History of Mathematics

MATHEMATICS, HISTORY OF. The science of mathe matics had its origin in the practical needs of men for number, names and simple measures, and in their mental thirst for knowl edge. Intuitively the early man knew that a straight line is the shortest path between two points; he took pleasure in symmetric decorations; he developed the names of small groups of objects; in time he felt the need for counting within a very limited range, and he later came to be conscious of the mystery of numbers and to set aside the lower primes as having specially mysterious properties. It was the age of mathematical intuition. It extended throughout the long prehistoric era and characterized the science of the early Chinese, Hindus, Babylonians and Egyptians.

Mathematical Proof.

Simple observations, the induction which leads mankind to draw conclusions from a similarity of results, faith in one's vision and touch—these characterized the world's mathematics down to about the 7th century B.C. Some of the results in Egypt, of a thousand years earlier, have been thought to give evidence of the existence of logical proofs, but the assertion rests on too uncertain foundations to be generally accepted. Thales (c. 64o–c. 546 B.c.) of Miletus, on the Ionian coast of Asia Minor, is credited with having proved five or six theorems in plane geometry. In themselves, they were of the simplest nature, but the fact that they were proved in some kind of a logical manner is so significant as to render the achievement epoch-making. The greatest disciple of Thales was Pythagoras (c. 572–c. 501 B.c.). He taught that mathematics was the basis of all sciences and gave to it a position which it has held sub stantially ever since. It was largely due to his influence that men like Oenopides, Hippocrates of Chios, Antiphon, Archytas, and Theodorus of Cyrene, all of the 5th century B.C., and Plato, Eudoxus, and Aristotle of the century following, were enabled to perfect Greek mathematics and to prepare the way for such later writers as Euclid (c. 30o B.c.), Archimedes (c. 240 B.c.), and Apollonius (c. 225 B.c.). It was the beginning of the epoch of deduction, and the position of mathematics in the scheme of knowledge is due almost entirely to Greek influence.

Number Symbolism.

No ancient civilization had a number symbolism that, for convenience and ability to secure results, approaches the one which we commonly use to-day. The Greeks came nearer to it than other peoples, but it was the Hindu-Arabic system which began to attract the attention of European scholars about A.D. I000, that rendered possible the great advance of non geometric mathematics. (See NUMERALS.) Mediaeval Mathematics.—The Muslim civilization, particu larly as represented at Baghdad, c. 800–c. I000, developed a type

of mathematics which combined the characteristic features of the Greek and Hindu treatments of the science. Eastern faith in astrology and skill in number met with Western faith in philos ophy and skill in geometry, and the Baghdad scholars, absorbing each, produced text books in general algebra, elementary number, astronomy, and trigonometry which, through the efforts of Latin translators, gave new life to mathematics in Europe.

The Period of the Renaissance.—The century found Europe ready to carry on the work of the Arab mathematicians, particularly in algebra and astronomy. The next two centuries saw astronomy more distinctly placed among the exact sciences, largely through the efforts of men like Copernicus (c. 152o) and Kepler (c. 161o) ; saw algebra develop into something besides a set of number puzzles, largely as a result of the labours of such scholars as Tartaglia, Cardan, and Viete; and saw the first steps taken to give the world the disciplines of analytic geometry (Fermat and Descartes, c. 163o-4o), logarithms (Napier and Briggs, c. 1615), and the calculus (Newton and Leibnitz, c. 168o). Modern Mathematics.—The new geometry and the calculus, together with the work of such men as Fermat, Mersenne, and Pascal in the theory of numbers, prepared the way for the re markable advance which has taken place in mathematics in the last two centuries. The applications of the calculus to mechanics and astronomy; the forming of such special branches as differ ential equations; the invention of elliptic functions, hyperbolic trigonometry, descriptive geometry, and modern projective geom etry; the enlarging of the doctrine of probability to include the modern theory of statistics and the development of the theory of least squares; the advance in the theory of numbers since the time of Gauss (c. 1800) ; the development of the non-Euclidean geometries; the invention of new number systems; the creation of the theory of functions with its many ramifications; the en richment of the theory of equations through the introduction of the Galois theory ; the modern study of polynomials; the appli cations of much of this work to the study of electricity, the wave theory, optics, physics, and the nature of the universe; and the laying of more secure foundations of mathematics in its various branches, all this has led to a science of such extent as to make the work of the earlier centuries seem almost insignificant. Certain it is that the 19th and the first part of the loth centuries have shown a development in pure mathematics and its applica tions that promises well for its continued growth.