The Origin of the "Impossible" Proof Claim

On page 244, Loomis stated the following:

Basically, Loomis claimed that all trigonometric formulae are based on the Pythagorean theorem, and, as a result, one cannot prove the Pythagorean theorem using trigonometry as this leads to circular reasoning.

Whether this is true depends on your point of view. Some believed that because all fundamental formulae can ONLY be proved through the use of the Pythagorean theorem, and, hence using any trigonometric formulae to prove the Pythagorean theorem causes circular reasoning, i.e., citing a theorem to prove the theorem itself. The opposite camp belived that Loomis did not take the ONLY approach. Instead, Loomis only stated a POSSIBILITY and did not rule out that there may be other possible proofs of the fundamental formulae without using the Pythagorean throem. Therefore, using fundamental focumlae in trigonometriy to prove the Pythagorean theorem is possible and does not involve circular reasoning. It is your decision because Loomis passed away many decades ago and no one can interprete his claim in Loomis' way accurately.

Because of Loomis' claim, it is not a surprise that when Ne’Kiya D. Jackson and Calcea Rujean Johnson presented their work at the AMS Spring Southeastern Sectional Meeting the title was An Impossible Proof of Pythagoras.

Do You Believe Loomis' Claim?

The media also reported Jackson-Johnson's 2023 work very extensively. For example, Popular Mechanics's article indicated that proving the Pythagorean theorem using trigonometry was something that mathematicians have never been able to do.

The Guardian said the Jackson and Johnson claimed to "have new proof for 2,000-year-old mathematical theorem." The reporter Ramon Antonio Vargas did not say the proof is "impossible."

Inatead, the reporter said the following, which suggests that academics for two millennia have thought to be impossible.

Scientific American took a more neutral tone suggesting that an approach that SOME once considered impossible and also mentioned Zimba's article which was mentioned by Bogomilny.

By now, you perhaps have realized that the proof of Jackson-Johnson of the Pythagorean theorem using trigonometry is impossible because of Loomis' claim. Due to Jackson-Johnson's proof, we know that Loomis' impossible claim is not true. But, there are several important questions to be answered before the impossible claim can be settled completely. Because of the title of the section NO TRIGONOMETRIC PROOF, I do believe that Loomis took the ONLY approach.

Let me side step a little by showing two quotes. I posted this topic at a QA site in the Geometry and History section and hopefully I could get some help from those who have the understanding, the knowledge and further documentation. People suggested that I post my stuff under the guise of the question and indicated the following:

Then, another people stated this:

The "correctness" is this person's mind seems to be fixed to the "presentation order" of "usual mathematics." This is exactly what I said on the home page of this site. People at the beginning of the 19th century rejected the non-Euclidean geometry because the non-Euclidean geometry is un-usual. A mathematical proposition should be TRUE UNCONDITIONALLY and the condition for a proposition to be true must be stated properly and accurately. If we look at Loomis' statement carefully in a neutral way, no condition was stated and hence one can assume safely that this is a universal statement. As a result, there is no controversy and the discussion of proving the Pythagorean Theorem and the Pythagorean Identity using trigonometry is not a conspiracy theory but a very reasonable and legitimate topic.

All roads lead to Rome. There are different ways going from Euclidean geometry to trigonometry. One may choose to get there using the Pythagorean Theorem, while some others may bypass the use of the Pythagorean Theorem. If you take route A, you should not deny the existence of route B on which other people are using. There are no controversies for people to choose different ways to reach the ultimate goals. This is the beauty of mathematics and in general in science and enginnering. Isn't it? By the way, the unsolvability by radicals of quintic polynomials can be proved using Galois theory which is commonly taught in colleges, but there is a topological proof by Vladimir Arnold using topology and complex analysis (later becoming topological Galois theory). There are many more such examples in mathematics. Consequently, it is just different ways of doing the same thing rather than usual vs. unusual.

Therefore, I conclude this page with the following:

Some Questions to Be Answered

• Apparently, proving the Pythagirean theorem using trigonometry is not very difficult. Why the mathematics community took more than 2,000 years to get Jackson-Johnson's proof?
• Are there other proofs appearing before or long before Jackson-Johnson's proof?
• Given the existing and available documents, can we identity the earliest possible proof or proofs?
• Before 2023, the year Jackson-Johnson's proof appeared, are there any proofs in recent decades? Or, are there similar proofs earlier than Zimba's 2009 proof of the angle difference identities for the sine and cosine functions?