300 Notices respecting New Books. 



Supplement to the First Boole of Euclid's Elements, containing the 

 Sixth-Booh Propositions proved independently of the Fifth Book, 

 and the Elementary Propositions of Modem Geometry. By Edward 

 Butler, M.A.T.C.D. Dublin: Alexander Thorn. 1872 (12mo, 

 pp. 60). 



Euclidian Geometry. By Era^cis Cuthbertson, M.A., late Fellow 

 of Corpus Christi College, Cambridge, Head Mathematical Master 

 of the City of London School. London : Macmillan and Co. 

 1874 (fcap. 8vo, pp. 266). 



The titlepages of these works sufficiently indicate their pur- 

 pose ; they are intended to be substitutes for Euclid's Geometry, 

 or for part of it, and while retaining the form and spirit of the 

 original, to improve on it in detail and to supplement its supposed 

 deficiencies. Both books are, to all appearance, Avritten by mathe- 

 maticians of average competency ; and one, at least, of the authors 

 (Mr. Cuthbertson) is in a position which makes it highly probable 

 that he is a teacher of considerable experience*. But though this 

 is the case, we regret to have to add that, after having looked into 

 these books with some care, we do not see why they have been pub- 

 lished. 



With Mr. Butler's book the difficulty is not so great as with 

 Mr. Cuthbertson's. We may surmise that it is adapted to some 

 course of instruction sanctioned by the Commissioners of National 

 Education in Ireland ; and if so, its somewhat fragmentary form is 

 explained. It consists of a number of propositions designed as a 

 substitute for Euclid's sixth book, and a selection of elementary 

 propositions on Harmonic and Anharmonic Section and some allied 

 subjects. It is hardly necessary to notice this book further; and 

 were we asked for the reason, we should regard it as a sufficient 

 answer to state that Mr. Butler's definition of proportion runs 

 thus : — " Eour straight lines are said to be proportionals, that is, 

 the same ratio the first to the second as the third to the fourth, 

 when the rectangle contained by the first and fourth is equal to 

 the rectangle contained by the second and third." Putting out of 

 the question the typographical error which may be presumed to 

 exist in the passage, the sentence betrays a view of the function 

 of definition which is above or below criticism. 



Mr. Cuthbertson's book covers just the same ground as Euclid's 

 Books 1-6, and the first twenty-one propositions of Book 11. He 

 has manifestly expended a great deal of care and thought on its 

 composition, and yet we are constrained to say that his attempt 

 to improve on Euclid is a failure. In the first place, his book is 

 about as long and quite as abstruse as Euclid's. In the next, he 

 has increased the difficulty of his task by adapting his book to a 

 certain form of examination which our limits will not allow us to 



* We do not know what position Mi'. Butler holds ; he calls himself " Pro- 

 fessor &c. under the Commissioners of ^National Education in Ireland." Wo 

 are wholly in the dark as to the meaning of the "&c." 



