established by Euler. He invented a new algorithm for continued fractions, which he employed in the solution of
the indeterminate equation . We now know that substantially the same solution of this equation was given years earlier, by the Hindoos. By giving the factors of the number when , 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 is expressible as the sum of two squares in one and only one way. A third theorem of Fermat, that , has no integral solution for values of greater than , was proved by Euler to be correct when . 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.
Of no little importance are Euler's labours in analytical mechanics. Says Whewell: "The person who did most to
give to analysis the generality and symmetry which are now its pride, was also the person who made mechanics analytical; I mean Euler."11 He worked out the theory of the rotation of a body around a fixed point, established the general equations of motion of a free body, and the general equation of hydrodynamics. He solved an immense number and variety of mechanical problems, which arose in his mind on all occasions. Thus, on reading Virgil's lines, "The anchor drops, the rushing keel is staid," he could not help inquiring what would be the ship's motion in such a case. About the same time as Daniel Bernoulli he published the Principle of the Conservation of
Areas and defended the principle of "least action," advanced
by Maupertius. He wrote also on tides and on sound.
Astronomy owes to Euler the method of the variation of
arbitrary constants. By it he attacked the problem of perturbations, explaining, in case of two planets, the secular variations of eccentricities, nodes, etc. He was one of the first to take up with success the theory of the moon's motion by giving approximate solutions to the "problem of three bodies."
He laid a sound basis for the calculation of tables of the moon. These researches on the moon's motion, which captured two prizes, were carried on while he was blind, with the assistance of his sons and two of his pupils.
Most of his memoirs are contained in the transactions of the Academy of Sciences at St. Petersburg, and in those of the Academy at Berlin. From 1728 to 1783 a large portion of the Petropolitan transactions were