ter Luftre to the Mathematical Sciences. He enlarg'd Geometry with great and notable Additions, bestowing incredible Study upon it. And above all, the Art Analytick, or of Resolution, was found out by him, the most certain way of Invention and Reasoning. He fet off and illustrated his Books of Philofophy in a Mathematical way, and encourag'd whatsoever was admirable in Mathematical Philofophy. Upon the Door of his Academy was read this Inscription : ἐδεὶς ἀγεωμέτρηςΘ εἰείτω : Let no one ignorant of Geometry enter here; an illustrious Instance to demonstrate, how the Mathematicks are not foreign but proper, not unufeful, or unbecoming, but honourable and profitable to found and certain Philofophy. In a word, how great both Admirer and Master of the Mathematicks Plato was, that Man will of himself easily understand, who shall read his Monuments thro'.

Out of Plato's Academy, almost innumerable Mathematicians came forth. Thirteen of Plato's familiar Acquaintance are commemorated by Proclus, as Men by whose Studies the Mathematicks were improv'd. From hence were Leodamus the Thasian, Archytas the Tarentine, Theatetus the Athenian, by whom the Mathematicks were notably enlarged. Leodamus practifed the Analysis received from Plato, and

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and is faid by Laertius to have found out many things by the Help of it. As for Theatetus, both his own Inventions, amongst which are the Elements written by him, and the Inscription of regular Bodies; and Plato's Encomiums, who also inscribed a Dialogue to his Name, do make him fa

mous.

Archytas alfo wrote Elements himself; and his Doubling of the Cube is mentioned by Eutocius; whose singular Commendation it likewife was, that he was almost the First that brought down the Mathematicks to humane Ufes; by whose Contrivance also a wooden Pigeon was made to fly, as Gellius reports; he being followed by Dedalus, and other Artificers, yielded Matter for the Fables of the Poets. Moreover, Archytas was both a Mathematician and General of an Army: He five times commanded the Forces of his own Citizens, in the Wars of his Country, and five times overcame their Enemies. The meer Name of Neoclides is only Famous, he being more illustrious for his Scholar Leon perhaps, than for his own Inventions. Leon certainly wrote Elements of all the Mathematicks, improv'd them, and made them more fit for Ufe. Wherefore he is deservedly to be reckon'd amongst the chief Compilers of Elements.

Eudoxus

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Eudoxus of Cnidos was not inferior to Leon: A Man great in Arithmetick, and to him (if we may believe the Greek Scholiaft) we owe the whole fifth Book. He likewife wrote Elements, and made them more general, and increased the Sections begun by Plato; over and above this he was the first Framer of Aftronomical Hypothefes, and derived down the Springs of Geometry, as Archytas had done before, to Mechanicks. Amyclas the Heracleot, and Menæchmus, and his Brother Dinoftratus, Helicon of Cyzium, Theudius, Hermotimus the Colophonian, Philippus the Medmaan, all Platonists, rendered Geometry much more perfect. Menæchmus alfo found out the Conick Sections, and by the help of them, two mean Proportionals; whose Invention in this Cafe is preferr'd by Eutocius before any other. Theudius and Hermotimus made the Elements more universal and full. And all these, who were of Plato's Academy, brought Mathematick Philofophy to Perfection, as Proclus faith. Xenocrates alfo, one of Plato's Auditors, and Master of Aristotle, as well as Ariftotle himself, were famous for the Knowledge of the Mathematicks. When a certain Person, who knew nothing of Geometry, was defirous to be his Auditor, Go thy way, faith he, for thou wantest the very Handles of Philosophy.

But of Aristotle, what can I say? All his Books are filled with Mathematical Arguments, from a Collection of which Blancane hath made a Book. Two of Ariftotle's School are especially celebrated, Eudemus and Theophraftus: This latter wrote two Books of Numbers, four of Geometry, and one of indivisible Lines: The other, compofed a Mathematical History; and from him Proclus, and others have borrowed theirs. To Aristeus, Isidore, Hypsicles, most fubtle Geometricians, we are especially indebted for the Books of Solids. Lastly, Euclid gathered together the Inventions of others, disposed them into Order, improv'd them, and demonstrated them more accurately, and left to us those Elements, by which Youth is every where instructed in the Mathematicks. He died in the Year before Christ 284. There follow'd Euclid almost an 100 Years afterwards Eratosthenes and Archimedes. The Name of Eratofthenes was very famous, but his Writings are loft. Many Remains we have of Archimedes, and many we have loft.

-. But when I name Archimedes, I conceive in my Mind the very Top of humane Subtilty, and the Perfection of the whole Mathematical Sciences. His wonderful Inventions have been delivered to us by Polybius, Plutarch, Tzetzes, and others. Conan was Contemporary to Archimedes,

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one who was both a Geometrician and an Astronomer, whose Death Archimedes laments in his Book of the Quadrature of the Parabola. There followed Archimedes and Conon, and that at no great Distance, Apollonius of Perga, another Prince in Geometry, who was called by way of high Encomium, The Great Geometrician. There are extant Four [now Seven] most subtle Books of his Conicks. To the same Perfon are afcribed, the 14 and 15 Books of Euclid, which were contracted by Hypficles. Hipparchus and Menelaus wrote, this latter 6, the other 12 Books of Subtenses in a Circle; for which Invention, so very profitable and necessary, great Commendations and Thanks are due to both. There are alfo extant three Books of Menelaus concerning Spherical Triangles. Three most ufetul Books of Sphericks of Theodofius the Tripolite are also in the Hands of all. And these indeed, if you except Menelaus, lived all of them before Christ.

In the Year after Christ 70, there appeared in the World Claudius Ptolemæus, the Prince of Astronomers, a Man certainly wonderful, and (as Pliny faith) above the Nature of Mortals. But he was not only most skilful in Astronomy, but in Geometry also; which as many other Things written by him do witness, so especially do the Books of Subtenses: Those of Menelaus