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merely as satellites or secondaries, which, in his system, Venus and Mercury would thus have become.
In considering the theory of the sun, Ptolemy adopted without alteration the elements of Hipparchus. But to the theory of the moon he made several important additions. We have seen that Hipparchus explained the irregularity of the sun's motion by the hypothesis of an excentric circle. There is another way, however, of explaining this irregularity, by the hypothesis of an epicycle. In this case the planet was supposed to move in a small circle, called the epicycle, the centre of which revolved uniformly round the earth: in the case of the sun, this epicycle had for its radius the observed excentricity of the orbit; and the sun's motion in it was such, that during the interval between the apogee and perigee, that planet had approached the earth by exactly the diameter of its epicycle.*
Thus let E be the earth; M, which is the centre of the epicycle D BCN, represents the sun's mean place, and describes uniformly the circumference of the deferent MGAH; while the real sun describes the circumference of the epicycle; at the apogee the sun is in N; the true and mean places coincide, and the distance of the sun from the earth is EM+MN; MN being equal to the excentricity. At the perigee S is in P; the true and mean places again coincide, and the distance is EA-AP-EM-MN. In any intermediate position the true and mean places will differ by the angle GES, and the distance will take every value between the limits EA+MN.
There is a slight mistake in the figure given above: ES should be drawn on the other side of
EG, so that the point S should fall within the angle MEG: otherwise the point S would not appear to revolve in the epicycle with a contrary direction to
that of G in the deferent.
This angle GES, which in fact is the equation of the centre, may easily be calculated for any given value of the arc MG. For EG, and GS the radii of the deferent and epicycle are known, and the angle OGS is equal to MEG: the motion in the epicycle being supposed in a contrary direction to that in the deferent. This hypothesis of the epicycle coincides with that of the excentric, when the radius of the epicycle in the one case is equal to the excentricity in the other.
But as the motion of the moon is much more complicated than that of the sun, it was necessary to have recourse to a combination of excentrics and epicycles. Hipparchus had discovered in the moon's motion an inequality similar to that of the sun, and depending on the same cause, the excentricity of its orbit: Ptolemy detected another depending on the angular distance between the moon and sun. This inequality, usually called the evection, is greatest in the quadratures, and least in the syzygies; but its magnitude also depends on the combination of the places of the lunar apsides with those of the conjunctions. When the conjunctions happen in the moon's apogee, the inequality we are speaking of becomes the greatest possible in quadratures, and amounts to about 2° 40'. It is then negative in the first two quarters (that is to, say, the moon is behind her calculated place), and positive in the last two. When the conjunction takes place in the perigee, the inequality in the quadratures is also at its maximum; but it is positive in the first two quarters, and negative in the latter two. In intermediate positions of the lunar apsides, the inequality diminishes; when they are in quadratures, it is reduced to nothing. Finally it is negative in the first two quarters, and positive in the last two, or the converse, according as the conjunctions happen in the first or second quadrant of a circle, counting from either of the apsides in the direction of the moon's motion. The detection of the law existing between these complicated phenomena reflects great credit on the sagacity of Ptolemy.
To explain the first inequality of the moon, that depending on the excentricity of its orbit, he imagined an epicycle carried on an excentric; an hypothesis which is the same as that of a simple excentric, if the excentricity and radius of the epicycle together are equal to the excentricity of the simple excentric..
Let ABGD be a circle homocentric with the ecliptic, let E be the place of the earth's centre, AEG is the diameter. Suppose the moon, when in conjunction, to be in apogee, and let her place at that time be A; let the centre of the excentric be Z. Now if the moon in one day revolve through the arc AB, the apogee in the same time will move through the arc AD, equal to AB, and Z will have moved to Z'. In a quarter of a month, the points D and B will be diametrically opposed, and the centre Z will be at Z"; and the centre of the epicycle will be at A', its nearest point to the earth. The inequality will be at a maximum, and its general effect will be to augment the first inequality, by making the radius of the epicycle appear larger: consequently in the first two quarters the moon will be retarded. At the end of half a month, D and B will be in conjunction at G; the inequality will vanish. After this it will augment again gradually till the quadrature, and then diminish till conjunction; but in these two last quarters the moon will be accelerated. Were the moon in perigee in conjunction, the same phenomena would take place, with the difference, that the acceleration would be in the first two quarters, and the retardation in the two last. If the moon's perigee and apogee were in quadratures, the inequality would altogether vanish; as at the end of each quarter, the centre of the epicycle would always be ninety degrees from the apogee. This hypo
thesis of Ptolemy represents pretty well the greatest of the moon's inequalities; and it certainly was a very ingenious effort for the time; but it had several defects, the principal of which was, that in consequence of the proportion that Ptolemy was obliged to establish between the excentricity of the moveable orbit or deferent, and the radius of the epicycle, the moon's distance from the earth in quadratures would sometimes be only half of what it is in syzigies, which is entirely contradicted by observation; the variations of the moon's distance are comprised within limits comparatively very small.
From the theory of the moon, Ptolemy proceeded to that of the planets, which it appears that Hipparchus had not ventured to touch, deterred in all probability by the apparent complication of their motions. Ptolemy, however, attempted to represent them, by a combination of epicycles and excentrics. For the superior planets, he supposed the centre of the epicycle to make a revolution on its deferent in the time of a mean revolution of the planet, while this latter revolved in its epicycle in such a way, that it was always at the lowest point of the epicycle at the instant of mean opposition with the sun. The deferent itself was an excentric. It is evident that by determining properly the magnitudes of the epicycles, he could represent all the phenomena observed. For when the planet was in the superior part of its epicycle, its motion was direct; when in the inferior part, it moved in a contrary direction to that of the centre of the epicycle, and its motion, seen from the earth, would appear direct, stationary, or retrograde, according as the motion in the epicycle was less rapid, equal to, or greater than that of the centre on the deferent. We see too that each retrogradation was preceded and followed by a station, and that the place of the opposition sensibly bisected the arc of retrogradation. Finally, the excentricity of the deferent explained the inequality of the intervals between the oppositions and of the arcs of retrogradation. But even this was not sufficient to satisfy all the phenomena observed; Ptolemy was compelled to make the centre of the epicycle revolve with a motion that was uniform, not round the centre of the excentric it described, but round a point as far beyond this centre in one direction, as the earth was from it in the opposite direction: thus virtually abandoning the perfect regularity which was long
thought necessary in the orbits of the heavenly bodies.
For the inferior planets the same hypothesis of an excentric and epicycle was employed. In this case the centre of the epicycle always coincided with the mean place of the sun, while the planet described its circumference with a velocity proportional to the time employed in going from one point of greatest digression to another. As the ellipse of Mercury is much more sensibly excentric than that of the larger planets, Ptolemy found the hypotheses which satisfied the others insufficient in this case. He was compelled to suppose that the point, which was the centre of the uniform motion called the centre of the equant, instead of remaining fixed, revolved in a small circle round the centre of the excentric; the radius of this circle being equal to the distance between these two centres, and the direction of the motion against the order of the signs. But it is impossible, in this place, to follow him into all the artifices he was forced to have recourse to in explaining the irregularities of the planetary motions.
The extreme complication of this system arose in a great measure from the law he had imposed upon himself of admitting none but circular motions in the heavens: "uniform and circular motions," says he, "belonging by their nature to celestial bodies." (Syntax. lib. ix. c. ii.) That astronomers should have attempted to represent all the celestial motions by circles, was natural enough in the infancy of the science; and as long as the apparent inequalities could be represented by a combination of these circles, they were justified in so doing but it is lamentable to observe that men of talent could mistake gratuitous and arbitrary assumptions of their own for laws of nature;-these metaphysical fancies, principally borrowed from Aristotle, about the perfection and incorruptibility of circular motion, long retarded the progress of science.
But though the Aristotelian physics of Ptolemy form a strange contrast with the geometrical knowledge displayed in his work, it would be unfair to charge him with having admitted the monstrous doctrine of solid transparent spheres, revolving the one within the other, and each carrying a planet attached to it, which was promulgated by Eudoxus. To these he makes no allusion; and it is but justice to him to suppose that he
himself considered his system of defer-
Ptolemy was unsuccessful in his researches on the quantity of precession. Hipparchus had supposed it to be about one degree in 75 years: Ptolemy, undertaking to correct this determination, went much wider from the truth; he fixed it at one degree in 100 years, whereas the real value is one degree in 72 years. But there is a heavier charge against him: that of having appropriated and published as his own the catalogue of fixed stars, formed by Hipparchus. This seems to be but too well proved. He states the quantity of precession in the 265 years between himself and Hipparchus at 2° 40': this alone would show that he had not observed, as he would have found it considerably more ; but if we subtract from all his longitudes 2° 40', the precession he supposed for 265 years, we get exactly the longitudes, such as they were in the time of Hipparchus, and such, in fact, as that astronomer seems to have fixed them, judging from the positions given in the commentary on Aratus.
As all astronomy must be founded on observation, Ptolemy has not neglected to describe the instruments used for that purpose at Alexandria. To determine the sun's altitude, the Eastern nations had long been in the habit of measuring the shadow of a vertical gnomon; and, if a few simple and obvious precautions be attended to, this method may give very accurate results. learned the use of the gnomon from the Chaldeans at a very early period; and we have seen that it was employed by Meton, Pytheas, and others. But at Alexandria it seems to have been but little used; the astronomers of that place substituting for it armillary spheres of different kinds. To observe the sun's passage through the equinox they used two circles, firmly attached to each other, and placed one in the plane of the meridian, the other in that of the equator: at the moment of the equinox this latter was not illuminated by the sun on either side. For the solstice they used two concentric circles in the plane of the meridian, the one revolving within the other, and carrying two small prisms at right angles to the limb, and fixed at points diametrically opposed on the circle. To observe the sun's meridian altitude with this, the inner circle was
turned till the shadow of one prism completely covered the other: the shadow of this second fell on the graduated limb of the outer circle, and the middle of it being marked, gave the altitude of the sun's centre. For this solstitial circle Ptolemy substituted a quadrant, on which the observation was made in a manner very similar: but his most important invention was that of the parallactic rulers. These rulers formed an isosceles triangle, susceptible of being opened to any angle at the vertex; one of the equal sides was always vertical, the other being pointed on the star; the observer read off on the graduation of the base the length of the chord: a table of chords gave him the value of the angle at the vertex, that is, of the zenith distance.
The construction of the astrolabium, with which the longitudes and latitudes of the planets or fixed stars were observed, was rather more complicated than that of the solstitial or equatorial armillæ. They carried circles representing the equator, the ecliptic, the meridian, &c., and placed respectively in the planes of the celestial great circles they represented. Two other circles, moveable on the poles of the ecliptic, were made to pass through two stars: the observer then read off on the graduation of the ecliptic and circles in question respectively, the latitudes of the two stars and their difference in longitude.
As the Greeks had no means of measuring time with any accuracy, they were obliged, when they wished to compare the place of the sun with that of the fixed stars, to measure in the daytime the difference of longitude between the sun and moon, and at night that between the moon and a fixed star. The moon's rapid and variable proper motion necessarily rendering this method very inexact, the Arabs improved it considerably by substituting the planet Venus for the moon. A still greater improvement will be noticed when we con
sider the observations of El-Batani.
Ptolemy was the author of a most important discovery not recorded in the Syntaxis, the effect of refraction in augmenting the apparent altitudes of the heavenly bodies. This is clearly shewn in his Optics*, where he investigates the theory of refraction in general. He was
Vid. Delambre, Astr. Anc., vol. ii. This work
of Ptolemy, though known to the Arabs and to Roger Bacon, was for a long time lost in Europe. A Latin translation of it was fortunately disco ered by La Place in the Royal Library at Paris,
aware of the existence of a certain constant relation between the angle of incidence and that of refraction, and made several experiments to determine the value of the latter when a ray passes from air into water. Though he perceived clearly the nature of the effect produced on the altitudes of the stars, and that it diminished with the zenith distance, yet he declared himself unable to determine the absolute quantity of refraction, from not knowing the height of the terrestrial atmosphere. However, this treatise is extremely remarkable, and one of those that reflect the greatest honour on its author. To point out the existence of refraction, even without measuring it, was to render an important service to astronomy; to which we must add, that this is the only work of the ancients in which there is anything resembling the experimental philosophy of the moderns. We also find here an ingenious explanation of the optical illusion which makes the disks of the sun and moon apparently much larger when near the horizon; and this explanation is the one generally received at present, though there still seems to be some doubt on the subject.
Astronomy of the Arabs.-The Persians.-The Chinese.
WITH the Syntaxis we take our leave of the astronomy of the Greeks. The interval between the publication of this work and the conquest of Egypt and Syria by the Arabs did not produce a single astronomer; for we cannot give that name to one or two commentators on Ptolemy, of whom Theon is the most generally known. But when the Arabs had firmly established themselves in the East, they began to cultivate all the branches of mathematical science, and astronomy in particular, with extraor dinary zeal.
This revolution in the character of the Arabs, the beginning of which dates from the Caliphs, El-Mansour and Haroun-el-Reschid, at the end of the eighth century after Christ, was finally accomplished under El-Mamoun, who reigned in the beginning of the ninth. Ibu Jounis has recorded several observations made by the astronomers of this prince, the most interesting of which are those instituted to determine the obliquity of the ecliptic. This was found by some 23° 33′, by others 23° 33′ 52", which is exact within 3 minutes, and more correct than any of the
determinations made by the Greeks. Justly dissatisfied with the rough attempts of the Greek astronomers to measure an arc of the meridian, ElMamoun ordered his astronomers to proceed to a new measurement. The method they followed is sufficiently simple. Having chosen a large plain in Mesopotamia, they divided themselves into two parties; then, starting from a given point, each party measured in a right line an arc of one degree, the one towards the north, the other towards the south. The former found for the length of a degree fifty-six miles, the latter fiftysix and two-thirds; the mile being equal to 4000 cubits. But here arises the question, what was the length of these cubits? Unfortunately this is not easy to decide. Two Arabian authors agree, that the cubit employed was the black cubit of twenty-seven inches; but one says that the inch was determined by six grains of barley placed in contact sideways; the other makes it equivalent to five similar grains *. The latter seems to agree better with the real length of the degree; but the error is still very considerable, being between three and four miles in excess. But if we suppose the cubit employed to be the royal cubit of twenty-four similar inches, the length of the degree will then be brought within about a third of a mile of its real value.
The two centuries immediately following the reign of El-Mamoun were extremely fertile in astronomers, and particularly in observers; forming thus an advantageous contrast with the Greeks, who seem, with very few exceptions, to have had little taste for observation and experiment in any of the sciences. In this respect the Arabs effected a complete reform in astronomy. They have left behind them an immense mass of recorded observations, of which the greater part has never been printed; and which might be of great service to astronomy, did not the superiority of our instruments render the modern observations so much more accurate, as to compensate for the smaller interval of time existing between them.
The most distinguished of the Arabian astronomers is Albategnius or El-Batani, who rectified, in many points, the determinations of Ptolemy, and added the important discovery of the motion
According to Thevenot, 144 grains of Oriental barley, placed side by side, are exactly equal to one foot and a half of the old French measure.
of the solar apogee*. Ptolemy had fixed the precession at one degree in one hundred years, instead of seventy-two, the real value: El-Batani corrected this mistake, but made it, on the other hand, a little too rapid; namely, one degree in sixty-six years. Similarly he made the length of the solar year about two minutes and a half too small; but it is just to remark, that the errors of El-Batani proceed from the confidence he placed in the observations of Ptolemy,-observations which, as we have seen before, appear to be fictitious; had the Arabian astronomer compared his observations directly with those of Hipparchus, he would have approached much nearer to the truth. The excentricity of the solar orbit was determined by him with great accuracy, the equation of the centre fixed at 1° 58', and the obliquity of the ecliptic at 23° 35'. The observations used to determine these quantities seem to have been made with great care, and are much superior to any recorded by the Greeks. El-Batani, who gives in his writings many proofs of a sound judgment, rejects, with reason, a pretended motion of the fixed stars, by which they appeared to oscillate about a certain point, their motion in longitude becoming sometimes direct, and sometimes retrograde. To explain this pretended motion, which was called trepidation, the equinoctial points were supposed to revolve in a circle of 4° 18' 43" radius round their mean places, which retrograded along the ecliptic, according to the laws of precession. This, at least, is the way in which the theory was represented subsequently. El-Batani merely states that the stars were supposed to move directly through 8°, then to retrograde through the same arc. The Arabian astronomer, while refuting this theory, attributes it distinctly to Ptolemy. It is remarkable that none of the extant works of Ptolemy make the slightest allusion to trepidation; the first mention
This is not expressly stated by El-Batani, but it is an evident consequence, from his discovery, that the apogee, which Ptolemy found to be in 65° 30′, was now in 82° 17'. This gives an annual motion of 79". Now the Arabian astronomer allowed 54" for the annual effects of precession; there would remain, therefore, about 25" for the annual proper motion of the apogee.
The equinox taken by El-Batani to compare with his own observations, is recorded by Ptolemy with the mistake of a whole day on its date. On the observations of Ptolemy in general, Halley has expressed a severe but just opinion. V. Delamb. Ast. du Moyen Age, pp. 61, 62. Delambre is of opinion that Ptolemy never observed at all. his catalogue of fixed stars, see what has been said above, p. 31. D