130 SCIENCE AND THE HUMAN MIND 



with what lies beyond, and the clearer are the gaps 

 within it. 



One of the immediate results of the application of 

 mathematical mechanics to the problems of astro- 

 nomy was the need of improvement in the mathe- 

 matical tools used in the researches. Hence the same 

 period which saw the labours of Kepler, Galileo, 

 Huygens and Newton was marked also by a great 

 increase in mathematical knowledge and skill. 



Perhaps the most noteworthy of these achievements 

 was the invention of the infinitesimal calculus, 

 developed by Newton and Leibniz. Algebra and 

 geometry had begun to assume their modern shapes, 

 trigonometry had been extended to imaginary quan- 

 tities, but the introduction of the idea of varying 

 velocity demanded a method of dealing with the 

 rates of variation of changing quantities. A con- 

 stant velocity is measured by the space s described 

 in a time t, and the quantity s/t will be the same 

 however great or small s and t may be. But, if the 

 velocity vary, its value at any instant can only be 

 found by taking a time so short that the velocity does 

 not change appreciably, and measuring the space 

 described in that short time. When s and t are re- 

 duced without limit and become infinitesimal, their 

 quotient gives the velocity at the instant, and was 

 written by Leibniz as ds/dt, which is called the differ- 

 ential coefficient of s with regard to t. Newton, in 

 his method of fluxions, wrote the same quantity as 

 S, a notation which is less convenient and is now 

 superseded by that of Leibniz. We have taken space 

 and time as an example, but any two quantities which 

 depend on each other may be treated in the same 



