The Pythagorean Theorem - Educational Outreach

Even though the Cartesian coordinate finds its way intomany calculus problems, this backdrop is not necessary inorder for calculus to function since the primary purpose ofa Cartesian coordinate system is to enhance ourvisualization capability with respect to functional and otheralgebraic relationships. In the same regard, calculus mostdefinitely does not require a metric of distance—as definedby the Distance Formula, another Pythagorean derivate—inorder to function. There are many ways for one to metricizeEuclidean n-space that will lead to the establishment ofrigorous limit and continuity theorems. Table 2.2 lists thePythagorean metric and two alternatives. Reference 19presents a complete and rigorous development of thedifferential calculus for one and two independent variablesusing the rectangular metric depicted in Table 2.2.METRIC SET DEFINITION SHAPEPythagoreanTaxi CabRectangular2( x x0 ) ( y y0)|0y0x x | | y | |02 x x | and y y | |0CircleDiamondSquareTable 2.2 Three Euclidean MetricsLastly, the derivative concept—albeit enhanced viathe geometric concept of slope introduced with a touch ofmetrics—is actually a much broader notion thaninstantaneous “rise over run”. So what mathematicalprinciple may have prompted Elisha Loomis, our early 20 thcentury Ohioan, to discount the methods of calculus as aviable means for proving the Pythagorean Theorem? Onlythat calculus requires geometry as a substrate. The implicitand untrue assumption is that all reality-based geometry isPythagorean. For a realty-based geometric counterexample,the reader is encouraged to examine Taxicab Geometry: anAdventure in Non-Euclidean Geometry by Eugene Krauss,(Reference 20).69