Adrien-Marie Legendre (caricature above) is best known as the author of Éléments de géométrie, which was published in 1794 and was the leading elementary text on the topic for around 100 years.
John Playfair, according to John A. Parker’s book, The Quadrature of the Circle (1874 edition), was co-author with Legendre of the widespread book, Elements of Geometry (Éléments de géométrie).
Something called “the Playfair/Legendre method” creeps into some of the Propositions contained in Parker’s book on “squaring the circle.” Hence a slight detour here into a possible glimpse of the aforementioned “Playfair/Legendre method.”
“Foremost among those who have thrown discredit on the pursuit of the quadrature,” wrote Parker, “has been Legendre, the eminent French geometer; and I confess to an abundant surprise at finding, that the professors of our own day and in our own country [the United States] particularly, have received what Legendre and a few others have said as established facts, and have adopted their opinions without investigation.”
It is the old case of the “we now know” – except that we do not “now know”! In the strange case of the Cardiff Giant, the “we now know” crowd endlessly clucks that it was a hoax – except it wasn’t a hoax! (Details in “Cardiff Giant Lives!”, Ersjdamoo’s Blog entry of August 30, 2012) And also we find these dumb clucks who will drone about the impossibility of squaring the circle. As long as we are not limited to straightedge and compass in attempting a solution, the truth is, we do not “now know.”
One such dumb cluck mentioned by Parker was a “distinguished professor of one of our own distinguished colleges.” Parker had sent the distinguished professor some of his early papers dealing with the quadrature of the circle. The distinguished professor wrote back, and referred Parker to Legendre’s note on squaring the circle in his Elements of Geometry. “Ahem,” wrote the distinguished professor, “after seeing what Legendre has said, I presume you will think no more about the quadrature.”
Ah-hah! Got you there, Parker! Legendre has decreed against the subject, and all good ducks must line up in a row!
“About the era of the publication of Legendre’s work [Elements of Geometry], the Academy of Science in Paris, instigated perhaps by Legendre himself, passed a resolution, that, in order to discourage such futile attempts, the Academy would not thereafter receive or consider any paper purporting to be on the subject of the quadrature! And within a few years thereafter the Royal Society of London passed a similar resolution!! And under the influence of these tyrannical proceedings it soon became disreputable in learned circles for any one to talk of finding the quadrature.” (Parker, Quadrature of the Circle)
We do not know. “The only demonstration ever made or ever pretended to be made, by any body , of the impossibility of finding the quadrature, is only a demonstration that the thing is impossible to be done by means of straight lines alone…” (Ibid.)
In spite of the clucking gallery of naysayers, Parker was sustained in his efforts to square the circle by “the opinions of all ancient geometers, that the quadrature of the circle lies at the foundation of all geometry, and hence it is an elementary truth, necessary to be known for the perfection of mathematical sciences.” (Ibid.)