MATHEMATICS IN EVOLUTION. 205 



found relief in the charms of music, and strangely enough dreaded an 

 exhaustion of it, just as many other people who have not the excuse 

 of morbid ailment think that all the greatest possible discoveries have 

 been made, and that all the finest things in prose and verse have been 

 said. Such notions are denied by the laws which have been stated, 

 as exemplified not only in the diversity and might of modern achieve- 

 ment, but also in the deep relations between the elements of natural 

 action divulged by their very multiplication of effects ; the generali- 

 zations of this age have never been equaled in scope and force the 

 persistence of force and the theory of evolution. 



As sciences advance, their essential unity becomes more and more 

 evident; methods that at first view would seem utterly unconnected 

 are being constantly found to have a secret and helpful family tie. 

 The comparative value of various types of bridges has been investi- 

 gated by submitting glass models duly weighted to polarized light, 

 which shows at once the distributions of strain and pressure. A 

 common magnetic needle has been successfully employed in finding 

 weak places in iron and steel axles by its unequal deflection at such 

 points, due to internal heterogeneity in the mass examined. At Paris 

 recently an underground pneumatic tube became obstructed at an 

 unknown point ; excavation was correctly guided by the adoption of 

 an acoustic principle ; a loud sound was made at the tube's entrance, 

 and the time occupied before the reflected wave returned was care- 

 fully noted, from which was inferred the distance traversed by it to 

 and from the obstacle. Many instruments at first made for purely 

 philosophical study have been drafted into the world's practical 

 uses. Applications of the rheostat and Wheatstone's bridge serve to 

 locate the oft-recurring breaks in ocean-cables and telegraph-lines, 

 and have very lately yielded the marvellous duplex and quadruplex 

 telegraphs. The spectroscope, originally directed to the heavens, has 

 now found uses on earth of great value; it detects adulteration, 

 marks defectiveness in drainage, and points out impurities in water- 

 supplies. 1 



1 A proposition in pure mathematics may receive elucidation and extension by an 

 illustration taken from optics. In Newton's "Principia," book i., section xii., prop. 70, 

 he proves, in a manner very difficult to follow, that a corpuscle placed within a hollow 

 sphere, if attracted as the square of the distance by all the points in the concave sur- 

 face, will remain unmoved wherever placed, as the sums of attraction always balance. 



This may be made clear not only of a spherical surface, but the closed interior of any 

 surface whatever, provided it has no reentrant angles, as a pyramid or an obliquely- 

 truncated cone. 



For, imagine the corpuscle to be luminous and to be bisected by any plane extended 

 so as tg cut the containing hollow surface into two parts, it is evident that equal 

 amounts of light are radiated by each half of the corpuscle on each of the two parts of 

 the surface containing it. Now, these rays diminish in intensity as the square of the 

 distance, and so reciprocally correspond with a force emanating according to the same 

 law from the surface and affecting the corpuscle. Hence, the area of the surface of any 

 hollow body, having no reentrant angles, varies as the square of its average distance 

 from any point within it. 



