 ## Thursday, December 29, 2011

### 153: Cracking Pythagoras

A few days ago I was browsing the infotainment site cracked.com, and saw an amusing article.  It was titled "6 Famous Firsts That You Learned In History Class (And Are Total BS)."  One of these 'famous firsts' was Pythogras's discovery of the Pythagorean Theorem.  As you probably recall, this is the theorem that if the legs of a right triangle are A and B, and the hypotenuse is C, A squared plus B squared equals C squared.  Pythagoras is known for having first proven this theorem around 550 BC.  Is it really the case that this is total BS?

Actually, I wasn't too surprised to see this theorem on such a list.  This theorem is critical for measuring distances and amounts of materials in construction, real estate boundaries, and many other real cases where life doesn't always provide you with straight lines.  Because of this, it was discovered much earlier than Pythagoras-- ancient Babylonians had been using it for at least 1000 years by Pythagoras's time, and there are also references to it in ancient India, China, and Egypt.  Cracked.com even included a photo of an ancient Babylonian stone tablet inscribed with Pythagorean triples, sets of integers such that A squared plus B squared equals C squared.

But there's a huge difference between having observed the theorem experimentally and actually knowing for sure that it will always be true.  The history of math and science is full of examples of supposedly true facts that were later overturned; just ask your doctor if he has any leeches in stock.  If you read your high school math textbook a little more carefully, you'll see that Pythagoras is credited not with discovering the theorem, but with *proving* it.  So if you really want to claim it is illegitimate to cite Pythagoras, you need to supply an earlier proof of the theorem.  The Cracked guys are on their toes though-- they have done this as well, pointing to an earlier Chinese text known as the Chou Pei Suan Ting.

This book does indeed contain a diagram which seems to illustrate a common geometric proof of the theorem.  You can see the picture if you follow the links in the show notes.  The way it works is that you put four copies of the triangle together to form a large square, in such a way that each side contains one of each of the A and B legs of the original triangle.  This forms a large square whose sides are each of length (A+B), and in its center is a smaller square whose side is of length C.  The total area of the large square is (A+B), the quantity, squared, or A squared + 2AB + B squared.  The four triangles are each of area AB/2, using the standard fomula for area of a right triangle, so together their area adds up to 2AB.  But the remaining square in the middle has sides each of length C, with total area C squared.   So we have A squared + 2AB + B squared equals C squared + 2AB--- or A squared + B squared equals C squared!   Does this illustration thus establish that the Chinese proved the theorem before Pythagoras?  Cracked seems to think so.

But not so fast-- there are a few problems with the Chinese illustration.  Most glaringly, it has grid markings that show it is addressing a particular case, of 3-4-5 right triangles, with no clear evidence that it was thought of as a general proof.  There is even some dispute as to whether the diagram was actually part of the original book, or transcribed by later commentators.  While the book was begun as early as 1046 BC, annotations and additions continued until 220 A.D.  And the book is not part of any kind of general document created to supply axiomatic proofs of mathematical discoveries; it's a collection of 246 specific problems encountered by the Duke of Zhou and his astrologer.

But most importantly, we need to keep in mind that the Pythagorean Theorem is not taught as a standalone discovery, a single isolated contribution of the Greeks.  It was one of the beginnings of a systematic approach to mathematics, of not just being satisfied with observations, but of proving hypotheses based on fundamental assumptions, that culminated in Euclid's Elements.  Whether earlier societies knew some of the facts discussed by Greek mathematicians, or even came up with a few isolated proofs, is beside the point.  And if you want to unseat the Greeks as founders of modern mathematics, you need to point out systematic efforts to prove theorems on the basis of fundamental axioms, not grab a few random factoids out of other societies' texts.   I love cracked.com, but I think they truly are cracked on this one.

And this has been your math mutation for today.

• Cracked article
• Chou Pei Suan Ching at Wikipedia
• Pythagoream Theorem at Wikipedia
• Another Pythagorean Theorem article