In regard to the practical value of the speculations of the ancient astronomers, it may be said that had they possessed clocks and telescopes, their scientific methods would have sufficed for all practical purposes.  The greatness of modern discoveries lies in the great stretch of the perceptive powers, and the magnificent field they afford for sublime contemplation.  “But,” as Sir G. Cornewall Lewis remarks, “modern astronomy is a science of pure curiosity, and is directed exclusively to the extension of knowledge in a field which human interests can never enter.  The periodic time of Uranus, the nature of Saturn’s ring, and the occultation of Jupiter’s satellites are as far removed from the concerns of mankind as the heliacal rising of Sirius, or the northern position of the Great Bear.”  This may seem to be a utilitarian view, with which those philosophers who have cultivated science for its own sake, finding in the same a sufficient reward, can have no sympathy.

The upshot of the scientific attainments of the ancients, in the magnificent realm of the heavenly bodies, would seem to be that they laid the foundation of all the definite knowledge which is useful to mankind; while in the field of abstract calculation they evinced reasoning and mathematical powers that have never been surpassed.  Eratosthenes, Archimedes, and Hipparchus were geniuses worthy to be placed by the side of Kepler, Newton, and La Place, and all ages will reverence their efforts and their memory.  It is truly surprising that with their imperfect instruments, and the absence of definite data, they reached a height so sublime and grand.  They explained the doctrine of the sphere and the apparent motions of the planets, but they had no instruments capable of measuring angular distances.  The ingenious epicycles of Ptolemy prepared the way for the elliptic orbits and laws of Kepler, which in turn conducted Newton to the discovery of the law of gravitation,—­the grandest scientific discovery in the annals of our race.

Closely connected with astronomical science was geometry, which was first taught in Egypt,—­the nurse and cradle of ancient wisdom.  It arose from the necessity of adjusting the landmarks disturbed by the inundations of the Nile.  There is hardly any trace of geometry among the Hebrews.  Among the Hindus there are some works on this science, of great antiquity.  Their mathematicians knew the rule for finding the area of a triangle from its sides, and also the celebrated proposition concerning the squares on the sides of the right-angled triangle.  The Chinese, it is said, also knew this proposition before it was known to the Greeks, among whom it was first propounded by Thales.  He applied a circle to the measurement of angles.  Anaximander made geographical charts, which required considerable geometrical knowledge.  Anaxagoras employed himself in prison in attempting to square the circle.  Thales, as has been said, discovered the important theorem that in a right-angled