*a ^{2} + b^{2} = c^{2}*

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.

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.

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.

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

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

"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.

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

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.

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.

*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.