Maybe there is an interesting connection between algebra and scissors, and maybe not, but there surely is an interesting connection between squares and right triangles. It was discovered by Pythagoras of Samos, sometime near 500 B.C., and there is a proof of it by Euclid - a proof I never could like well enough to memorize. Fortunately, there are many other ways to prove the Pythagorean Theorem, including some that use algebra and others that use scissors. And it seems that the way we state the theorem just might influence the way we go about trying to prove it.

Here are two statements of the Pythagorean theorem.

• From the Merriam-Webster's Collegiate Dictionary:

  Pythagorean theorem n: a theorem in geometry: the square of the length of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the other two sides.

• From the Oxford English Dictionary:

  Pythagorean proposition or theorem, the 47th of the 1st book of Euclid, namely, that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides: said to have been discovered by Pythagoras.

While the MWCD speaks of the square of the length of ... and reminds us of algebra, the OED speaks of the square on ..., and that reminds us of geometrical objects that are not necessarily defined by measurements and numbers. So here are the outlines of three proofs in a sequence that runs from the very algebraic to the very geometric.

• Algebraic:

  

  This proof relies on a result from geometry: the construction line that is perpendicular to the hypotenuse divides the right triangle in two smaller triangles that are geometrically similar to one another and to the larger one. After that, the theorem is established by strictly alegbraic manipulations of symbols.

• Both:

  

  Here the four right triangles are congruent, and the formulas for areas of squares and rectangles are used in a simpler algebraic argument. If the theorem were restated to fit this proof it might speak of the area of the square on ...

• Geometric:

  

  This one has neither areas nor algebra - just scissors. The proofs that the cuts and moves can be made as shown all come down to showing the congruence of simple geometric figures. The trick was to draw the square on the hypotenuse below it, rather than above it.