The Pythagorean Theorem - Educational Outreach

2.10] Ohio and the Elusive Calculus ProofElisha Loomis (1852-1940), was a Professor ofMathematics, active Mason, and contemporary of PresidentGarfield. Loomis taught at a number of Ohio colleges andhigh schools, finally retiring as mathematics departmenthead for Cleveland West High School in 1923. In 1927,Loomis published a still-actively-cited book entitled ThePythagorean Proposition, a compendium of over 250 proofsof the Pythagorean Theorem—increased to 365 proofs inlater editions. The Pythagorean Proposition was reissued in1940 and finally reprinted by the National Council ofTeachers of Mathematics in 1968, 2 nd printing 1972, as partof its “Classics in Mathematics Education” Series.Per the Pythagorean Proposition, Loomis is creditedwith the following statement; there can be no proof of thePythagorean Theorem using either the methods oftrigonometry or calculus. Even today, this statementremains largely unchallenged as it is still found with sourcecitation on at least two academic-style websites 1 . Forexample, Jim Loy states on his website, “The book ThePythagorean Proposition, by Elisha Scott Loomis, is a fairlyamazing book. It contains 256 proofs of the PythagoreanTheorem. It shows that you can devise an infinite number ofalgebraic proofs, like the first proof above. It shows that youcan devise an infinite number of geometric proofs, likeEuclid's proof. And it shows that there can be no proofusing trigonometry, analytic geometry, or calculus. Thebook is out of print, by the way.”That the Pythagorean Theorem is not provable usingthe methods of trigonometry is obvious since trigonometricrelationships have their origin in a presupposedPythagorean right-triangle condition. Hence, any proof bytrigonometry would be a circular proof and logically invalid.However, calculus is a different matter.1 See Math Forum@ Drexel,http://mathforum.org/library/drmath/view/6259.html ;Jim Loy website,http://www.jimloy.com/geometry/pythag.htm .68