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á real science, we shall at once transport ourselves to Greece and the age of Thales.

born at Mile

After receiv

This illustrious philosopher was tum about 640 years before Christ. ing the usual learning of his own country he travelled into Egypt, where he became eminent in Astronomy, Geometry, Philosophy, &c. Whatever instructions he might receive from them in many branches of knowledge, it does not seem probable that he obtained much information from them in Geometry, as all writers agree that he was the first who measured the height of the pyramids by the extent of their shadows. It is said that he also applied the circumference of the circle in measuring angles. There can be little doubt that he made many other discoveries, which have not been directly handed down to us, but which might have been inserted in elementary books, and ranked among discoveries of later times ; from him also Astronomy made a very considerable advance; and he is generally reputed to be the father of the Greek philosophy, being the first that made any researches into natural knowledge.

From Thales we pass on to Pythagoras, a philosopher no less distinguished than the former for the variety and extent of his discoveries; among the most eminent of these, at least in Geometry, which he made, may be mentioned, that the square of the hypothenuse of a right angled triangle is equal to the sum of the squares of the other two sides (see note to the 47th proposition of the 1st

book); he is also said to be the inventor of the 32d proposition of the same book, viz. that the three angles of any triangle are together equal to two right angles; as likewise to have shown that only three polygons, or regular plane figures, can fill up the space about a point; viz. the equilateral triangle, the square, and the hexagon; these, however, when compared with his other inventions, will appear but trifles: in Astronomy he is reported to have maintained the true system of the world, which places the sun at the centre, and makes the planets to revolve round him; and from him it is called the Pythagorean system, which was revived by Copernicus. Whether we consider the variety of his discoveries, or the extent of his attainments; whether we reflect upon his inventive genius, which distinguishes him in all his pursuits, or upon that amazing assiduity so conspicuous amongst the whole race of Grecian philosophers, few men will be found to possess a greater claim to the honour of posterity than Pythagoras. As a mathematician, he was decidedly the first of his time. As a philosopher, we find him delivering many excellent things concerning God and the human soul, and a great variety of precepts relating to the conduct of life both political and civil.

Although it is doubtful whether Geometry at this time had been founded into a regular system, yet, from what has been said, it appears that it must certainly have made considerable advances, and that many of its detached parts were known; for not more than a century had elapsed, from the

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age of Pythagoras, when Zenodorus, a man of great parts, arose, and whose writings are the first amongst the ancients, which have survived the wreck of time, a geometrical tract of his having been preserved by Theon in his Commentary upon Ptolemy's Almagest, wherein he has shown the falsity of the opinions then entertained that figures, with equal peripheries, have equal areas: a problem not easy of solution, and shows that Geometry must have then made a great progress. The ingenious theory of the five regular bodies originated also about the same time in the Pythagorean school.

Next in order comes Hippocrates, a man possessing a very brilliant genius, and who rendered Geometry essential services by his diligence and assiduity. Among his discoveries the quadrature of the celebrated lunulæ of the circles, which bear his name, may be placed foremost in the list. Having described three semicircles on the three sides of a right angled triangle considered as diameters, the one on the hypothenuse being in the same direction as the others, he found that the sum of the areas of the two equal lines comprised between the two quadrants to the hypothenuse, and the semicircles answering to the other two sides, was equal to the area of the triangle. He also wrote Elements of Geometry; which, from the account given by Proclus, were much esteemed in his time; although, having been superseded by the Elements of Euclid, those of Hippocrates were consigned to oblivion. He also appears with

honour among the list of Geometers, who attempted to solve the celebrated problem of the duplication of the cube, which at that period began to be pursued with ardour. The circumstance of this problem is well known; its solution at first sight appeared easy; but the mistake was soon perceived, and all the geometricians of Greece were baffled in attempting to solve it. Notwithstanding the failure of this, Geometry still continued to advance, and was cultivated with great care and attention by Plato (B. C. 300); although we have no work of his written upon this subject, yet we learn from other writers, and indeed from many passages in his works, that he was well acquainted with its different branches, and had even enriched it with many of his discoveries. Indeed so profound a veneration did he entertain for the science here spoken of, that he made it the principal object of instruction among his scholars. He had written over the door of his academy, Let no one enter here who is ignorant of Geometry."

The problem before-mentioned, viz. the duplication of the cube, particularly engaged his attention; and although he was unable to resolve it by a method purely geometrical, or at least as considered by the ancients, that is, by the rule and compasses only, yet he invented an ingenious, though mechanical solution, by means of an instrument consisting of two rules, one of them moved in the grooves of two arms at right angles with the other, so as always to continue parallel

with it. But though Plato was unfortunate in his attempts to double the cube, yet we find him more successful in another speculation of a kind entirely new. Before his time the circle was the only curve admitted into geometry, Plato, however, discovered the conic sections, or those curves which are found on the surface of a coné by a plane cutting it in different directions; and by attentively examining the generation of those curves, he discovered some of their most remarkable properties, which being made the continual study of his scholars and successors, it at length became a distinct science from the common elements, and received the appellation of the higher or sublime geometry.

The important addition of conic sections to the mathematical sciences being, as before observed, particularly cultivated by the geometers of that time; Aristeus, a friend and disciple of Plato, composed five books on that subject, which are spoken of with great eulogies by the ancients; but either from the despotism of ignorant barbarians, or from the ravages of time, they have unfortunately not reached us, and nothing more is known of them than the little that is mentioned by Pappus, in his Mathematical Collections. Of Menechmus, we have two learned applications of the same theory to the problem of the duplication of the cube; and from the result of his labours, it appears that if we possessed the means of describing conic sections by one continued motion, in as simple a way as we trace a circle with the

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