REVIEWS 697 



always to co-operate with the author, and it is very rarely that authors, in writing 

 their manuscripts, reflect upon the effect that it will have when it is set up by the 

 compositor. It would seem as if there was room for a third party who could take 

 the complicated formula as it leaves the writer's pen, and without altering its 

 meaning, yet by help of solidus or negative indices arrange the masses of symbols 

 into a more attractive and artistic form than that in which they generally issue 

 from the press. 



C. 



Ruler and Compasses. By Hilda P. Hudson, M.A., Sc.D. [Pp. vi + 144, 

 with Diagrams.] (London : Longmans, Green & Co., 1916. Price 6s. net.) 



IT is well known that Euclid put at the beginning of his Elements the postulates 

 that a straight line might be drawn between any two points, that such a straight 

 line might be produced to any length in the same straight line, and that a circle 

 might be described from any centre and at any distance from that centre ; and 

 that all the constructions used in the first six Books are made with these three 

 operations only. The many attempts to solve the great problems of the dupli- 

 cation of a cube, trisection of an angle, and quadrature of a circle by what were 

 thus called M Euclidean methods " failed, and such attempts were analytically 

 proved in more modern times to be necessarily vain. On the other hand, certain 

 constructions can be carried out by Euclidean methods which were not given by 

 Euclid. It is the purpose of this excellent little book to investigate, both 

 analytically and geometrically, how far Euclidean constructions can carry us, al- 

 though no attempt is made to give an account of work on transcendental numbers. 

 It is well remarked (p. 3) that " the ancient or classical geometry lends itself 

 curiously little to any general treatment ; and even modern geometry lacks a 

 notation or calculus by which to examine its own powers and limitations " ; thus 

 it has to be by using co-ordinate methods that Miss Hudson attacks the general 

 question as to the problems which can be solved by ruler and compass (Chap. II.). 

 The second chapter also contains an answer to the question as to what regular 

 polygons are within our powers of construction, and Richmond's (1909) construc- 

 tion of one of seventeen sides is given on p. 34. Chaps. III. and IV. are devoted 

 to ruler constructions and ruler and compass constructions respectively ; Chap. V. 

 is concerned with a classification of methods ; and Chap. VI. is devoted to a 

 comparison of methods. The last two chapters (VII. and VIII.) are concerned 

 with the solution of Euclidean problems by drawing only one circle and the 

 requisite number of straight lines (Poncelet and Steiner), or else by drawing 

 circles only (Mascheroni and Adler). 



This is an exceedingly competently written book, and there are only a few 

 little criticisms that might be offered. In the first place, the way of printing 

 analytical formulae does not strike one as wholly pleasant, though this is probably 

 due to the fact that one is used to small italic letters for the purpose. In the 

 second place, it is surely rather misleading to the student continually to speak 

 as if the geometry of the ancient Greeks were intimately concerned with drawing 

 instruments and the material on which figures were drawn (cf. pp. 1, 6, 70): 

 although the solution of the problem as to why Euclid stopped at a postulate that 

 lengths may be carried about from place to place, when he had admitted that one 

 end of a straight line may be carried about if the other end is fixed, may, perhaps, 

 be answered historically by such considerations. This book is a very valuable 

 addition to " Longmans' Modern Mathematical Series." 



Philip E. B. Jourdain, 



