February i, 1894] 



NATURE 



327 



through an arc of 90°. This introduces a new element into 

 the problem, and enables us to change the cube from any one 

 position to any other position, that is, to effect any interchange 

 among the sides which would be consistent with their remain- 

 ing sides of the same cube. Here we have a series of rota- 

 tions which, in the case of the cube, are homologous with 

 certain linear transformations which have been developed by 

 Klein in his very beautiful book on the Icosahedron. 



But it is also obvious that in introducing these rotations we 

 are practically operating with quaternions, the operator being 

 a unit vector. Thus we have a homology between certain 

 forms of quaternion multiplication and linear transformations 

 involving the imaginary unit. Moreover, since these rotations 

 are also homologous with substitutions, performed on six sym- 

 bols representing the six sides of the cube, it follows that there 

 is also a homology between certain groups of substitutions and 

 certain linear transformations involving two quantities, a 

 numerator and a denominator, and quaternion multiplication 

 by unit vectors. 



I have taken a cube as the simplest illustration. Evidently 

 we can construct a great number of groups of substitutions of 

 the same sort between the sides of any regular solid, as Klein 

 has done in the work I have already cited. The relation be- 

 tween the linear substitutions thus found and the solution of 

 corresponding algebraic equations forms one of the most 

 beautiful branches of our modern mathematics. 



We have in all these cases a very simple illustration of a law 

 of thought, the application of which forms the basis of an im- 

 portant part of modern mathematical research. We may call it 

 the law of homology. I am not sure of my ability to define it 

 rigorously, but I think we may express it in some such form as 

 this : If we have two sets of concepts, say A and B, such that 

 to every concept of the one set shall correspond a concept of the 

 other, and to every relation between any two of one set a cor- 

 responding relation between the corresponding two of the other, 

 then all language, reasoning, and conclusions as to the one set 

 may be applied to the other set^ We may, of course, extend 

 the law to a correspondence between things or concepts, and 

 symbols, or other forms of language. 



This law is, I think, more universal than might at first sight 

 appear. Not only the progress, but the very existence of our race 

 depends upon that coordination between our mental processes 

 and the processes of the external universe, which has gradually 

 been brought about by the attrition between man and nature 

 through unnumbered generations. A man is perfect, powerful, 

 and effective 'in proportion as his thoughts of nature coincide 

 with the processes of nature herself ; each process of nature 

 having its image in his thought, and vice versa. Now, language 

 consists in coordination oetween words and conceptions. 

 Thus we pass from nature to what corresponds to it in thought, 

 and from thought to what corresponds to it in language, and 

 thus bring about a correspondence between language and 

 nature. 



Modern scientific research affords many examoles of the 

 application of this law, which would be very marvellous if they 

 were not so familiar. We are so accustomed to the prediction 

 of an eclipse that we see no philosophy in it. And yet might 

 not a very intellectual being from another sphere see something 

 wonderful in the fact that by a process of making symbols with 

 pen and ink upon sheets of paper, and combining them accord- 

 ing to certain simple rules, it is possible to predict with un- 

 erring certainty that the shadow of the moon, on a given day and 

 at a given hour and minute, will pass over a certain place on 

 the earth's surface? Surely the being might ask with surprise 

 how such aresult could be attained. Our reply would be simply 

 this : There is a one-to-one correspondence between the sym- 

 bols which the mathematician makes on his paper, and the 

 laws of motion of the heavenly bodies. His symbols embody 

 the methods of nature itself. 



The introduction and application of homologies such as 1 have 

 pointed out have, perhaps, their greatest value as thought- 

 savers. In the field of mathematical thought they bear some 

 resemblance to labour-saving machines in the field of economics. 

 They enable the results of ratiocination to be reached without 

 going through the process of reasoning in the particular case. 

 Much that I have said illustrates this use of the method, but 

 there is yet another case which has been so fruitful as to be 

 worthy of special mention : I mean the general theory of 

 functions of an imaginary variable. We may regard such 

 functions as being in reality representative of a pair of functions 



NO, 1266, VOL. 49J 



of a certain class involving a pair of real variables ; but the 

 difficulty of conceiving the various ways in which the two 

 variables might be related, and the results of the changes which 

 they might go through, in such a way as to clearly follow out all 

 possible results, would have rendered their direct study impos- 

 sible. 



But when Gauss and Cauchy conceived the happy idea of 

 representing two such variables, the real and the imaginary 

 one, by the rectangular coordinates of a point in a plane, those 

 relations which before taxed the powers of conception became 

 comparatively simple. Considered as a magnitude, the com- 

 plex variable, or the sum of a real quantity and a purely 

 imaginary one, the latter being considered as one measured in 

 imaginary units, was represented by the length and position of 

 a straight line drawn from an origin of coordinates to the point 

 whose coordinates were represented by the values of the variable. 

 Such a line, when both length and direction are considered, is 

 now familiarly known as a vector. Theconceptionof the vector 

 would, however, in many cases be laborious. But the vector is 

 completely determined by its terminal point ; to every vector 

 corresponds one and only one terminal point, and to every ter- 

 minal point one and only one vector. Hence we may make 

 abstraction of the vector entirely, and in thought attend only to 

 the terminal point. Since for every pair of values we assign to 

 our original variables there is one point, and only one, we may 

 in thought make abstraction of both of these variables, and of 

 the vectors which they represent, and consider only the point 

 whose coordinates they are. Thus the continuous variation of 

 the two quantities, how complex soever it may be, is repre- 

 sented by a motion of the point. Now such a motion is very 

 easy to conceive. We may consider it as performing a number 

 of revolutions around some fixed position without the slightest 

 difficulty, whereas to conceive the corresponding variations in 

 the algebraic variables themselves would need considerable 

 mental effort. Thus, and thus alone, has the beautiful theory, 

 first largely developed by Cauchy, and afterward continued by 

 Riemann, been brought to its present state of perfection. 



Another example of the principle in question, where the two 

 objections of reasoning are so nearly of a kind that no thought is 

 saved, is afforded by the principle of duality in projective geo- 

 metry. Here a one-to-one correspondence is established be- 

 tween the mutual relations of points and lines, with the result 

 that in demonstrating any proposition relating to these concepts 

 we at the same time demonstrate a correlative proposition 

 formed from the original one by simply interchanging the words 

 " point " and " line." 



The subjects of which I have heretofore spoken belong [con- 

 jointly to algebra and geometry. Indeed, one of the great 

 results of bringing homologous interpretation into modern 

 mathematics has been to unify the treatment of algebra and 

 geometry, and almost fuse them into a single science. To a 

 large class of theorems of algebra belong corresponding theo- 

 rems of geometry, each of one class proving one of the other 

 class. Thus the two sciences become mutually helpful. In 

 geometry we have a visible representation of algebraic theorems; 

 by algebraic operations we reach geometrical conclusions which 

 it might be much more difficult to reach by direct reasoning. A 

 remarkable example is afforded by the geometrical application 

 of the theory of invariants. These are perhaps the last kind 

 of algebraic conclusion which the student, when they are first 

 presented to his attention, would conceive to have a geometrical 

 application, yet a very litl le study suffices to establish a complete 

 homology between them and the distribution of points upon 

 a straight line. 



This use of homologies does not mark the only line by which 

 we have advanced beyond our predecessors. Progress has been 

 possible only by emancipating ourselves from certain of the con- 

 ceptions of ancient geometry which are still uppermost in all our 

 elementary teaching. The illustration I have already given is 

 here much to the point. The expression of a relation between 

 two straight lines by the multiplier which would change one 

 into the other is now familiar to every schoolboy, and the rela- 

 tion itself was familiar to Euclid. But the yet simpler relation 

 of a line to another of equal length standing at right angles to 

 it, and the corresponding operator which will change one into 

 the other, was never thought of by Euclid, and is unfamiliar in 

 our schools. Why is this ? It seems to me that it grows out 

 of the ancestral idea that mathematics concerns itself with 

 measurement and that the object of measurement is to express 

 all magnitudes in one-dimensional measure. So completely has 



