12 REPOBT— 1878. 



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 simdarly 

 be deduced from mechanical principles. For many things move me to 

 suspect that everything depends upon certain forces in virtue of which 

 the particles of bodies, through forces not yet understood, are either 

 impelled together so as to cohere 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 

 itself the higher side or aspect of the subjects themselves ; and it would 

 be little more than a translation of his notions into other language, little 

 more than a paraphrase of his own words, if we were to describe the 

 mathematical 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 inaccu- 

 rate artificer. Mr. Burrowes, in his Preface to the first volume of the 

 ' Transactions of the Royal Irish Academy,' has carried out the same 

 argument, approaching it from the other side. " No one science," 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 in- 

 capable of being applied to practical purposes. There is no apparent 

 connexion between duration and the cycloidal arch, the properties of 

 which have furnished us with the best method of measuring time ; and 

 he who has made himself master of the nature and affections of the loga- 

 rithmic curve has advanced considerably towards ascertaining the propor- 

 tionable 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 interests is evinced by the success of manufactures in 

 countries where the hand of the artificer has taken its direction from 

 the philosopher. Every manufacture is in reality but a chemical process, 

 and the machinery requisite for carrying it on but the right application 

 of certain propositions in rational mechanics." So far your Academician. 

 Every subject, therefore, whether in its usual acceptation scientific or 

 otherwise, may have a mathematical aspect ; as soon, in fact, as it 

 becomes a matter of strict measurement, or of numerical statement, so 

 soon does it enter upon a mathematical phase. This phase may, or it 

 may not, be a prelude to another in which the laws of the subject are 

 expressed in algebraical formulae or represented by geometrical figures. 

 But the real gist of the business does not always lie in the mode of 

 expression, and the fascination of the formulas or other mathematical 



