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 which 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 laws of gravitation—the grandest scientific discovery in the annals of our race.
[Sidenote: Geometry.]
[Sidenote: Ancient Greek geometers.]
[Sidenote: Euclid.]
[Sidenote: Archimedes.]
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. Thales introduced the science to the Greeks. He applied a circle to the measurement of angles. Anaximander invented the sphere, the gnomon, and geographical charts, which required considerable geometrical knowledge. Anaxagoras employed himself in prison in attempting to square the circle. Pythagoras 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. He also discovered that of all figures having the same boundary, the circle among plane figures and the sphere among solids, are the most capacious. The theory of the regular solids was taught in his school, and his disciple, Archytas, was the author of a solution of the problem of two mean proportionals. Democritus of Abdera treated of the contact of circles and spheres, and of irrational lines and solids. 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 loci. They also attempted to resolve the problems of the trisection of an angle and the duplication of a cube. To Leon is ascribed that part of the solution of a problem, called its determination, which treats of the cases in which the problem is possible, and of those in which it cannot be resolved. Euclid has almost given his name to the science of geometry. He was born B.C. 323, 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 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 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. But, perhaps, a greater even than Euclid was Archimedes, born 287 B.C., who wrote on the sphere and cylinder, which terminate in the discovery that the solidity and surface of a sphere are respectively two thirds 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 only reached by prodigious mathematical power. Archimedes is popularly better known as the inventor of engines of war, and various ingenious machines, than as a mathematician, great as were his attainments. 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. He 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 he built. He used also a 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. [Footnote: See article in Smith's Dictionary, by Prof. Darkin, of Oxford.]
[Sidenote: Eratosthenes.]
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—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.
[Sidenote: Apollonius of Perga.]
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, regarded with unbounded admiration by contemporaries, and, in some respects, 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. 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? and if Thales had been contemporaneous with Plato, he might have added to his sublime science even more than 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. [Footnote: See Bayle's Dict.; Bossuet, Essai sur L'Hist. Gen. des Math.; Simson's Sectiones Conicae.]