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That EUCLID taught mathematics in the school of Alexandria under the first of the Ptolemies, is all that is known with certainty of his life. Pappus speaks emphatically of his friendliness to other students of mathematics, contrasting him in this respect, rightly or wrongly, with Apollonius. Euclid wrote on several mathematical subjects, notably on the Data for determining the possibility of a problem, and on Conic Sections; but his work on elementary mathematics, which has had the singular fortune, in this country at least, to identify a writer with the science of which he treats, can alone be here considered.
Broadly speaking, this work consists of four divisions. The first, which includes the first six books, treats of such plane figures as can be described with rule and compass; dealing, first with equal magnitudes, subsequently with magnitudes that are unequal but similar. The second book, establishing equations, and the 5th and 6th dealing with proportions, may be regarded as containing the essential principles of Algebra. The second part, including the 7th, 8th, and 9th books, is a treatise on arithmetic. The third part, corresponding to the 10th book, discusses incommensurable magnitudes. The fourth division, including the 11th, 12th, and 13th books, discusses the geometry of solids. The books called 15th and 16th are by a later writer, probably Hypsicles.
Euclid was a compiler and arranger, not a discoverer. Thales had discovered the constancy in the angles of all triangles, and the proportionality of sides where the triangles were similar. The school of Pythagoras studied the regular solids, and the relation of their bounding lines and surfaces. This led them to investigate proportion, and especially the famous problem of the Golden Section (Euc. Ele. ii. 11) leading to the discussion of incommensurable magnitudes, and of the duplication of the cube; this last involving the insertion of two mean proportionals, a problem which Euclid's Elements do not enable us to solve. In these researches they evolved the analytical method, erroneously attributed to Plato. Archytas, the last of the Pythagoreans, was the tutor of Eudoxus, who is expressly stated by Archimedes to be the discoverer of certain proportions on the similarity of solids (Euc. Ele. xii. 7 and 10), which imply possession of the Method of Exhaustions (Euc. Ele. x. 1). Euclid is spoken of by Proclus as the arranger of many works of Eudoxus and the completer of many of Theætetus. These latter were on incommensurables, discussed by Euclid in his 10th book--the one therefore in which he has the strongest claims to originality.
Though the claims of Euclid as a discoverer are thus reduced to narrow limits, his function as a systematic arranger of mathematical truths was one of great importance. Doubtless exaggerated value has been attributed to it. The degeneracy of philosophic speculations in his time led to circuitous demonstrations of much that might have been more shortly stated; and most mathematicians are agreed that his arrangement admits of improvement. But as an historical monument of the claims established by geometry in the fourth century B.C. to be an instrument of education, the Elements of Euclid will remain forever memorable.
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| This biography is
reprinted from The New Calendar of Great Men. Ed. Frederic
Harrison. London: Macmillan and Co., 1920. |
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