•46 



NA TURE 



[Dec. 15, 1887 



Valentin Balbin, delivered to an audience comprising 

 several of his colleagues. The volume, written in 

 Spanish, has been printed at Buenos Ayres, and in 

 size (xix. and 359 pages), and in quality of paper and 

 print, presents a very handsome appearance. 



In his preface the author informs us that in his opinion 

 the calculus of quaternions is the best vehicle for the 

 teaching of applied mathematics, and that therefore he has 

 had recourse to Sir William Rowan Hamilton's beautiful 

 invention. The author is aware of the fact that he is 

 the first to introduce quaternions into the Spanish scien- 

 tific literature, and for this reason he aims at presenting 

 the theory from its very elements up to its higher branches 

 of application. 



In the matter of notations we are also informed that 

 those of Hamilton and of Prof. P. G. Tait have been 

 scrupulously adhered to, and that, in one word, the 

 author has not found it advisable to follow those of M. 

 Hoiiel and of M. Laisant. It may not be known to 

 everybody that these two French mathematicians have in 

 their publications (1874, 1877, 1881) adopted a thorough 

 reversal of Hamilton's lettering. In the place of the 

 inventor's Greek letters, they use Roman characters 

 (X for p, Y for a-, A for a, &c.) ; and in the place of the handy 

 S, y, T, U, they put black-letter symbols, which are at 

 once difficult to write, and tiring to the eyesight. These 

 are what they call " slight alterations " or " improve- 

 ments." 



Again, they upset the rule about the relative appellation 

 of the factors of a product. Our author (p. 40) states the 

 Hamiltonian rule, and justifies it by the simple example 

 drawn from a -\- a = 2a, where (according to everybody's 

 ideas) the coefficient 2 is the multiplier, and a the multi- 

 plicand. According to the " innovated" rule one ought 

 to write a -\- a = aX 2. 



The rule just named makes its influence felt more 

 particularly in the establishment of the operator of conical 

 rotation, and here we are sorry to find that our author 

 falls a victim to a delusion. Instead of Hamilton's well- 

 established q{ )q'^, he arrives at the inverse q-^{ )q 

 (at p. 296), and uses it under this form through several 

 pages (up to p. 303). This comes from following M. 

 Laisant's text, and forgetting his own rule. In M. Hoiiel's 

 opinion, " nothing is easier than to pass from one system 

 (his own system) to the other" ; nevertheless, such passage 

 requires to be nicely managed, because by it the expres- 

 sion for the instantaneous axis is affected, and we might 

 ask whether it be fair to introduce a source of confusion 

 into a theory which in itself is difficult enough. Our 

 author does not introduce us to the searching treatment 

 which Prof. Tait has devoted to the question of the move- 

 ment of a solid about its centre of mass (" Elementary 

 Treatise," &c., second edition, §§ 383-400). M. Balbin's 

 treatment of that question is very curtailed, and we might 

 be inclined to attribute this shortness to a feeling of 

 distrust, otherwise how could we understand his utter- 

 ance, at p. 87, where he says forcibly, " Some simplifications, 

 particularly in the physico-mathematical applications, 

 must be made in the future as to the matter of symbols " 

 (.f(? hagan, imperative oi haccr). 



The more we consider the innovations, the more are 

 we convinced that their proposer and his follower, publish- 

 ing in 1874 and i8Sr, had not fully realized the extent and 



importance of the researches which, during many years, 

 had been expressed in what we may term the Hamil- 

 tonian notations. In 1862 no less a Frenchman than M. 

 Allegret set the example of following these last-named, 

 and that precedent ought to have been adhered to. As 

 it is, students of MM. Hoiiel and Laisant will be 

 hampered by the French notations when they approach 

 those rich mines of information contained in such unique 

 classics of the quaternion method as Hamilton's " Lec- 

 tures " and " Elements," and the " Elementary Treatise on 

 Quaternions," by Prof. Tait. 



Let us now try to give some idea of the contents of the 

 volume. For the English student these are all contained 

 in the sources known to him. First " The Introduction to 

 Quaternions," by Kelland and Tait. This work has been 

 reproduced in its whole extent, with the exception of 

 Chapter X.,due to Prof. Tait alone. The author acknow- 

 ledges in several instances (pp. 1 14, 120," 252) his 

 special indebtedness to the English authors ; and his 

 translations are adequate. Perhaps, however, he knows 

 them best through the medium of M. Laisant's reproduc- 

 tion of a great part of Kelland's work (with acknowledg- 

 ments in the preface, tempered by the praise of the new 

 notations). 



In the second place, in the treatment of linear vector- 

 functions and the resolution of equations involving them, 

 which were originally given by Hamilton, there are clear 

 indications that our author has taken his text from works 

 where the innovatred notations reign supreme ; some 

 traces of x (at pp. 183, 184, 193), for instance, are left 

 standing in the place of p, and are contained concurrently 

 with p in one and the same equation, in several cases ; no 

 explanation about the signification of x being given. A 

 similar fate befell the vector p, at p. 13S, where a- is put 

 into its place by being defined: x—ix^-\-jx2-\-kx.^. 

 Under this form x is introduced into the operator v, 

 which in its turn undergoes a little adaptation. But all 

 this is not the promised adhering to Hamiltonian nota- 

 tions. 



The solution of the vector-equation 2aS3p = y is gone 

 partially into ; but the calculation of the coefficient 7n of 

 the cubic (at p. 192) contains an inexact intermediate 

 step, and the coefficient Wi is given with the wrong sign. 

 Finally the solution of the proposed equation (p. 193) is 

 incorrect, owing to the absence of the factor y in the first 

 term of the second member. These three inaccuracies 

 can be traced to one of the French texts. 



In the third place, curves in space, and centres of 

 curvature of those curves, and of plane sections'of sur- 

 faces, subjects exhausted by Hamilton and by Prof. Tait, 

 have been treated by our author with the help of Ur. 

 Graefe's little volume on Quaternions (Leipzig, i88j). We 

 might ta'^e exception to Dr. Graefe's deduction (p. 236 of 

 Balbin's) of Meusnier's theorem, as well as of that of the 

 curvature of a normal section of a surface. To replace a 

 scalar, say Sa^, by l{aS-\-^a), in order to procure an 

 expression of the product a,3 separately, seems to us to be 

 forsaking the spirit of the method of quaternions ; the 

 expression for Sa,3 being given, and that of Va^ being 

 deducible from other considerations, it would have been 

 far simpler to deduce a,3 by forming the sum Sa,3 -f Va8 

 straight forward. Some reticences (p. 236), and even 

 some inaccuracies, in the text of Dr. Graefe, have been 



