6 9 6 SCIENCE PROGRESS 



mathematics that anyone, be he physicist or chemist, can hope to use mathematics 



as a tool. M. Zoretti is right in setting forth to give a treatise suitable for all ; 



he is right also in not giving us a treatise over-elaborated with detail, and written 



with that repellent meticulousness which some mathematicians regard as essential 



to any statement of their subject. I think, however, that he would have produced 



a more readable, and therefore a better book, if he had grouped his chapters 



around certain central ideas and not attempted a comprehensiveness which has 



been acquired at the expense of that quality of luminous power in the exhibition 



of which his countrymen have so often set their English brethren such an excellent 



example. 



C. 



Projective Geometry. By G. B. Mathews, M.A., F.R.S. [Pp. xiv + 349, 

 with diagrams.] (London : Longmans, Green & Co., 1914. Price 5.?.) 



The publication of this book commemorates the centenary of the conception of 

 the subject. For although Poncelet did not publish his Proprictes Projectives 

 before 1822, he tells us that the ideas of the subject were conceived in his 

 brain in a prison in Russia after the retreat from Moscow. In that most un- 

 likely place he demonstrated the greatness of the human spirit by achieving 

 an intellectual triumph more lasting in its results than the political success and 

 failure of the millions who were engaged in that Russian campaign of 181 3-4. 

 In the hundred years which have succeeded, Projective Geometry has been 

 elaborated by German, French, English, and Italian mathematicians, until it is 

 now perhaps the most perfect monument in Pure Logic which has been raised 

 by the genius of man. 



It is one of the few omissions in Mr. Mathews' book that he does not 

 briefly unfold the story of the development of Projective Geometry, placing 

 before us the stages by which the subject was first freed from the metrical 

 conditions under which Poncelet conceived it, and then describing the parts 

 which von Staudt, Laguerre, Cayley, and others played in bringing within its 

 sphere the wider realms which it has conquered in the last sixty years. 



In the English language we have in Projective Geometry a translation of 

 one volume out of three of Reye's Geometrie der Lage, the first volume of a 

 brilliant book by Messrs. Veblen and Young, the second volume of which is 

 awaited with eager expectancy, and two admirable tracts by Dr. Whitehead 

 upon the axioms. But this is the first book in the language which gives a 

 serious account of the real scope of Projective Geometry. 



The book is not one of those which follow the syllabus of examinations ; 

 examinations will, if English mathematicians wake up, follow it. But whether 

 they follow it or not, no student will in future have to be satisfied with the 

 jumble of disconnected theorems and haphazard results which have hitherto 

 masqueraded as Projective Geometry in English text-books. 



Mr. Mathews has not started with the first verse of the first chapter of the 

 first book of geometry as taught by the abstract logical school, and in this he 

 is wise. He does not, however, shirk the genuine difficulties of the subject, 

 as those who follow him will find when they arrive at Chap. VII. and tackle 

 the Fundamental Theorem of Projectivity. In the elementary portions of the 

 subject, which are covered by the first eighteen chapters, the author shows a 

 wise discretion in avoiding the duplicity or quadriplicity of which the various 

 theorems are susceptible. Students of the subject should, however, follow his 

 precept and not his example, and state these theorems completely, as without 



