180 



SCIENCE. 



[Vol. VUL, No. 186 



MULTIPLE ALGEBRA. 



Professor Gibbs's masterly address upon the 

 subject of ' multiple algebra ' was too long and of 

 too technical a nature for presentation in full to 

 our readers, and the quotation of a few passages, 

 and a brief summary of the bearings of the re- 

 mainder, must suffice to acquaint them with its 

 general drift and importance. His opening re- 

 marks w^ere as follows : — 



" It has been said that ' the human mind has 

 never invented a labor-saving machine equal to 

 algebra.' If this be true, it is but natural and 

 proper that an age like our own, characterized by 

 the multiplication of labor-saving machinery, 

 should be distinguished by an unexampled devel- 

 opment of this most refined and most beautiful of 

 machines. That such has been the case, no one 

 will question. The improvement has been in 

 every part. Even to enumerate the principal lines 

 of advance, would be a task for any one, — for 

 me, an imijossibility. But if we should ask in 

 what direction the advance has been made, what 

 is to characterize the development of algebra in 

 our day, we may, I think, point to that broaden- 

 ing of its fields and methods which gives us 

 'multiple algebra.' Of the importance of this 

 change in the conception of the office of algebra, 

 it is hardly necessary to speak : that it is really 

 characteristic of our time, will be most evident if 

 we go back some two or three score years, to the 

 time when the seeds were sown which are now 

 yielding so abundant a harvest. The failure of 

 Mobius, Hamilton, Grassmann, Saint- Venant, to 

 make an immediate impression upon the course 

 of mathematical thought in any way commen- 

 surate with the importance of their discoveries, is 

 the most conspicuous evidence that the times were 

 not ripe for the methods which they sought to in- 

 troduce. A satisfactory theory of the imaginary 

 quantities of ordinary algebra, which is essentially 

 a simple case of multiple algebra, with difficulty 

 obtained recognition in the lirst third of this cen- 

 tury. We must observe that this ' double algebra,' 

 as it has been called, was not sought for or in- 

 vented, — it forced itself, unbidden, upon the at- 

 tention of mathematicians, and with its rules 

 already formed." 



The speaker then gave a critical historical re- 

 view of the different contributions of Hamilton, 

 Mobius, Grassmann, Saint- Venant, Cauchy, Cay- 

 ley, Hankel, the Peirces, father and son, and 

 Sylvester, to these new methods of mathematical 



Abstract of an address before the section of mathe- 

 matics and astronomy of the American association for the 

 advancement of science at Buffalo, Aug. 19, 1886, by Prof. 

 J. Willard Qibbs, of New Haven, Cone, vice-president of 

 the section. 



analysis, showing the additions and developments, 

 made by each to the various subjects. 



In the second part of the paper, Professor Gibbs 

 criticised the methods of some modern writers on 

 these subjects, showing how they failed to grasp 

 the full significance and bearings of the matters 

 they were dealing with, being too much hampered 

 by the old ideas and methods of simple algebra. 

 We quote here a few sentences : — 



" This fault has been denounced by Sylvester ; 

 and if any one thinks that I make too much of 

 the stand-point from which we view the subject, 

 I will refer him to the opening paragraphs of the 

 lectures on ' universal algebra ' in the sixth vol- 

 ume of the American journal of mathematics, 

 where, with a wealth of illustration and an energy 

 of diction which I cannot emulate, the most elo- 

 quent of mathematicians expresses his sense of the 

 importance of the substitution of the idea of 

 the matrix for that of the determinant. If this 

 is so important, why was the idea of the matrix 

 let slip ? Of course, the writers on this subject 

 had it to commence with. One cannot even de- 

 fine a determinant without the idea of a matrix. 

 The simple fact is, that the writers on this sub- 

 ject have especially developed those ideas which 

 are naturally expressed in simple algebra, and 

 have postponed, or slurred over, or omitted alto- 

 gether, those ideas which find their natural ex- 

 pression in multiple algebra. But in this subject, 

 the latter happened to be the fundamental ideas, 

 and those which ought to direct the whole course 

 of thought." Many illustrations were then given 

 of the applications of these ideas to cases in , 

 point. 



The author introduced the third part of his pa- 

 per as follows : " We have considered the subject 

 a good while from the outside ; we have glanced 

 at the principal events in the history of multiple 

 algebra ; we have seen how the course of modern 

 thought seems to demand its aid, how it is actual- 

 ly leaning toward it, and beginning to adopt its 

 methods. It may be worth while to direct our 

 attention more critically to multiple algebra itself, 

 and to inquire into its essential character and its 

 most important principles. I do not know that 

 anything useful or interesting, which relates to 

 multiple quantity, and can be symbolically ex- 

 pressed, falls outside the domain of multiple alge- 

 bra. But if it is asked, what notions are to be 

 regarded as fundamental? we must answer, here 

 as elsewhere, those which are most simple and 

 fruitful. Unquestionably, no relations are more 

 so than those which are known by the names of 

 addition and multiplication." 



Then followed a long discussion of the funda- 

 mental conceptions and methods of modern math- 



