Notices respecting New Books. 



317 



easily. The title is " A Drill-book in Algebra." The authors are the 

 publishers of their works, which are well printed and sold at a low 

 price. Those we have noticed above are excellent of their kind, 

 and are evidently written by men who are experienced in teaching 

 young men of ordinary ability. 



The Number- System of Algebra treated Theoretically and Historically. 

 By Henry B. Fine, Ph.D. (Boston and j\ T ew York : Leach, 

 Shewell, and Sanborn; pp. x + 131.) 



We simply call attention to this small work, which presents in a 

 readable form much of what is given in the larger works of Hankel 

 (Complete Zahlensystenien), M. Cantor (Vorlesungen iiber die 

 Geschichte der Maihematih), Weierstrass, Thomse, and other Conti- 

 nental writers on the nature of the number-concept and the 

 number-system of Algebra. The discussion follows the same lines, 

 in the main, as the author's previous pamphlet ' On the Forms of 

 IS T umbers arising in Common Algebra.' 



A Treatise on Plane Trigonometry. By E. "W. Hobson. (Cam- 

 bridge : University Press, 1891; pp. xvi-|-356.) 

 The gratification we have derived from a perusal of this work is 

 akin to that we felt on reading Chrystal's 'Algebra/ The thorough- 

 ness and yet withal the freshness with which our author has 

 treated his subject puts his book in the first rank of Mathematical 

 publications lately issued by the University Press. The theory of 

 the circular functions is treated from the modern point of view, 

 and yet the elementary parts are worked out in sufficient detail 

 for students " whose range of reading is to be more limited " than 

 that of those readers to whom the work is primarily addressed. 

 It is pre-eminently a strong book, and will be of prime value to 

 advanced University men, who will find the principles of the theory 

 of Complex Quantities clearly discussed in a manner similar to that 

 of the 'Algebra' above referred to. Even in the most recent works 

 Analytical Trigonometry "has been too frequently presented to 

 the student in the state in which it was left by Euler, before the 

 researches of Cauchy, Abel, Grauss, and others had placed the use 

 of imaginary quantities, and especially the theory of infinite series 

 and products, where real or complex quantities are involved, on a 

 firm scientific basis." There are eighteen chapters in all. 



Tn a short chapter (n.) the method of projections is introduced, 

 and this method is subsequently effectively used in conjunction 

 with the definitions of the circular functions employed by 

 De Morgan (see 'Double Algebra'). These definitions appear to 

 the author to be " those from which the fundamental properties of 

 the functions may be most easily deduced in such a way that the 

 proofs may be quite general, in that they apply to angles of all 

 magnitudes. It will be seen that this method exhibits the formula? 

 for the sine and cosine of the sum of two angles, in the simplest 

 light, merely as the expression of the fact that the projection of 

 the hvpothenuse of a right-angled triangle on any straight line in 

 its plane is equal to the sum of the projections of the sides on the 

 same line." The periodicity of these functions is clearly shown, 



