REVIEWS 669 



they are often drawn from physics, chemistry, radio-activity, and probability, 

 while there is a problem about finding Sir Ronald Ross's equation in his theory of 

 the infection of a population (p. 296), and problems about the rate at which eggs 

 go bad (pp. 298, 303). The only mistakes noticed are the wrong spelling of the 

 names of Bendixson (p. 252) and Leibniz (e.g. on p. 262). 



Philip E. B. Jourdain. 



Projective Geometry. By Oswald Veblen, Professor of Mathematics, 

 Princetown University, and John Wesley Young, Professor of Mathe- 

 matics, Dartmouth College. [Vol. I. pp. x + 344, 1910 ; reprinted 1916. 

 Vol. II. pp. xii+ 511, 1918.] (Boston, New York, and London : Ginn&Co.) 



The first volume of this important book was reprinted in 1916, and the second 

 volume, which was published in 1918, is due almost exclusively to Professor Veblen. 

 In recent times the science of geometry has been put into a purely deductive form 

 principally by Peano and other Italian mathematicians, and, later on and by 

 rather different methods, by Hilbert and his school. In this transformation of 

 geometry, Prof. Veblen has taken a leading place among American mathematicians 

 by his original work, and now he and Prof. Young have rendered a very great 

 service to the mathematical world by this treatise : little original work and no 

 treatises of great importance on deductive geometry have as yet been produced 

 by British logicians. 



In giving a complete foundation for geometry, it is necessary to study linear 

 order and continuity, and this is deferred to the second volume : "the more 

 elementary part of the subject rests on a very simple set of assumptions which 

 characterise what may be called ' general projective geometry ' " (Vol. I. p. iii). 



After an introduction explaining the logical foundations of the subject and 

 dealing with the " consistency," " categoricalness," and " independence " of a set of 

 postulates, the chapters begin with one on theorems of alignment and the principle 

 of duality. The second to the fifth chapters are on projection, section, perspec- 

 tivity, and elementary constructions ; projectivities of the primitive projective forms 

 of one, two, and three dimensions ; harmonic constructions and the fundamental 

 theorem of projective geometry ; and conic sections. In the sixth chapter, on an 

 algebra of points and one-dimensional co-ordinate sytems, analytic methods are 

 introduced on a purely projective basis by what are, in essentials, the processes of 

 von Staudt and his modern successors, and the introduction "brings clearly to 

 light the generality of the set of assumptions used in this volume. What we call 

 'general projective geometry' is, analytically, the geometry associated with a 

 general number field. All the theorems of this volume are valid, not alone in the 

 ordinary real and the ordinary complex projective spaces, but also in the ordinary 

 rational space and in the finite spaces" (Vol. I. p. iv). The other chapters are 

 on coordinate systems in two- and three-dimensional forms ; projectivities in one- 

 dimensional forms ; geometrical constructions and invariants ; projective trans- 

 formations of two-dimensional forms ; and families of lines. 



The second volume can be read starting from after the eighth chapter of the 

 first volume, and contains chapters on foundations ( including discussions of order 

 and continuity) ; elementary theorems on order ; the affine group in the plane ; 

 Euclidean plane geometry ; ordinal and metric properties of conies 5 inversion 

 geometry and related topics ; affine and Euclidean geometry of three dimensions ; 

 non-Euclidean geometries ; and theorems on sense and separation. Both volumes 

 contain indexes. 



