given in the Relation, was used by Fermat for the decomposition of large numbers into prime factors. (11) If the integers a, b, c represent the sides of a right triangle, then its area cannot be a square number. This was proved by Lagrange. (12) Fermat's solution of ax2+1=y, where a is integral but not a square, has come down in only the broadest outline, as given in the Relation. He proposed the problem to the Frenchman, Bernhard Frenicle de Bessy, and in 1657 to all living mathematicians. In England, Wallis and Lord Brounker conjointly found a laborious solution, which was published in 1658, and also in 1668, in an algebraical work brought out by John Pell. Though Pell had no other connection with the problem, it went by the name of "Pell's problem." The first solution was given by the Hindoos. We are not sure that Fermat subjected all his theorems to rigorous proof. His methods of proof were entirely lost until 1879, when a document was found buried among the manuscripts of Huygens in the library of Leyden, entitled Relation des découvertes en la science des nombres. It appears from it that he used an inductive method, called by him la descente infinie ou indefinie. He says that this was particularly applicable in proving the impossibility of certain relations, as, for instance, Theorem 11, given above, but that he succeeded in using the method also in proving affirmative statements. Thus he proved Theorem 3 by showing that if we suppose there be a prime 4 n + 1 which does not possess this property, then there will be a smaller prime of the form 4n+1 not possessing it; and a third one smaller than the second, not possessing it; and so on. Thus descending indefinitely, he arrives at the number 5, which is the smallest prime factor of the form 4n+1. From the above supposition it would follow that 5 is not the sum of two squares - a conclusion contrary to fact. Hence the supposition is false, and the theorem is established. Fermat applied this method of descent with success in a large number of theorems. By this method Euler, Legendre, Dirichlet, proved several of his enunciations and many other numerical propositions. A correspondence between Pascal and Fermat relating to a certain game of chance was the germ of the theory of probabilities, which has since attained a vast growth. Chevalier de Méré proposed to Pascal the fundamental problem, to determine the probability which each player has, at any given stage of the game, of winning the game. Pascal and Fermat supposed that the players have equal chances of winning a single point. The former communicated this problem to Fermat, who studied it with lively interest and solved it by the theory of combinations, a theory which was diligently studied both by him and Pascal. The calculus of probabilities engaged the attention also of Huygens. The most important theorem reached by him was that, if A has p chances of winning a sum a, and q chances of winning a sum b, then he may expect to ap+bq. The next great work on the theory of p + q win the sum Among the ancients, Archimedes was the only one who attained clear and correct notions on theoretical statics. He had acquired firm possession of the idea of pressure, which lies at the root of mechanical science. But his ideas slept nearly twenty centuries, until the time of Stevin and Galileo. Stevin determined accurately the force necessary to sustain a body on a plane inclined at any angle to the horizon. He was in possession of a complete doctrine of equilibrium. While Stevin investigated statics, Galileo pursued principally dynamics. Galileo was the first to abandon the Aristotelian idea that bodies descend more quickly in proportion as they are heavier; he established the first law of motion; determined the laws of falling bodies; and, having obtained a clear notion of acceleration and of the independence of different motions, was able to prove that projectiles move in parabolic curves. Up to his time it was believed that a cannon-ball moved forward at first in a straight line and then suddenly fell vertically to the ground. Galileo had an understanding of centrifugal forces, and gave a correct definition of momentum. Though he formulated the fundamental principle of statics, known as the parallelogram of forces, yet he did not fully recognise its scope. The principle of virtual velocities was partly conceived by Guido Ubaldo (died 1607), and afterwards more fully by Galileo. Galileo is the founder of the science of dynamics. Among his contemporaries it was chiefly the novelties he detected in the sky that made him celebrated, but Lagrange claims that his astronomical discoveries required only a telescope and perseverance, while it took an extraordinary genius to discover laws from phenomena, which we see constantly and of which the true explanation escaped all earlier philosophers. The first contributor to the science of mechanics after Galileo was Descartes. DESCARTES TO NEWTON. Among the earliest thinkers of the seventeenth and eighteenth centuries, who employed their mental powers toward the destruction of old ideas and the up-building of new ones, ranks René Descartes (1596-1650). Though he professed orthodoxy in faith all his life, yet in science he was a profound sceptic. He found that the world's brightest thinkers had been long exercised in metaphysics, yet they had discovered nothing certain; nay, had even flatly contradicted each other. This led him to the gigantic resolution of taking nothing whatever on authority, but of subjecting everything to scrutinous examination, according to new methods of inquiry. The certainty of the conclusions in geometry and arithmetic brought out in his mind the contrast between the true and false ways of seeking the truth. He thereupon attempted to apply mathematical reasoning to all sciences. "Comparing the mysteries of nature with the laws of mathematics, he dared to hope that the secrets of both could be unlocked with the same key." Thus he built up a system of philosophy called Cartesianism. Great as was Descartes' celebrity as a metaphysician, it may be fairly questioned whether his claim to be remembered by posterity as a mathematician is not greater. His philosophy has long since been superseded by other systems, but the analytical geometry of Descartes will remain a valuable possession forever. At the age of twenty-one, Descartes enlisted in the army of Prince Maurice of Orange. His years of soldiering were years of leisure, in which he had time to pursue his studies. At that time mathematics was his favourite science. But in 1625 he ceased to devote himself to pure mathematics. Sir William Hamilton is in error when he states that Descartes considered mathematical studies absolutely pernicious as a means of internal culture. In a letter to Mersenne, Descartes says: "M. Desargues puts me under obligations on account of the pains that it has pleased him to have in me, in that he shows that he is sorry that I do not wish to study more in geometry, but I have resolved to quit only abstract geometry, that is to say, the consideration of questions which serve only to exercise the mind, and this, in order to study another kind of geometry, which has for its object the explanation of the phenomena of nature. . . . You know that all my physics is nothing else than geometry." The years between 1629 and 1649 were passed by him in Holland in the study, principally, of physics and metaphysics. His residence in Holland was during the most brilliant days of the Dutch state. In 1637 he published his Discours de la Méthode, containing among others an essay of 106 pages on geometry. His Geometry is not easy reading. An edition appeared subsequently with notes by his friend De Beaune, which were intended to remove the difficulties. It is frequently stated that Descartes was the first to apply algebra to geometry. This statement is inaccurate, for Vieta and others had done this before him. Even the Arabs sometimes used algebra in connection with geometry. The new step that Descartes did take was the introduction into geometry of an analytical method based on the notion of variables and constants, which enabled him to represent curves by algebraic equations. In the Greek geometry, the idea of motion was wanting, but with Descartes it became a very fruitful conception. By him a point on a plane was determined in position by its distances from two fixed right lines or axes. These distances varied with every change of position in the point. This geometric idea of co-ordinate representation, together with the algebraic idea of two variables in one equation having an indefinite number of simultaneous values, furnished a method for the study of loci, which is admirable for the generality of its solutions. Thus the entire conic sections of Apollonius is wrapped up and contained in a single equation of the second degree. The Latin term for "ordinate," used by Descartes comes from the expression lineæ ordinate, employed by Roman surveyors for parallel lines. The term abscissa occurs for the first time in a Latin work of 1659, written by Stefano degli Angeli (1623-1697), a professor of mathematics in Rome.3 Descartes' geometry was called "analytical geometry," partly |