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opinion of the early astronomers of Greece. However when they began to make observations with the gnomon, they could not help perceiving a considerable difference between the intervals of the equinoxes and solstices; intervals which must be equal were the motion of the sun round the earth uniform. Hipparchus undertook to investigate this point. He observed that the interval between the vernal equinox and the summer solstice was 944 days; between the summer solstice and the autumnal equinox 924. Thus the sun took 187 days to describe the northern half of the ecliptic, and only 178 for the southern half; indicating a considerable increase of velocity during the latter. To explain this irregularity, Hipparchus supposed the sun to move round the earth in an excentric circle; that is in a circle, whose centre did not coincide with that of the earth. It is clear, that in this case the sun, though moving uniformly in its orbit, would appear to a spectator at the earth to move with an unequal velocity, on account of the variation of its distance. The question was to determine the quantity of this excentricity, that is to say, the distance of the earth from the centre of the solar orbit; and the position of the apogee and perigee, or of the points of greatest and least
distance. Let ADB F represent the circle in which the sun is supposed to revolve; let the centre of this circle be at C, and the earth at E: according to Hipparchus the sun revolves with an uniform motion round C: it is evident that, seen from E, his motion will appear unequal it will be fastest at the point B or the perigee; slowest at A, the apogee: let M N be the line joining the sun's places, at the two solstices; PQ at the equinoxes: the object of Hippar
Ptol. Syntax. Lib. iii.
éhus was to ascertain the ratio of EC to BC, and the arc Q A which determines the position of the apogee. By combining his observations of the equinoxes and solstices, he found the excentricity equal toth part of the radius, and the longitude of the apogee, or the are QA, equal to 65° 30. This value of the excentricity is, however, too great by about one-sixth. The excentricity and place of the apogee being once known, it was easy to construct Tables which should give the sun's position at any time. For, suppose the sun to be at S, then as he is supposed to revolve uniformly round C, we can find from the time taken to describe the arc AS, the value of the angle ACS, and therefore S CE; and in the triangle S CE, CE, and CS are known, whence we may find CSE, which is the difference between the angles ACS and A E S, or between the mean and true anomaly. This difference is called the equation of the centre. From what has been just said, we may see how Hipparchus calculated the values of the equation of the centre corresponding to successive values of A S, or the angle A CS, in his solar Tables. We must recollect that the instant of the sun's passage through the equinox at Q may always be supposed known: the arc QS is proportional to the time elapsed since the equinox, and is soon found: QA is known: hence we find A S, and looking into the Tables, find the corresponding equation of the centre. This gives us AES, and consequently QES, or the sun's apparent longitude for any given time.
From the theory of the sun Hipparchus proceeded to that of the moon. By comparing some ancient eclipses with those observed by himself, and dividing the interval of time by the number of revolutions, he obtained the value of a synodic revolution of the moon. By methods similar to those employed for the sun, he determined the excentricity of the lunar orbit, and its inclination to the ecliptic, which latter he fixed at 5°. Finally, he is said to have measured the motions of the lunar apogee and node. With these data he calculated the first Tables of the sun and moon of which history makes mention. This alone would have secured for him the gratitude and admiration of posterity. The want of observations, and perhaps the difficulty of their theory in his system, prevented
him from attempting a similar under taking with regard to the planets.
But the most important, perhaps, of all the services rendered to astronomy by Hipparchus, was the formation of a catalogue of the fixed stars. If we consider the boldness of the attempt, the labour of the execution, and the importance of the result, the author of it seems not undeserving the enthusiastic praises of Pliny. Such a catalogue is, in fact, the foundation of all astronomy, The fixed stars are so many standard points to which the celestial motions are referred, and the determination of their relative distances is of the utmost importance. By comparing their positions at distant periods, we may detect those small variations which require centuries to become sensible; and there is every reason to believe that, if we possessed a really accurate catalogue of twenty or thirty centuries back, we should be in possession of many valua ble discoveries, which perhaps are destined to lie hid for ages. It was, indeed, in this way that Hipparchus was led to his great discovery of the precession of the equinoxes. On comparing his own observations with those of Aristillus and Timocharis, made 150 years previously, he perceived that all the fixed stars, while they retained their latitudes sensibly unaltered, had advanced about two degrees in longitude; or what comes to the same, the equinoctial points appeared to have retrograded along the ecliptic by the same quantity. It was reserved for Newton to explain the causes of this singular phenomenon.
Such is a brief account of the astronomical discoveries of Hipparchus: we have already seen that he was the inventor of trigonometry; it also appears that he was the first who suggested the method of fixing the positions of places on the earth's surface by their longitudes and latitudes, and that he proposed to determine the former by means of lunar eclipses; a method excellent in its principle, though now abandoned on account of some practical objections.
As nothing connected with astronomy seems to have escaped the sagacity of Hipparchus, he did not overlook the correction of the Calendar. We have seen that the period of Callippus was far from exact: according to the calculations of Hipparchus, the error at
Hipparchus nunquam satis laudatus, ut quo nemo magis comprobaverit cognationem cum homine syderum, animasque nostras partem esse cœll,..... ausus rem etiam Deo improbam, an. numerare posteris stellas.-Hist. Nat. ii. 26.
the end of a period was about one-fourth of a day. He proposed to quadruple the period of Callippus, and then to sub tract a day. This new period brought the moon again to the same place pretty exactly: the error on the sun's motion was about a day and a quarter, which is one-fourth of the error of Callippus in the same time.
We have some reason to be surprised that the discoveries of Hipparchus were not followed up by succeeding astrohomers. One might have imagined that such brilliant success would have stimulated others to the further development of the science; but, extraordinary as it may appear, history records not one astronomer of note in the three centuries between Hipparchus and Ptolemy. The attempt made by Posidonius to measure a degree of the meridian has been already noticed: a few authors on spherical trigonometry flourished in this interval, among whom may be distinguished Theodosius and Menelaus; but astronomy itself seems to have made no progress fill the time of Ptolemy. This eminent and laborious philosopher felt the necessity of uniting all the scattered materials existing in the works of Hipparchus and others, which, combined with his own discoveries, formed, as far as the knowledge of the time allowed, a complete system of astronomy: by so doing he rendered a distinguished service to science; and the publication of his μαθηματικη σύνταξις forms an important epoch. This work, which has fortunately survived the barbarism of the middle ages, formed the basis of all the astronomy of the Arabians, and for a considerable time that of modern Europe. Its importance requires here a concise analysis.
Ptolemy begins his work with a discussion of the relative positions of the earth, sun, and planets. We have already seen that the Greek astronomers were divided on the subject of the earth's motion. Though many distinguished philosophers held the opinions of Pythagoras, the majority seem to have embraced the opposite doctrines. Ptolemy followed these latter, and, unfortunately for him, his name has become attached to a system now universally admitted to be erroneous. It is true that the ancients wanted some decisive and convincing proofs of the earth's motion, which we possess; but though much has been said to excuse Ptolemy, his justification remains very incomplete. The arguments that he urges
against the earth's motion, such as that in this case the poles would not be immoveable points on the celestial sphere, that the fixed stars would not always preserve the same apparent distances from one another, and other objections of a similar kind, are all obviated by the single remark made by Aristarchus, four centuries previously, that the earth's orbit was a point in comparison with the distance of the fixed stars. On the other hand, the motions of the planets, so complicated and almost inexplicable in the one hypothesis, are accounted for so simply in the system of Pythagoras, that one cannot but feel astonished that Ptolemy should have felt so little hesitation in rejecting it. "The same reasons," says he, "which show that the earth is a point in magnitude compared with the heavens, will show the impossibility of its having a motion of translation:" and the only argument he combats at any length, is that which appears to have been urged by some Pythago reans, that the earth being spherical and unsupported, could not remain at rest in the centre of the heavenly motions. Having discussed this point, with a singular mixture of truth and error, he adds these remarkable words: "But if there were any motion of the earth common to it and all other heavy bodies, it would certainly precede them all by the excess of its mass, being so great; and animals and a certain portion of heavy bodies would be left behind, riding upon the air, and the earth itself would very soon be completely carried out of the heavens. But such things are most ridiculous, even only to imagine." This passage is remarkable, because it shows how little the Greeks had studied natural and experimental philosophy, and how falsely their geometers could reason on purely physical subjects. A heavy body in vacuo does not, as Ptolemy supposes, move faster than a lighter one, as may be verified by direct experiment; yet this he clearly considered a self-evident truth, and founded on it arguments which must be classed among the weakest ever urged against the Pythagorean system of the world.
After rejecting the motion of translation assigned by some to the earth, he proceeds to examine the probability of its diurnal motion on its axis. This system he confesses simplifies very much the appearances of the heavens; but it appears to him equally ridiculous with the former; as in this case, the earth revolving with great rapidity from west
to east, would leave behind it the clouds, birds flying in the air, and, generally, all objects suspended in the atmosphere. A stone thrown to the east would not advance, the earth constantly preceding it by the excess of its velocity. These objections are all founded on an ignorance of the principles of mechanics, and seem to be on a par with those urged by some against the roundness of the earth, and the possibility of the existence of Antipodes; arguments which he has himself successfullyrefuted.
The earth then, according to Ptolemy, was fixed and motionless in the centre of the heavens; he supposed the different planets to revolve round it, arranged in the following order, according to their distances: first, the Moon, then Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and, lastly, the sphere of the fixed stars. With regard to Venus and Mercury, Ptolemy remarks, that some astronomers had placed them beyond the sun, while others made them nearer: the most ancient writers had adopted the latter opinion, which had been rejected by subsequent authors, because these two planets had never been seen on the sun's disk. This reason Ptolemy rightly rejects as insufficient; for such passages over the sun's disk would not happen, unless the planes of the orbits coincided with the ecliptic, or else the nodes happened to coincide nearly with the sun's place at the time of inferior conjunction. He does not seem to be aware that these passages or transits really do take place; and sufficiently often in the case of Mercury, though but rarely in that of Venus. But the difficulty of observing these phenomena renders it by no means extraordinary that they should not have been noticed, though it might have taught caution to those who affirmed positively their non-existence. It is much more remarkable that Ptolemy should not have perceived that it was possible to conciliate the two hypotheses in question, by making these two planets revolve round the sun; in which case it is clear they would be sometimes more distant, and sometimes nearer, than that body. And this inadvertence is the more singular, as the doctrine just mentioned is said to have been maintained by the ancient Egyptians. It seems probable that the systematic ideas of Ptolemy made him unwilling to place the sun in the centre of any of the heavenly motions; or he might have been repugnant to consider any of the planets
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 DBCN, 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
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 à 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 opposi tion 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