MATHEMATICS 49 



certain unconscious use of geometrical knowledge is almost 

 as widespread. This use totals more than all the use in all 

 the sciences, but it is perhaps not so important, and not so 

 far reaching. In science the practical uses of mathematics 

 are apparently unlimited; beyond the mere question of reck- 

 oning, the fundamental idea of rate of change as expressed 

 in the calculus, the fundamental idea of dependence, as ex- 

 pressed in functions, the fundamental notion of expression 

 of results by mathematical symbolism, enters, almost domi- 

 nates, physics, engineering, astronomy, and, to a minor ex- 

 tent, some other sciences. To commence to give a list of 

 these applications would be nothing but foolhardy. As a 

 striking illustration, I might mention that Halley's comet has 

 just been seen in the heavens. It has been gone these seventy 

 years, and its return has been predicted absolutely. This 

 spectacular instance of our knowledge of the heavens is not 

 really so important as many less startling, but more vital 

 phenomena in astronomy which today are absolutely under- 

 stood through the use of mathematical knowledge. The cal- 

 culation of a comet's path, however, unites in itself all of the 

 various points which I have made : the acceptance from science 

 of fundamental assumptions, of which the greatest is New- 

 ton's law of gravitation, the formulation in the language of 

 mathematical symbolism, checked and re-checked by reasoning, 

 the reduction of functional expression, the application of 

 the notion of the rate of motion expressed by means of the 

 calculus, the final unraveling of all attendant mysteries. At 

 last mathematics returned to science its conclusion, and this 

 conclusion proved useful to the race, — in that people are quite 

 interested in seeing this comet as soon as possible when it 

 comes back. But it is not only in such far away things that 

 mathematics has been absolutely essential to progress. Every 

 electric light which shines is a visible sign of. the mathematical 

 attainment of the race; and from electric lights to north pole 

 discovery; from airships on the one hand to life insurance 



