The Pythagorean Theorem: a Wonder for all Ages

  The Pythagorean Proposition:  A Theorem for all Ages  C  onsider old Pythagoras, A Greek of long ago, And all that he did give to us,Three sides whose squares now show In houses, fields and highways straight;In buildings standing tall;In mighty planes that leave the gate; And, micro-systems smallYes, all because he got it right When angles equal ninety—One geek (BC), his plain delight—One world changed aplenty!  January 2002  1) Pythagoras and the First Proof Pythagoras was not the first in antiquity to know about theremarkable theorem that bears his name, but he was the first toformally prove it using deductive geometry and the first to actively‘market’ it (using today’s terms) throughout the ancient world Oneof the earliest indicators showing knowledge of the relationshipbetween right triangles and side lengths is a hieroglyphic-stylepicture, Figure 1 , of a knotted rope having twelve equally spacedknots Figure 1: Egyptian Knotted Rope, Circa 2000 BCE 1  The rope was shown in a context suggesting its use as a worker’stool for creating right angles, done via the fashioning of a 3-4-5right triangle Thus, the Egyptians had a mechanical device for demonstrating the converse of the Pythagorean Theorem for the3-4-5 special case: 0222 90543 =⇒=+ γ   Not only did the Egyptians know of specific instances of the Pythagorean Theorem, but also the Babylonians and Chinesesome 1000 years before Pythagoras definitively institutionalizedthe general result circa 500 BCE And to be fair to the Egyptians,Pythagoras himself, who was born on the island of Samos in 572BCE, traveled to Egypt at the age of 23 and spent 21 years thereas a student before returning to Greece While in Egypt,Pythagoras studied a number of things under the guidance of Egyptian priests, including geometry Table 1 briefly summarizeswhat is known about the Pythagorean Theorem beforePythagoras DateCulturePersonEvidence 2000 BCEEgyptianUnknownWorkman’s rope for fashioning a3-4-5 triangle1500 BCEBabylonian CaldeanUnknownRules for right triangleswritten on clay tabletsalong with geometricdiagrams1100 BCEChineseTschou-GunWritten geometriccharacterizations of right angles520 BCEGreekPythagorasGeneralized result anddeductively proved Table 31: Prior to Pythagoras The proof Pythagoras is thought to have actually used isshown in Figure 2 It is a visual proof in that no algebraiclanguage is used to support numerically the deductive argument 2  In the top diagram, the ancient observer would note that removingthe eight congruent right triangles, four from each identical master square, brings the magnificent sum-of-squares equality intoimmediate view Figure 2: The First Proof by PythagorasFigure 3 is another original, visual proof attributed toPythagoras Modern mathematicians would say that this proof ismore ‘elegant’ in that the same deductive message is conveyedusing one less triangle Even today, ‘elegance’ in proof ismeasured in terms of logical conciseness coupled with the amountinsight provided by the conciseness ⇓ + 3