94 GREAT MEN OF SCIENCE 



atmosphere, it was necessary to undertake a similar division 

 and execute a large number of single calculations, and this 

 could only really be done satisfactorily to any desired degree 

 of accuracy, if the division had been carried to an infinite 

 number of infinitely small parts. Newton recognised that 

 as we proceed to a finer and finer division, the relationship 

 between two corresponding magnitudes, such as distance 

 and time, approaches a fixed limit, although the magnitudes 

 themselves, taken singly, continually become smaller. The 

 essential point of these problems is always of this nature. 

 His special mathematical studies, which we cannot discuss 

 here, but which in themselves were fruitful in results, 

 enabled him to propound rules for the direct calculation 

 of such limiting values, in the case when the division becomes 

 infinitely small. This is the fundamental idea of Newton's 

 calculus of fluxions, which gave him a correct solution of 

 many mathematical problems hitherto only soluble with 

 great difficulty, or not at all.^ 



The Principia is not by any means confined to the 

 mechanics of the heavens. It develops and increases our 

 whole knowledge of nature quite generally, and in a fashion 

 fundamental for a long time subsequently, so that many 

 later achievements in single departments appear unimportant 

 by comparison, since they did not bring anything essentially 

 new, but only refined upon what was already known. In 

 particular, the fundamental lines of the phenomena of motion 

 in liquids and gases (hydrodynamics and aerodynamics), 

 in which department only Toricelli's law existed, were 

 developed almost exhaustively by Newton as regards 

 characteristic single cases.* He investigates phenomena of 

 flow and formation of eddies in liquids, oscillations of liquids 



^ The method used by Newton for this purpose is not given in detail 

 in the Principia. Since our present-day, easily learned, method of calcu- 

 lation was only later fully developed by Leibniz, what Newton managed 

 to achieve at that time is the more astonishing. 



^ Principia, book 2. 



