April 20, 1893] 



NATURE 



591 



ulvocated by Gibbs and Heaviside. Two products of vectors 

 ue defined, which correspond to Hamilton's Vafl and — Sa/3 ; 

 ind applications are given of the linear and vector function and 

 of the operator ady + 39-2 + "i^i which somewhat resembles 

 the quaternion v. 



The broad argument advanced by Gibbs in his letter to 

 Nature is that, in comparison with the quantities Vafl and 

 SyVojS, which symbolise an area and a volume which "are the 

 very foundations of geometry," anything that can be urged in 

 favour - f the quaternion product or quotient as a '■'■fundamental 

 notion in vector analysis " is " trivial or artificial." These are 

 brave words. Let us examine them by considering what is the 

 purpose of a vector analysi*;. Clearly such a calculus is intended 

 to show forth the properties of vectors in a form suitable for 

 ase. 



Having formed the conception of a vector, we have next to 

 find what relations exist between any two vectors. We have to 

 compare one with another, and this we may do by taking either 

 their difference or their ratio. The geometry of displacements 

 and velocities suggests the well-known addition theorem — 



o + 5 = /3 



in which by adding the vector 5 we pass from the vector a to the 

 vector )8. 



But this method is not more fundamental geometrically than 

 the other method which gives us the quaternion. When we wish 

 to compare two len>;th,> a and h, we divide the one by the other. 

 We form the quotient ajb, and this quotient is defined as the 

 factor which changes b into a. Now a vector is a directed 

 length. By an obvious generalisation, therefore, we compare 

 two vectors by taking their quoient and by defining this 

 quotient o//8 as the factor which changes the vector & 

 into the vector o. This is the germ out of which the whole of 

 vector analysis naturally grows. A more fundamental concep- 

 tion it is hardly possible to make. Yet Gibbs calls it trivial and 

 artificial ! Far more fundamental, we are told, are the concep- 

 tions of a vector bounded area and a vector bounded volume, 

 whose bounding vectors may have an infinity of values. Or 

 take the more general case of a body strained homogeneously. 

 Ttie relative vector of any two of its points passes into its new 

 position by a process which is a combination of stretching and 

 turning. A simpler and more complete description cannot be 

 imagined. It is perfectly symbolised by the quaternion with its 

 tensor and versor factors. And this is trivial and artificial — as 

 trivial, say, as the versor operation which every one performs when 

 estimating the time that must be allowed to catch a train. . . . 



Another a-gument advanced by Willard Gibbs is in the para- 

 graph beginning : " How much more deeply rooted in the nature 

 of things are the functions So/3 and Va^S than any which depend 

 on the definition of a quaternion, will appear in a strong light 

 if we try to extend our formulfe to space of four or more dimen- 

 sions." To elucidate the "nature of things" by an appeal to 

 the fourth dimension — to solve the Irish question by a discussion 

 of social life in Mars — it is a grand conception, worthy of the 

 scorner of the trivial and artificial quaternion of three dimen- 

 sions. Further on we are told that there " must be vectors in 

 such a space " ; that is, space of four or more dimensions. 

 True, and if there be vectors, must there not be operations for 

 changing one vector into another? .... 



"Vectors must be treated vectorially " is a high-sounding 

 phrase uttered by Prof. Henrici and Mr. Heaviside What 

 does it mean? On the same sapient principle, I suppose, 

 scalars must be treated scalarially, rotors rotorially, algebra 

 algebraically, geometry geometrically. That is to say the 

 remark is a very loose statement of a truism, or it is profound 

 nonsense. S rictly speaking, to treat vectorially is to treat 

 after the manner of vectors, or to treat as vectors do. 



Now what does a vector do? Prof. Gibbs, the prince of 

 vector purists, says on page 6 of his pamphlet that "the eftect 

 of the skew [or vector] multiplication by a [any unit vector] 

 upon vectors in a plane perpendicular to o is simply to rotate 

 them all 90° in that plane." Hence a vector ?> a versor. To 

 which Mr. Heaviside in fierce denunciation : " In a given 

 equation [in quaternion-vector analysis] one vector may be a 

 vector and another a quaternion. Or the same vector in one 

 and the same equation may be a vector in one place and a 

 quaternion (versor or turner) in another. This amalgamation 

 of the vectorial and quaternionic functions is very puzzling. 

 You never know how things may turn out." Puzzling? 

 Then must Heaviside find his own system as puzzling as any. 



NO. 1225. VOL. 47] 



For when he writes the vector product ij=k, he is simply acting 

 on / by / or on i byy, and turning it through a right angle. It 

 is impossible to get rid of this versorial effect of a vector. It 

 stares you in the face from the very beginning. 



A very sore grievance with Heaviside and Macfarlane — 

 although Gibbs cautiously steers clear of the whole question — is 

 that Hamilton puts i-, j\ k-, each equal to negative unity, with 

 the consequence that Saj8 is equal to - ab cos 9, where a and b 

 are the lengths of o and 8, and d the angle between them. Thb 

 putting the square of a vector equal to minus the square of its 

 length vexes their souls mightily. It is so "unnatural," so trouble- 

 some. 



Now Prof. Kelland, in Kelland and Tail's "Introduction to 

 Quaternions," chap, iii., shows that if we assume, as do Heavi- 

 side and Macfarlane, the cyclic relations 



ij=zk—-ji jk = i=—kj ki=j= -ik, 



and if in addition we desire an associative algebra, then of neces- 

 sity we must have f-=/2=^! = — i. If then, following these 

 O'Brienites, we put what they consider to be so much simpler 

 and more natural, namely, t"- = y- = /6'= -f- 1, we get a non- 

 associative algebra of appalling complexity, which in the long 

 run gives us no more than the associative quaternion algebra. 



Heaviside apparently is unaware of the non-associative 

 beauties of his system, which he believes "to represent what 

 the physicist wants ; " for he says, much to the credit of the 

 Philosophical Transnctions, that his system (which is 

 demonstrably not quaternions) is "simply the elements of 

 quaternions without the quaternions, with the notation simplified 

 to the uttermost, and with the very inconvenient minus sign 

 before scalar products done away with" {Phil. Trans., vol. 

 clxxxiii. 1892, p. 428). 



We have seen how perfectly natural is the geometric concep- 

 tion of a quaternion as the quotient of two vectors ; and the 

 quaternion product is as simply conceived of as the operator 

 (aj8) which turns the vector &~' into a. Space considerations 

 quickly lead us to consider quaternions which rotate a given 

 vector through a right angle. If we take two such right or 

 qnadrantal quaternions I I' and operate severally on the vector 

 a that is perpendicular to the axes of both, it is easy to show 

 that 



la -J- I'a = (14- I') a 



gives a right quaternion (I-fT) bearing to I and I' the same 

 relation which would exist were I and I' vectors. Tnat is, right 

 or quadrantal quaternions are added and subtracted according to 

 the recognised rules for vector addition and subtraction, which 

 so far, be it noted^ are all we know about vectors. Hence in 

 combinations other than addition and subtraction we may treat 

 vectors as quadrantal quaternions, exactly as Gibbs, Heaviside, 

 and Macfarlane do, although in a half-hearted fashion. 



It remains now to consider wherein the systems advocated by 

 these vector analysts improve upon Hamilton's. Do they give 

 us anything of value not contained in quaternions? 



Prof. Gibbs, having objected in toto to the quaternion pro- 

 duct a&, is for consistency's sake bound to object to Hamilton's 

 selective principle of notation. His own notation is very similar 

 in appearance to O'Brien's of old. He defines two products, 

 the </«■;-.?<:/ product (a . 6) and the skew product (a x 3). The 

 direct product is Grassmann's inner product or Hamilton's 

 - Saj3; and the skew product is Va/S, so called probably 

 because it has a value "nly when a and 8 are skew, or 

 inclined to one another. Now there is a serious objection at 

 the very outset to such a form as o x )8 for the vector product 

 of a and /3. There corresponds to it no quotient amenable to 

 symbolic treatment. The reason is, of course, that a x 8 is not 

 a complete product. Given the quaternion equation a/S = //, 

 any one quantity is uniquely determined if the other two are 

 given. But it is impossible, in spite of the suggestiveness of the 

 form, to throw Prof. Gibhs's o x /3 = 7 into any such shape as 

 a = ^ -^ j8. The point is that Hamilton's notation does not even 

 suggest the possibility of such a transformation. It is certainly 

 inexpedient, to say the lea-t, to use a notation strongly resem- 

 bling that for multiplication of ordinary algebraic quantities, but 

 having no corresponding process by which either factor can be 

 carried over as a generalised divisor to the other side of the 

 equation. 



One peculiar perspicuity of Hamilton's notation arises from 

 the fact that S and V are thrown out in bold relief from amongst 

 the small Greek letters used for vectors and the small 



