414 



NATURE 



{August 15, 1878 



well as of negation, may have been in early times transferred 

 from logic to mathematics. But the connection of our ideas of 

 number is probably anterior to the connection of any of our 

 other ideas. And as a matter of fact, geometry and arithmetic 

 had already made considerable progress when Aristotle invented 

 the syllogism. 



General Ideas. 



General ideas there were beside those of mathematics — true 

 flashes of genius which saw that there must be general laws to 

 which the universe conforms, but which saw them only by 

 occasional glimpses, and through the distortion of imperfect 

 knowledge ; and although the only records of them now remain- 

 ing are the inadequate representations of later writers, yet we 

 must still remember that to the existence of such ideas is due 

 not only the conception but even the possibility of physical 

 science. But these general ideas were too vrfde in their grasp, 

 and in early days at least were connected to their subjects of 

 application by links too shadowy to be thoroughly apprehended 

 by most minds, and so it came to pass that one form of such an 

 idea was taken as its only form, one application of it as the 

 idea itself ; and philosophy, unable to maintain itself at the 

 level of ideas, fell back upon the abstractions of sense, and, by 

 preference, upon those which were most ready to hand, namely, 

 those of mathematics. Plato's ideas relapsed into a doctrine of 

 numbers ; mathematics into mysticism, into Neo-Platonism, and 

 the like. And so, through many long ages, through good report 

 and evil report, mathematics have always held an unsought-for 

 sway. It has happened to this science, as to many other sub- 

 jects, that its warmest adherents have not always been its best 

 friends. Mathematics have often been brought in to matters 

 where their presence has been of doubtful utility. If they have 

 gfiven precision to literary style, that precision has sometimes 

 been carried to excess, as in Spinoza and perhaps Descartes ; if 

 they have tended to clearness of expression in philosophy, that 

 very clearness has sometimes given an appearance of finality not 

 always true ; ^ if they have contributed to definition in theologj', 

 that definitiveness has often been fictitious, and has been attained 

 at the cost of spiritual meaning." And, coming to recent 

 times, although we may admire the ingenuity displayed in the 

 logical machines of Earl Stanhope and of Stanley Jevons, in 

 the formal logic of De Morgan, and in the calculus of Boole ; 

 although as mathematicians we may feel satisfaction that these 

 feats (the possibility of which was clear h. priori) have been 

 actually accomplished; yet we must bear in mind that their 

 application is really confined to cases where the subject matter 

 is perfectly uniform in character, and that beyond this range 

 they are liable to encumber rather than to assist thought. 



Not unconnected with this intimate association of ideas and 

 their expression is the fact that, which ever may have been 

 cause, which ever effect, or whether both may not in turn have 

 acted as cause and effect, the culminating age of classic art was 

 contemporaneous with the first gi-cat development of mathe- 

 matical science.^ In an earlier part of this discourse I have 

 alluded to the importance of mathematical precision recognised 

 in the technique of art during the Cinquecento ; and I have now 

 time only to add that, on looking still further back, it would 

 seem that sculpture and painting, architecture and music, nay 

 even poetry itself, received a new, if not their first true, im- 

 pulse at the period when geometric form appeared fresh 



' For example, in Herbart's "Psychologic." 



« A specimen will be found in the Moralia of Gregory the Great, Lib. 

 I., c. xiv., of which I quote only the arithmetical part: — 



'' Quid in septenario numero, nisi summa perfectionls accipitur ? Ut 

 enim humanae rationis causas de septenario numero taceamus, quae afferunt, 

 qu6d idcirco perfectus sit, quia exprima pari constat, et primo imparl ; ex 

 primo, qui dividi potest, et primo, qui dividi non potest; certissime scimus, 

 qu6d septenarium numerum Scriptura Sacia pro perfectione pcnere cor.- 

 suevit. . . . . A septenario qulppe numero in duodenarium surgitur. Nam 

 septenarius suis in se partibus multiplicatus, ad duodenarium tenditur. Sive 

 enim quatuor per tria, sive per quatuor tria ducantur, septem in duodecim 

 vertuntur. .... Jam superiiis dictum est. quod in quinquagenario numero, 

 qui septem hebdomadibus ac monade addita impletur, requ.es designatur ; 

 denario autem numero summa perfecticnis e.xprimeLur." 



3 Approximate dates B.C. of — 



Sculptors, Painters, and Poets. Mathematicians. 



Stesichorus, 600. 



Pindar, 522-442. 



.iEschylus, 500-450. 



Sophocles, 495-400. 



Eunpides, 480-400. 



Phidias, 4S8-432. 



Praxiteles, 450-400. 



Zeuxis, 400. 



Apelles, 350. 



Scopas, 350. 



Thales, 6co. 



Pythagoras, 550. 

 Anaxagoras, 500-450. 

 Hippocrates, 460. 



Thejetetus, 440. 

 Archytas, 400. 



Euclid, 



323-2S3. 



chiselled by the hand of the mathematician, and when the first 

 ideas of harmony and proportion rang joyously together in the 

 morning tide of art. 



Relations of Science to Literature and Art. 



Whether the views on which I have here insisted be in any 

 way novel, or whether they be merely such as from habit or 

 from inclination are usually kept out of sight, matters little. 

 But whichever be the case, they may still furnish a solvent of 

 that ri^d aversion which both literature and art are too often 

 inclined to maintain towards science of all kinds. It is a very 

 old story that, to know one another better, to dwell upon 

 similarities rather than upon diversitie.';, are the first stages 

 towards a better understanding between two parties ; but in few 

 cases has it a truer application than in that here discussed. To 

 recognise the common growth of scientific and other instincts 

 until the time of harvest is not only conducive to a rich crop, 

 but it is also a matter of prudence, lest, in trying to root up 

 weeds from among the wheat, we should at the same time root 

 up that which is as valuable as wheat. When Pascal's father 

 had shut the door of his son's study to mathematics, and 

 closeted him with Latin and Greek, he found on his return 

 that the walls were teeming with formulae and figures, the 

 more congenial product of the boy's mind. Fortunately for 

 the boy, and fortunately also for science, the mathematics 

 were not torn up, but were suffered to grow together with other 

 subjects. And all said and done, the lad was not the worse 

 scholar or man of letters in the end. But, truth to tell, con- 

 sidering the severance which still subsists in education and 

 during our early years between literature and science, we can 

 hardly wonder if, when thrown together in the afterwork of 

 life, they should meet as strangers ; or if the severe garb, the 

 curious implements, and the strange wares of the latter, should 

 seem little attractive when contrasted with the light companion- 

 ship of the former. The day is yet young, and in the early 

 dawn many things look weird and fantastic which in fuller light 

 prove to be familiar and useful. The outcomings of science, 

 which at one time have been deemed to be but stumbling-blocks 

 scattered in the way, may ultimately prove stepping-stones which 

 have been carefully laid to form a pathway over difficult places 

 for the children of " sweetness and of light." 



Concluding Remarks, 



The instances on which we have dwelt are only a few out of 

 many in which mathematics may be found ruling and governing 

 a variety of subjects. It is as the supreme result of all expe- 

 rience, the framework in which all the varied manifestations of 

 nature have been set, that our science has laid claim to be the arbiter 

 of all knowledge. She does not indeed contribute elements of 

 fact, which must be sought elsewhere ; but she sifts and regu- 

 lates them ; she proclaims the laws to which they must conform, 

 if those elements are to issue in precise results. From the data 

 of a problem she can infallibly extract all possible consequences, 

 whether they be those first sought, or others not anticipated ; 

 but she can introduce nothing which was not latent in the origi- 

 nal statement. Mathematics cannot tell us whether there be ov 

 be not limits to time or space ; but to her they are both of in- 

 definite extent, and this in a sense which neither affirms nor 

 denies that they arc either infinite or finite. Mathematics can- 

 not tell us whether matter be continuous or discrete in its struc- 

 ture ; but to her it is indifferent whether it be one or the other,, 

 and her conclusions are independent of either particular hypo- 

 thesis. Mathematics can tell us nothing of the origin of matter, 

 of its creation or its annihilation ; she deals only with it in a 

 state of existence ; but within that state its modes of existence 

 may vary from our most elementary conception to our most 

 complex experience. Mathematics can tell us nothing beyond 

 the problems which she specifically undertakes ; she will carry 

 them to their limit, but there she stops, and upon the great 

 region beyond which she is imperturbably silent. 



Conterminous with space and coeval with time is the kingdom 

 of mathematics ; within this range her dominion is supreme ; 

 otherwise than according to her order nothing can exist ; in con- 

 tradiction to her laws nothing takes place. On her myste- 

 rious fcroU is to be found written for those who can read it that 

 which has been, that which is, and that which is to come.^ 

 Everything material which is the .subject of knowledge has 

 number, order, or position ; and those are her first outlines for 

 a sketch of the uuiverf.e. If our more feeble hands cannot fol- 

 low out the details, still her part has been drawn with an un- 

 erring pen, and her work cannot be gainsaid. So wide is the 



