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.