Why Did Lagrange "Prove" the Parallel Postulate?

1. 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, 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. In another version of the story, told by Jean-Baptiste Biot, who claims to have been there (though the minutes do not list his name), everybody there could see that something was wrong, so Lagrange's talk was followed by a moment of complete silence [2, p. 84]. Still, Lagrange kept the manuscript with his papers for posterity to read. This episode raises the three questions I will address in this article. First, what did Lagrange actually say in this paper? Second, once we have seen how he " proved " the parallel postulate, why did he do it the way he did? And last, above all, why did Joseph-Louis Lagrange, the consummate analyst, creator of the Analytical Mechanics , of Lagrange's theorem in group theory …