Pythagoras' Theorem

Pythagoras Theorem asserts that for a right triangle with short sides of length a and b and long side of length c

a2 + b2 = c2

Of course it has a direct geometric formulation.

Click on & move the node to change the shape of the triangle.

For many of us, this is the first result in geometry that does not seem to be self-evident. This has apparently been a common experience throughout history, and proofs of this result, of varying rigour, have appeared early in several civilizations. We present a selection of proofs, dividing roughly into three types, depending on what geometrical transformations are involved.

The oldest known proof

Proofs that use shears (including Euclid's). These work because shears of a figure preserve its area. Some of these proofs use rotations, which are also area-preserving.

Proofs that use translations. These dissect the large square into pieces which one can rearrange to form the smaller squares. Some of these are among the oldest proofs known.

Proofs that use similarity. These are in some ways the simplest. They rely on the concept of ratio, which although intuitively clear, in a rigourous form has to deal with the problem of incommensurable quantities (like the sides and the diagonal of a square). For this reason they are not as elementary as the others.

References

Oliver Byrne, The first six books of the Elements of Euclid, in which coloured diagrams and symbols are used instead of letters for the greater ease of learners, published by Pickering, London, 1847. An extraordinary attempt to convey Euclid's reasoning in pictures. You may view on line a copy of Byrne's title page.

Thomas L. Heath, The thirteen books of Euclid's Elements in three volumes, Dover, 1956. The Elements of Euclid are available on the Internet as are all of Heath's comments.

Elisha Loomis, The Pythagorean Proposition, National Council of Teachers of Mathematics, 1968. This eccentric book was first compiled in 1907, first published in 1928 (at a price of $2.00!), and reissued in this edition. It contains 365 more or less distinct proofs of Pythagoras' Theorem. The total effect is perhaps a bit overwhelming, and the quality of the figures is very poor, but nonetheless there are a few gems distributed throughout.

One man's experience

He was 40 years old before he looked in on Geometry; which happened accidentally. Being in a Gentleman's Library, Euclid's Elements lay open, and 'twas the 47 El. libri I. He read the Proposition. By God, sayd he (he would now and then swear an emphaticall Oath by way of emphasis) this is impossible! So he reads the Demonstration of it, which referred him back to such a Proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps that at last he was demonstratively convinced of that trueth. This made him in love with Geometry.

From the life of Thomas Hobbes in John Aubrey's Brief lives, about 1694.

Another man's experience

At the age of 12 I experienced a second wonder of a totally different nature: in a little book dealing with Euclidean plane geometry, which came into my hands at the beginning of a schoolyear. Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which --- though by no means evident --- could nevertheless be proved with such certainty that any doubt appeared to be out of the question. This lucidity and certainty made an indescribable impression upon me. For example I remember that an uncle told me the Pythagorean theorem before the holy geometry booklet had come into my hands. After much effort I succeeded in ``proving'' this theorem on the basis of the similarity of triangles ... for anyone who experiences [these feelings] for the first time, it is marvellous enough that man is capable at all to reach such a degree of certainty and purity in pure thinking as the Greeks showed us for the first time to be possible in geometry.

From pp. 9-11 in the opening autobiographical sketch of Albert Einstein: Philosopher-Scientist, edited by Paul Arthur Schilpp, published in 1951.

And in war ...

... However, if he wanted eccentric diversion ...

"You mentioned isosceles triangles. Will it do if I prove Pythagoras for you?"

"Jesus," he said. "The square on the hypotenuse. I'll bet you can't."

I did it with a bayonet, on the earth beside my pit - which may have been how Pythagoras himself did it originally, for all I know. I went wrong once, having forgotten where to drop the perpendicular, but in the end there it was ... He followed it so intently that I felt slightly worried; after all, it's hardly normal to be utterly absorbed in triangles and circles when the surrounding night may be stiff with Japanese.

From p. 150 of Quartered safe out here by George MacDonald Fraser, (that's right, the author of the Flashman books) an account of his experiences in the Burmese fighting in 1944-45 against the Japanese. Published by Harper-Collins in 1992.

To understand is to invent

It is not by knowing the Pythagorean Theorem that free exercise of personal reasoning power is assured; it is in having rediscovered its existence and its usage.

From p. 106 of the Penguin edition of To understand is to invent by Jean Piaget.

To love Euclid ...

Euclid brings us grace and beauty and elegance. What does he bring us, boys?

Grace, sir.

And?

Beauty, sir.

And?

Elegance, sir.

Euclid is compete in himself and divine in application. Do you understand that, boys?

We do, sir.

I doubt it boys, I doubt it. To love Euclid is to be alone in this world.

He opens his eyes and sighs and you can see the eyes are a little bleary.

The opening to Chapter VI of Angelas's Ashes by Frank McCourt, Touchstone, 1996.

... or not

He slaved away at Euclid and algebra, and finished up with an abhorrence of Euclid only equalled by his detestation of algebra.

From Conan Doyle by Hesketh Pearson, a biography of the creator of the Sherlock Holmes. Although irrelevant to mathematics, there are a few other quotable lines in the book: To any but the simplest and most easily gulled individual the average professor is about as far from one's conception of a superhuman or an electric force as it is possible for a man to be.

Doyle's attitude towards mathematics was certainly embodied in unusual form: ... Professor Moriarty ... a man of good birth and excellent education, endowed by Nature with a phenomenal mathematical faculty ... had hereditary tendencies of the most diabolical kind (from `The Final Problem' in the Memoirs of Sherlock Holmes).

Mathematics had a terrible time of it among the literati of the late nineteenth century. Mahaffy had suggested in his book that intelligent questions addressed to a pure mathematician would elicit many curious facts that would pleasantly beguile the time. Wilde drew the line at this: `Here, in the interest of Society, we feel bound to enter a formal protest. Nobody, even in the provinces, should ever be allowed to ask an intelligent question about pure mathematics across a dinner table.' This from another biography by Hesketh Pearson, The life of Oscar Wilde, pp. 129-30 of the fourth edition, published by Methuen in 1949.

Squaring the lodge

The 47th proposition of Euclid's first book of the "Elements", also known as "The Pythagorean Theorem", stands as one of Masonry's premier symbols, though it is little discussed and less understood today. That fact is made the more unfortunate, since the 47th proposition may well be the principal symbol and truth upon which Freemasonry is based.

From Circumambulation and Euclid's 47th Proposition, by Reid McInvale, at http://www.io.com/~janebm/summa.html.