The latest mathematical news involves two high school students from New Orleans who have found a new way to prove the Pythagorean Theorem, which of course states that for a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. Now of course there are many proofs of this theorem (the link above gives some), but, according to the Guardian article referenced below (click on screenshot), and the POCIT article below that (also click on screenshot), one form of proof is missing: a proof based on trigonometry. The Guardian article reports that this proof was supplied by Calcea Johnson and Ne-Kiya Jackson from St Mary’s Academy.
Here’s a tweet from the American Mathematical Society:
Click to read the Guardian article:
Now this is somewhat above my pay grade, but perhaps a mathematician will weigh in. The Guardian notes this:
The 2,000-year-old theorem established that the sum of the squares of a right triangle’s two shorter sides equals the square of the hypotenuse – the third, longest side opposite the shape’s right angle. Legions of schoolchildren have learned the notation summarizing the theorem in their geometry classes: a2+b2=c2.
As mentioned in the abstract of Johnson and Jackson’s 18 March mathematical society presentation, trigonometry – the study of triangles – depends on the theorem. And since that particular field of study was discovered, mathematicians have maintained that any alleged proof of the Pythagorean theorem which uses trigonometry constitutes a logical fallacy known as circular reasoning, a term used when someone tries to validate an idea with the idea itself.
Johnson and Jackson’s abstract adds that the book with the largest known collection of proofs for the theorem – Elisha Loomis’s The Pythagorean Proposition – “flatly states that ‘there are no trigonometric proofs because all the fundamental formulae of trigonometry are themselves based upon the truth of the Pythagorean theorem’.”
What is the circularity? To me (and again I may be wrong) it’s instantiated in the section below from the Wikipedia article. The supposed trigonometric “proofs” of the Theorem depends on an identity:
The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions.
The identity is
. . . and that is proven simply from the observation of right triangles and the definition of sines and cosines (see here). Thus, since this equation is derived from definitions and the use of a right triangle, it cannot itself be used to prove the theorem. Solid proofs have depended on arguments from geometry, rearranging similar triangles, algebra, and differentials. What is said to be new here is the proof of the Theorem using trigonometry, but not in a circular way. Because I haven’t seen the students’ proof, I can’t comment on it, and probably don’t have the chops, either. I’ve put one mathematician’s video interpretation below.
A lot of the press has gotten this mixed up, like this article below (click to read):
It says this:
The Pythagorean theorem is a fundamental theorem in trigonometry that describes the relationship between the three sides of a right-angled triangle. It is expressed with the formula a² + b² = c².
The theorem holds true in every plausible example and has been around since the days of the Ancient Greeks.
American mathematician Elisha Loomis argued that no mathematician has been able to establish its truth without using circular logic i.e. without using Pythagorean Theorem. However, others have argued it can be proved using the notion of similar triangles.
Well, that’s a bit misleading, as you can establish the truth of the Pythagorean theorem without using circular logic—so long as the proof is not a trigonometric one. And that’s what the teens are said to have overcome. However, I noticed this morning that an addendum had been added to the article.
This article was corrected on March 28, 2023. The article previously stated that the Pythagorean theorem had yet to be proven without using circular logic. The article has been amended to acknowledge that this argument was posited by American mathematician Elisha Loomis and is not accepted by all mathematicians.
Even so, the correction (which doesn’t itself invalidate what the students did) still leaves out the fact that even if Loomis was wrong, his claim was not that it couldn’t be proven without using circular logic, for it can be. What he said is that it can’t be proven trigonometrically without using circular logic.
Here’s a video which is one person’s interpretation of Johnson and Jackson’s proof based on the slides he saw of their presentation at the AMS meeting:
It’s quite clever, and doesn’t depend on the use of the Pythagorean Identity, merely the definition of “sine”. Note that the person who made this video isn’t 100% sure that he’s presenting Johnson and Jackson’s proof.
So, were they the first people to prove the Pythagorean theorem without trigonometry? I did some Googling, and found this at the site cut-the-knot.org (click to read), which apparently came out before 2019.
It merely reprises another trigonometric proof of the Pythagorean Theorem, one given in the paper below. This is from the article above:
Now, Jason Zimba has showed that the theorem can be derived from the subtraction formulas for sine and cosine without a recourse to sin²α + cos²α = 1. [JAC: the latter is of course the Pythagorean Trigonometric Identity.]
The paper it refers to is this:
J. Zimba, On the Possibility of Trigonometric Proofs of the Pythagorean Theorem, Forum Geometricorum, Volume 9 (2009) 275-278
And I found it online. You can read Zimba’s paper by clicking on the screenshot:
Zimba says he can derive the Pythagorean Trigonometric Identity independently of the Pythagorean Theorem, and thus you can use the former to prove the latter without circularity. He defines sine and cosine without using right angles, and then employs a subtraction procedure to come up with the Pythagorean Identity. That identity then establishes the truth of the Pythagorean Theorem. Here’s a bit of the ending, but it’s a short paper.
The author of the cut-the-knot.org piece says that Zimba’s proof is correct. But what do I know? If it is, then Johnson and Jackson were not the first people to produce a non-circular trigonometric proof of the Pythagorean theorem. But even if that’s the case, if the video gives an accurate take of the two students’ proof, what they did is still remarkably clever.
I suppose mathematicians will weigh in on the Internet in the next few days, so stay tuned.