MATHEMATICS (Lat, mathematics, front ilk. patlrhawruiri, mathi.matikC, mathematics, from pfifhipa, mat hi'ma, leaming, scienee, f1.0111 pay mantlwacin, to learn). The technical meaning of the word is due to the Pythagoreans, who distinguished four branches: "There art' four degrees of mathematics: arithmetic, mush', geometry, In modern times attempts have frequently been made to frame a satisfac tory definition of the scope of the science. Des cartes asserts that "all sciences which have for their object the search after order and measure ladong to mathematics." D'Alembert in the EttcycioyMic defines it as the science siders the properties of magnitude in so far tis this is calculable or measurable. Comte, in his posit ire. speaks of it as the seienee which proposes to determine certain magnitudes front certain others from the exact relations that exist them. Sagnet has proposed the following: ":\itillientafies have for their object the study of exact and necessary relations con cerning the magnitude, the form. and the relative position of various objects, material or imma terial. which appeal to our senses." With re gard to these definitions it may be observed that they are all based on concepts such as `magni tude,' order,"ineasure.' that are themselves ..x trentely difficult to define.
Alatheniatics as a seience makes its first definite appearance among the Egyptians. There arc evidences of its antiquity among the C'hinese. Hindus. and Babylonians, but the earliest written records of considerable mathe matical progress are found in Egypt. and give all interesting view of the state of the science 89 early as the latter part of the third millennium before Christ. At that time arithinetie was ,?11 111'11.11i ly III.V011t1.11 to include a fair numerical system. a cumbersome but elaborate treatment of eommon fractions, and some work in finite series. A limited and imperfect system of mensuration was known, a beginning was made in a Igebra ic symbolism, and the simple equation was solved.
Of the several mathematical papyri that have come to light in recent years, the most elaborate is that transcribed by Ahmes about n.c. 1700, from one written probalrly some six or eight cen turies earlier. in Egypt, however, made but slight progress beyond this point until the Greek ascendency in Alexandria. The Baby lonians were the next to show signs of mathe matical power, particularly in the application of arithmetic and geometry to astronomy. To
them is due the development of the sexagesimal system of fractions still commonly used in angle and time measurements. The extensive trade of the Plnenicians also developed a commercial arithmetic among them and their neighbors, but it did not lead to any general scientific progress.
The real beginning of mathematics as a stead ily progressing science is to be found in and in particular in the establishing of the Ionian school of Thales about we. 600. Geometry as a science here makes its appearance. The next great step in the progress of mathematics was taken by Pythagoras in founding his famous school at Croton, in Southern Italy. Under his influence a considerable part of elenwntary geom etry became developed, and a beginning was made in creating a theory of numbers. (See NUM n1:.) ConsideralAle progress had been made in geometry before the third epoch-making step was taken, the founding of the Athenian school about n.c. 420. Hippocrates of Chios began the move ment that made Athens the mathematical centre for the next century and a half. It was Plato, hewever, who brought the school to the zenith of its fame. Although he was not, strictly speak ing, a mathematician, his ideas concerning the methods of establishing truths in philosophy and science gave a powerful impulse to the progress of mathematics. The third century n.c. saw the rise of the great Alexandrian school, where Euclid taught, and Archimedes, Apollonins. and Eratosthenes studied. With that century closes the Hellenic ascendency in mathematics and philosophy, and thenceforth we find scientific progress sporadic and short-lived. By the second century of our era progress had practically ceased. Hero and Ptolemy were the greatest of the later Greek writers on applied geometry. The only new movement in mathematics made by the post-Christian Greeks was that of Diophan His, whose work on equations is the first of any pretensions ever composed. The Romans did almost nothing in mathematics except in a pure ly mercantile way, their only contribution being to the practical work of surveying. Among the later Romans the name of the philosopher Boethius stands out with some prominence for his text-book work in elementary mathematics, but he displayed no originality. The same must be said for such mediaeval writers as Alcuin, Gerbert (see SYLVESTER), and Bede.