West by the labours of those persons who were educated in the Crotonian school, it continued to be cultivated in the states of Greece; where, about one hundred years after the time of Pythagoras, Democritus of Abdera made himself celebrated by his efforts to assign a physical cause for the origin of the universe. This philosopher, whose notions appear to have been subsequently adopted by Leucippus and Epicurus, and are fully developed in the poem of Lucretius, and whose system of nature resembles, in some respects, that of Descartes, is said by Aristotle to have taught, that innumerable atoms, coming from every part of space and striking each other obliquely, formed vortices in which the lighter particles ascended towards the surface, or upper regions of each vortex, while the more gross concreted together about the centre, and thus constituted the sun, planets and earth; the latter he supposed to remain at rest in the central vortex where it was formed, but the others were conceived to revolve about it at various distances, the nearer planets moving with less velocity than those more remote.

That astronomical observations were occasionally made at this time in Greece, and that they were registered and compared together for the purpose of regulating the calendar, and perhaps, of forming tables of the movements of the sun and moon, is rendered probable by the consideration that the lunar cycles now began to be employed in that country in order to make the festivals which depend upon the moon fall in the same season of the year, and to render their commencement an epoch from whence the times of the observed phenomena of the heavens might be reckoned. The Philolaus above mentioned is stated to have invented a cycle of fifty-nine years which he called a great year and in which he said were contained twenty-one intercalary moons C. It is uncertain by what estimate of the length of the solar year and of the time of a lunar revolution he formed this period; but, if he considered the first as equal to 365 days, and the latter to 29.75 days, there will be about 729 complete lunations in that number of years, and then, the lunar year being equal to twelve such lunations, there must be added b De Cœlo, Lib. I.

a De Rerum Natura.

• Plutarch. de Placitis, Lib. II. cap. 20.

twenty-one of these to make the number in 59 lunar years equal to the number in 59 solar years; and it is not improbable that the cycle may have been so determined.

b

It appears from Diogenes Laertius and Censorinus that either Cleostratus or Eudoxus was the author of the Octaëteris, or cycle of eight solar years; but it is easy to conceive that, from its simplicity and inaccuracy, it is likely to have been in use before the times of those philosophers; and we conclude, therefore, that its invention may with propriety be assigned to the age of which we are speaking. According to Geminus, the period consists of 2922 days, and contains very nearly 99 months, or synodical revolutions of the moon; he says each of these was estimated at 29+33 days, (29.5303 days,) and that ninety-nine such months are equal to 2923 days; which, therefore, exceed the above value of 8 years by one day and a half. Considering, consequently, the lunar year to consist of 12 months, or 8 lunar years to consist of 96 months, they intercalated, in each octaëteris, 3 months, in order to make the lunar calendar agree with the solar: the error of one day and a half must, however, have remained, and this, in 160 years, must have amounted to about one month .

d

But the most famous cycle of antiquity is that which was invented, or improved, and promulgated by Meton and Euctemon, at Athens, and which began to be employed on a day corresponding to the 16th of July in the year 433 Before Christ. From a passage in Ptolemy's Mathematical Syntax we learn that, according to Hipparchus, in his work on Embolismic Months and Days, (since lost,) the above philosophers had found, by observing the epoch of a solstice at Athens in the year 316 of Nabonassar, or 430 Before Christ, and comparing it with one of more ancient date, the length of the solar tropical year to be 365+ days, or 365.2368 days; and, assuming the duration of a synodical lunar revolution to be 29.53 days, Meton is said to have found that 19 solar years contained 235 lunar revolutions, or 19 lunar years together with 7 intercalary revolutions; the lunar year being understood to be equal to twelve revolub De Die Natali, cap. 18.

a Lib. VIII.

C

De Apparentiis Coelestibus, cap. De Mensibus.

d Lib. III.

tions, or 354.358 days. This cycle of nineteen years, which is called by the name of its inventor, has been ever since his time employed for the regulation of the public festivals depending upon the moon: it is found in the astronomical works of the Arabians and Hindus; but whether either of these people received it from the Greeks, or the Greeks from the Hindus, or whether it was the result of independent observations made by either people, it is impossible to determine. The arrangement of the calendar, according to Meton was, that of the 235 months of the cycle, 110 months should have 29 days each, and the remaining 125 months, 30 days each; but as this number of days exceeds the length of nineteen solar tropical years by about one quarter of a day, at the end of four times nineteen years, or 76 years, the excess amounts to one solar day and the cycle soon became erroneous. To remedy this evil Calippus subsequently quadrupled the former cycle, and proposed another of 76 years, at the end of which time one day was to be omitted; this cycle began to be employed in the year 330 Before Christ, and it was expected that it would produce an exact agreement between the calendar and the seasons; but the values assigned to the solar and lunar years not being quite accurate, such is not the fact; the error thence arising in the course of many ages, is found, however, to be of small amount.

1 1

CHAPTER IX.

CELESTIAL SPHERES IMAGINED BY THE GREEKS.

The constellations described by Eudoxus-Uncertainty of the positions assigned by this philosopher to the equinoctial points. Probability that Eudoxus made observations with instruments.-The system of concentric spheres supposed to have been invented by him.-Dispositions and movements of the planetary spheres according to Eudoxus. The number of spheres increased by Calippus.-Nature and general laws of the motions of concentric spheres.-Investigation of the planetary orbits recommended by Plato. His disposition of the orbits.—The eternity of the universe asserted by Aristotle. His opinion that the planets move with equal velocity.- His estimated magnitude of the earth and sun. He augments the number of planetary spheres.-Celestial observations made by Pytheas.—Opinions of the ancients concerning the tides.

ABOUT sixty years after the adoption of the Metonic cycle a description of the face of the heavens and a system of the universe appear to have been made public by Eudoxus of Cnidos, who lived about 370 years before Christ and was certainly one of the most celebrated mathematicians of his time, since he is acknowledged to be the author of one of the most important books in that collection of the elements of Geometry which is ascribed to Euclid. The astronomical works composed by him are lost, but one of them, which seems to have contained a description of the constellations, was paraphrased by Aratus about one hundred years after his time in a poem still extant: from this it appears that the constellations into which the heavens were then divided were nearly the same as those represented on our present celestial globes. It is found, however, that the relative positions assigned by Eudoxus to the fixed stars present very numerous discrepancies, and it has been attempted to explain them by supposing that the philosopher had copied the places of the stars from registers of ancient observations, which, having been made at different times, could not agree with each other on account of the movement of the equinoctial points in the intervals; but the irregularity of the errors is so great as to render it impossible to avoid concluding that the places of the stars were estimated by the eye alone, and without any attempt at

precision. Aratus remarks, and his observations may be considered as those of Eudoxus himself, that the stars revolve regularly, meaning with equal angular velocities, because they are permanently fixed in the celestial sphere; but he adds, there are some among them, meaning the planets, which change their places, performing revolutions and returning to conjunction in the same part of the heavens at the end of various intervals of time; and he directs that their places should be observed by referring them to the fixed stars which they occasionally approach. From this precept Delambre concludes that the Greeks then possessed no instruments for determining the longitudes and latitudes of stars, and that they were almost entirely ignorant of any theory of the planetary movements: both these consequences, however, do not necessarily follow, for it is possible that Eudoxus may have used instruments though Aratus has not noticed any observations made with them, which might be because such details do not form fit subjects for a poem.

It would be desirable to ascertain, if possible, from the work of Aratus, the position which Eudoxus assigned to the equinoctial or solstitial points, in order that the subsequent movement of those points might be determined with precision; or, assuming this to be known, that the time might be found at which the observations were made from whence the position so assigned was discovered; but it is to be regretted that the description he has given does not enable us to arrive at any satisfactory conclusion on either of these heads. Eudoxus mentions particularly the two tropics and the equator, observing, as is related by Hipparchus in his commentary on Aratus, that the northern tropic touches the zodiac in, or about the constellation Cancer, and that the southern tropic cuts the middle of Capricornus: hence he must have supposed the equinoctial colure to pass through the middle of the constellation Aries. But the difficulty is to ascertain where is the middle of that constellation; for if we suppose its whole extent to be equal to 30 degrees, the value now given to it, and that it commenced with y Arietis, which is the first remarkable star in it, and whose longitude is at present equal to about 30 degrees, since the equinoctial colure passes near a Andromeda; it would follow that the precession