Caesar was a student of astronomy, and always found time for its contemplation. He is said even to have written a treatise on the motion of the stars. He was assisted in his reform of the calendar by Sosigines, an Alexandrian astronomer. He took it out of the hands of the priests, and made it a matter of pure civil regulation. The year was defined by the sun, and not as before by the moon.
Thus the Romans were the first to bring the scientific knowledge of the Greeks into practical use; but while they measured the year with a great approximation to accuracy, they still used sun-dials and water-clocks to measure diurnal time. Yet even these were not constructed as they should have been. The hour-marks on the sun-dial were all made equal, instead of varying with the periods of the day,--so that the length of the hour varied with the length of the day. The illuminated interval was divided into twelve equal parts; so that if the sun rose at five A.M., and set at eight P.M., each hour was equal to eighty minutes. And this rude method of measurement of diurnal time remained in use till the sixth century. Clocks, with wheels and weights, were not invented till the twelfth century.
The last great light among the ancients in astronomical science was Ptolemy, who lived from 100 to 170 A.D., in Alexandria. He was acquainted with the writings of all the previous astronomers, but accepted Hipparchus as his guide. He held that the heaven is spherical and revolves upon its axis; that the earth is a sphere, and is situated within the celestial sphere, and nearly at its centre; that it is a mere point in reference to the distance and magnitude of the fixed stars, and that it has no motion. He adopted the views of the ancient astronomers, who placed Saturn, Jupiter, and Mars next under the sphere of the fixed stars, then the sun above Venus and Mercury, and lastly the moon next to the earth. But he differed from Aristotle, who conceived that the earth revolves in an orbit around the centre of the planetary system, and turns upon its axis,--two ideas in common with the doctrines which Copernicus afterward unfolded. But even Ptolemy did not conceive the heliocentric theory,--the sun the centre of our system. Archimedes and Hipparchus both rejected this theory.
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 triangle the squares on the sides containing the right angle are together equal to the square on the opposite side of it. Pythagoras discovered that of all figures having the same boundary, the circle among plane figures and the sphere among solids are the most capacious. Hippocrates treated of the duplication of the cube, and wrote elements of geometry, and knew that the area of a circle was equal to a triangle whose base is equal to its circumference and altitude equal to its radius. The disciples of Plato invented conic sections, and discovered the geometrical foci.
It was however reserved for Euclid to make his name almost synonymous with geometry. He was born 323 B.C., and belonged to the Platonic sect, which ever attached great importance to mathematics. His "Elements" are still in use, as nearly perfect as any human production can be. They consist of thirteen books. The first four are on plane geometry; the fifth is on the theory of proportion, and applies to magnitude in general; the seventh, eighth, and ninth are on arithmetic; the tenth on the arithmetical characteristics of the division of a straight line; the eleventh and twelfth on the elements of solid geometry; the thirteenth on the regular solids. These "Elements" soon became the universal study of geometers throughout the civilized world; they were translated into the Arabic, and through the Arabians were made known to mediaeval Europe. There can be no doubt that this work is one of the highest triumphs of human genius, and it has been valued more than any single monument of antiquity; it is still a text-book, in various English translations, in all our schools. Euclid also wrote various other works, showing great mathematical talent.
Perhaps a greater even than Euclid was Archimedes, born 287 B.C. He wrote on the sphere and cylinder, terminating in the discovery that the solidity and surface of a sphere are two thirds respectively of the solidity and surface of the circumscribing cylinder. He also wrote on conoids and spheroids. "The properties of the spiral and the quadrature of the parabola were added to ancient geometry by Archimedes, the last being a great step in the progress of the science, since it was the first curvilineal space legitimately squared." Modern mathematicians may not have the patience to go through his investigations, since the conclusions he arrived at may now be reached by shorter methods; but the great conclusions of the old geometers were reached by only prodigious mathematical power. Archimedes is popularly better known as the inventor of engines of war and of various ingenious machines than as a mathematician, great as were his attainments in this direction. His theory of the lever was the foundation of statics till the discovery of the composition of forces in the time of Newton, and no essential addition was made to the principles of the equilibrium of fluids and floating bodies till the time of Stevin, in 1608. Archimedes detected the mixture of silver in a crown of gold which his patron, Hiero of Syracuse, ordered to be made; and he invented a water-screw for pumping water out of the hold of a great ship which he had built. He contrived also the combination of pulleys, and he constructed an orrery to represent the movement of the heavenly bodies. He had an extraordinary inventive genius for discovering new provinces of inquiry and new points of view for old and familiar objects. Like Newton, he had a habit of abstraction from outward things, and would forget to take his meals. He was killed by Roman soldiers when Syracuse was taken; and the Sicilians so soon forgot his greatness that in the time of Cicero they did not know where his tomb was.
Eratosthenes was another of the famous geometers of antiquity, and did much to improve geometrical analysis. He was also a philosopher and geographer. He gave a solution of the problem of the duplication of the cube, and applied his geometrical knowledge to the measurement of the magnitude of the earth,--being one of the first who brought mathematical methods to the aid of astronomy, which in our day is almost exclusively the province of the mathematician.
Apollonius of Perga, probably about forty years younger than Archimedes, and his equal in mathematical genius, was the most fertile and profound writer among the ancients who treated of geometry. He was called the Great Geometer. His most important work is a treatise on conic sections, which was regarded with unbounded admiration by contemporaries, and in some respects is unsurpassed by any thing produced by modern mathematicians. He however made use of the labors of his predecessors, so that it is difficult to tell how far he is original. But all men of science must necessarily be indebted to those who have preceded them. Even Homer, in the field of poetry, made use of the bards who had sung for a thousand years before him; and in the realms of philosophy the great men of all ages have built up new systems on the foundations which others have established. If Plato or Aristotle had been contemporaries with Thales, would they have matured so wonderful a system of dialectics? Yet if Thales had been contemporaneous with Plato, he might have added to the great Athenian's sublime science even more than did Aristotle. So of the great mathematicians of antiquity; they were all wonderful men, and worthy to be classed with the Newtons and Keplers of our times. Considering their means and the state of science, they made as great though not as fortunate discoveries,--discoveries which show patience, genius, and power of calculation. Apollonius was one of these,--one of the master intellects of antiquity, like Euclid and Archimedes; one of the master intellects of all ages, like Newton himself. I might mention the subjects of his various works, but they would not be understood except by those familiar with mathematics.