266 



NA TURE 



[January 19, 1893 



these are brought into line, and are shown to be capable 

 of an explanation in the theory of germ plasma. Di- 

 morphism and seasonal dimorphism are explained by 

 the assumed existence of double determinants, one 

 corresponding with each form, and remaining inactive 

 during the life-time of the organism controlled by the 

 other. 



Such changes as affect the cell body cannot be trans- 

 mitted according to this view, since alterations in the cell 

 have no effect upon the biophore which in the next 

 generation will dominate the corresponding cell of the 

 offspring. But, since variation must be ultimately de- 

 pendent on external circumstances, influences such as 

 climate, change of food, &c., are considered to have in 

 the course of time some effect upon the determinants, 

 and a corresponding change in the organism results. In 

 this lies one of the chief differences between the germ- 

 plasma theory of Prof. Weismann and the Pangenesis of 

 Darwin. They both give a possible explanation of heredity ; 

 but in the latter case the gemmules, coming from every cell 

 of the body, afford an explanation of the transmission of 

 acquired somatic changes; in the former case the biophores, 

 arising only from other biophores, would be uninfluenced 

 by any such change. A. E. S. 



THE BASIS OF ALGEBRA. 

 The Algebra of Co-planar Vectors and Trigonometry. By 

 R Baldwin Hayward, M.A., F.R.S., Senior Mathe- 

 matical Master in Harrow School, formerly Fellow of 

 St. John's College, Cambridge. (London : Macmillan 

 & Co., 1892.) 



THIS work is constructed on the methods of the 

 school of mathematicians who derived their in- 

 spiration from the teaching of De Morgan, a school 

 which is represented by many of the most influential of 

 our recent writers on mathematical subjects. It is in- 

 tended to occupy the place of the " Trigonometry and 

 Double Algebra," published in 1849 and now a long time 

 out of print, at the same time incorporating such im- 

 provements in elementary treatment as have been 

 evolved out of half a century's discussion of the founda- 

 tions of Algebra. Those who are acquainted with Mr. 

 Hayward's other writings, such as his " Elements of 

 Solid Geometry," will expect a fresh and interesting 

 treatment of his subject ; and they will not be disap- 

 pointed. On turning over the pages we constantly come 

 across elegant touches and happy turns of expression, 

 and historical appreciations — the stuff which constitutes 

 the basis of literary excellence in mathematical writings. 

 The treatise is primarily concerned with the logical 

 exposition and illustration of the principles on which 

 Algebraic Analysis, including Analytical Trigonometry, 

 is founded. The utility of this subject in its practical 

 applications renders it a necessary part of even an ele- 

 mentary course of reading ; while a very refined treatment 

 of it may lead so far into the notions of the Theory of 

 Functions and algebraic continuity in general, as to some- 

 what overlay the really simple matters with which it is 

 concerned. In this country the tendency in elementary 

 books has, until recently, been rather to take the funda- 

 mental formulae on credit, and to make the subject 

 NO. I 21 2, VOL. 47] 



consist of the development of their analytical and prac- 

 tical consequences. The philosophical principles which 

 bind them into an organic whole have retained so much 

 the aspect of a posteriori developments, that there is- 

 some temptation to proceed in the view that Mathe- 

 matical Analysis is an inductive science like Natural 

 Philosophy ; that it is one part of the science to invent 

 and verify the formulae, while the logical calculus which 

 gives them precision and limitation is quite another de- 

 partment. The great majority of readers of the elements 

 of Algebra have no time for an exhaustive discussion of 

 the nature of continuous quantity and all the types of 

 singularity to which it is liable ; while on the other hand' 

 complete neglect of the logical basis of Analysis deprives 

 it both of a main source of its interest, and of a large 

 part of its value as an intellectual training. Hence arises 

 the importance, even in presence of the complete theory 

 with which a specialist must be acquainted, of the sim- 

 plification and improvement of methods of exact treat- 

 ment within the domain of elementary ideas. 



The author's method starts off from an a priori dis- 

 cussion of the Algebra of Co-planar Vectors, which leads 

 him to the two modes of specification of a vector, as a 

 power of the fundamental vector, and as a sum of com- 

 ponents. The identification of these two expressions 

 leads to the analytical definitions of the sine and cosine,. 

 and by way of certain theorems in algebraic limits [whose 

 explicit enunciation is by the way not essential] to the- 

 orderly development of the subject. The theorems con- 

 cerning series which involve the complex variable are 

 strikingly illustrated by corresponding vector chains, and 

 geometrical interpretations are throughout very copious. 

 The treatment is here so full and many-sided, that it 

 would form an interesting occupation for the reader to- 

 take up the other aspect of the matter, and try to pick 

 out the simplest and briefest analytical foundation on 

 which the formulas required for practical applications 

 may be built. 



The remarkable formula of p. 115, (4-810475 ....)' = £,, 

 if removed from the context in which it is set, might be 

 propounded as a puzzle in interpretation. The author 

 introduces us straight off to an expression with a com- 

 plex index, and proceeds to ascertain whether any mean- 

 ing can be assigned to it, which will allow the inclusion 

 of such expressions in the algebraic calculus. The 

 geometrical method gives him very neatly and definitely 

 an expression for the values of A^, where A and B are 

 both complex, by means of the vector ribs of a fan of 

 equiangular spiral form. But when the conception of 

 logometers (analytical logarithms) to a vector base comes- 

 up for interpretation, the answer proves to be some one 

 of an infinito-infinite series of vectors related to one 

 another in a manner so complicated as to elude definite 

 grasp ; and we have arrived at a case in which the in- 

 clusion of the function in our calculus would, in the 

 absence of special machinery of representation like a 

 Riemann's surface, be best avoided. 



The origin of all difficulty in the treatment of complex, 

 algebra lies in fact in the multiplicity of values of the 

 functions with which it deals. If each function can be 

 defined as spread out in a multiple sheet so as to be 

 single-valued at each point on each leaf of the sheet, a 

 great part of the trouble disappears. We can then if we 



