MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

$$\frac{\pi}{4} = 2 \tan^{-1} \frac{1}{3} + \tan^{-1} \frac{1}{7} \;.$$ Is there some geometric construction that explains this beautiful equation (known as Hutton's formula)? Perhaps a "proof without words" figure that makes it self-evident?

Here is Figure 1a from the reference Henry provided:

Nelsen, Roger B. "Proof Without Words: The Formulas of Hutton and Strassnitzky." Mathematics Magazine 86 5 (2013): 350-350.

share|cite|improve this question
Roger Nelsen gave a proof without words in Math Magazine 86 (2013), 350. See – Henry Cohn Apr 12 '14 at 22:12
What a lovely proof! – Lucia Apr 13 '14 at 3:53
up vote 14 down vote accepted

All of these Machin-like formulas have proofs-without-words. First, notice that there is a one line proof using the Gaussian integers: $$(3+i)^2 (7+i) = 50 + 50 i.$$ Taking arguments of both sides proves the result (modulo $2 \pi$).

Now, plot the products $$3 \times 3 \times 7=63,\qquad (3+i) \times 3 \times 7=63+21 i,$$ $$(3+i)^2 \times 7 = 56+42i,\qquad (3+i)^2 (7+i) = 50+50 i.$$ Here the $3$'s and $7$'s are the real parts of $3+i$ and $7+i$.

Each consecutive pair of complex numbers forms a right triangle with the third vertex at $0$: $$(0,63, 63+21i),\ (0, 63+21i, 56+42i),\ (0, 56+42i, 50+50 i).$$ Draw each of those triangles and you have a proof without words.

          (Image added by J.O'Rourke)

share|cite|improve this answer
Thank you, David, this is a beautifully clarifying viewpoint! – Joseph O'Rourke Apr 13 '14 at 16:19
You're welcome! Sorry for the missing scalar factors in the first version; added now. – David Speyer Apr 13 '14 at 17:49

A similar argument was found by pappus on the French forum

enter image description here

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.