Page:A History of Mathematics (1893).djvu/271

 chapters of his Institutiones calculi differentialis, and then deduced the differential calculus from it. He established a theorem on homogeneous functions, known by his name, and contributed largely to the theory of differential equations, a subject which had received the attention of Newton, Leibniz, and the Bernoullis, but was still undeveloped. Clairaut, Fontaine, and Euler about the same time observed criteria of integrability, but Euler in addition showed how to employ them to determine integrating factors. The principles on which the criteria rested involved some degree of obscurity. The celebrated addition-theorem for elliptic integrals was first established by Euler. He invented a new algorithm for continued fractions, which he employed in the solution of the indeterminate equation $\scriptstyle{ax+by=c}$. We now know that substantially the same solution of this equation was given 1000 years earlier, by the Hindoos. By giving the factors of the number $$\scriptstyle{2^{2^n}+1}$$ when $\scriptstyle{n=5}$, he pointed out that this expression did not always represent primes, as was supposed by Fermat. He first supplied the proof to "Fermat's theorem," and to a second theorem of Fermat, which states that every prime of the form $$\scriptstyle{4n+1}$$ is expressible as the sum of two squares in one and only one way. A third theorem of Fermat, that $\scriptstyle{x^n+y^n=z^n}$, has no integral solution for values of $$\scriptstyle{n}$$ greater than 2, was proved by Euler to be correct when $\scriptstyle{n=3}$. Euler discovered four theorems which taken together make out the great law of quadratic reciprocity, a law independently discovered by Legendre.[48] Euler enunciated and proved a well-known theorem, giving the relation between the number of vertices, faces, and edges of certain polyhedra, which, however, appears to have been known to Descartes. The powers of Euler were directed also towards the fascinating subject of the theory of probability, in which he solved some difficult problems.