflects the rays proceeding from beneath the surface of water, 36.

Alembert, M. d', his discoveries concerning achromatic telescopes, p. 177, col. 1. Alluxen's discoveries concerning the refraction of the atmosphere, No 6. His conjectures about the cause of it, ib. He gave the first hint of the magnifying power of glasses, ib. Alkaline salt diminishes the mean refraction, ib.

but not the dispersive power of glass, No. 18.

Angles refracted, tables of, published by Kepler and Kircher, 11.

Antonio de Dominis, bishop of Spalatro, discovered the nature of the rainbow, 213.

Apparatus for measuring light, p. 281.

Apparent place of objects seen by reflection, first discovered by Kepler, 27. Barrow's theory respecting, 181. M. de la Hire's observations, 182. Berkeley's hypothesis on distance by confused vision, 184. Objected to by Dr Smith, 185. The objection obviated by Robins, 186. M. Bouguer adopts Barrow's maxim, 187. Porterfield's view of this subject, 188. Atmosphere varies in its refractive power at different times, 20. Illumination of the shadow of the earth by the refraction of the atmosphere, 236.

Attractive force supposed to be the cause of reflection, 163. The supposition objected to, 164. Obviated, 165. Another hypothesis, 166. Sir Isaac Newton's hypothesis, 167. Untenable, 168.

Bacon, Mr., makes an object-glass of an extraordinary focal length, 82. On the apertures of refracting telescopes, 84.

Bacon, Roger; his discoveries, 6, 8.

Baron, Lord, his mistake concerning the possibility of making images appear in the air, 26.

Barrow's, Dr, reflecting microscope, 101. Barrow's theory of the apparent place of objects, 181. Adopted by Bouguer, 187.

Beams of light, the phenomenon of diverging, more frequent in summer than in winter, 235.

Bennett, Mr., cannot fire inflammable liquids with hot iron or a burning coal, unless those substances be of a white heat, 44.

Berkeley's theory of vision, 72. His hypothesis concerning the apparent place of objects, 166. Objected to by Dr Smith, 185. The objection obviated by Mr Robins, 186.

Binocular telescope invented by Father Rheita, 80.

Black marble reflects very powerfully, 35.

Blair, Dr Robert, makes an important discovery, 19. Blair and Dollond's refracting telescopes superior to others, 263.

Bodies which seem to touch one another are not in actual contact, 45.

Bouguer's experiments on the quantity of light lost by reflection, 33. His discoveries concerning the reflection of glass, &c. 34. His observations on the apparent place of objects, 187. Throws light on the fallacies of vision, 191. Explains the green and blue shadows seen in the sky, No. 227. Contrivances for measuring light, 266. Calculations concerning the light of the moon, 270.

Boyle's experiments on the light of coloured substances, 28.

Brewster, Dr, his fluid microscopes of varnish, 100. Improvement on the camera obscura, p. 269, col. 2. New finder for Newtonian telescopes, No. 258. Tables for microscopes, p. 273. Tables for telescopes, 276, 279.

Briggs's theory of single vision, No. 146.

Brilliant, the cut in diamonds, produces total reflection, 116.

Bristle, curious appearance of the shadow of one, 55.

Buffon's experiments on the reflection of light, 33. Observed green and blue shadows in the sky, 224, 225.

Burning glasses of the ancients described, 25.

C.

Camera obscura improved by Dr Brewster, p. 269. Portable one, ib.

Campagni's telescope, No. 81.

Candle, rays of light extended from, in several directions, like the tails of comets, 50.

Cassegrainian telescope, 262.

Cat, M. le, explains the magnifying of objects by the inflection of light, 62. Accounts for the large appearance of objects in mist, 183. Explains a remarkable deception of vision, 197.

Clairaut's calculations respecting telescopes, 17.

Colours discovered to arise from refraction, 15. Supposed by Dechales to arise from the inflection of light, 49. Produced by a mixture of shadows, 56. Colours simple or compound, 266.

Concave glasses: an object seen through a concave lens is seen nearer, smaller, and less bright than with the naked eye, 157. Law of reflection from a concave surface, 170. Proved, 172. Concave mirrors, 241.

Convex lens, an object seen through appears brighter, larger, and more distant, than when seen by the naked eye, 155. In certain circumstances it appears inverted and pendulous in the air, 156. Law of reflexion from a convex surface, 171. Proved mathematically, 172. Method of finding the focal distance of rays reflected from a convex surface, 176.

Contact of bodies in many cases apparent without being real, 45.

Coronas, p. 262.

Cylinders: experiments by Maraldi concerning their shadows, No. 53.

D.

Deception in vision; a remarkable one explained by M. le Cat, 197.

Dechales's observations on the inflection of light, 49.

Descartes: his discoveries concerning vision, No. 65. Account of the invention of telescopes, 68.

Diamonds, the brilliant cut in, produces total reflection, 116.

Diaphanometer, Saussure's, p. 288.

Dioptric instruments; difficulties attending the construction of them, No. 108.

Distance of objects, p. 240, &c. Berkeley's account of the judgment formed concerning distance by confused vision, No. 184. Smith's account, 185. Objected to by Robins, 186. Bouguer adopts Barrow's maxim, 187. Porterfield's view of it, 188.

Dieini, a celebrated maker of telescopes, 81. His microscope, 94.

Diverging beams more frequent in summer than in winter, 235.

Dollond, Mr, discovers a method of correcting the errors arising from refraction, 17. He discovers a mistake in one of Newton's experiments, ib. Discovers the different refractive and dispersive power of glass, ib. Difficulties in the execution of his plan, p. 176. His improvements in the refracting telescope, No. 88. Dollond and Blair's refracting telescopes superior to all others, 263.

Dominis, De, discovered the cause of the colours of the rainbow, 213.

E.

Edward's, Mr, improvements in the reflecting telescope, 86.

Emergent rays, the focus of, found, 131.

Equatorial telescope, or portable observatory, 89. New one invented by Ramsden, ib.

Euler, Mr, first suggested the thought of improving refracting telescopes, 17. His controversy with Clairaut, &c. ib. His scheme for introducing vision by reflected light into the solar microscope and magic lantern, 104. His theory of undulation contrary to fact, 123; and therefore misleads artists, 124.

Eye: the density and refractive powers of its humours first ascertained by Scheiner, 64. Description of it, 132. Dimensions of the insensible spot of it, 138. Seat of vision in, dispute about, 137. Arguments for the retina being the seat of vision in, 139.

Eyes, single vision with two, 145. Various hypotheses concerning it, 146, 147, 148, &c. Brightness of objects greater when seen with two eyes than only with one, 150. When one eye is closed, the pupil of the other is enlarged, 151.

F.

Fallacies, several, of vision explained, 190. Great light thrown on this subject by M. Bouguer, 191.

Focus, the, of rays refracted by spherical surfaces ascertained, 128. Focus of pa... parallel rays falling perpendicular upon any lens, No. 130. Focus of emergent rays found, 121. Proportional distance of the focus of rays reflected from a spherical surface, 175. Method of finding the focal distance of rays reflected from a convex surface, 176.

Force, repulsive, supposed to be the cause of reflection, 161. The supposition objected to, 162. Attractive, supposed, 163. The supposition objected to, 164. The objection obviated, 165.

Funk, Baron Alexander, his observation concerning the light in mines, 46.

G.

Galilean telescope, more difficult of construction than others, 76.

Galileo made a telescope without a pattern, 71. An account of his discoveries with it, 72. Account of his telescopes, 73. Was not acquainted with their rationale, 74. His telescope, 253. Magnifying power of, 254.

Glass globes, their magnifying powers known to the ancients, 3. Different kinds of them, ib. Table of the different compositions of glass for correcting the errors in refracting telescopes, 18. Shows various colours when split into thin laminae, 30. Table of the quantities of light reflected from glass not quicksilvered, at different angles of incidence, p. 138. Glass, multiplying, phenomena of, No. 238.

Glasses, difference in their powers of refraction and dispersion of the light, p. 176.

Globes have shorter shadows than cylinders, No. 53. And more light in their shadows, 54.

Globules used for microscopes by Hartsoeker, 95. Adams's method of making them, 98.

Gregory's invention of the reflecting telescope, 85. Gregorian telescope, 260. Magnifying power of, 261. Gregorian telescope superior for common uses to the Newtonian, 265.

Grey, Mr., observation on aerial speculums, 46. Temporary microscopes, 99.

Grimaldi first observes that colours arise from refraction, 15. Inflection of light first discovered by him, p. 186. His discoveries concerning inflection, No. 48.

H.

Hairs, remarkable appearance of their shadows, 51.

Hall, Mr., discovers the achromatic telescope, 18.

Hartsoeker's microscope, 95.

Herschel's improvements on reflecting telescopes, 87.

Hire, M. de la, his reason why rays of light seem to proceed from luminous bodies when viewed with the eyes half shut, No. 50. Observations on the apparent place of objects, 182.

Hooker, Dr., his discoveries concerning the inflection of light, 50.

Horizon, an object situated in, appears above its true place, 153. Extent of the visible horizon on a plane surface, 220.

Horizontal moon. Ptolemy's hypothesis concerning it, 5.

Huygens greatly improves the telescopes of Scheiner and Rhetta, 79. Improves the Newtonian telescope, 258.

I.

Jansen, Zacharias, the first inventor of the telescope, 69. Made the first microscope, 92.

Images, Lord Bacon's mistake concerning the possibility of making them appear in the air, 26. Another mistake in the same subject by Vitellio, ib. B. Porta's method of producing this appearance, ib. Kircher's method, ib. Images, aerial, formed by concave mirrors, 241.

Incidence, ratio of the sine of, to that of refraction, 113.

Incident velocity, increase of, diminishes refraction, 117.

Inflection of light, discoveries concerning it, p. 186. Dr Hooke's discoveries concerning it, No. 47. Grimaldi's observations, 48. Dechales's observations, 49. Newton's discoveries, 51. Maraldi's, 52. Probably produced by the same forces with reflection and refraction, 61.

Inversion, a curious instance of it observed by Mr Grey, 46.

Irradiations of the sun's light appearing through the interstices of the clouds, p. 265, &c. Converging, observed by Dr Smith, No. 232. Explained by him, 233. Not observed by moonlight, 234.

Jupiter's satellites discovered by Jansen, 70. By Galileo, and called him by Medicean planets, 72.

K.

Kepler first discovered the true reason of the apparent place of objects seen by reflecting mirrors, 27. His discoveries concerning vision, 63. Improved the construction of telescopes, 77. His method first put in practice by Scheiner, 78.

Kircher attempted a rational theory of refraction, 11.

L.

Lambert on light, 41.

Lead increases the dispersive power of glass, 17.

Leeuwenhoek's microscope, 96.

Lenses, their effects first discovered by Kepler, 74. Lenses, how many, 129. The focus of parallel rays falling perpendicular upon any lens, 130. Convex, an object seen through, appears larger, brighter, and more distant, than by the naked eye, No. 155. In some circumstances it appears inverted, and pendulous in the air, 156. An object seen through a concave lens is seen nearer, smaller, and less bright, than with the naked eye, 157. Method of grinding and polishing them, 276.

Leslie's photometer, p. 288.

Light discovered not to be homogeneous, No. 16. Quantity of, reflected by different substances, 39. Quantity of it absorbed by plaster of Paris, 40. By the moon, ib. Observations on the manner in which bodies are heated by it, 42. No heat produced by it on a transparent medium, unless it is reflected from the surface, ib. Newton's experiments with respect to its inflection, 51. Reflected, refracted, and inflected by the same forces, 61. Different opinions concerning the nature of, 129. It issues in straight lines from each point of a luminous surface, 110. In what case the rays of light describe a curve, 111. Its motion accelerated or retarded by refraction, 114. Light of all kinds subject to the same laws, 119. The law of refraction when light passes out of one transparent body into another contiguous to it, 120. Some portion of light always reflected from transparent bodies, 128. Light is not reflected by impinging on the solid part of bodies at the first surface, 139. Nor at the second, 162. Light consists of several sorts of coloured rays differently refrangible, 204. Reflected light differently refrangible, 205. Bouguer's contrivances for measuring light, 266. These instruments measure only the intensity of light, 267. Great variation of the light of the moon at different altitudes, 268. Variation in different parts of the disks of the sun and planets, ib. Bouguer's calculations concerning the light of the moon, 270. Dr Smith's, 271. Mr Michell's, 262. Density of, in different points of refraction, 262.

Lignum nephriticum, remarkable properties of its infusion, 29.

Lines can be seen under smaller angles than spots, and why, 144.

Liquid substances cannot be fired by the solar rays concentrated, 44.

Long-sightedness, 142.

M.

Magic lantern, Mr Euler's attempt to introduce vision by reflected light into, 104.

Magnitudes of objects, p. 240, &c.

Mairan, M. his observations on the inflection of light, No. 47.

Maraldi's discoveries concerning the inflection... Index.

Optics.

N.

Newton, Sir Isaac, his discovery concerning colours, 16. Mistaken in one of his experiments, 18. His discoveries concerning the inflection of light, 51. Theory of refraction objected to, 121. These objections are the necessary consequences of the theory, and therefore confirm it, 122. Reflecting telescope, 257. Magnifying power of, 259. Inferior to Gregorian, 265. Nollet, Abbé, cannot fire inflammable liquids by burning-glasses, 43.

Objects on the retina of the eye appear inverted, 133. Why seen upright, 134. An object when viewed with both eyes does not appear double, because the optic nerve is insensible of light, 135. Proved by experiments, 136. Seen with both eyes brighter than when seen only with one, 150. The various appearances of objects seen through different media stated and investigated, 152. An object situated in the horizon appears above its true plane, 153. An object seen through a plane medium appears nearer and brighter than seen by the naked eye, 154. Object seen through a convex lens appears larger, brighter, and more distant, 157. In some circumstances an object through a convex lens appears inverted and pendulous in the air, 156. Barrow's theory respecting the apparent place of objects, 181. M. de la Hire's observations, 182. M. le Cat's account of the largeness of objects in mist, 183. Why objects seen from a high building appear smaller than they are, 189. Dr Porterfield's account of objects appearing to move to a giddy person when they are both at rest, 193. Wells's account, 194. Upon what data we judge visible objects to be in motion or at rest, 195. Experiments to ascertain it, 196.

Object-glasses improved by Dollond, 17, and by Blair, 19.

Observatory, portable. See Equatorial Telescope.

Opaque objects, microscope for, 103.

Optic nerve insensible of light; and therefore an object viewed by both eyes is not seen double, 135. Proved by experiments, 136.

Optical instruments, p. 267.

O.

first treatise of, by Claudius Ptolemy, No. 4. Vitellio's treatise, 7. Treatise attributed to Euclid, 24.

P.

Parallel rays falling perpendicularly upon any lens, the focus of, found, 130.

Parhilion, p. 262.

Photometer, Romford's, No. 273. Sansure's, p. 287. Leslie's, p. 288.

Plane medium, an object seen through, appears nearer and brighter than by the naked eye, No. 154.

Plane surfaces, laws of refraction in, 127. Extent of the visible horizon on, 220.

Planets more luminous at their edges than in the middle of their disks, 40, 269.

Plates. Maraldi's experiments concerning their shadows, 55.

Porta, Joannes Baptista, his discoveries, 10.

Porterfield's solution of single vision with two eyes, 147. Of the judging of the distance of objects, 188. Fallacies of vision explained, 190. Porterfield's account of objects appearing to move to a giddy person when they are both at rest, 193.

Primary rainbow never greater than a semicircle, and why, 217. Its colours stronger than those of the secondary, and ranged in contrary order, 219.

Prisms in some cases reflect as strongly as quicksilver, 38. Why the image of the sun by heterogeneous rays passing through a prism is oblong, 207.

Ptolemy first treated of refraction scientifically, 4.

R.

Rainbow, knowledge of the nature of, a modern discovery, 211. Approach towards it by Fletcher of Breslaw, 212. The discovery of, made by Antonio de Dominis bishop of Spalatro 213. True cause of its colours, 214. Phenomena of the rainbow explained on the principles of Sir Isaac Newton, 215. Two rainbows seen at once, 216. Why the arc of the primary rainbow is never greater than a semicircle, 217. The secondary rainbow produced by two reflections and two refractions, 218. Why the colours of the secondary rainbow are fainter than those of the primary, and ranged in a contrary order, 219.

Ramside's, Mr., new equatorial telescope, 89.

Rays of light extinguished at the surface of transparent bodies, 37. Why they seem to proceed from any luminous object when viewed with the eyes half shut, 50. Rays at a certain obliquity are wholly reflected by transparent substances, 115. The focus of rays refracted by by spherical surfaces ascertained, No. 128. The focus of parallel rays falling perpendicularly upon any lens, 130. Emergent rays, the focus of, found, 131. Rays proceeding from one point and falling on a parabolic concave surface are all reflected from one point, 174. Proportional distance of the focus of rays reflected from a spherical surface, 175. Several sorts of coloured rays differently refrangible, 204. Why the image of the sun by heterogeneous rays passing through a prism is oblong, 207. Every homogeneous ray is refracted according to one and the same rule, 210.

Reflected light, table of its quantity from different substances, 39.

Reflecting telescope of Newton, 257. Magnifying power of, 259.

Reflection of light, opinions of the ancients concerning it, 23. Bouguer's experiments concerning the quantity of light lost by it, 32. Method of ascertaining the quantity lost in all the varieties of reflection, ib. Buffon's experiments on the same subject, 33. Bouguer's discoveries concerning the reflection of glass and polished metal, 34. Great difference of the quantity of light reflected at different angles of incidence, 35. No reflection but at the surface of a medium, 42. Rays at a certain obliquity are wholly reflected by transparent substances, 115. Total reflection produced by the brilliant cut in diamonds, 116. Some portion of light always reflected from transparent bodies, 138. Light is not reflected by impinging on the solid parts of bodies at the first surface, 159; nor at the second, 160. Fundamental law of reflection, 169. Laws of, from a concave surface, 170. From a convex, 171. These preceding propositions proved mathematically, 172. Reflected rays from a spherical surface never proceed from the same point, 173. Rays proceeding from one point and falling on a parabolic concave surface are all reflected from one point, 174. Proportional distance of the focus of rays reflected from a spherical surface, 175. Method of finding the focal distances of rays reflected from a convex surface, 176. The appearance of objects reflected from plane surfaces, 177; from convex, 178; from concave, 179. The apparent magnitude of an object seen by reflection from concave surface, 180. Reflected light differently refrangible, 205.

Refracting telescopes improved by Mr Dollond, 17. By Dr Blair, 19. Magnify in proportion to their lengths, 255. Imperfections in, remedied, 256.

Refraction, known to the ancients, 2. Its laws discovered by Snellius, No. 11. Explained by Descartes, 12. Fallacy of his hypothesis, 13. Experiments of the Royal Society for determining the refractive powers of different substances, ib. —M. de la Hire's experiments on the same subject, ib. Refraction of air accurately determined, 13, 14. Mistake of the Academy of Sciences concerning the refraction of air, 13. Allowance for refraction in computing the height of mountains, first thought of by Dr Hooke, 14. Mr Dollond discovers how to correct the errors of telescopes arising from refraction, 17. The same discovery made by Mr Hall, 18. Important discovery of Dr Blair for this purpose, 19. Refraction defined, 111. Phenomena of refraction solved by an attractive power in the medium, 112. Refraction explained and illustrated, pages 206, 207, &c.—Ratio of the sine of incidence to the sine of refraction, No. 113. Refraction accelerates or retards the motion of light, 114. Refraction diminishes as the incident velocity increases, 117. Refraction of a star greater in the evening than in the morning, 118. Laws of refraction when light passes out of one transparent body into another contiguous to it, 120. The Newtonian theory of refraction objected to, 121. Which objections, as they are the necessary consequences of that theory, confirm it, 122. Laws of refraction in plane surfaces, 127. The focus of rays refracted by spherical surfaces ascertained, 128. Light consists of several sorts of coloured rays differently refrangible, 194.—Reflected light differently refrangible, 205. Every homogeneous ray is refracted according to one and the same rule, 210.

Reid's solution of single vision with two eyes, 148.

Repulsive force supposed to be the cause of reflection, 161. Objected to, 162. Another hypothesis, 266. Sir Isaac Newton's, 167. untenable, 168.

Retina of the eye, objects on, inverted, 133. Why seen upright, 134. When viewed with both eyes, not seen double, because the optic nerve is insensible of light, 135. Arguments for the retina's being the seat of vision, 139.

Rheita's telescope improved by Huygens, 79. His binocular telescope, 80.

Robins's, Mr, objection to Smith's account of the apparent place of objects, 186.

S.

Saturn's ring discovered by Galileo, 72. Secondary rainbow produced by two reflections and two refractions, 218. Its colours why fainter than those of the primary, and ranged in contrary order, No. 219.

Schreiner completes the discoveries concerning vision, 64. Puts the improvements of the telescope by Kepler in practice, 78.

Shadows of bodies, observations concerning them, 47, 48, 49. Green shadows observed by Buffon, 224. Blue ones, 225. Explained by Abbé Mazeas, 226.—Explained by Melville and Bouger, 227. Curious observations relative to this subject, 228. Blue shadows not confined to the morning and evenings, 227.—Another kind of shadows, 230. Illumination of the shadow of the earth by the refraction of the atmosphere, 236.

Short's, Mr, equatorial telescope, 89.

Short-sightedness, 142.

Sky, concave figure of, p. 262, &c. Why the concavity of the sky appears less than a semicircle, No. 222. Opinions of the ancients respecting the colour of the sky, 223. No explanation of its blue colour, 231.

Smith's, Dr, reflecting microscope superior to all others, 102. Account of the apparent place of objects, 185. Objected to, 186. Converging irradiation of the sun observed and explained by, 232, 233. He never observed them by moon-light, 234. Diverging beams more frequent in summer than in winter, 235. Calculation concerning the light of the moon, 271. His microscope, magnifying power of, 244.

Solar microscope, 103. Mr Euler's attempt to introduce vision by reflected light into the solar microscope, 104. Martin's improvement, 105. Magnifying power of, 248.

Spectacles, when first invented, 67.

Specula for reflecting telescopes, how to grind and polish them, 285.

Spots of the sun discovered by Galileo, 72. Not seen under so small an angle as lines, 144.

Stars, how to be observed in the daytime, 90. The refraction of a star greater in the evening than in the morning, 118.

Sun, image of, by heterogeneous rays passing through a prism, why oblong, 207. The image of, by simple and homogeneous light, circular, 208. Variation of light in different parts of the sun's disk, 269.

Surfaces of transparent bodies have the property of extinguishing light, and why, 37. Supposed to consist of small transparent planes, 39, 41. Laws of refraction in plane surfaces, 127. The focus of rays refracted by spherical surfaces ascertained, 128. Reflected rays from

Rainbow, knowledge of the nature of, a modern discovery, 211. Approach towards it by Fletcher of Breslaw, 212. The discovery of, made by Antonio de Dominis bishop of Spalatro 213. True cause of its colours, 214. Phenomena of the rainbow explained on the principles of Sir Isaac Newton, 215. Two rainbows seen at once, 216. Why the arc of the primary rainbow is never greater than a semicircle, 217. The secondary rainbow produced by two reflections and two refractions, 218. Why the colours of the secondary rainbow are fainter than those of the primary, and ranged in a contrary order, 219.

Ramside's, Mr., new equatorial telescope, 89.

Rays of light extinguished at the surface of transparent bodies, 37. Why they seem to proceed from any luminous object when viewed with the eyes half shut, 50. Rays at a certain obliquity are wholly reflected by transparent substances, 115. The focus of rays refracted by by spherical surfaces ascertained, No. 128. The focus of parallel rays falling perpendicularly upon any lens, 130. Emergent rays, the focus of, found, 131. Rays proceeding from one point and falling on a parabolic concave surface are all reflected from one point, 174. Proportional distance of the focus of rays reflected from a spherical surface, 175. Several sorts of coloured rays differently refrangible, 204. Why the image of the sun by heterogeneous rays passing through a prism is oblong, 207. Every homogeneous ray is refracted according to one and the same rule, 210.

Reflected light, table of its quantity from different substances, 39.

Reflecting telescope of Newton, 257. Magnifying power of, 259.

Reflection of light, opinions of the ancients concerning it, 23. Bouguer's experiments concerning the quantity of light lost by it, 32. Method of ascertaining the quantity lost in all the varieties of reflection, ib. Buffon's experiments on the same subject, 33. Bouguer's discoveries concerning the reflection of glass and polished metal, 34. Great difference of the quantity of light reflected at different angles of incidence, 35. No reflection but at the surface of a medium, 42. Rays at a certain obliquity are wholly reflected by transparent substances, 115. Total reflection produced by the brilliant cut in diamonds, 116. Some portion of light always reflected from transparent bodies, 138. Light is not reflected by impinging on the solid parts of bodies at the first surface, 159; nor at the second, 160. Fundamental law of reflection, 169. Laws of, from a concave surface, 170. From a convex, 171. These preceding propositions proved mathematically, 172. Reflected rays from a spherical surface never proceed from the same point, 173. Rays proceeding from one point and falling on a parabolic concave surface are all reflected from one point, 174. Proportional distance of the focus of rays reflected from a spherical surface, 175. Method of finding the focal distances of rays reflected from a convex surface, 176. The appearance of objects reflected from plane surfaces, 177; from convex, 178; from concave, 179. The apparent magnitude of an object seen by reflection from concave surface, 180. Reflected light differently refrangible, 205.

Refracting telescopes improved by Mr Dollond, 17. By Dr Blair, 19. Magnify in proportion to their lengths, 255. Imperfections in, remedied, 256.

Refraction, known to the ancients, 2. Its laws discovered by Snellius, No. 11. Explained by Descartes, 12. Fallacy of his hypothesis, 13. Experiments of the Royal Society for determining the refractive powers of different substances, ib. —M. de la Hire's experiments on the same subject, ib. Refraction of air accurately determined, 13, 14. Mistake of the Academy of Sciences concerning the refraction of air, 13. Allowance for refraction in computing the height of mountains, first thought of by Dr Hooke, 14. Mr Dollond discovers how to correct the errors of telescopes arising from refraction, 17. The same discovery made by Mr Hall, 18. Important discovery of Dr Blair for this purpose, 19. Refraction defined, 111. Phenomena of refraction solved by an attractive power in the medium, 112. Refraction explained and illustrated, pages 206, 207, &c.—Ratio of the sine of incidence to the sine of refraction, No. 113. Refraction accelerates or retards the motion of light, 114. Refraction diminishes as the incident velocity increases, 117. Refraction of a star greater in the evening than in the morning, 118. Laws of refraction when light passes out of one transparent body into another contiguous to it, 120. The Newtonian theory of refraction objected to, 121. Which objections, as they are the necessary consequences of that theory, confirm it, 122. Laws of refraction in plane surfaces, 127. The focus of rays refracted by spherical surfaces ascertained, 128. Light consists of several sorts of coloured rays differently refrangible, 194.—Reflected light differently refrangible, 205. Every homogeneous ray is refracted according to one and the same rule, 210.

Reid's solution of single vision with two eyes, 148.

Repulsive force supposed to be the cause of reflection, 161. Objected to, 162. Another hypothesis, 266. Sir Isaac Newton's, 167. untenable, 168.

Retina of the eye, objects on, inverted, 133. Why seen upright, 134. When viewed with both eyes, not seen double, because the optic nerve is insensible of light, 135. Arguments for the retina's being the seat of vision, 139.

Rheita's telescope improved by Huygens, 79. His binocular telescope, 80.

Robins's, Mr, objection to Smith's account of the apparent place of objects, 186.

S.

Saturn's ring discovered by Galileo, 72. Secondary rainbow produced by two reflections and two refractions, 218. Its colours why fainter than those of the primary, and ranged in contrary order, No. 219.

Schreiner completes the discoveries concerning vision, 64. Puts the improvements of the telescope by Kepler in practice, 78.

Shadows of bodies, observations concerning them, 47, 48, 49. Green shadows observed by Buffon, 224. Blue ones, 225. Explained by Abbé Mazeas, 226.—Explained by Melville and Bouger, 227. Curious observations relative to this subject, 228. Blue shadows not confined to the morning and evenings, 227.—Another kind of shadows, 230. Illumination of the shadow of the earth by the refraction of the atmosphere, 236.

Short's, Mr, equatorial telescope, 89.

Short-sightedness, 142.

Sky, concave figure of, p. 262, &c. Why the concavity of the sky appears less than a semicircle, No. 222. Opinions of the ancients respecting the colour of the sky, 223. No explanation of its blue colour, 231.

Smith's, Dr, reflecting microscope superior to all others, 102. Account of the apparent place of objects, 185. Objected to, 186. Converging irradiation of the sun observed and explained by, 232, 233. He never observed them by moon-light, 234. Diverging beams more frequent in summer than in winter, 235. Calculation concerning the light of the moon, 271. His microscope, magnifying power of, 244.

Solar microscope, 103. Mr Euler's attempt to introduce vision by reflected light into the solar microscope, 104. Martin's improvement, 105. Magnifying power of, 248.

Spectacles, when first invented, 67.

Specula for reflecting telescopes, how to grind and polish them, 285.

Spots of the sun discovered by Galileo, 72. Not seen under so small an angle as lines, 144.

Stars, how to be observed in the daytime, 90. The refraction of a star greater in the evening than in the morning, 118.

Sun, image of, by heterogeneous rays passing through a prism, why oblong, 207. The image of, by simple and homogeneous light, circular, 208. Variation of light in different parts of the sun's disk, 269.

Surfaces of transparent bodies have the property of extinguishing light, and why, 37. Supposed to consist of small transparent planes, 39, 41. Laws of refraction in plane surfaces, 127. The focus of rays refracted by spherical surfaces ascertained, 128. Reflected rays from OPTIMATES, one of the divisions of the Roman people, opposed to populares. It is not easy to ascertain the characteristic differences betwixt these two parties. Some say the optimates were warm supporters of the dignity of the chief magistrate, and promoters of the grandeur of the state, who cared not if the inferior members suffered, provided the commanding powers were advanced: Whereas the populares boldly stood up for the rights of the people, pleaded for larger privileges, and laboured to bring matters nearer to a level. In short, they resembled, according to this account, the court and country parties amongst the people of this island.

Tully says, that the optimates were the best citizens, who wished to deserve the approbation of the better sort; and that the populares courted the favour of the populace, not so much considering what was right, as what would please the people and gratify their own thirst of vain glory and empty applause.