534 



NA TURE 



[April 6, 1893 



iournal, I have delayed replying to the letters of Profs. MacAulay 

 and Tait, from an expectation that Prof. Gibbs would have 

 something to say. In this I have not been mistaken ; and, as 

 there is a general agreement between us on the whole, I have 

 merely to add some supplementary remarks. Prof. MacAulay 

 refers to me as having raised the question again. I can assure 

 him it has never been dropped. Apart from the one-sided dis- 

 cussion, it has been a live question with Prof. Gibbs and myself 

 since about 1882, and is now more alive than ever. I cannot 

 help thinking that Prof. MacAulay's letter was overhastily 

 written, and feel sure that if he knew as much about the views 

 and methods of those to whom he appeals as he does about 

 Quaternions, he would have written it somewhat differently, or 

 perhaps not have written it at all, from a conviction of the use- 

 lessness of his appeal. There is no question of suicide with us ; 

 on the contrary, quite the reverse. I am asked whether the 

 "spoonfeeding," as he terms it, ofMixweli, Fitzgerald, &c., 

 is not good enough for me. Why, of course not. It is quater- 

 nionic, and that is the real point concerned. Again, he thinks 

 nothing of the inscrutable negativity of the square of a vector in 

 Quaternions ; here, again, is the root of ihe evil. As regards a 

 uniformity of notation amongst antiquaternionists, I dare say that 

 will come in time, but the proposal is premature. We have 

 fiist to get people to study the matter and think about it. I 

 have developed my system, such as it is, quite independently of 

 Prof. Gibbs. Nevertheless, I would willingly adopt his notation 

 (as I have adopted his dyadical notion of the linear operator) 

 if I found it better. But I do not. I have been particularly 

 careful in my notation to harmonise as closely as possible 

 with ordinaVy mathematical ideas, processes, and notation ; I 

 do not think Gibbs has succeeded so well. But that matters 

 little now ; ihe really important thing is to depose the quater- 

 nion from the masterful position it has so long usurped, whereby 

 the diffusion of vector analysis has been so lamentably impeded. 

 I have been, until lately, very tender and merciful towards 

 quaternionic fads, thinking it possible that Prof. Tait might 

 modify his obstructive attitude. But there is seeminsjly no 

 chance of that. Whether this be so or not, I think it is prac- 

 tically certain that there is no chance whatever for Quaternions 

 as a practical system of mathematics for the use of physicists. 

 How is it possible, when ic is so utterly discordant with physical 

 notions, besides being at variance with common mathematics? 

 A vector is not a quaternion ; it never was, and never will be, 

 and its square is not negative ; the supposed proofs are perfectly 

 rotten at the core. Vector-analysis should have a purely 

 vectorial basis, and the quaternion will then, if wanted at all, 

 merely come in as an occasional auxiliary, as a special kind of 

 operator. It is to Prof. Tait's devotion to his master that we 

 should look for the reason of the little progress made in the last 

 20 years in spreading vector-analysis. 



Now I have, in my turn, an appeal to make to Prof. MacAulay. 

 I have been much interested in his recent R. S. paper. As 

 the heart knoweth its own wickedness, he will not be surprised 

 when I say that I seem to see in his mathematical powers the 

 "promise and potency" of much future valuable work of a 

 hard-headed kind. This being so, I think it a great pity that 

 he should waste his talents on such an anomaly as the quater- 

 nionic .system of vector analysis. I have examined a good deal 

 of his paper, and can find nothing quaternionic about it except 

 the language concerned in his symbols. On conversion to 

 purely vectorial form, I find that it is greatly improved. I 

 would suggest that he give up the quaternion. If he does not 

 like my notation, or Prof. Gibbs's, or Prof. Macfarlane's, and 

 will invent one for himself, it will receive proper consideration. 

 He will greatly extend the sphere of his usefulness by the con- 

 version. A difficulty in the way is that he has got used to 

 quaternions. I know what it is, as I was in the quaternionic 

 slough myself once. But I made an effort, and recovered my- 

 self, and have little doubt that Prof. MacAulay can do the same. 



Passing to Prof. Tait's letter, it seems to be very significant. 

 The quaternionic calm and peace have been disturbed. There is 

 confusion in the quaternionic citadel ; alarms and excursions, 

 and hurling of stones and pouring of boiling water upon the 

 invading host. What else is the meaning of his letter, and 

 more especially of the concluding paragraph ? But the worm 

 may turn ; and turn the tables. 



It would appear that Prof. Tait, being unable to bring his 

 massive intellect to understand my vectors, or Gibbs's, or 

 Macfarlane's, has delegated to Prof. Knott the task of examin- 

 ing them, apparently just upon the remote chance that there 



NO. 1223. VOL. 47] 



might possibly be something in them that was not utterly 

 despicable. Prof. Knott has examined them, and has made 

 some remarkable discoveries. One of them is that those vector 

 methods in which the quaternion is not the master lead to 

 formulfe of the most prodigious and alarming complexity. He 

 has counted up the number of symbols in certain equations. 

 Admirable critic ! 



Now, since this discovery, and Prof. Tait's remarks, are cal- 

 culated to discourage learners, I beg leave to say, distinctly and 

 emphatically, that there is no foundation for the imputation. 

 Prof. Knott seems to have found a mare's nest of the first 

 magnitude ; unless, indeed, he is a practical joker, and has beerv 

 hoaxing his venerated friend. Speaking from a personal know- 

 ledge of the quaternionic formulae of mathematical physics, and 

 of the corresponding formulae in my notation and in Prof. 

 Gibbs's, I can say definitely that there is very little to choose 

 between them, so far as mere length goes. Perhaps Prof. Knott 

 has been counting the symbols in a Cartesian formula, or in a 

 semi cartesian one, or some kind of expanded form. I do not 

 write for experts who delight in the most condensed symbolism. 

 I do not even claim to be an expert myself. I have to make my 

 readers, and therefore frequently, of set purpose, give expanded 

 forms rather than the most condensed. 



But so far as regards the brief vector formulae, I find that the 

 advantage is actually in my favour. I attach no importance t& 

 this, but state it merely as a fact which upsets Profs. Knott and 

 and Tait's conclusions. It is desirable that I should point out 

 the reason, otherwise the fact may not be believed. In common 

 algebra there is but one kind of product of a pair of quantities, 

 say F and v, which is denoted by Yv. In vector algebra there 

 are two kinds of products. One of these closely resembles the 

 usual product, whilst the other is widely different, being a vector 

 itself. Accordingly, to harmonise with common algebra, I 

 denote the scalar product by Fv. It degenerates to Yv when 

 the vectors have the same direction. Now, since the 

 quaternionists denote this function by - SFv, which is double 

 as long, whilst ± Fv becomes :p SFv, it is clear that there 

 must be an appreciable saving of space from this cause alone, 

 because the scalar product is usually the most frequently 

 occurring function. 



But there are other causes. The quaternionic ways of 

 specialising formulae are sometimes both hard to read and 

 lengthy in execution. Look at S . UaUpS . U)8Up, which I see 

 in Tait's book. I denote this by (oiPj) (;8ip^), or else by 

 «iPi • ^\9\- Tait is twice as long. But the mere shortness is not 

 important. It is distinctness that should be aimed at, and 

 that is also secured by departing from quaternionic usage. 

 Examples of shortening and clarifying by- adopting my notation 

 may be found on nearly every page of Tait's book. 



Consider, for example, rotations. Quaternionists, I believe, 

 rather pride themselves upon their power of representing a 

 rotation by means of a quaternion. Thus, b = q&q'^. The 

 continued product of a quaternion (/, a vector a, and another 

 quaternion </"!, produces a vector b, which is a turned round a 

 certain axis through a certain angle. It is striking that it 

 should turn out so ; but is it not also a very clumsy way of re- 

 presenting a rotation, to have to use two quaternions, one to 

 pull and the other to push, in order to turn round the vector lodged 

 between them ? Is it not plainer to say b = ra, where r is the 

 rotator ? Then we shall have ac = arr'c = ''arc = &c. , if r' is 

 the reciprocal of?-. Then Prof. Tait's 'Vq&q'^q(\> {q'^Xtq) q~^ is- 

 represented by Yra.r^r'\i. See his treatise, p. 326, 3rd edition,, 

 and note how badly the q ( ) q~^ system works out there and) 

 in the neighbouring pages. 



What, then, is this rotator? It is simply a linear operator,, 

 like (^. It is, however, of a special kind, since its conjugate 

 and its reciprocal are one, thus r-/=i, or r'=r~''-. Far be it 

 from me to follow Prof. Tait's example (see his letter) and im- 

 pute to him an " imperfect assimilation " of the linear and 

 vector operator. What I should prefer to suggest is that his 

 admiration for the quaternionic mantle is so extreme that he 

 will wear it in preference to a better-fitting and neater garment. 

 If we like we can express the rotator in terms of a quaternion,, 

 in another way than above, though involving direct operations 

 only. But I am here merely illustrating the clumsiness of the 

 quaternionic formulae in physical investigations, and their un- 

 naturalness, by way of emphasising my denial and disproof of 

 the charge made by Prof. Tait against vectorial methods. The 

 general anti-quaternionic question I have considered elsewhere. 



Paignton, Devon, March 24. Oliver Heavisiue. 



