Read This! The MAA Online book review column Early History of Mathematics by Tom M. Apostol Reviewed by David Graves

The introductory section surveys the origins of mathematics from the primitive need to count, through calendar-making, commercial requirements of measuring and counting, and astronomical measurement, to the Pythagorean abstractions of numbers and geometrical figures ("playthings of the mind") as laying a foundation for subsequent technology. Included is Galileo's observation that the book of nature is written in mathematical language.

Section 2 ("From Euclid to the Seventeenth Century") describes the "Elements" as a turning point in human history, and in swift steps takes us through Hindu-Arabic numerals and the development of algebra, which when introduced into Europe led to the Fermat/Descartes creation of analytic geometry.

Section 3 ("From Scratch Marks to Number Systems") introduces some humor into the narrative, listing uses of numbers as identifying tools (the telephone number of the Caltech bookstore; purported social security numbers of such figures as Apollonius, Bhaskara, and Kepler; the Project MATHEMATICS! zip code), and at other levels (including a Rose Bowl scoreboard showing Caltech leading M.I.T. 38 to 9). Illustrations of various number systems are given: Babylonian, Egyptian, Chinese, Mayan, Greek, Roman, Hindu-Arabic, binary, and bar codes ("modern scratch marks"). The development of arithmetic is outlined, including the tardy acceptance of zero and negative numbers.

Section 4 ("From Numerology to Number Theory") focuses mainly on Pythagorean matters: musical intervals and ratios of frequencies; even/odd and prime/composite classification; perfect numbers; and figurate numbers, including a jump to Fermat's uncovering "one of the deep secrets of number theory" about expressing positive integers as sums of at most three triangular numbers, four squares, five pentagonal numbers, etc.

Section 5 ("The Pythagorean Theorem") represents a transition in style from a nicely-illustrated narrative approach (with absolutely no "talking heads") to some superb animated sequences. The ancient Chinese 3-4-5 argument becomes a general proof through the use of sliding triangles, and Euclid's proof is made to seem easy and natural in about 40 seconds. In between, Pythaogrean triples are defined, and the early Babylonian knowledge of them is illustrated with a quick glimpse of Plimpton 322 (not identified as such).

Section 6 ("A Shocking Discovery") presents another terrific animation, this one showing the irrationality of sqrt(2). The proof (indirect, of course) looks at the smallest isosceles right triangle with integral hypotenuse and sides, then quickly constructs an even smaller one inside. The Pythagoreans' consternation at this unwelcome news is noted, as is Eudoxus' theory of proportions, and for good measure we leap to a relevant result in Hilbert's "Grundlagen."

Section 7 ("Pi Through the Ages") presents two animations on the area of a circle. One cuts the circle into a number of sectors that are rearranged to form a "sort of parallelogram" that becomes more rectangular as the number of sectors is increased. The other divides the circle into a number of concentric rings that are unrolled to form a figure that approximates a right triangle when the number of rings is large. Numerical approximations to Pi are traced from the Babylonian 3.125 through the Egyptian 256/81 and the Archimedean 96-sided polygonal approximations to the Chinese 355/113, and the section concludes by mentioning Lambert's proof of Pi's irrationality.

Section 8 ("From Astronomy to Trigonometry") notes the origins of trigonometry in early astronomical calculations involving chords of assumed circular orbits. Hipparchus' lost work on chords is described as surviving (in part) in Ptolemy's "Almagest." A quick review of modern trigonometric functions is followed by mention of their use in describing various kinds of wave motion.

Section 9 ("From Archimedes to Fermat and Descartes") samples some high points in mathematical development. The geometric achievements of Archimedes are seen as requiring nearly 18 centuries to proceed further, with the priniting press, Hindu-Arabic numerals, a symbol for 0, and decimal notation aiding a "slow but revolutionary change in the development of mathematics." The Italian triumph over cubic and quartic equations spurs acceptance of algebraic language, leading to the numerical and algebraic description of geometric figures in analytic geometry. The importance of the conic sections is illustrated for the parabola with shots of discus, shot put, long jump, and a home run, and for the ellipse with planetary orbits.

The final section ("The Race for the Calculus") sees the revival of the Greek method of exhaustion for estimating areas and volumes as a natural outgrowth of analytic geometry, considers the investigation of velocity and acceleration as leading to the ideas of differential calculus, and gives another fine animated illustration, this time of the fundamental theorem of calculus. Newton and Leibniz are described as ending th calculus race "in a controversial photo finish," and the video concludes with a characterization of calculus becoming "the collective language of science."

The videotape is accompanied by a booklet that sometimes repeats the video narration, sometimes supplements the video with factual or historical material (e.g., about Alexandria), and that includes some exercises for the fifth and sixth sections. Tom Apostol is the guiding spirit behind the project, Al Hibbs skillfully provides the narration, Jim Barry did the sqrt(2) animation, and James Blinn all other animation. The generosity of Caltech alumnus Irving Reed in providing financial support should also be mentioned. This is a great project that is producing lively and useful material for classroom use. I look forward to showing it (and probably re-showing it) the next time I have an opportunity to teach history of mathematics.

Publication Data: Early History of Mathematics, by Tom M. Apostol. Project MATHEMATICS!, 2000. Video, 30 min., $34.90.

David Graves (dgraves@elmira.edu) is Associate Professor of Mathematics at Elmira College, where he is active as a pianist, and has taught courses in cryptology, opera, and history of astronomy as well as the usual run of mathematics courses.

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