Can you Prove the Pythagorean Theorem?</t> </head> <body> <font size="-1">Nick Brooke - <em>A History of Mathematical Proof</EM> - <a href="../episode1">Episode 1</a> - <a href="../episode2">Episode 2</a> - <a href="../conclude.html">Conclusions</a></font> <hr> <p align="center"><font size="+3"><b>The Pythagorean Theorem</b></font><br>You proabably know it, but are you able to prove it?</p> <p>Many of the youngest schoolchildren are familiar with “a squared plus b squared is c squared”, even before they even understand its meaning. The <a href="http://en.wikipedia.org/wiki/Pythagoreanism">Pythagoreans</a> knew of the idea as well, although they never would have put it that way; rather, in a right triangle, the sum of the squares of the sides adjacent to the right angle is equal to the square of the hypotenuse. <p>First, we must remember that algebra was not developed until well into the 6th century CE. Any notions of squaring numbers, that is, the length of the sides, to prove the idea was unhelpful to the Pythagoreans. (Struik) They, like the other mathematicians of their times, thought geometrically. “The sum of the squares” refers to the sum of the areas, the shapes themselves, rather than any number representing the area. The most compelling arguments for them involved <a href="http://en.wikipedia.org/wiki/Geometric_construction">geometric construction</a> with a compass and straightedge, dissecting the areas of the squares and rearranging them to show the relationship. <p>Today there are hundreds of known proofs of the <a href="http://en.wikipedia.org/wiki/Pythagorean_Theorem">Pythagorean Theorem</a>, some geometric, algebraic, trigonometric, and even a few using calculus. A famous proof appeared in Euclid’s <em>Elements</em> (Book One, proposition 47), known today as the bride’s chair proof, which, while complete and correct, is a little bit needlessly complicated by modern standards. President James Garfield, a few years before his national election, published a novel geometric proof to the theorem using the geometry of trapezoids. <p><table width="50%" align="center"><td align="center"><img src="chair.gif"><br><font size="-1"><em>The bride's chair proof. You know, if you kind of turn your head sideways and squint, you might be able to make it out. Remember that these are the same people who thought that the Big Dipper looks like a bear.</em></font></td></table> <p>One of most accessible proofs is attributed to <a href="http://en.wikipedia.org/wiki/Bhaskara_II">Bhaskara</A>, a twelfth century Indian mathematician. While Bhaskara was very familiar with algebraic methods (he was in fact the first to generalize the solution to the quadratic equation, something you may remember from high school algebra as the quadratic formula), his famous proof of the Pythagorean theorem is wholly geometric and would have been quickly understood by mathematicians of ancient times. <p><table width="50%" align="center"><td align="center"><img src="Bhaskara.gif"><br><font size="-1"><em>Behold! Oh so the story goes anyway.</em></font></td></table> <p>The apocryphical story goes that Bhaskara presented this figure alone with the caption “Behold!” and left working the reasoning out as an exercise to the reader. Below, though, is a bit of a hint: each figure with matching colors is the same shape. <p align="center"><img src="Bhaskara_colored.gif"></p> <hr> <p>Euclid, translated by Fitzpatrick, Richard. <em>Elements of Geometry</em>. Self-published: 2008. ISBN 978-0615179841. http://farside.ph.utexas.edu/euclid.html. Accessed 9/22/08. <p>Struik, Dirk. <em>A Concise History of Mathematics</em>. 4th ed. Courier Dover Publications: 1987. ISBN 978-0486602554. <p>All public domain figures are from <a href="http://mathworld.wolfram.com/PythagoreanTheorem.html">Wolfram Mathworld</a> and are ineligible for copyright. </body> </html><div style="font-family: Arial, Helvetica, sans-serif; font-size: xx-small; color: #333333; padding-top: 15px;"><a href="http://www.ou.edu">OU Home</a> | <a href="http://www.ou.edu/publicaffairs/WebPolicies/termsofuse.html">Disclaimer</a> | <a href="http://www.ou.edu/publicaffairs/WebPolicies/copyright.html">Copyright</a> | <a href="http://www.ouhsc.edu/eoaa/">Equal Opportunity</a> | <a href="http://www.ou.edu/committees/itc/policy/web