High school students Calcea Johnson and Ne’Kiya Jackson did something in 2023 that most mathematicians had thought impossible: They discovered a way to prove Pythagoras’ theorem using trigonometry that did not depend on circular reasoning. Now, they have published their work in a peer-reviewed academic paper, while also providing another four such proofs.

Pythagoras’ theorem (that the square of the longest side of a right-angled triangle equals the square of the other two sides) predates Pythagoras himself by at least a thousand years. In the time since its discovery, mathematicians have found hundreds of ways to prove its validity. Euclid, Einstein, and even a former US President are credited with contributing new ones We don’t need more proofs to use the theorem to build our cities or send rockets to space – however, finding additional proofs is not just a way for mathematicians to keep their brains sharp or show off. New proofs sometimes lead to advances in our understanding of other areas of mathematics.

These proofs come from several branches of mathematics, but the most comprehensive collection of them claims none come from trigonometry, which has taken the theorem as one of its starting points. Then, in 2023, Ne’Kiya Jackson and Calcea Rujean Johnson changed that. 

The abstract of the talk Johnson and Jackson gave about their discovery started with; “In the 2,000 years since trigonometry was discovered it’s always been assumed that any alleged proof of Pythagoras’s Theorem based on trigonometry must be circular,” As they went on to show, in the talk, this isn’t true.

Such a finding would inevitably have created excitement among the niche group who are delighted by mathematical proofs. It broke into the wider community because the discoverers were both still in high school and had been motivated by a bonus question in a math contest with a $500 prize. Videos reconstructing the work based on some slides from their talk attracted unusual numbers of views for something abstract math-related.

The announcement received some pushback. Some claimed the truth was not trigonometric, others said trigonometric proofs had been made before, and still others that this particular proof was not original. Draw your own conclusions as to whether the hostility had anything to do with the fact that Jackson and Johnson are Black women. 

While others were carping, the audience at the mathematical conference where Jackson and Johnson spoke were encouraging, and the pair submitted their work to a peer-reviewed journal. Before they did so, they went looking for more proofs. 

Remaining critics now need to establish why they are more qualified to assess the work’s originality than the journal’s editors and peer reviewers.

The paper starts by providing a specific proof for the case of an isosceles triangle (two sides the same length). It then details the original proof presented at the conference, which applies to all non-isosceles right-angled triangles, and adds four others also made using trigonometry. The authors demonstrate that none of these are circular, that is, relying on theorems that are in turn built on Pythagoras’ theorem. The fifth proof even captures both isosceles and non-isosceles triangles in one go.

As if that was not enough, the pair also claim to have found five additional proofs using the same basic method, and leave these to the reader to work out.

Jackson and Johnson acknowledge some trigonometric proofs predate their work, but argue most of these are just rewritings of proofs made in other ways. In these cases, they say; “Using trigonometric terminology here adds nothing—in fact it only complicates a simpler view of the same exact approach.” However, they consider two previously published proofs qualify as trigonometric.

Even with that acknowledgment, it seems clear their work represents a major advance in producing trigonometric proofs, which generations of mathematicians have overlooked.

“To have a paper published at such a young age — it’s really mind blowing,” Johnson said in a statement. “It’s very exciting for me, because I know when I was growing up, STEM [science, technology, engineering, and math] wasn’t really a cool thing. So the fact that all these people actually are interested in STEM and mathematics really warms my heart and makes me really excited for how far STEM has come.”

As well as advancing mathematics, Jackson and Johnson make a contribution to mathematics education. The paper includes a sharp criticism of the way trigonometry is taught in schools, noting; “There are two different ways to define the same trigonometric terms.” For those who do not understand the difference; “Trying to make sense of trigonometry can be like trying to make sense of a picture where two different images have been printed on top of each other.”

The authors say their work was possible because they distinguished between these two ways of defining trigonometric expressions, and focussed on one that meets the original definition of the word.

Although the initial work was done while the pair were in high school, contributing to the popular interest, Jackson and Johnson are now at university, studying pharmacy and environmental engineering respectively. They note the challenges of trying to publish one’s first academic paper while also negotiating the adjustment to university life.

“I am very proud that we are both able to be such a positive influence in showing that young women and women of color can do these things, and to let other young women know that they are able to do whatever they want to do. So that makes me very proud to be able to be in that position,” Johnson said,

As excited as their universities must be to have them, Jackson and Johnson may still have something to learn about academia, where promotion is often based on the number of papers published. That leads many academics to break work down into “minimum publishable units” to maximize the number of papers to their name. Still, when your first paper comes out in your teens and gets this much attention, you probably don’t need to make each proof a separate paper to launch your career.

The work is published in the American Mathematical Monthly, whose editor-in-chief, Della Dumbaugh said; “The Monthly is honored and delighted to publish the work of these two students on its pages. This work echoes the spirit of Benjamin Finkel when he founded the Monthly in 1894 to feature mathematics within reach of teachers and students of mathematics.”