BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 301 



LEIBNITZ. Newton's great contemporary and scientific rival 

 Leibnitz has been called the Aristotle of the seventeenth century. 

 Born in 1646 at Leipsic, he took his doctor's degree at 20 and was 

 immediately offered a university professorship. Alchemy, diplo- 

 macy, philosophy, mathematics, all shared his energetic attention. 

 In the last he invented a calculating machine and the differential 

 calculus. In regard to the machine Klein says: 



merely the formal rules of computation are essential, for only these 

 can be followed by the machine. It cannot possibly have an intuitive 

 conception of the meaning of the numbers. It is thus no accident that 

 a man so great as Leibnitz was both father of purely formal mathe- 

 matics and inventor of the first calculating machine. 



He became librarian at Hannover, founded the Academy of 

 Sciences in Berlin, and was instrumental in the organization of 

 similar bodies in St. Petersburg, Dresden, and Vienna. His ad- 

 vanced ideas on education may be inferred from his remark 



We force our youths first to undertake the Herculean labor of 

 mastering different languages, whereby the keenness of the intellect 

 is often dulled, and condemn to ignorance all who lack knowledge of 

 Latin. 



But for the overshadowing genius of Newton, Leibnitz' service 

 to the progress of science would have been even greater than it 

 actually was. Comparing their work in mathematics where their 

 competition was keenest, it should be appreciated that while New- 

 ton's work in mathematical science was incomparably greater in 

 range, it was Leibnitz who gave to the differential calculus the 

 better form and notation out of which our own has grown. 



It appears that Fermat, the true inventor of the differential cal- 

 culus, considered that calculus as derived from the calculus of finite 

 differences by neglecting infinitesimals of higher orders as compared 

 with those of a lower order . . . Newton, through his method of 

 fluxions, has since rendered the calculus more analytical, he also sim- 

 plified and generalized the method by the invention of his binomial 

 theorem. Leibnitz has enriched the differential calculus by a very 

 happy notation. Laplace. 



