PRESIDENTIAL ADDRESS SECTION A. 4I 



To mention one other fruitful idea contained in Newton's 

 Corollaries to his Laws — the misnamed " parallelogram " of 

 forces. This everyday j)rinci])le of mechanics is an immediate 

 deduction from the principle of the iiidcpouioicc of forces acting 

 simultaneously, and attempts to establish it on purely statical 

 principles are unsatisfactory. We realize it nowadays as having 

 its origin deeper even than the laws of force: it is the law of 

 addition of " vectors '" — that is, if AB, BC represent any two 

 " directed " quantities — any two similar conce])ts involving only 

 magnitude and direction in space — then the two combined (added 

 in the simplest sense) are rei)resented by the short-cut AC. 



One word on the Calculus. Up to Newton's time, mathe- 

 matical analysis, such as it was — i.e., symbolical arithmetic and 

 algebra — lacked an essential qualilication for a language of 

 Science: it was essentially discrete; coiifiiiiilfx was inexpressible 

 by means of it. It has been aptly suggested that the Greeks, 

 subtle, powerful intellects as they were, expressed their clearest, 

 most scientific reasoning wholly in geometry ( leaving " analysis " 

 to be devised by the Hindoos and .\rabs), because of this lack 

 of continuity in analytical language. They loved numbers on 

 their own account, but as a separate subject of thought, uncon- 

 nected with space and time. When Newton, with that character- 

 istic reluctance of the highest minds to publicity, ultimately 

 consented to give his ideas to the world, he chose to present 

 it in the old Greek geometrical form, with all the beauty of a 

 complete picture, as contrasted with the business-like methods of 

 a tape machine. Yet it is sujjposed that he obtained his results 

 more as we do now, by the use_ of Descartes' co-ordinates and 

 the methods of the Calculus, which he himself invented. Thi^ 

 preference of Newton's for geometrical presentation 'had a 

 curious result. His disciples in England clung to his prejudice : 

 and so, though possessing fine intellects like Maclaurin, had 

 little influence on the progress of Science, while Lagrange. 

 Laplace and Legendre, the great French trio, and Euler and 

 Gauss, filled a whole century with masterpieces of efifective 

 reasoning on the applications of the Law of Gravitation to the 

 solar system and the wider universe of matter. 



The dominating idea of the Laws of Motion is " rate of 

 change." No one following this reasoning can fail to feel the 

 need of some simple symbolism to express this idea. Newton 

 did it by placing a dot over the symbol that expressed the chang- 

 ing thing; but he seems to have tised it only in a mechanical 

 .sense for change in time. Leibnitz, who invented the system 

 independently, seems to have approached it mathematically and 

 purely symbolically, with no special reference to time ; he and 

 his followers applied the idea to every mathematical expression 

 in their vocabulary — to every conceivable fuuction, as we say^ — 

 and so built up a mass of results, which were essential when 

 the mechanical reasoning took, as it must, the form of differen- 

 tial e(|uations needing the inverse process for their solution. 



It has l)een said that " Nature expresses herself in differen- 



