Notices respecting New Books. 435 



d'une Variable seule. There is a good analysis of the contents, in 

 which the subject-matter of each article is carefully stated. 



An extract from the last article, in which Mr. Davis rapidly 

 glances over the route pursued, will most concisely represent what 

 he has attempted. 



" The commutative, associative, distributive, and index laws 

 that held with the first simplest numbers have been made to hold 

 throughout. Finally, by our extensions, we have made the three 

 primary operations and their four inverses always possible, i. e. 

 always resulting in a number belonging to our scheme. In the 

 reasoning first employed for the handling of incommensurables, 

 and further developed in the treating of growths and rates, we 

 have the germ of ... . the Infinitesimal Calculus. The Graphic 

 representation leads to the Analytical Geometry of Des Cartes. 

 The theory of sines and cosines with its geometric applications is 

 Trigonometry, and leads, by the introduction of new numbers and 

 conceptions, to the Function Theory. The theory of double 

 numbers is simple and restricted, and but a faint suggestion of 

 what is to be found in the beautiful developments proposed by 

 Hamilton and Grassmann, the Peirces, and Sylvester. These are 

 a few only of the lines of thought open to the student. Yet, in 

 whatsoever direction investigation may carry him, he will find his 

 work essentially the same in character. Definitions and conven- 

 tions and their logical consequences and relations make up the 

 whole of it. These relations form the universe wherein the Mathe- 

 matician lives ; a universe, to be sure, of his own construction, a 

 product of his brain, but none the less real and substantial to 

 him. Here he observes and compares, and experiments ; here he 

 reasons out connections, discovers causes, and foretells results." 



It is only in these closing words that Mr. Davis leaves the beaten 

 track, his feelings at having finished his task, we suppose, getting 

 the upper hand of his usual sobriety. The rest of hi3 work is 

 characterized by great clearness in exposition and by fertility of 

 illustration. His Argand diagrams make many points clear. The 

 printing of such a work as this makes great demands upon the 

 author and the printers, yet we have detected but few errors. On 

 p. 7, line 8 up, b has been omitted : p. 10, line 7 up, for first a 

 read b : p. 12, line 13 up, for d read b : p. 32, line 2 up, last state- 

 ment appears to be wrong: p. 42, line 6, ?last statement: p. 45, 

 line 6 up, " +10," should be "— 10": p. 67, line 14 up, same 

 error: p. 73, line 6, for " -2" read " =2": p. 74, line 5, dele 

 last 2. We have noticed some other trivial errors which are 

 easily corrected. 



The student may consult, when using this book, Prof. ChrystaFs 

 ' Text-book of Algebra ' for further illustrations on the subject of 

 Graphs. 



