It is probable that the greater number of the disciples of Tholes were acquainted with geometry ; but the names of Ameristus and Anaximander only have reached our times. The first is said to have been a skilful geometer ; the other composed a kind of elementary treatise or intro duction to geometry, the earliest on record. Tholes was succeeded in his school by Anaximander, who is said to have invented the sphere, the gnomon, geographical charts, and sun-dials ; he was succeeded by Anaximenes; and this philosopher again was succeeded by his scholar Anaxago ras, who, being cast into prison on account of his opinions relating to astronomy, employed himself in attempting to square the circle. This is the earliest effort on record, to resolve the most celebrated problem in geometry.
Pythagoras was one of the earliest and most successful cultivators of geometry. He was born about 580 years be fore the Christian xra; he studied under Tholes, and by his advice travelled into Egypt. Here he is said to have consulted the columns of Sothis, on which that celebrated person had engraven the principles of geometry, and which were deposited in subterranean vases. A learned curiosity induced him to travel also into India ; and it is far from being improbable, that he was more indebted for his knowledge to the Brahmins, on the banks of the Gan ges, than to the priests of Egypt. On his return, finding his native country a prey to tyranny, he settled in Italy, and there founded one of the most celebrated schools of antiquity. He is said to have discovered that in any right angled triangle, the square on the side opposite the right angle, is equal to the two squares on the sides containing it ; and, on this account, to have sacrificed one hundred oxen, to express his gratitude to the muses. This, how ever, was incompatible with his moral principles, which led him to abhor the shedding of blood on any account whatever ; and besides, the moderate fortune of a philoso pher would not admit of such an expensive proof of his piety. The application which the Pythagoreans made of geometry gave birth to several new theories, such as the incommensurability of certain lines, for example, the side of a square, and its diagonal, also the doctrine of the regu lar solids, which, although of little use in itself, must have led to the discovery of many propositions in geometry. Diogenes Laertius has attributed to Pythagoras the merit of having discovered, that of all figures having the same boundary, the circle among plane figures, and the sphere among solid figures, are the most capacious: if this was so, he is the first on record that has treated of isoperime trical problems.
The Pythagorean school sent forth many mathema ticians ; of these Archytas claims attention, because of his solution of the problem of finding two mean pl'oportionals ; also on account of his being one of the first that employed the geometrical analysis, which he had learnt from Plato, and by means of which he made many discoveries. He is
said to have applied geometry to mechanics, for which he was blamed by Plato; but probably it was rather for apply ing, on the contrary, mechanics to geometry, as he employ ed motion in geometrical resolutions and constructions.
Democritus of Abdera studied geometry, and was a profound mathematician. From the titles of his works, It has been conjectured that he was one of the principal pro moters of the elementary doctrine respecting the contact of circles and spheres, and concerning irrational numbers and solids. He treated besides of some of the principles of optics and perspective.
Hippocrates was originally a merchant, but having no turn for commerce, his affairs went into disorder ; to re pair them, he came to Athens, and was one day led by cu riosity to visit the schools of philosophy. There he heard of geometry for the first time ; and as probably there is a natural adaptation of certain minds to particular studies, he was instantly captivated with the subject, and became one of the hest geometers of his time. He discovered the quadrature of a space bounded by half the circumference of one circle, and one fourth the circumference of another, their convexities being turned the same way. This figure, called a lune, he sheaved to be equal to a right angled tri angle having its sides about the right angle equal, and the remaining side equal to the common chord of the two arcs; and thus he was the first that proved a curvilineal to be equal to a rectilineal space. But although a kind of quad rature, it cannot be compared as a discovery with the quad rature of the parabola found afterwards by Archimedes: the former is merely a geometrical trick, which leads to nothing further; but the latter was an important step in the progress of the science. Hippocrates attempted the quadrature of the circle, but if his mode of reasoning has been correctly handed down to us, he committed a blunder : this is the oldest paralogism in geometry upon record. On the other hand it must be mentioned to his credit, that he first proved the duplication of the cube to depend on the finding of two mean proportionals between two given lines: (See Introduction to CONIC SECTIONS.) He was also the first that composed Elements of Geometry, which, how ever, have been lost, and are only to be regretted, because we might have learnt from them the state of the science at that period. It has been said that, notwithstanding his want of success in commerce, he retained something of the mercantile spirit : he accepted money for teaching geometry, and for this lie was expelled the school of the Pythagoreans. This offence we think might have been forgiven in consideration of his misfortunes.