MATHEMATICS. This article comprises a compact survey of the development of mathematics from ancient times until the early twentieth century. The treatment is broadly chronological, and most of it is concerned with Europe.

    Unknown Origins
    It seems unavoidable that mathematical thinking played a role in human theorizing from the start of the race, and in various ways. Arithmetic (as the later branch of mathematics became known) would have been one of them, motivated initially by forming integers in connection with counting. But other branches surely include geometry, linked to the appreciation of line, surface, and space; trigonometry, inspired by awareness of angles; mechanics, related to the motion of bodies large and small and the (in)stability of structures; part-whole theory, from consideration of collections of things; and probability, coming from judging and guessing about situations. In all cases the thinking would have started out as very intuitive, gradually becoming more explicit and less particular. 

    Some of the associated contexts would have been provided by study of the environment (such as days) and the heavens (such as new and full moons), which was a major concern of ancient cultures in all parts of the world; in those times mathematics and astronomy were linked closely. For example, the oldest recognized artifact is a bone from Africa, thought to be about thirty-seven thousand years old, upon which phases of the moon seem to have been recorded.

    Among the various ancient cultures, the Babylonians have left the earliest extant evidence of their mathematical practice. They counted with tokens from the eighth millennium; and from the late fourth millennium they expressed numbers and properties of arithmetic in a numeral system to base 10 and handled fractions in expansions of powers of 1/60. Many surviving artifacts seem to relate to education—for example, exercises requiring calculations of unknown quantities, which correspond to the solution of equations but are not to be so identified. They also developed geometry, largely for terrestrial purposes. The Egyptians pursued similar studies, even also finding a formula (not the same one) for the volume of the rectangular base of a pyramid of given sides. They also took up the interesting mathematical problem of representing a fraction as the sum of reciprocals.

    A major mathematical question for these cultures concerned the relationship between circles and spheres and rectilinear objects such as lines and cubes. They involve the quantity that we symbolize by , and ancient evidence survives of methods of approximating to its value. But it is not clear that these cultures knew that the same quantity occurs in all the relationships.

    On Greek Mathematics
    The refinement of mathematics was effected especially by the ancient Greeks, who flourished for about a millennium from the sixth century B.C.E. Pythagoras and his clan are credited with many things, starting with their later compatriots: the eternality of integers; the connection between ratios of integers and musical intervals; the theorem relating the sides of a right-angled triangle; and so on. Their contemporary Thales (c. 625–c. 547 B.C.E.) is said to have launched trigonometry with his appreciation of the angle. However, nothing survives directly from either man.

    A much luckier figure concerning survival is Euclid (fl. c. 300 B.C.E.), especially with his Elements. While no explanatory preface survives, it appears that most of the mathematics presented was his rendition of predecessors’ work, but that (some of) the systematic organization that won him so many later admirers might be his. He stated explicitly the axioms and assumptions that he noticed; one of them, the parallel axiom, lacked the intuitive clarity of the others, and so was to receive much attention in later cultures.

    The Elements comprised thirteen Books: Books 7–9 dealt with arithmetic, and the others presented basic plane (Books 1–6) and solid (Books 11–13) geometry of rectilinear and circular figures. The extraordinary Book 10 explored properties of ratios of smaller to longer lines, akin to a theory of irrational numbers but again not to be so identified. A notable feature is that Euclid confined the role of arithmetic within geometry to multiples of lines (say, “twice this line is . . . ”), to a role in stating ratios, and to using reciprocals (such as 1/5); he was not concerned with lengths—that is, lines measured arithmetically. Thus, he said nothing about the value of, for it relates to measurement.

    The Greeks were aware of the limitations of straight line and circle. In particular, they found many properties and applications of the “conic sections”: parabola, hyperbola, and ellipse. Hippocrates of Chios (fl. c. 600 B.C.E.) is credited with three “classical problems” (a later name) that his compatriots (rightly) suspected could not be solved by ruler and compass alone: (1) construct a square equal in area to a given circle; (2)
    divide any angle into three equal parts; and (3) construct a cube twice the volume of a given one. The solutions that they did find enlarged their repertoire of curves. 

    Among later Greeks, Archimedes (c. 287–212 B.C.E.) stands out for the range and depth of his work. His work on circular and spherical geometry shows that he knew all four roles for ; but he also wrote extensively on mechanics, including floating bodies (the “eureka!” tale) and balancing the lever, and focusing parabolic mirrors. Other figures developed astronomy, partly as applied trigonometry, both planar and spherical; in particular, Ptolemy (late second century) “compiled” much knowledge in his Almgest, dealing with both the orbits and the distances of the heavenly bodies from the central and stationary Earth.

    (Continued .......)


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