4§4 



NA TURE 



[Sept. 20, 1883 



tyle of art which will attract round it the largest circle of 

 admirers. So the fact that it is a very limited circle 

 which is capable of appreciating the beauty of the work 

 done by a great mathematician should not prevent men 

 from understanding that it is like the work done by a 

 poet or a painter, work done entirely for its own sake, 

 and capable of affording lively pleasure both to the 

 worker himself and his admirers, without any thought of 

 material benefit to be derived from it. 



But in point of fact mathematics stand midway be- 

 tween the arts which minister to man's sense of beauty 

 and those which supply his material comforts. The name 

 " pure mathematics" suggests that there is such a thing 

 as " applied mathematics," and it is well known that the 

 mathematician furnishes the instruments employed by 

 cultivators of sciences whose practical utility is beyond 

 dispute. If the mathematician did no more than manu- 

 facture such instruments precisely as the demand arose 

 for them, his might count as one of the arts which are 

 valued only for their practical utility. But actually the 

 invention of the mathematical instruments usually comes 

 first, and the use to be made of them is found out after- 

 wards. The stock example of the kind is the debt which 

 physical astronomy owes to the labours of the early geo- 

 meters on the theory of conic sections, a theory cultivated 

 without any suspicion that it could be turned to practical 

 account. Yet it was because Newton was in his day the 

 greatest master of this as of every other branch of pure 

 mathematics that he was able to bring all the motions of 

 the heavenly bodies under the dominion of mathematical 

 calculation, and to convert the moon into a timepiece by 

 which the mariner can ascertain his position on the seas. 

 With the advance of physical science greater refinement 

 and power in the mathematical instruments of investiga- 

 tion have become necessary ; but pure mathematicians 

 have ever outrun the demands of the practical workers, 

 for instrument-making has delights of its own. The late 

 Lord Rosse I have no doubt found more pleasure in de- 

 vising the innumerable ingenious and beautiful contriv- 

 ances necessary for the manufacture of his huge telescope 

 ihan he ever did from observing with it after it was 

 made. It is impossible for any one now to say what 

 advantages future investigators will derive from the per- 

 fection to which the mathematical instruments have been 

 brought by the labours of such men as Cayley, who have 

 invented mathematical steam hammers by which pon- 

 derous masses of formula; can be manipulated with ease 

 and calculations made simple which in former times were 

 looked on as impracticable. 



There is hardly anything that comes under the head of 

 pure mathematics at which Cayley has not worked, but it 

 will be enough if I try to say something as to that by 

 which his name is likely to be best remembered— his 

 creation of an entirely new branch of mathematics by his 

 discovery of the theory of invariants, which has given 

 quite a new aspect to several departments of mathematics. 

 It has introduced such a host of new ideas, and conse- 

 quently of new words, that a Senior Wrangler of forty 

 years ago, who had not kept pace with modern investiga- 

 tions, would find, on taking up a book of the present day 

 on geometry or algebra, that he could not read it without 

 a glossary, and must go to school again to learn what the 

 writer was speaking of. It would be out of place if I 



were to enter into a very long technical exposition here, 

 but it is possible, without assuming in the reader more than 

 a moderate knowledge of analytic geometry, to make him 

 at least understand what the word " invariant " means. 

 Suppose that we have written down the general equation 

 of a curve of any degree, and also have found the relation 

 that must subsist between the coefficients in order that 

 the curve should assume some special form. For sim- 

 plicity I suppose the equation to be of the second degree, 

 and I take the well known relation between the coeffi- 

 cients which is satisfied when the curve represented reduces 

 itself to two right lines. Now imagine the equation to be 

 transformed to any new coordinates whatever, this can 

 make no change in the form of the curve represented. If 

 the relation in question were satisfied by the coefficients 

 of the original equation, it must also be satisfied by the 

 coefficients of the transformed equation. But by actually 

 performing the transformation we can express these new 

 coefficients in terms of the old ones and of the constants 

 introduced in the process of transformation. The ex- 

 pression will be complicated enough, and that of the 

 relation of which I am speaking still more so. But 

 since the relation must vanish whenever the correspond- 

 ing relation expressed in terms of the old coefficients 

 vanishes, the one must contain the other as a factor. 

 The remaining factor, it will be seen on examination, 

 contains nothing but the constants introduced by trans- 

 formation. All this can be verified by actual work ; but 

 the result which I have stated can be foreseen without 

 any calculation. 



The principle which I have described has proved to be 

 very fertile in applications. The late Dr. Boole made, in 

 1 841, some interesting use of a simple case of the same 

 principle. But it was Cayley who set himself the problem 

 to determine a priori what functions of the coeffi- 

 cients of a given equation possess this property of 

 invariance, viz., that when the equation is linearly trans- 

 formed the same function of the new coefficients is equal 

 to the given function multiplied by a quantity independent 

 of the coefficients. The result of his investigations was 

 to bring to light a number of important functions (some 

 of them involving the variables as well as the coefficients) 

 whose relations to the given equation are unaffected by 

 linear transformation. And the effect has been that the 

 knowledge which mathematicians now possess of the 

 structure of algebraic forms is as different from what it 

 was before Cayley's time as the knowledge of the human 

 body possessed by one who has dissected it and knows 

 its internal structure is different from that of one who has 

 only seen it from the outside. 



In an age when the work of mathematical research is 

 so actively carried on, whenever one worker finds a 

 nugget there is an immediate rush to the spot of other 

 searchers. In the present case Cayley's friend Sylvester 

 was one of the first on the spot, and both being resident 

 in London were able by frequent oral communication to 

 stimulate each other's ideas. As I am not relating the 

 history of mathematical science, I need not name the 

 foreign mathematicians who rapidly came in to labour in 

 the same field ; but it is agreed on all hands that it was 

 Cayley who both discovered the "diggins" and got out 

 some of the biggest nuggets. It is not always the case 

 that the history of a mathematical discovery has not to 



