NOVEMBEE 12, 1915] 



SCIENCE 



669 



things to things of sense — much like the 

 difference between one who greets a new- 

 born day because of its glory and one who 

 regards it as a time for doing chores and 

 values its light only as showing the way. 

 For the former, mathematics is justified by 

 its supreme beauty; for the latter, by its 

 manifold use. But are the two kinds of 

 value essentially incompatible? They are 

 certainly not. The difference is essen- 

 tially a difference of authority — a dif- 

 ference, that is, of worth, of elevation, 

 of excellence. The pure mathematician 

 and the applied mathematician some- 

 times may, indeed they not infrequently 

 do, dwell together harmoniously in a 

 single personality. If our spokesman be 

 such a one — and I will not suppose the 

 shame of having the utilities of the science 

 represented on such an occasion by one in- 

 capable of regarding it as anything but a 

 tool, for that would be disgraceful — if, 

 then, our spokesman be such a one as I have 

 supposed, he might properly begin as fol- 

 lows: In speaking to you of the applica- 

 tions of mathematics I would not have you 

 suppose, ladies and gentlemen, that I am 

 thus presenting the highest claims of the 

 science to your regard; for its highest 

 justification is the charm of its immanent 

 beauty; I do not mean, he will say, the 

 beauty of appearances — the fleeting beau- 

 ties of sense, though these, too, are precious 

 — even the outer garment, the changeful 

 robe, of reality is a lovely thing; I mean 

 the eternal beauty of the world of pure 

 thought; I mean intellectual beauty; in 

 mathematics this nearly attains perfection ; 

 and "intellectual beauty is self-sufficing"; 

 uses, on the other hand, are not ; they wear 

 an aspect of apology; uses resemble ex- 

 cuses, they savor a little of a plea in miti- 

 gation. Do you ask: Why, then, plead 

 them? Because, he will say, many good 

 people have a natural incapacity to ap- 



preciate anything else; because, also, many 

 of the applications, especially the higher 

 ones, are themselves matters of exceeding 

 beauty; and especially because I wish to 

 show, not only that use and beauty are 

 compatible forms of worth, but that the 

 more mathematics has been cultivated for 

 the sake of its inner charm, the fitter has 

 it become for external service. 



Having thus at the outset put himself in 

 proper light and given his auditors a 

 scholar's warning against what would else, 

 he fears, foster a disproportionment of 

 values, what will he go on to signalize 

 among the utilities of a science whose pri- 

 mary allegiance to logical rectitude allies 

 it to art, and which only incidentally and 

 secondarily shapes itself to the ends of in- 

 strumental service? He knows that the 

 applications of mathematics, if one will 

 but trace them out in their multifarious 

 ramifications, are as many-sided as the in- 

 dustries and as manifold as the sciences of 

 men, penetrating everywhere throughout 

 the full round of life. What will he select ? 

 He will not dwell long upon its homely uses 

 in the rude computations and mensurations 

 of counting-house and shop and factory and 

 field, for this indispensable yet humble 

 manner of world-wide and perpetual serv- 

 ice is known of all men and women. He 

 will quickly pass to higher considerations 

 — to navigation, to the designing of ships, 

 to the surveying of lands and seas, and the 

 charting of the world, to the construction 

 of reservoirs and aqueducts, canals, tun- 

 nels and railroads, to the modern miracles 

 of the marine cable, the telegraph, the tele- 

 phone, to the multiform achievements of 

 every manner of modern engineering, civil, 

 mechanical, mining, electrical, by which, 

 through the advancing conquest of land 

 and sea and air and space and time, the 

 conveniences and the prowess of man have 

 been multiplied a billionfold. It need not 



