Maech 3, 1899.] 



SCIENCE. 



327 



the two parts add to form the fiual result, but 

 in the case where the square root of minus one 

 is present the sign must be preserved in the 

 intermediate processes of calculation. A com- 

 plex index (both terms involving a sign of direc- 

 tion) has its meaning in an angle which is partly 

 circular, partly hyperbolic; and a scalar complex 

 quantity expresses the cosine or siue of such 

 complex angle. It follows that the functions 

 of a complex quantity can be defined really. 

 It has been the practice of writers to follow the 

 formal view, and define, for instance, the cosine 

 of a complex quantity as the sum of a certain 

 infinite series. Let z denote a complex quan- 

 tity, then, according to that view, by cos z is 

 meant the sum of the series 





-etc. 



But when the cosine of a complex angle is de- 

 fined in the same manner as the cosine of a cir- 

 cular angle or of a hyperbolic angle, namely, 

 as the ratio of the projection of the radius-vec- 

 tor to the initial line, then 



■- + 

 2 ' 



4 ! 



■ etc. , 



becomes not a dead convention, but a living 

 truth. 



In the first book the author states more fully 

 the principles of universal algebra: "There 

 are certain general definitions which hold for 

 any process of addition and others which hold 

 for any process of multiplication. These are 

 the general principles of any branch of uni- 

 versal algebra. But beyond these general defi- 

 nitions there are other special definitions which 

 define special kinds of addition or multiplica- 

 tion. The development and comparison of 

 these special kinds of addition or of multipli- 

 cation form special branches of universal al- 

 gebra," p. 18. The general principles are as 

 follows : Addition follows the commutative and 

 associative laws, viz : a -'r b^b -\- a and {a -\- b) 

 -f c = a -t- (6 -F c). Multiplication follows the 

 distributive law, viz : a (c -\- d) ^= ac -\- ad and 

 (a -|- 6) c = ac -\- be. Multiplication does not nec- 

 essarily follow the commutative and associative 

 laws, that is, ab = ba and (ab) c = a (be) are laws 

 of special branches only. It has been main- 

 tained by followers of Hamilton that the asso- 



ciative law is essential to multiplication. It is 

 true of spherical c[uaternions, but is not true of 

 the complementary branch of vector analysis. 

 It is satisfactory to find that Mr. Whitehead 

 adopts the latter view, and, indeed, it is in- 

 vcilved in his detailed exposition of vector 

 analysis in the concluding book of his first 

 volume. 



But one who looks upon algebraic analysis 

 not as the sum of several correlated branches, but 

 as one logical whole, must consider the above 

 principles or so-called definitions as arbitrary. ■ 

 For let p and q denote two quaternions, then 

 e^e' is not in general equal to e''e^\ conse- 

 quently 6' + ' is not equal to e« + ^; hence the 

 commutative law does not hold in the addition 

 of these indices. Thus to define addition as 

 necessarily following the commutative law, and 

 multiplication as not necessarily following it, is 

 an arbitrary procedure. 



In expounding the algebra of logic the author 

 follows largely the exposition of Dr. Schroeder 

 in his learned treatise, ' Vorlesungen iiber die 

 Algebra der Logik,' but he does not take up 

 the most valuable part of that work, namely, 

 the Algebra of Relatives. Symbolic Logic as 

 expounded by Schroeder differs essentially from 

 the calculus devised by Boole in his ' Laws of 

 Thought.' It was Boole's aim to keep as close 

 as possible to ordinary algebra, and to make 

 his method the foundation of a calculus of prob- 

 abilities. In fact, the full title of his famous 

 book is ' An Investigation of the Laws of 

 Thought on which are founded the mathemat- 

 ical theories of Logic and Probabilities.' Ac- 

 cording to Boole the special peculiarity of the 

 algebra is that x~ = x, when x is an elementary 

 elective symbol. Jevons is said to have intro- 

 duced the further supposed law that x-\- x=:x, 

 which destroys the quantitative character of 

 the calculus. Indeed, Mr. Whitehead says that 

 the algebra is non-numerical, and in Dr. 

 Schroeder' s elaborate work no mention is made 

 of probabilities. According to the more recent 

 school a — --6 supposes that b is included in a 

 (p. 82), whereas Boole made no such limitation. 

 It is a step backwards, just as it would be a 

 step backwards in ordinary algebra to hold 

 that a — b carries the supposition that 6 is less 

 than a. 



