89 



treated than in any prior work and that it Is the only worli in which 

 general equations beyond the fourth degree are solved. It is also the 

 only book that shows the fallacies in Abel's proof that equations of higher 

 degree than the fourth can not be solved by radicals. 



That the book is interesting goes without saying. No one who prom- 

 ises so much can fail to write in an interesting manner. One follows 

 breathlessly to see the kind of a paradox that will be produced. 



A number of simple theorems in the theory of numbers and the 

 theory of equations are stated as though they were new. 



On page 53, article 164, we read: "The roots of quadratics represent 

 the sides of right triangles when Real Quantities; the sides of isosceles 

 triangles when Real Imaginaries; and when Pure Imaginnries may be 

 represented by lines." His argument for the latter part of the statement, 

 it is needless to say, is not convincing. 



A number of special numerical problems in equations of various de- 

 grees are solved. In many of these some very ingenious special methods 

 are exhibited. 



One chapter is devoted to the discussion of Wantzel's modification of 

 Abel's proof of the impossibility of an algebraic solution of equations 

 of higher degree than the fourth. The character of the discussion can 

 be best understood by quoting the conclusion. "If we should accept 

 his (Wantzel's) demonstration as true, we would be forced to the conclu- 

 sion that the general equation of a degree higher than four was destitute 

 of roots. The concluson of Wantzel that the roots can not be indicated in 

 algebraical language is equivalent to saying that there are no roots, 

 since it is absurd to say that finite quantities exist which can not be 

 expressed in any function of other finite quantities, which are themselves 

 symmetrical functions of the first, however complicated." 



The author's notion of the imaginary is summed up in a general 

 theorem, as follows: "An Imaginary Quantity is the indicated square 

 root of the difference of the squares (with its sign changed) of the bases 

 of two right triangles having a common perpendicular which is the 

 radius of a circle; two of such triangles lying wholly within the semi- 

 circle, and two partly within and partly without the semicircle." What 

 the theorem or the demonstration means would be hard to tell. 



V— A. OF Science. 



