JSotices respecting New Boohs. 407 



A New Analysis of Plane Geometry, Finite and Differential, with 

 numerous Examples. By A. W. H. Thompson, B.A. Cambridge 

 University Press, 1914. 



IjS" this analysis points are represented by small Roman letters 

 and straight lines by small Greek letters. The symbol ab repre- 

 sents the line joining a and b ; and aft represents the point of 

 intersection of the lines a and ft. The distance between a and b is 

 denoted by (ab) ; the angle between a and ft by («/3) ; the perpen- 

 dicular distance from a on ft by (a/3). The expression (abyd) 

 means the distance of the point d from the intersection of 



the line ab with the line y. The expression sin (otjScB) means the 

 sine of the angle which the line £ makes with the line joining the 

 intersection of the lines ocft and the point c. The symbol (abc) is 

 twice the area of the triangle formed by the points ; it may be 

 written j (ab) | (abc)» which means the product of the length of the 

 line joining a and b and the perpendicular distance from the 

 point c on_the line ab. Also the symbol («/3y) is denned as 

 sin (a/3) . (a/3y). A suffix notation is introduced to represent the 

 vectorial construction by which one point is got from another ; 

 and by consideration of the consecutive positions of a moving 

 point and of a moving line a new Differential Geometry is 

 obtained. The tangential line to the curve at the point x is 

 represented by rx, and the normal to the curve at the point x 

 by vx, so that (rx*/x)=7r/2. Similarly the expression p£ represents 

 the point of intersection of the line t, and its consecutive position £'. 

 The curvature at the point x may be written either as the differ- 

 ential coefficient of rx or of vx with respect to x, and is represented 

 by px. Some of these notations are suggestive, as the author 

 himself points out, of Grassmaun's methods ; but the whole deve- 

 lopment is quite different and essentially new. The author is to 

 be congratulated upon the invention of a system of symbolic plane 

 geometry which expresses results and demonstrates theorems by 

 a general process built upon the analytical recognition of the 

 elements of position and displacement. The book is beautifully 

 printed, and into its 118 pages there is packed a vast amount of 

 geometrical fact. 



T7ie Algebra of Logic. By Louis Couturat. Authorized English 

 Translation by Lydia Gillingham Kobinsox, B.A. "With a 

 Preface by Philip E. B. Jouedain, M.A. (Cantab.). Chicago 

 and London. The Open Press Publishing Company. 1914. 



The mathematical analysis of logic originated with George Boole 

 (1847) and was developed in his later work on the Laws of Thought 

 (1854). Ernst Schroder in 1877 and Alexander Macfarlane in 

 1879 continued the investigations along somewhat similar lines, 



