Why Did Lagrange "Prove" the Parallel Postulate?

Abstract

Introduction We begin with an often-told story from the Budget of Paradoxes by Augustus de Morgan: “Lagrange, in one of the later years of his life, imagined” that he had solved the problem of proving Euclid's parallel postulate. “He went so far as to write a paper, which he took with him to the [Institut de France], and began to read it.” But, De Morgan continues, “something struck him which he had not observed: he muttered ‘Il faut que j'y songe encore’ [I've got to think about this some more] and put the paper in his pocket” [ 8 , p. 288]. Is De Morgan's story true? Not quite in that form.But, as Bernard Cohen used to say, “Truth is more interesting than fiction.” First, according to the published minutes of the Institut for 3 February 1806, “M. Delagrange read an analysis of the theory of parallels” [ 25 , p. 314; italics added]. Those present are listed in the minutes: Lacroix, Cuvier, Bossut, Delambre, Legendre, Jussieu, Lamarck, Charles, Monge, Laplace, Hauy, Berthollet, Fourcroy—a most distinguished audience! Furthermore: Lagrange did not throw his manuscript away. It survives in the library of the Institut de France [ 32 ]. There is a title page that says, in what looks to me like Lagrange's handwriting, “On the theory of parallels: memoir read in 1806,” together with the signatures of yet more distinguished people: Prony and Poisson, along with Legendre and Lacroix. The first page of text says, again in Lagrange's handwriting, that it was “read at the Institut in the meeting of 3 February 1806.” It is true that Lagrange never did publish it, so he must have realized there was something wrong.