A Mathematical and Philosphical Dictionary
, in Astronomy, an apparent motion of the celestial bodies, occasioned by the progressive motion of light, and the earth's annual motion in her orbit.
This effect may be explained and familiarized by the motion of a line parallel to itself, much after the manner that the composition and resolution of forces are explained. If light have a progressive motion, let the proportion of its velocity to that of the earth in her orbit, be as the line BC to the line AC; then, by the composition of these two motions, the particle of light will seem to describe the line BA or DC, instead of its real course BC; and will appear in the direction AB or CD, instead of its true direction CB. So that if AB represent a tube, carried with a parallel motion by an observer along the line AC, in the time that a particle of light would move over the space BC, the different places of the tube being AB, ab, cd, CD; and when the eye, or end of the tube, is at A, let a particle of light enter the other end at B; then when the tube is at ab, the particle of light will be at e, exactly in the axis of the tube; and when the tube is at cd, the particle of light will arrive at f, still in the axis of the tube; and lastly, when the tube arrives at CD, the particle of light will arrive at the eye or point C, and consequently will appear to come in the direction DC of the tube, instead of the true direction BC. And so on, one particle succeeding another, and forming a continued stream or ray of light in the apparent direction DC. So that the apparent angle made by the ray of light with the line AE, is the angle DCE, instead of the true angle BCE; and the difference, BCD or ABC, is the quantity of the aberration.
M. de Maupertuis, in his Elements of Geography, gives also a familiar and ingenious idea of the aberration, in this manner: “It is thus,” says he, “concerning the direction in which a gun must be pointed to strike a bird in its flight; instead of pointing it straight to the bird, the fowler will point a little before it, in the path of its flight, and that so much the more as the flight of the bird is more rapid, with respect to the flight of the shot.” In this way of considering the matter, the flight of the bird represents the motion of the earth, or the line AC, in our scheme above, and the flight of the shot represents the motion of the ray of light, or the line BC.
Mr. Clairaut too, in the Memoires of the Academy of Sciences for the year 1746, illustrates this effect in a familiar way, by supposing drops of rain to fall rapidly and quickly after each other from a cloud, under which a person moves with a very narrow tube; in which case it is evident that the tube must have a certain inclination, in order that a drop which enters at the top, may fall freely through the axis of the tube, without touching the sides of it; which inclination must be more or less according to the velocity of the drops in respect to that of the tube: then the angle made by the direction of the tube and of the falling drops, is the aberration arising from the combination of those two motions.
This discovery, which is one of the brightest that have been made in the present age, we owe to the accuracy and ingenuity of the late Dr. Bradley, Astronomer Royal; to which he was occasionally led by the result of some accurate observations which he had made with another view, namely, to determine the annual parallax of the fixed stars, or that which arises from the motion of the earth in its annual orbit about the sun.
The annual motion of the earth about the sun had been much doubted, and warmly contested. The defenders of that motion, among other proofs of the reality of it, conceived the idea of adducing an incontestable one from the annual parallax of the fixed stars, if the stars should be within such a distance, or if instruments and observations could be made with such accuracy, as to render that parallax sensible. And with this view various attempts have been made. Before the observations of M. Picard, made in 1672, it was the general opinion, that the stars did not change their position during the course of a year. Tycho Brahe and Ricciolus fancied that they had assured themselves of it from their observations; and from thence they concluded that the earth did not move round the sun, and that there was no annual parallax in the fixed stars. M. Picard, in the account of his Voyage d'Uranibourg, made in 1672, says that the pole star, at different times of the year, has certain variations which he had observed for about 10 years, and which amounted to about 40″ a year: from whence some who favoured the annual motion of the earth were led to conclude that these variations were the effect of the parallax of the earth's orbit. But it was impossible to explain it by that parallax; because this motion was in a manner contrary to what ought to follow only from the motion of the earth in her orbit.
In 1674 Dr. Hook published an account of observations which he said he had made in 1669, and by which he had found that the star g Draconis was 23 more northerly in July than in October: observations which, for the present, seemed to favour the opinion of the earth's motion, although it be now known that there could not be any truth or accuracy in them.
Flamsteed having observed the pole star with his mural quadrant, in 1689 and the following years, found that its declination was 40″ less in July than in December; which observations, although very just, were yet however improper for proving the annual parallax: and he recommended the making of an instrument of 15 or 20 feet radius, to be firmly fixed on a strong foundation, for deciding a doubt which was otherwise not soon likely to be brought to a conclusion.
In this state of uncertainty and doubt, then, Dr. Bradley, in conjunction with Mr. Samuel Molineux, in the year 1725, formed the project of verifying, by a series of new observations, those which Dr. Hook had communicated to the public almost 50 years before. And as it was his attempt that chiefly gave rise to this, so it was his method in making the observations, in some measure, that they followed; for they made choice of the same star, and their instrument was constructed upon nearly the same principles: but had it not greatly exceeded the former in exactness, they might still have continued in great uncertainty as to the parallax of the fixed stars. And this was chiefly owing to the accuracy of the ingenious Mr. George Graham, to whom the lovers of astronomy are also indebted for several other exact and convenient instruments.
The success then of the intended experiment, evidently depending very much on the accuracy of the instrument, that leading object was first to be well secured. Mr. Molineux's apparatus then having been completed, and fitted for observing, about the end of November 1725, on the third day of December following, the bright star in the head of Draco, marked g by Bayer, was for the first time observed, as it passed near the zenith, and its situation carefully taken with the instrument. The like observations were made on the fifth, eleventh, and twelfth days of the same month; and there appearing no material difference in the place of the star, a farther repetition of them, at that season, seemed needless, it being a time of the year in which no sensible alteration of parallax, in this star, could soon be expected. It was therefore curiosity that chiefly urged Dr. Bradley, being then at Kew, where the in strument was fixed, to prepare for observing the star again on the 17th of the same month; when, having adjusted the instrument as usual, he perceived that it passed a little more southerly this day than it had done before. Not suspecting any other cause of this appearance, they first concluded that it was owing to the uncertainty of the observations, and that either this, or the foregoing, was not so exact as they had before supposed. For which reason they proposed to repeat the observation again, to determine from what cause this difference might proceed: and upon doing it, on the 20th of December, the doctor found that the star passed still more southerly than at the preceding observation. This sensible alteration surprised them the more, as it was the contrary way from what it would have been, had it proceeded from an annual parallax of the star. But being now pretty well satisfied, that it could not be entirely owing to the want of exactness in the observations, and having no notion of any thing else that could cause such an apparent motion as this in the star; they began to suspect that some change in the materials, or fabric of the instrument itself, might have occasioned it. Under these uncertainties they remained for some time; but being at length fully convinced, by several trials, of the great exactness of the instrument; and finding, by the gradual increase of the star's distance from the pole, that there must be some regular cause that produced it; they took care to examine very micely, at the time of each observation, how much the variation was; till about the beginning of March 1726, the star was found to be 20″ more southerly than at the time of the first observation: it now indeed seemed to have arrived at its utmost limit southward, as in several trials, made about this time, no sensible difference was observed in its situation. By the middle of April it appeared to be returning back again towards the north; and about the beginning of June, it passed at the same distance from the zenith, as it had done in December, when it was first observed.
From the quick alteration in the declination of the star about this time, increasing about one second in three days, it was conjectured that it would now proceed northward, as it had before gone southward, of its present situation; and it happened accordingly; for the star continued to move northward till September following, when it became stationary again; being then near 20″ more northerly than in June, and upwards of 39″ more northerly than it had been in March. From September the star again returned towards the south, till, in December, it arrived at the same situation in which it had been observed twelve months before, allowing for the difference of declination on account of the precession of the equinox.
This was a sufficient proof that the instrument had not been the cause of this apparent motion of the star; and yet it seemed difficult to devise one that should be adequate to such an unusual effect. A nutation of the earth's axis was one of the first things that offered itself on this occasion; but it was soon found to be insufficient; for though it might have accounted for the change of declination in g Draconis, yet it would not at the same time accord with the phenomena observed in the other stars, particularly in a small one almost opposite in right ascension to g Draconis; and at about the same distance from the north pole of the equator: for though this star seemed to move the same way, as a nutation of the earth's axis would have made it; yet changing its declination but about half as much as g Draconis in the same time, as a peared on comparing the observations of both made on the same days, at different seasons of the year, this plainly proved that the apparent motion of the star was not occasioned by a real nutation; since, had that been the case, the alteration in both stars would have been nearly equal.
The great regularity of the observations left no room to doubt, but that there was some uniform cause by which this unexpected motion was produced, and which did not depend on the uncertainty or variety of the seasons of the year. Upon comparing the observations with each other, it was discovered that, in both the stars above mentioned, the apparent difference of declination from the maxima, was always nearly proportional to the versed sine of the sun's distance from the equinoctial points. This was an inducement to think that the cause, whatever it was, had some relation to the sun's situation with respect to those points. But not being able to frame any hypothesis, sufficient to account for all the phenomena, and being very desivous to search a little farther into this matter, Dr. Bradley began to think of erecting an instrument for himself at Wanstead; that, having it always at hand, he might with the more ease and certainty enquire into the laws of this new motion. The consideration likewise of being able, by another instrument, to confirm the truth of the observations hitherto made with that of Mr. Molineux, was no small inducement to the undertaking; but the chief of all was, the opportunity he should thereby have of trying in what manner other stars should be affected by the same cause, whatever it might be. For Mr. Molineux's instrument being originally designed for observing g Draconis, to try whether it had any sensible parallax, it was so contrived, as to be capable of but little alteration in its direction; not above seven or eight minutes of a degree: and there being but few stars, within half that distance from the zenith of Kew, bright enough to be well observed, he could not, with his instrument, thoroughly examine how this cause affected stars that were differently situated, with respect to the equinoctial and solsticial points of the ecliptic.
These considerations determined him; and by the contrivance and direction of the same ingenious person, Mr. Graham, his instrument was fixed up the 19th of August 1727. As he had no convenient place where he could make use of so long a telescope as Mr. Molineux's, he contented himself with one of but little more than half the length, namely of 12 feet and a half, the other being 24 feet and a half long, judging from the experience he had already had, that this radius would be long enough to adjust the instrument to a sufficient degree of exactness: and he had no reason afterwards to change his opinion; for by all his trials he was very well satisfied, that when it was caresully rectisied, its situation might be securely depended on to half a second. As the place where his instrument was hung, in some measure determined its radius; so did it also the length of the arc or limb, on which the divisions were made, to adjust it: for the arc could not conveniently be extended farther, than to reach to about 6 1/4 degrees on each side of his zenith. This however was sufficient, as it gave him an opportunity of making choice of several stars, very different both in magnitude and situation; there being more than two hundred, inserted in the British Catalogue, that might be observed with it. He needed not indeed to have extended the limb so far, but that he was willing to take in Capella, the only star of the first magnitude that came so near his zenith.
His instrument being fixed, he immediately began to observe such stars as he judged most proper to give him any light into the cause of the motion already mentioned. There was a sufficient variety of small ones, and not less than twelve that he could observe through all seasons of the year, as they were bright enough to be seen in the day-time, when nearest the sun. He had not been long observing, before he perceived that the notion they had before entertained, that the stars were farthest north and south when the sun was near the equinoxes, was only true of those stars which are near the solsticial colure. And after continuing his observations a few months, he discovered what he then apprehended to be a general law observed by all the stars, namely, that each of them became stationary, or was farthest north or south, when it passed over his zenith at six of the clock, either in the evening or morning. He perceived also that whatever situation the stars were in, with respect to the cardinal points of the ecliptic, the apparent motion of every one of them tended the same way, when they passed his instrument about the same hour of the day or night; for they all moved southward when they passed in the day, and northward when in the night; so that each of them was farthest north, when it came in the evening about six of the clock, and farthest south when it came about six in the morning.
Though he afterwards discovered that the maxima, in most of these stars, do not happen exactly when they pass at those hours; yet, not being able at that time to prove the contrary, and supposing that they did, he endeavoured to find out what proportion the greatest alterations of declination, in different stars, bore to each other; it being very evident that they did not all change their declination equally. It has been before noticed, that it appeared from Mr. Molineux's observations, that g Draconis changed its declination above twice as much as the before-mentioned small star that was nearly op- posite to it; but examining the matter more nicely, he found that the greatest change in the declination of these stars, was as the sine of the latitude of each star respectively. This led him to suspect that there might be the like proportion between the maxima of other stars; but finding that the observations of some of them would not perfectly correspond with such an hypothesis, and not knowing whether the small difference he met with might not be owing to the uncertainty and error of the observations, he deferred the farther examination into the truth of this hypothesis, till he should be sarther furnished with a series of observations made in all parts of the year; which would enable him not only to determine what errors the observations might be liable to, or how far they might safely be depended on; but also to judge, whether there had been any sensible change in the parts of the instrument itself.
When the year was completed, he began to examine and compare his observations; and having pretty well satisfied himself as to the general laws of the phenomena, he then endeavoured to sind out the cause of them. He was already convinced that the apparent motion of the stars was not owing to a nutation of the earth's axis. The next that occurred to him, was an alteration in the direction of the plumb-line, by which the instrument was constantly adjusted; but this, upon trial, proved insufficient. Then he considered what refraction might do; but here also he met with no satisfaction. At last, through an amazing sagacity, he conjectured that all the phenomena hitherto mentioned, proceeded from the progressive motion of light, and the earth's annual motion in her orbit: for he perceived, that if light were propagated in time, the apparent place of a fixed object would not be the same when the eye is at rest, as when it is moving in any other direction but that of the line passing through the object and the eye; and that when the eye is moving in different directions, the apparent place of the object would be different.
He considered this matter in the following manner. He imagined CA to be a ray of light, falling perpendicularly upon the line BD: then, if the eye be at rest at A, the object must appear in the direction AC, whether light be propagated in time, or in an instant. But if the eye be moving from B towards A, and light be propagated in time, with a velocity that is to the velocity of the eye, as AC to AB; then, light moving from C to A, whilst the eye moves from B to A, that particle of it by which the object will be discerned, when the eye in its motion comes to A, is at C when the eye is at B. Joining the points B, C, he supposed the line BC to be a tube, inclined to the line BD in the angle DBC, and of such a diameter as to admit of but one particle of light: then it was easy to conceive, that the particle of light at C, by which the object must be seen when the eye arrives at A, would pass through the tube BC, so inclined to the line BD, and accompanying the eye in its motion from B to A; and that it would not come to the eye, placed behind such a tube, if it had any other inclination to the line BD. If, instead of supposing BC so small a tube, we conceive it to be the axis of a larger; then, for the same reason, the particle of light at C cannot pass through that axis, unless it be inclined to BD in the same angle DBC.
In the like manner, if the eye move the contrary way, from D towards A, with the same velocity; then the tube must be inclined in the angle BDC. Although therefore the true or real place of an object, be perpendicular to the line in which the eye is moving, yet the visible place will not be so; since that must doubtless be in the direction of the tube. But the difference between the true and apparent place, will be, cæteris paribus, greater or less, according to the different proportions between the velocity of light and that of the eye: so that if we could suppose light to be propagated in an instant, then there would be no difference between the real and visible place of an object, although the eye were in motion; for in that case, AC being infinite with respect to AB, the angle ACB, which is the difference between the true and visible place, vanishes. But if light be propagated in time, which was then allowed by most philosophers, then it is evident from the foregoing considerations, that there will always be a difference between the true and visible place of an object, except when the eye is moving either directly towards, or from the object. And in all cases, the sine of the difference between the true and visible place of the object, will be to the sine of the visible inclination of the object to the line in which the eye is moving, as the velocity of the eye, is to the velocity of light.
If light moved only 1000 times faster than the eye, and an object, supposed to be at an infinite distance, were really placed perpendicularly over the plane in which the eye is moving; it follows, from what has been saíd, that the apparent place of such object will always be inclined to that plane, in an angle of 89° 56′ 1/2; so that it will constantly appear 3′ 1/2 from its true place, and will seem so much less inclined to the plane, that way towards which the eye tends. That is, if AC be to AB or AD, as 1000 to 1, the angle ABC will be 89° 56′ 1/2, and the angle ACB 3′ 1/2, and BCD or 2ACB will be 7′, if the direction of the motion of the eye be contrary at one time to what it is at another.
If the earth revolve about the sun annually, and the velocity of light were to the velocity of the earth's motion in its orbit, as 1000 is to 1; then it is easy to conceive, that a star really placed in the pole of the ecliptic, would to an eye carried along with the earth, seem to change its place continually; and, neglecting the small difference on account of the earth's diurnal revolution on its axis, it would seem to describe a circle about that pole, every where distant from it by 3′ 1/2. So that its longitude would be varied through all the points of the ecliptic every year, but its latitude would always remain the same. Its right ascension would also change, and its declination, according to the different situation of the sun in respect of the equinoctial points; and its apparent distance from the north pole of the equator, would be 7′ less at the autumnal, than at the vernal equinox.
The greatest alteration of the place of a star, in the pole of the ecliptic, or, which in effect amounts to the same, the proportion between the velocity of light and the earth's motion in its orbit, being known, it will not be difficult to find what would be the difference, on this account, between the true and apparent place of any other star at any time; and, on the contrary, the difference between the true and apparent place being given, the proportion between the velocity of light, and the earth's motion in her orbit, may be found.
After the history of this curious discovery, related by the author nearly in the terms above, he gives the results of a multitude of accurate observations, made on a great number of stars, at all seasons of the year. From all which observations, and the theory as related above, he found that every star, in consequence of the earth's motion in her orbit and the progressive motion of light, appears to describe a small ellipse in the heavens, the transverse axis of which is equal to the same quantity for every star, namely 40″, nearly; and that the conjugate axis of the ellipse, for different stars, varies in this proportion, namely, as the right sine of the star's latitude; that is, radius is to the sine of the star's latitude, as the transverse axis to the conjugate axis: and consequently a star in the pole of the ecliptic, its latitude being there 90°, whose sine is equal to the radius, will appear to describe a small circle about that pole as a centre, whose radius is equal to 20″. He also gives the following law of the variation of the star's declination: if A denote the angle of position, or the angle at the star made by two great circles drawn from it through the poles of the ecliptic and equator, and B another angle, whose tangent is to the tangent of A, as radius is to the sine of the star's latitude; then B will be equal to the difference of longitude between the sun and the star, when the true and apparent declination of the star are the same. And if the sun's longitude in the ecliptic be reckoned from that point in which it is when this happens; then the difference between the true and apparent declination of the star, will be always as the sine of the sun's longitude from that point. It will also be found that the greatest difference of declination that can be between the true and apparent place of the star, will be to 20″, the semitransverse axis of the ellipse, as the sine of A to the sine of B.
The author then shews, by the comparison of a number of observations made on different stars, that they exactly agree with the theory deduced from the progressive motion of light, and that consequently it is highly probable that such motion is the cause of those variations in the situation of the stars. From which he infers, that the parallax of the fixed stars is much smaller, than hath been hitherto supposed by those, who have pretended to deduce it from their observations. He thinks he may venture to say, that in the stars he had observed, the parallax does not amount to 2″; nay, that if it had amounted to 1″, he should certainly have perceived it, in the great number of observations that he made, especially of g Draconis; which agreeing with the hypothesis, without allowing any thing for parallax, nearly as well when the sun was in conjunction with, as in opposition to, this star, it seems very pro- bable that the parallax of it is not so much as one single second; and consequently that it is above 400000 times farther from us than the sun.
From the greatest variation in the place of the stars, namely 40″, Dr. Bradley deduces the ratio of the velocity of light in comparison with that of the earth in her orbit. In the preceding figure, AC is to AB, as the velocity of light to that of the earth in her or bit, the angle ACB being equal to 20″; so that the ratio of those velocities is that of radius to the tangent of 20″, or of radius to 20″, since the tangent has no sensible difference from so small an are: but the radius of a circle is equal to the arc of 57° 3/10 nearly, or equal to 206260″; therefore the velocity of light is to the velocity of the earth, as 206260 to 20, or as 10313 to 1.
And hence also the time in which light passes over the space from the sun to the earth, is easily deduced; for this time will be to one year, as AB or 20″ to 360° or the whole circle; that is, 360°: 20″ :: 365 1/4 days: 8m 7s, namely, light will pass from the sun to the earth in the time of 8 minutes, 7 seconds; and this will be the same, whatever the distance of the sun is.
Dr. Bradley having annexed to his theory the rules or formulæ for computing the aberration of the fixed stars in declination and right ascension; these rules have been variously demonstrated, and reduced to other practical forms, by Mr. Clairaut in the Memoirs of the Academy of Sciences for 1737; by Mr. Simpson in his Essays in 1740; by M. Fontaine des Crutes in 1744; and several other persons. The results of these rules are as follow: Every star appears to describe in the course of a year, by means of the aberration, a small ellipse, whose greater axis is 40″, and the less axis, perpendicular to the ecliptic, is equal to 40″ multiplied by the sine of the star's latitude, the radius being 1. The eastern extremity of the longer axis, marks the apparent place of the star, the day of the opposition; and the extremity of the less axe, which is farthest from the ecliptic, marks its situation three months after.
The greatest aberration in longitude, is equal to 20″ divided by the cosine of its latitude. And the aberration for any time, is equal to 20″ multiplied by the cosine of the elongation of the star found for the same time, and divided by the cosine of its latitude. This aberration is subtractive in the first and last quadrant of the argument, or of the difference between the longitudes of the sun and star; and additive in the second and third quadrants. The greatest aberration in latitude, is equal to 20″ multiplied by the sine of the star's latitude. And the aberration in latitude for any time, is equal to 20″ multiplied by the sine of the star's latitude, and multiplied also by the sine of the elongation. The aberration is subtractive before the opposition, and additive after it.
The greatest aberration in declination, is equal to 20″ multiplied by the sine of the angle of position A, and divided by the sine of B the difference of longitude between the sun and star when the aberration in declination is nothing. And the aberration in declination at any other time, will be equal to the greatest aberration multiplied by the sine of the difference between the sun's place at the given time and his place when the aberration is nothing. Also the sine of the latitude of the star is to radius, as the tangent of A the angle of position at the star, is to the tangent of B, the difference of longitude between the sun and star when the aberration in declination is nothing. The greatest aberration in right-ascension, is equal to 20″ multiplied by the cosine of A the angle of position, and divided by the sine of C the difference in longitude between the sun and star when the aberration in right ascension is nothing. And the aberration in right-ascension at any other time, is equal to the greatest aberration multiplied by the sine of the difference between the sun's place at the given time, and his place when the aberration is nothing. Also the sine of the latitude of the star is to radius, as the cotangent of A the angle of position, to the tangent of C.
Aberration of the Planets, is equal to the geocentric motion of the planet, the space it appears to move as seen from the earth, during the time that light employs in passing from the planet to the earth. Thus, in the sun, the aberration in longitude is constantly 20″, that being the space moved by the sun, or, which is the same thing, by the earth, in the time of 8m 7s, which is the time in which light passes from the sun to the earth, as we have seen in the foregoing article. In like manner, knowing the distance of any planet from the earth, by proportion it will be, as the distance of the sun is to the distance of the planet, so is 8m 7s to the time of light passing from the planet to the earth: then computing the planet's geocentric motion in this time, that will be the aberration of the planet, whether it be in longitude, latitude, right-ascension, or declination.
It is evident that the aberration will be greatest in the longitude, and very small in latitude, because the planets deviate very little from the plane of the ecliptic, or path of the earth; so that the aberration in the latitudes of the planets, is commonly neglected, as insensible; the greatest in Mercury being only 4″ 1/3, and much less in the other planets. As to the aberrations in declination and right-ascension, they must depend on the situation of the planet in the zodiac. The aberration in longitude, being equal to the geocentric motion, will be more or less according as that motion is; it will therefore be least, or nothing at all, when the planet is stationary; and greatest in the superior planets Mars, Jupiter, Saturn, &c, when they are in opposition to the sun; but in the inferior planets Venus and Mercury, the aberration is greatest at the time of their superior conjunction. These maxima of aberration for the several planets, when their distance from the sun is least, are as below: viz, for Saturn27″.0Jupiter29.8Mars37.8Venus43.2Mercury59.0The Moon 2/3 And between these numbers and nothing the aberrations of the planets, in longitude, vary according to their situations. But that of the sun varies not, being constantly 20″, as has been before observed. And this may alter his declination by a quantity, which varies from 0 to near 8″; being greatest or 8″ about the equinoxes, and vanishing in the solstices.
The methods of computing these, and the formulas for all cases, are given by M. Clairaut in the Memoirs of the Academy of Sciences for the year 1746, and by M. Euler in the Berlin Memoirs, vol. 2, for 1746.
Optic Aberration, the deviation or dispersion of the rays of light, when reflected by a speculum, or refracted by a lens, by which they are prevented from meeting or uniting in the same point, called the geometrical focus, but are spread over a small space, and produce a confusion of images. Aberration is either lateral or longitudinal: the lateral aberration is measured by a perpendicular to the axis of the speculum or lens, drawn from the focus to meet the refracted or reflected ray: the longitudinal aberration is the distance, on the axis, between the focus and the point where the ray meets the axis. The aberrations are very amply treated in Smith's Complete System of Opties, in 2 volumes 4to.
There are two species of aberration, distinguished according to their different causes: the one arises from the figure of the speculum or lens, producing a geometrical dispersion of the rays, when these are perfectly equal in all respects; the other arises from the unequal refrangibility of the rays of light themselves; a discovery that was made by Sir Isaac Newton, and for this reason it is often called the Newtonian aberration. As to the former species of aberration, or that arising from the sigure, it is well known that if rays issue from a point at a given distance; then they will be reflected into the other focus of an ellipse having the given luminous point for one focus, or directly from the other focus of an hyperbola; and will be variously dispersed by all other figures. But if the luminous point be infinitely distant, or, which is the same, the incident rays be parallel, then they will be reflected by a parabola into its focus, and variously dispersed by all other figures. But those figures are very difficult to make, and therefore curved specula are commonly made spherical, the figure of which is generated by the revolution of a circular arc, which produces an aberration of all rays, whether they are parallel or not, and therefore it has no accurate geometrical focus which is common to all the rays. Let BVF represent a concave spherical speculum, whose centre is C; and let AB, EF be incident rays parallel to the axis CV. Because the angle of incidence is equal to the angle of reflection in all cases, therefore if the radii CB, CF be drawn to the points of incidence, and thence BD making the angle CBD equal to the angle CBA, and FG making the angle CFG equal to the angle CFE; then BD, FG will be the reflected rays, and D, G, the points where they meet the axis. Hence it appears that the point of coincidence with the axis is equally distant from the point of incidence and the centre: for because the angle CBD is equal to the angle CBA, which is equal to the alternate angle BCD, therefore their opposite sides CD, DB are equal: and in like manner, in any other, GF is equal to GC. And hence it is evident that when B is indefinitely near the vertex V, then D is in the middle of the radius CV; and the nearer the incident ray is to the axis CV, the nearer will the reflected ray come to the middle point D; and the contrary. So that the aberration DG of any ray EFG, is always more and more, as the incident ray is farther from the axis, or the incident point F from the vertex V; till when the distance VI is 60 degrees, then the reflected ray falls in the vertex V, making the aberration equal to the whole length DV. And this shews the reason why specula are made of a very small segment of a sphere, namely, that all their reflected rays may arrive very near the middle point or focus D, to produce an image the most distinct, by the least aberration of the rays. And in like manner for rays refracted through lenses.
In spherical lenses, Mr. Huygens has demonstrated that the aberration from the figure, in different lenses, is as follows.
1. In all plano-convex lenses, having their plane surface exposed to parallel rays, the longitudinal aberration of the extreme ray, or that remotest from the axis, is equal to 9/2 of the thickness of the lens.
2. In all plano-convex lenses, having their convex surface exposed to parallel rays, the longitudinal aberration of the extreme ray, is equal to 7/6 of the thickness of the lens. So that in this position of the same planoconvex lens, the aberration is but about one-fourth of that in the former; being to it only as 7 to 27.
3. In all double convex lenses of equal spheres, the aberration of the extreme ray, is equal to 5/3 of the thickness of the lens.
4. In a double convex lens, the radii of whose spheres are as 1 to 6, if the more convex surface be exposed to parallel rays, the aberration from the figure is less than in any other spherical lens; being no more than 15/14 of its thickness.
But the foregoing species of aberration, arising from the figure, is very small, and easily remedied, in comparison with the other, arising from the unequal refrangibility of the rays of light; insomuch that Sir Isaac Newton shews in his Optics, pa. 84 of the 8vo. edition, that if the object-glass of a telescope be planoconvex, the plane side being turned towards the object, and the diameter of the sphere, to which the convex side is ground, be 100 seet, the diameter of the aperture being 4 inches, and the ratio of the sine of incidence out of glass into air, be to that of refraction, as 20 to 31; then the diameter of the circle of aberrations will in this case be only 961/72000000 parts of an inch: while the diameter of the little circle, through which the same rays are scattered by unequal refrangibility, will be about the 55th part of the aperture of the object-glass, which here is 4 inches. And therefore the error arising from the spherical figure of the glass, is to the error arising from the different refrangibility of the rays, as 961/72000000 to 4/55, that is as 1 to 5449.
So that it may seem strange that objects appear through telescopes so distinct as they do, considering that the error arising from the different refrangibility, is almost incomparably larger than that of the figure. Newton however solves the difficulty by observing that the rays, under their various aberrations, are not scattered uniformly over all the circular space, but collected insinitely more dense in the centre than in any other part of the circle; and that, in the way from the centre to the circumference, they grow more and more rare, so as at the circumference to become infinitely rare; and, by reason of their rarity, they are not strong enough to be visible, unless in the centre, and very near it.
In consequence of the discovery of the unequal refrangibility of light, and the apprehension that equal refractions must produce equal divergencies in every sort of medium, it was supposed that all spherical objectglasses of telescopes would be equally affected by the different refrangibility of light, in proportion to their aperture, of whatever materials they might be constructed: and therefore that the only improvement that could be made in refracting telescopes, was that of increasing their length. So that Sir Isaac Newton, and other persons after him, despairing of success in the use and fabric of lenses, directed their chief attention to the construction of reflecting telescopes.
However, about the year 1747, M. Euler applied himself to the subject of refraction; and pursued a hint suggested by Newton, for the design of making object-glasses with two lenses of glass inclosing water between them; hoping that, by constructing them of different materials, the refractions would balance one another, and so the usual aberration be prevented. Mr. John Dollond, an ingenious optician in London, minutely examined this scheme, and found that Mr. Euler's principles were not satisfactory. M. Clairaut likewise, whose attention had been excited to the same subject, concurred in opinion that Euler's speculations were more ingenious than useful. This controversy, which seemed to be of great importance in the science of optics, engaged also the attention of M. Klingenstierna of Sweden, who was led to make a careful examination of the 8th experiment in the second part of Newton's Optics, with the conclusions there drawn from it. The consequence was, that he found that the rays of light, in the circumstances there mentioned, did not lose their colour, as Sir Isaac had imagined. This hint of the Swedish philosopher led Mr. Dollond to re-examine the same experiment: and after several trials it appeared, that different substances caused the light to diverge very differently, in proportion to their general refractive powers. In the year 1757 therefore he procured wedges of different kinds of glass, and applied them together so that the refractions might be made in contrary directions, that he might discover whether the refraction and divergency of colour would vanish together. The result of his sirst trials encouraged him to persevere; for he discovered a difference far beyond his hopes in the qualities of different kinds of glass, with respect to their divergency of colours. The Venice glass and English crown glass were found to be nearly allied in this respect: the common English plate glass made the rays diverge more; and the English flint glass most of all. But without enquiring into the cause of this difference, he proceeded to adapt wedges of crown glass, and of white flint glass, ground to different angles, to each other, so as to refract in different directions; till the refracted light was entirely free from colours. Having measured the refractions of each wedge, he found that the refraction of the white glass was to that of the crown glass, nearly as 2 to 3: and he hence concluded in general, that any two wedges made in this proportion, and applied together so as to refract in contrary directions, would refract the light without any aberration of the rays.
Mr. Dollond's next object was to make similar trials with spherical glasses of different materials, with the view of applying his discovery to the improvement of telescopes: and here he perceived that, to obtain a refraction of light in contrary directions, the one glass must be concave, and the other convex; and the latter, which was to refract the most, that the rays might converge to a real focus, he made of crown glass, the other of white flint glass. And as the refractions of spherical glasses are inversely as their focal distances, it was necessary that the focal distances of the two glasses should be inversely as the ratios of the refractions of the wedges; because that, being thus proportioned, every ray of light that passes through this compound glass, at any distance 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.
But in the applications of this ingenious discovery to practice, Mr. Dollond met with many and great difficulties. At length, however, after many repeated trials, by a resolute perseverance, he succeeded so far as to construct refracting telescopes much superior to any that had hitherto been made; representing objects with great distinctness, and in their true colours.
Mr. Clairaut, who had interested himself from the beginning in this discovery, now endeavoured to ascertain the principles of Mr. Dollond's theory, and to lay down rules to facilitate the construction of these new telescopes. With this view he made several experiments, to determine the resractive power of different kinds of glass, and the proportions in which they separated the rays of light: and from these experiments he deduced several theorems of general use. M. D'Alembert made likewise a great variety of calculations to the same purpose; and he shewed how to correct the errors to which these telescopes are subject, sometimes by placing the object-glasses at a small distance from each other, and sometimes by using eye-glasses of different refractive powers. But though foreigners were hereby supplied with the most accurate calculations, they were very defective in practice. And the English telescopes, made, as they imagined, without any precise rule, were greatly superior to the best of their construction.
M. Euler, whose speculations had sirst given occasion to this important and useful enquiry, was very reluctant in admitting Mr. Dollond's improvements, because they militated against a pre-conceived theory of his own. At last however, after several altercations, being convinced of their reality and importance by M. Clair- aut, he assented; and he soon after received farther satisfaction from the experiments of M. Zeiher, of Petersburgh.
M. Zeiher shewed by experiments that it is the lead, in the composition of glass, which gives it this remarkable property, namely, that while the refraction of the mean rays is nearly the same, that of the extreme rays considerably differs. And, by increasing the lead, he produced a kind of glass, which occasioned a much greater separation of the extreme rays than that of the flint glass used by Mr. Dollond, and at the same time considerably increased the mean refraction. M. Zeiher, in the course of his experiments, made glass of minium and lead, with a mixture also of alkaline salts; and he found that this mixture greatly diminished the mean refraction, and yet made hardly any change in the dispersion: and he at length obtained a kind of glass greatly superior to the flint glass of Mr. Dollond for the construction of telescopes; as it occasioned three times as great a dispersion of the rays as the common glass, whilst the mean refraction was only as 1.61 to 1.
Other improvements were also made on the new or achromatic telescopes by the inventor Mr. John Dollond, and by his son Peter Dollond; which may be seen under the proper words. For various dissertations on the subject of the aberration of light, colours, and the figure of the glass, see Philos. Trans. vols. 35, 48, 50, 51, 52, 55, 60; Memoirs of the Academy of Sciences of Paris, for the years 1737, 1746, 1752, 1755, 1756, 1757, 1762, 1764, 1765, 1767, 1770; the Berlin Ac. 1746, 1762, 1766; Swed. Mem. vol. 16; Com. Nov. Petripol. 1762; M. Euler's Dioptrics; M. d'Alembert's Opuscules Math.; M. de Rochon Opuscules; &c, &c.