The Pythagorean Theorem - Educational Outreach

Whatever the original intent or implication, the PythagoreanProposition has most definitely discouraged the quest forcalculus-based proofs of the Pythagorean Theorem, for theyare rarely found or even mentioned on the worldwide web.This perplexing and fundamental void in elementarymathematics quickly became a personal challenge to searchfor a new calculus-based proof of the Pythagorean Theorem.Calculus excels in its power to analyze changing processesincorporating one or more independent variables. Thus, onewould think that there ought to be something of value inIsaac Newton and Gottfried Leibniz’s brainchild—hailed bymany as the greatest achievement of Western science andcertainly equal to the Pythagorean brainchild—that wouldallow for an independent metrics-free investigation of thePythagorean Proposition.Note: I personally remember a copy of the PythagoreanProposition—no doubt, the 1940 edition—sitting on my father’sbookshelf while yet a high-school student, Class of 1965.Initial thoughts/questions were twofold. Couldcalculus be used to analyze a general triangle as itdynamically changed into a right triangle? Furthermore,could calculus be used to analyze the relationship amongst2 2 2the squares of the three sides A , B , C throughout theprocess and establish the sweet spot of equality2 2 2A B C —the Pythagorean Theorem? Being a lifelongOhioan from the Greater Dayton area personallyhistoricized this quest in that I was well aware of thesignificant contributions Ohioans have made to technicalprogress in a variety of fields. By the end of 2004, a viableapproach seemed to be in hand, as the inherent power ofcalculus was unleashed on several ancient geometricstructures dating back to the time of Pythagoras himself.Figure 2.33, Carolyn’s Cauliflower (so named in honor ofmy wife who suggested that the geometric structure lookedlike a head of cauliflower) is the geometric anchor point fora calculus-based proof of the Pythagorean Theorem.70