him Euclid's Elements, which he, without assistance, mastered easily. His regular studies being languages, the boy employed only his hours of amusement on the study of geometry, yet he had so ready and lively a penetration that, at the age of sixteen, he wrote a treatise upon conics, which passed for such a surprising effort of genius, that it was said nothing equal to it in strength had been produced since the time of Archimedes. Descartes refused to believe that it was written by one so young as Pascal. This treatise was never published, and is now lost. Leibniz saw it in Paris and reported on a portion of its contents. The precocious youth made vast progress in all the sciences, but the constant application at so tender an age greatly impaired his health. Yet he continued working, and at nineteen invented his famous machine for performing arithmetical operations mechanically. This continued strain from overwork resulted in a permanent indisposition, and he would sometimes say that from the time he was eighteen, he never passed a day free from pain. At the age of twenty-four he resolved to lay aside the study of the human sciences and to consecrate his talents to religion. His Provincial Letters against the Jesuits are celebrated. But at times he returned to the favourite study of his youth. Being kept awake one night by a toothache, some thoughts undesignedly came into his head concerning the roulette or cycloid; one idea followed another; and he thus discovered properties of this curve even to demonstration. A corre

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spondence between him and Fermat on certain problems was the beginning of the theory of probability. Pascal's illness increased, and he died at Paris at the early age of thirty-nine years. By him the answer to the objection to Cavalieri's Method of Indivisibles was put in the clearest form. Like Roberval, he explained "the sum of right lines" to mean "the sum of infinitely small rectangles." Pascal greatly advanced

the knowledge of the cycloid. He determined the area of a section produced by any line parallel to the base; the volume generated by it revolving around its base or around the axis; and, finally, the centres of gravity of these volumes, and also of half these volumes cut by planes of symmetry. Before publishing his results, he sent, in 1658, to all mathematicians that famous challenge offering prizes for the first two solutions of these problems. Only Wallis and A. La Louère competed for them. The latter was quite unequal to the task; the former, being pressed for time, made numerous mistakes: neither got a prize. Pascal then published his own solutions, which produced a great sensation among scientific men. lis, too, published his, with the errors corrected. Though not competing for the prizes, Huygens, Wren, and Fermat solved. some of the questions. The chief discoveries of Christopher Wren (1632-1723), the celebrated architect of St. Paul's Cathedral in London, were the rectification of a cycloidal arc and the determination of its centre of gravity. Fermat found the area generated by an arc of the cycloid. Huygens invented the cycloidal pendulum.

Wal

The beginning of the seventeenth century witnessed also a revival of synthetic geometry. One who treated conics still by ancient methods, but who succeeded in greatly simplifying many prolix proofs of Apollonius, was Claude Mydorge in Paris (1585-1647), a friend of Descartes. But it remained for Girard Desargues (1593-1662) of Lyons, and for Pascal, to leave the beaten track and cut out fresh paths. They introduced the important method of Perspective. All conics on a cone with circular base appear circular to an eye at the apex. Hence Desargues and Pascal conceived the treatment of the conic sections as projections of circles. Two important and beautiful theorems were given by Desargues: The one is on the "involution of the six points," in which a transversal

meets a conic and an inscribed quadrangle; the other is that, if the vertices of two triangles, situated either in space or in a plane, lie on three lines meeting in a point, then their sides meet in three points lying on a line; and conversely. This last theorem has been employed in recent times by Branchion, Sturm, Gergonne, and Poncelet. Poncelet made it the basis of his beautiful theory of homoligical figures. We owe to Desargues the theory of involution and of transversals; also the beautiful conception that the two extremities of a straight line may be considered as meeting at infinity, and that parallels differ from other pairs of lines only in having their points of intersection at infinity. Pascal greatly admired Desargues' results, saying (in his Essais pour les Coniques), "I wish to acknowledge that I owe the little that I have discovered on this subject, to his writings." Pascal's and Desargues' writings contained the fundamental ideas of modern synthetic geometry. In Pascal's wonderful work on conics, written at the age of sixteen and now lost, were given the theorem on the anharmonic ratio, first found in Pappus, and also that celebrated proposition on the mystic hexagon, known as "Pascal's theorem," viz. that the opposite sides of a hexagon inscribed in a conic intersect in three points which are collinear. This theorem formed the keystone to his theory. He himself said that from this alone he deduced over 400 corollaries, embracing the conics of Apollonius and many other results. Thus the genius of Desargues and Pascal uncovered several of the rich treasures of modern synthetic geometry; but owing to the absorbing interest taken in the analytical geometry of Descartes and later in the differential calculus, the subject was almost entirely neglected until the present century.

In the theory of numbers no new results of scientific value had been reached for over 1000 years, extending from the

times of Diophantus and the Hindoos until the beginning of the seventeenth century. But the illustrious period we are now considering produced men who rescued this science from the realm of mysticism and superstition, in which it had been so long imprisoned; the properties of numbers began again to be studied scientifically. Not being in possession of the Hindoo indeterminate analysis, many beautiful results of the Brahmins had to be re-discovered by the Europeans. Thus a solution in integers of linear indeterminate equations was re-discovered by the Frenchman Bachet de Méziriac (15811638), who was the earliest noteworthy European Diophantist. In 1612 he published Problèmes plaisants et délectables qui se font par les nombres, and in 1621 a Greek edition of Diophantus with notes. The father of the modern theory of numbers is Fermat. He was so uncommunicative in disposition, that he generally concealed his methods and made known his results only. In some cases later analysts have been greatly puzzled in the attempt of supplying the proofs. Fermat owned a copy of Bachet's Diophantus, in which he entered numerous marginal notes. In 1670 these notes were incorporated in a new edition of Diophantus, brought out by his son. Other theorems on numbers, due to Fermat, were published in his Opera varia (edited by his son) and in Wallis's Commercium epistolicum of 1658. Of the following theorems, the first seven are found in the marginal notes:

(1) x2 + y2 = 2" is impossible for integral values of x, y, and z, when n>2. Remark: "I have found for this a truly wonderful proof, but the margin is too small to hold it." Repeatedly was this theorem made the prize question of learned societies. It has given rise to investigations of great interest and difficulty on the part of Euler, Lagrange, Dirichlet, and Kummer.

(2) A prime of the form 4 n+1 is only once the hypothenuse

of a right triangle; its square is twice; its cube is three times, etc. Example: 52 = 32 + 42; 252 = 152 + 202 = 72 + 242; 1252 = 752 + 1002 = 352 + 1202 = 442 + 1172.

(3) A prime of the form 4 n + 1 can be expressed once, and only once, as the sum of two squares. Proved by Euler.

(4) A number composed of two cubes can be resolved into two other cubes in an infinite multiplicity of ways.

(5) Every number is either a triangular number or the sum of two or three triangular numbers; either a square or the sum of two, three, or four squares; either a pentagonal number or the sum of two, three, four, or five pentagonal numbers; similarly for polygonal numbers in general. The proof of this and other theorems is promised by Fermat in a future work which never appeared. This theorem is also given, with others, in a letter of 1637(?) addressed to Pater Mersenne.

(6) As many numbers as you please may be found, such that the square of each remains a square on the addition to or subtraction from it of the sum of all the numbers.

(7) x1 + y1 = z2 is impossible.

(8) In a letter of 1640 he gives the celebrated theorem generally known as "Fermat's theorem," which we state in Gauss's notation: If p is prime, and a is prime to p, then ap−1 = 1 (mod p). It was proved by Euler.

(9) Fermat died with the belief that he had found a longsought-for law of prime numbers in the formula 22"+ 1 = a prime, but he admitted that he was unable to prove it rigorously. The law is not true, as was pointed out by Euler in the example 225+1 = 4,294,967,297 6,700,417 times 641.

=

The

American lightning calculator Zerah Colburn, when a boy, readily found the factors, but was unable to explain the method by which he made his marvellous mental computation. (10) An odd prime number can be expressed as the difference of two squares in one, and only one, way. This theorem,