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THE POPULAR SCLENCE MONTHLY.— SUPPLEMENT. 



aspect of the subject which he could present to 

 his hearers, he might well have given up the at- 

 tempt in despair. But although in its technical 

 character mathematical science suffers the incon- 

 veniences, while it enjoys the dignity, of its Olym- 

 pian position, still in a less formal garb, or in 

 disguise, if you are pleased so to call it, it is 

 found present at many an unexpected turn ; and 

 although some of us may never have learned its 

 special language, not a few have, all through our 

 scientific life, and even in almost every accurate 

 utterance, like Moliere's well-known character, 

 been talking mathematics without knowing it. It 

 is, moreover, a fact not to be overlooked that the 

 appearance of isolation, so conspicuous in mathe- 

 matics, appertains in a greater or less degree to 

 all other sciences, and perhaps also to all pursuits 

 in life. In its highest flight each soars to a dis- 

 tance from its fellows. Each is pursued alone 

 for its own sake, and without reference to its con- 

 nection with, or its application to, any other sub- 

 ject. The pioneer and the advanced guard are 

 of necessity separated from the main body, and 

 in this respect mathematics does not materially 

 differ from its neighbors. And, therefore, as the 

 solitariness of mathematics has been a frequent 

 theme of discourse, it may be not altogether un- 

 profitable to dwell for a short time upon the other 

 side of the question, and to inquire whether there 

 be not points of contact in method or in subject- 

 matter between mathematics and the outer world 

 which have been frequently overlooked ; whether 

 its lines do not in some cases run parallel to 

 those of other occupations and purposes of life ; 

 and, lastly, whether we may not hope for some 

 change in the attitude too often assumed toward 

 it by the representatives of other branches of 

 knowledge and of mental activity. 



In his preface to the " Principia," Newton 

 gives expression to some general ideas which may 

 well serve as the key-note for all future utterances 

 on the relation of mathematics to natural, includ- 

 ing also therein what are commonly called arti- 

 ficial, phenomena : 



" The ancients divided mechanics into two 

 parts, rational and practical ; and since artisans 

 often work inaccurately, it came to pass that me- 

 chanics and geometry were distinguished in this 

 way — that everything accurate was referred to 

 geometry, and everything inaccurate to mechan- 

 ics. But the inaccuracies appertain to the ar- 

 tisan and not to the art, and geometry itself has 

 its foundation in mechanical practice, and is in 

 fact nothing else than that part of universal 

 mechanics which accurately lays down and de- 



monstrates the art of measuring." ' He next ex- 

 plains that rational mechanics is the science of 

 motion resulting from forces, and adds : " The 

 whole difficulty of philosophy seems to me to lie 

 in investigating the forces of Nature from the 

 phenomena of motion ; and in demonstrating that 

 from these forces other phenomena will ensue." 

 Then, after stating the problems of which he has 

 treated in the work itself, he says : "I would that 

 all other natural phenomena might similarly be 

 deduced from mechanical principles. For many 

 things move me to suspect that everything de- 

 pends upon certain forces in virtue of which the 

 particles of bodies, through forces not yet under- 

 stood, are either impelled together so as to co- 

 here in regular figures, or are repelled and recede 

 from one another." 



Newton's views, then, are clear : he regards 

 mathematics not as a method independent of, 

 though applicable to, various subjects, but as it- 

 self the higher side or aspect of the subjects them- 

 selves ; and it would be little more than a trans- 

 lation of his notions into other language, little 

 more than a paraphrase of his own words, if we 

 were to describe the methematical as one aspect 

 of the material world itself, apart from which all 

 other aspects are but incomplete sketches, and, 

 however accurate after their own kind, are still 

 liable to the imperfections of the inaccurate ar- 

 tificer. Mr. Burrowes, in his preface to the first 

 volume of the "Transactions " of the Royal Irish 

 Academy, has carried out the same argument, ap- 

 proaching it from the other side. " No one sci- 

 ence," he says, " is so little connected with the 

 rest as not to afford many principles whose use 

 may extend considerably beyond the science to 

 which they primarily belong, and no proposition 

 is so purely theoretical as to be incapable of be- 

 ing applied to practical purposes. There is no 

 apparent connection between duration and the 

 cycloidal arch, the properties of which have fur- 

 nished us with the best method of measuring 

 time ; and he who has made himself master of 

 the nature and affections of the logarithmic curve 

 has advanced considerably toward ascertaining 

 the proportionable density of the air at various 

 distances from the earth. The researches of the 

 mathematician are the only sure ground on which 

 we can reason from experiments ; and how far 

 experimental science may assist commercial in- 

 terests is evinced by the success of manufactures 

 in countries where the hand of the artificer has 



1 Compare with this the latter part of Plato's " Phi- 

 lebus " on knowledge and the handicraft arts ; also 

 Prof. Jowett's introduction thereto. 



