Hidden Harmonies Reveals History of a Famous Theorem

January 12, 2011

"The Pythagorean Theorem deserves its own biography," wrote Alan Hirshfeld in a Wall Street Journal piece titled "All Hail the Hypotenuse."

Robert and Ellen Kaplan have responded with Hidden Harmonies: The Lives and Times of the Pythagorean Theorem.

Although born of geometry, the theorem appears in branches of mathematics from number theory to calculus. Demonstrating the theorem’s versatility, the logic- and symbol-filled text can engage all who delight in doing the mathematics. Geometric and algebraic proofs, radian measure, the law of cosines, irrational numbers, Euler's formula, Fourier series—all make appearances in the book.

The authors note that Pythagoras did not create his eponymous assertion. Rather, he provided the first real proof. Tracing its roots to the dawn of civilization and to a host of mathematicians, the authors state that Babylonian scribes left hints of its practicality on clay tablets in 1800 B.C.E. Around the same time, Egyptian surveyors may have used it to estimate areas of tracts flooded by the Nile. On the other hand, Pythagoras, who arrived on scene in the sixth century B. C.E., reveled in number theory, geometry, and the quantitative underpinnings of music.

"The authors place more emphasis on later contributors who re-proved and refined the theorem and subsequently elevated it from the realm of flat Euclidean geometry to that of the curved multidimensional space of Albert Einstein and his contemporaries. The sheer number of published proofs of the Pythagorean Theorem boggles the mind; there were 367 alone in a compilation by early 20th-century professor Elisha Scott Loomis. Einstein himself developed a proof when he was a youngster. So did his near contemporary, Indian mathematician Bhaskara, whose proof involved a rat being pursued by a Pythagorean-inspired hawk," wrote Hirshfeld (University of Massachusetts Dartmouth), who is the author of Eureka Man: The Life and Legacy of Archimedes.

Source: The Wall Street Journal (January 8, 2011)