Painting - Proof of the Pythagorean Theorem (Euclid)

The Pythagorean theorem states that in any right triangle, the square of the side opposite the right angle (the hypotenuse), is equal to the sum of the squares of the other two sides. This painting depicts the “windmill” figure found in Proposition 47 of Book I of Euclid’s Elements. Although the method of the proof depicted was written about 300 BC and is credited to Euclid, the theorem is named for Pythagoras, who lived 250 years earlier. It was known to the Babylonians centuries before then. However, knowing a theorem is different from demonstrating it, and the first surviving demonstration of this theorem is found in Euclid’s Elements.

Crockett Johnson based his painting on a diagram in Ivor Thomas’s article on Greek mathematics in The World of Mathematics, edited by James R. Newman (1956), p. 191. The proof is based on a comparison of areas. Euclid constructed a square on the hypotenuse BΓ of the right triangle ABΓ. The altitude of this triangle originating at right angle A is extended across this square. Euclid also constructed squares on the two shorter sides of the right triangle. He showed that the square on side AB was of equal area to the rectangle of sides BΔ and Δ;Λ. Similarly, the area of the square on side AΓ was of equal area to the rectangle of sides EΓ and EΛ. But then the square of the hypotenuse of the right triangle equals the sum of the squares of the shorter sides, as desired.

Crockett Johnson executed the right triangle in the neutral, yet highly contrasting, hues of white and black. Each square area that rests on the sides of the triangle is painted with a combination of one primary color and black. This draws the viewer’s attention to the areas that complete Euclid’s proof of the Pythagorean theorem.

Proof of the Pythagorean Theorem, painting #2 in the series, is one of Crockett Johnson’s earliest geometric paintings. It was completed in 1965 and is marked: CJ65. It also is signed on the back: Crockett Johnson 1965 (/) PROOF OF THE PYTHAGOREAN THEOREM (/) (EUCLID).

Currently not on view