NATURE 



385 



THURSDAY, AUGUST 25, 1898. 



COMPARATIVE ALGEBRA. 

 A Treatise on Universal Algebra^ with Applications. By 

 Alfred North Whitehead, M.A. Vol, I. Pp. xxvi 

 + 586. (Cambridge : at the University Press, 1898.) 



THIS work affords a sad illustration of the spirit of 

 lawlessness which has invaded one of our ancient 

 Universities since the time when she rashly began to 

 tamper with her Tripos Regulations. In the good old 

 times two and two were four, and two straight lines in a 

 plane would meet if produced, or, if not, they were 

 parallel ; but it would seem that we have changed all 

 that. Here is a large treatise, issued with the approval 

 of the Cambridge authorities, which appears to set every 

 rule and principle of algebra and geometry at defiance. 

 Sometimes ba is the same thing as <?^, sometimes it isn't ; 

 a -f- a may be 2a or a according to circumstances ; 

 straight lines in a plane may be produced to an infinite 

 distance without meeting, yet not be parallel ; and the 

 sum of the angles of a triangle appears to be capable of 

 assuming any value that suits the author's convenience. 

 It is a pity that we have not had an opportunity of show- 

 ing the book to some country rector who graduated with 

 mathematical honours, say, forty years ago ; it is easy 

 to imagine his feelings of surprise, bewilderment, possibly 

 of indignation, as he turned over the pages and en- 

 countered such a variety of paradoxical statements and 

 unfamiliar formulae. 



Seriously, Mr. Whitehead's work ought to be full of 

 interest, not only to specialists, but to the considerable 

 number of people who, with a fair knowledge of mathe- 

 matics, have never dreamt of the existence of any algebra 

 save one, or any geometry that is not Euclidean. Its 

 title, perhaps, hardly conveys a precise idea of its con- 

 tents. It is, in fact, a comparative study of special 

 algebras, exclusive of ordinary algebra, the results of 

 which are taken for granted throughout. Such an under- 

 taking has necessarily involved a very great deal of time 

 and labour ; for, in order to carry it out with any degree 

 of success, it is needful, not only to master each separate 

 algebra in detail, but also to adopt some general point of 

 view, so as to avoid the imminent risk of composing, not 

 one work, but a bundle of isolated treatises. Mr. White- 

 head has, happily, overcome this difficulty by viewing the 

 different algebras, in the main, in their relation to the 

 general abstract conception of space. Whether this plan 

 can be consistently followed throughout may be open 

 to question : it certainly works very well in this first 

 volume, the keynote of which is Grassmann's Extensive 

 Calculus. 



The first special algebra dealt with, however, appeals 

 to a much simpler range of spatial ideas ; it is the Algebra 

 of Symbolic Logic, which only requires the conception of 

 closed regions of space which may or may not overlap. 

 This algebra is charmingly simple : it does not involve 

 any arithmetical calculations, or even the use of digits, 

 because both a -\- a and aa are equivalent to a ; and it 

 enjoys a perfect dualism, so that from every proposition 

 (not self-reciprocal) another may be at once inferred. 

 On its value in its logical applications, it would be unwise 

 NO. 1504, VOL. 58] 



for a mere mathematician to express an opinion, and the 

 moral philosophers themselves appear to be of different 

 minds on this as on some other questions ; but this does 

 not detract from its merits as an algebra of extreme 

 simplicity, combined with symmetry and grace. 



The next three Books (II I. -V.) deal with positional 

 manifolds, the calculus of extension, and extensive 

 manifolds of three dimensions. In this very important 

 section the reader will find a systematic development of 

 the extensive calculus, with abundance of illustrative 

 applications ; so that English mathematicians will no 

 longer have any excuse for ignoring Grassmann's magni- 

 ficent conceptions. Time alone can show whether, as an 

 instrument of discovery, Grassmann's calculus will prove 

 superior to the ordinary methods ; but of its power as a 

 means of expression there can only be one opinion. To 

 see this the reader has only to turn, for example, to the 

 chapters on line geometry (Book V., Chapters i.-iii.), 

 where the properties of null systems, the linear complex, 

 and the invariants of groups of line systems (or, as the 

 author prefers to call them, systems of forces) are proved 

 with extreme directness and simplicity. The crux of 

 the calculus is the theory of regressive and inner multi- 

 plication, which is discussed in Book IV., Chapters ii., iii. : 

 the reader may be recommended to study these chapters 

 in connection with the applications which follow, especi- 

 ally in Book V., Chapter i., where the formuUe for three 

 dimensions are recapitulated. The idea of intensity is 

 introduced at the outset, and the exposition follows 

 mainly the Ausdehnungslehre of 1862 : this procedure 

 certainly has its advantages, but makes the extensive 

 calculus appear more closely allied to the barycentric 

 calculus than it naturally is. 



Book VII., on the application of the extensive calculus 

 to geometry, is largely concerned with vectors. From 

 Grassmann's point of view a vector, or, as he called it, 

 a "Strecke," is the difference between two extensive 

 magnitudes of equal weight ; with an appropriate law of 

 intensity, it may also be regarded, in a sense, as a point 

 at infinity. But there is a certain convenience, when 

 working with vectors, in regarding them as independent 

 elements, after the manner of Hamilton : this method is 

 explained in Chapter iv. of the Book, which contains a 

 number of kinematical and dynamical formula. Chapter 

 iii., on curves and surfaces, illustrates very fairly both the 

 strong and the weak points of the calculus. 



Book VI. contains a detailed account of the theory of 

 metrics. It is very refreshing to find that this theory 

 is treated by the author in a thoroughly satisfactory 

 way, without any of the sham metaphysics and faulty 

 psychology which so often disfigure it, especially when 

 an attempt is made to expound these abstract ideas to 

 a popular audience. Starting with the purely abstract 

 definition of a positional manifold, it is possible to con- 

 struct a theory in which there is associated with any two 

 elements of the manifold a numerical quantity called 

 their distance, which may be finite or infinite, real or 

 imaginary, but which only vanishes when the elements 

 coincide. In order to satisfy certain axioms which are 

 analogous to some of the assumptions tacitly or explicitly 

 made in ordinary geometry, and the fundamental theorem 

 of projective geometry that if three points of a row of 

 points are congruent to the three corresponding points 



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