4o8 



NATURE 



\Augiist 15, 1878 



foundation in mechanical practice, and is in fact nothing else 

 than that part of universal mechanics which accurately lays 

 down and demostrates 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 depends upon certain forces in virtue 

 of which the particles of bodies, through forces not yet under- 

 stood, are either impelled together so as to cohere in regular 

 figures, or are repelled and recede from one another." 



Burrowes^ Remarks. 

 ■Newton's views, then, are clear ; he regards mathematics not 

 -as a method independent of, though applicable to, various sub- 

 jects, but as itself the higher side or aspect of the subjects them- 

 selves ; 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 

 inaccurate 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- 

 -<:apable of being applied to practical purposes. There is no 

 apparent connection 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 logarithmic curve has advanced con- 

 siderably towards ascertaining the proportionable deiisity 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 direc- 

 tion 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, there- 

 fore, 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 formulce or repre- 

 sented by geometrical figures. But the real gist of the business 

 does not always lie in the mode of expression ; and the fascina- 

 tion of the formula: or other mathematical paraphernalia may 

 after all be little more than that of a theatrical transformation 

 scene. The process of reducing to formula; is really one of 

 abstraction, the results of which are not always wholly on the 

 side of gain ; in fact, through the process itself the subject may 

 lose in one respect even more than it gains in another. But 

 long before such abstraction is completely attained, and even in 

 cases where it is never attained at all, a subject may to all 

 intents and purposes become mathematical. It is not so much 

 ■elaboi-ate calculations or abstruse processes which characterise 

 this phase, as the principles of precision, of exactness, and of 

 proportion. But these are principles with which no true know- 

 ledge can entirely dispense. If it be the general scientific spirit 

 which at the outset moves upon the face of the waters, and out 

 of the unknown depth brings forth light and living forms ; it is 

 no less the mathematical spirit which breathes the breath of 

 life into what would otherwise have ever remained mere dry 

 bones of fact, which re-unites the scattered limbs and re-creates 

 from them a new and organic whole. 



And as a matter of fact, in the words used by Prof. Jellett at 

 our meeting at Belfast, viz., " Not only are we applying our 

 methods to many sciences already recognised as belonging to 

 the legitimate province of mathematics, but we ai-e learning to 



J Compare with this the latter part of Plato's " Philebus " on knowledge 

 and the handicraft arts ; also Prof. Jowett's Introduction thereto. 



apply the same instrument to sciences hitherto wholly or par- 

 tially independent of its authority. Physical science is learning 

 more and more every day to see in the phenomena of nature 

 modifications of that one phenomenon (namely, motion) which 

 is peculiarly under the power of mathematics." Echoes are 

 these, far off and faint perhaps, but still true echoes, in answer 

 to Newton's wish that all these phenomena may some day ' ' be 

 deduced from mechanical principles." 



Mathematics, Literature, and Art. 



If, turning from this aspect of the subject, it were my pur 

 pose to enumerate how the same tendency has evinced itself in 

 the arts, unconsciously it may be to the artists themselves, I 

 might call as witnesses each one in turn with full reliance on 

 the testimony which they would bear. And, having more 

 special reference to mathematics, I might confidently point to 

 the accuracy of measurement, to the truth of curve, which, 

 according to modern investigation, is the key to the perfection 

 of classic art. I might triumphantly cite not only the architects 

 of all ages, whose art so manifestly rests upon mathematical 

 principles, but I might cite also the literary as well as the 

 artistic remains of the great artists of the Cinquecento, both 

 painters and sculptors, in evidence of the geometry and the 

 mechanics which, having been laid at the foundation, appear to 

 have found their way upwards through the superstructure of 

 their works. ^ And in a less ambitious sphere, but nearer to 

 ourselves in both time and place, I might point with satisfaction 

 to the great school of English constructors of the eighteenth 

 century in the domestic arts ; and remind you that not only the 

 engineer and the architect, but even the cabinet-makers, devoted 

 half the space of their books to perspective and to the principles 

 whereby solid figures may be delineated on paper, or what is 

 now termed descriptive geometry.^ 



Nor perhaps would the sciences which concern themselves 

 with reasoning and speech, nor the kindred art of music, nor 

 even literature itself, if thoroughly probed, offer fewer points of 

 dependence upon the science of which I am speaking. What, 

 in fact, is logic bat that part of universal reasoning ; gram- 

 mar but that part of universal speech : harmony and counter- 

 point but that part of universal music, "which accurately lays 

 down," and demonstrates (so far as demonstration is possible) 

 precise methods appertaining to each of these arts? And I 

 might even appeal to the common consent which speaks of the 

 mathematical as the pattern form of reasoning and model of a 

 precise style. 



Taking, then, precision and exactness as the characteristics 

 which distinguish the mathematical phase of a subject, we are 

 naturally led to expect that the approach to such a phase will 

 be indicated by increasing application of the principle of mea- 

 surement, and by the importance which is attached to numerical 

 results. And this very necessary condition for progress may, I 

 think, be fairly described as one of the main features of scientific 

 advance in the present day. 



Measurements in Physics. 



If it were my purpose, by descending into the arena of special 

 sciences, to show how the most various investigations alike tend 

 to issue in measurement, and to that extent to assume a mathe- 

 matical phase, I should be embarrassed by the abundance of 

 instances which might be adduced. I will therefore confine 

 myself to a passing notice of a very few, selecting those which 

 exemplify not only the general tendency, but also the special 

 character of the measurements now particularly required, viz., 

 that of minuteness, and the indirect method by which alone we 

 can at present \\o^q to approach them. An object having a 

 diameter of an 8o,oo3th of an inch is perhaps the smallest of 

 which the microscope could give any well-defined representation; 

 and it is improbable that one of l20,cooth of an inch could be 

 singly discerned with the highest powers at our command. ^ But 

 the solar beams and the electric light reveal to us the presence 

 of bodies far smaller than these. And, in the absence of any 

 means of observing them singly. Prof. Tyndall has suggested a 

 scale of these minute objects in terms of the lengths of lumini- 

 ferous waves. To this he was led, not by any attempt at indi- 

 vidual measurement, but by taking account of them in the 



1 See " TrattatD della Pittura," by Leonardo da Vinci ; also the Memoir 

 on the MSS. of L. d. V., by Venturi, 1797. _, . 



2 " The Gentleman and Cabinet Maker's Director," by Thomas Chippen- 

 dale. London, 1754. " The Cabinet Maker and Upholsterer s Drawmg 

 Book," by Thomas Sheraton. LondoT?, i793' 



3 See Sorby's Address to the Microscopical Society, 1876. 



