THE HISTORY OF MATHEMATICS 397 



contemporaries. On the other hand, the circulation of knowledge was 

 greatly increased by the possibility of printing old and new work. 

 Books became regular articles of merchandise and could even be picked 

 up in out of the way places. 



The era of Newton is so important in the history of both pure and 

 applied mathematics that no excuse is necessary for dwelling on what 

 was achieved. If an attempt be made to characterize its results in a 

 single sentence, it may be perhaps best emphasized as the epoch of the 

 discovery of the fundamental laws of continuously varying magnitudes. 

 Before this time such solutions of dynamical problems as had been 

 obtained were isolated. Newton's formulation of the laws of motion 

 and proof of the law of gravitation were found in his hands and in 

 those of his successors sufficient to deal with all the problems of physics 

 which were then and later under consideration. Little progress, how- 

 ever, could have been made without the necessary complement, the 

 calculus, which permitted of the study of varying quantities by sym- 

 bolic methods. Rates of change, when uniform, presented little 

 difficulty; the real problem was to deal with them when they were 

 variable, as, for example, in the motion of a pendulum or the vibrations 

 of a string. To a limited degree, the geometry of the straight line and 

 the circle can deal with these questions as Newton showed in the classic 

 translation into geometry of his results for the motions of the moon. 

 But his manuscripts indicate that he failed beyond a certain point to 

 give geometrical proofs of other results obtained by means of the 

 calculus. 



But as I have emphasized before we must not only look to the 

 applications, the chief question in Newton's time, but also point out 

 that varying magnitudes have been studied for their own properties so 

 extensively that they form the larger part of mathematical develop- 

 ments up to the present day. If we except the science of discrete num- 

 bers, it is only in comparatively modern times that discontinuous 

 magnitudes have received any extended study and even these have been 

 advanced in many cases through developments of the calculus. 



Isaac Newton seems to have been one of those very rare cases of 

 genius breaking out without any very obvious stimulus from a par- 

 ticular teacher or school. His first introduction to mathematics was 

 accidental : a book in astrology picked up at a fair in 1661 containing 

 geometry and trigonometry induced him to study Euclid and then to 

 continue by reading such text books as were available. His discover)' 

 of the calculus or "fluxions" as he called it, was made within three 

 years and a half, the binominal theorem was formulated about the same 

 time, and a year later he began his first attempts to prove the gravita- 

 tion law. He was elected Professor of Mathematics in 1669, eight years 



