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SCIENCE 



[N. S. Vol. XXXIII. No. 845 



answer to the question of the extent of its 

 influences. Since astronomy is more thou- 

 sands of years old than most of the other 

 natural sciences are centuries, it has nat- 

 urally called forth most of those mathe- 

 matical processes which have been needed 

 in the others. About the only other nat- 

 ural science which has given rise to im- 

 portant mathematical theories is physics, 

 which has forced attention to certain 

 classes of partial differential equations and 

 to the statistical methods employed in the 

 kinetic theory of gases. Another impor- 

 tant advantage astronomy has enjoyed is 

 the delicate character of many of its ob- 

 servations and the high degree of precision 

 of many of its theories. These have nat- 

 urally directed attention to the questions 

 of logical rigor. It is probably known to 

 most of the members of this section that 

 the numerically most perfect theory in all 

 the range of physical science in all time is 

 the lunar theory of our retiring vice-presi- 

 dent. But the mathematics of the continu- 

 ous has not been inspired by astronomy 

 alone, or even by all the physical sciences 

 together. In geometry the questions of 

 tangents and areas have involved the same 

 principles and have given rise to some of 

 the same methods. Consequently we can 

 conclude only that the problems of astron- 

 omy have given rise to some of the theories 

 of the mathematics of the continuous. 



It will perhaps be worth while to descend 

 for a few minutes from the general to the 

 particular, and to consider more concretely 

 what contributions astronomy has actually 

 made to mathematics. It is agreed by all 

 that the invention of the calculus was one 

 of the most important steps ever made in 

 mathematics. It was founded first by 

 Newton and a little later independently by 

 Leibnitz. The work of either was sufficient 

 to open the way to all that which has fol- 

 lowed the invention of this important 

 branch of mathematics. Newton's ideas 



were largely inspired by the consideration 

 of physical phenomena, as is shown by the 

 terminology and notation he used as well as 

 by the problems to which he applied his 

 methods. He spoke of fluents and fluxions 

 and used the time as the independent 

 variable, though he knew this was not es- 

 sential. It simply indicates the stimulus 

 of his ideas. On the other hand, Leibnitz 

 used the terminology of geometry and 

 seemed to have arrived at his ideas of de- 

 rivatives through the consideration of tan- 

 gents to curves. These differences consti- 

 tute an internal evidence of the independ- 

 ence of the work of Newton and of Leib- 

 nitz. 



The history of the application of the cal- 

 culus in the century following its discovery 

 constitutes one of the most glorious records 

 of the achievements of the human mind. 

 Mathematicians had a new method of 

 enormous power and the greatest general- 

 ity, while the laws of motion and the law of 

 gravitation were the keys that imlocked a 

 new universe to them. The work of Clair- 

 aut, d'Alembert, Euler, Lagrange and La- 

 place was one succession of triumphs. 

 "With the enthusiasm of explorers they 

 traversed the worlds Newton and Leibnitz 

 had opened, and with Laplace it was sup- 

 posed the discoveries in them were about 

 exhausted. The point to be emphasized 

 here is that whatever may have been the 

 origin of the calciilus, its evolution into the 

 larger domain of analysis in the century 

 following its invention was due almost en- 

 tirely to the stimulus of physical, and in 

 particular astronomical, problems. There 

 does not seem room for doubt that the very 

 important place which analysis now occu- 

 pies in mathematics is to a large extent due 

 to its applications to astronomy. 



Astronomy not only turned the attention 

 of mathematicians to analysis, but it often 

 determined the precise form their theories 



