BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 289 



the first formulation (1669) of the most comprehensive law of 

 mechanics, the conservation of vis viva. 



Huygens visited England in 1689, but made no use of Newton's 

 new calculus in his published work. In his History of the Mathe- 

 matical Theories of Attraction and the Figure of the Earth, 

 Todhunter says of Huygens : 



To him we owe the important condition of fluid equilibrium, that the 

 resultant force at any point of the free surface must be normal to the 

 surface at that point ; and this has indirectly promoted the knowledge 

 of our subject. But Huygens never accepted the great principle of 

 the mutual attraction of particles of matter ; and thus he contributed 

 explicitly only the solution of a theoretical problem, namely the inves- 

 tigation of the form of the surface of rotating fluid under the action of 

 a force always directed to a fixed point. 



WALLIS AND BARROW. Before attempting to discuss the 

 extraordinary work of Sir Isaac Newton in the whole field of 

 mathematical science a few words should be added concerning two 

 slightly older English mathematicians, John Wallis (1616-1703), 

 Savilian professor at Oxford, and Isaac Barrow (1630-1677), 

 Lucasian professor at Cambridge. 



Wallis in his Arithmetic of The Infinites (1656) developed 

 Cavalieri's summation ideas effectively, employing the new 

 Cartesian geometry and a process equivalent to integration for 

 simple algebraic cases. In particular, he explains negative and 

 fractional exponents in the modern sense, and then proceeds to 

 find the area bounded by OX, the curve y = ax m , and any ordinate 

 x = h, or as we should say, he integrates the function ax m . 

 He develops ingenious methods of interpolation. 



In his Treatise on Algebra he says : 



It is to me a theory unquestionable, That the Ancients had some- 

 what of like nature with our Algebra ; from whence many of their pro- 

 lix and intricate Demonstrations were derived. . . . But this their 

 Art of Invention, they seem very studiously to have concealed : content- 

 ing themselves to demonstrate by Apagogical Demonstrations, (or re- 

 ducing to Absurdity, if denied,) without showing us the method, by 



