498 SCIENCE PROGRESS 



adhered to so strictly. In the present work, which embodies a course of lectures 

 delivered during the tenure of the now well-known Readership in the University 

 of Calcutta, the author confines his attention more especially to rectangular 

 matrices, and treats them in a manner which exhibits their advantages in a 

 comprehensive sense. 



It will be remembered that the first volume was devoted to a calculus of 

 matrices in which the only operations were addition, subtraction, and multiplica- 

 tion, together with further sections on the properties of the determinoid of a 

 matrix, and the solution of matrix equations of the first degree. The present 

 work still avoids the special properties of functional matrices, which are to form 

 the subject of a third volume, and is, in essence, concerned with the preparation 

 of the ground for their consideration. A very large part of the treatment is of 

 a geometrical type, which is a great advantage in that some very interesting 

 geometrical applications, of a kind not easily obtained by other methods, can 

 thereby be included. The definitions, it should be remarked, are not 

 geometrical. Chapter XII, which begins the work, is devoted mainly to 

 definitions and notations, and the following chapter exhibits the relations between 

 the elements and the minors of a matrix. The author then considers more 

 special properties of square matrices. These sections are, of course, mainly 

 introductory, and space does not permit of an account of the more considerable 

 parts of the theory developed. 



We are very pleased to notice that the author is expecting to be able to publish 

 the third volume at an early date, and so complete the theory in a form suitable 

 for the applications. The present volume worthily maintains the traditions of the 

 Cambridge University Press, and is a most valuable addition to the rapidly grow- 

 ing series of volumes for which the Readership at the University of Calcutta is 

 responsible. 



Dorothy Wrinch. 



The Analytical Geometry of the Straight Line and the Circle. By John 

 Milne, M.A., Senior Mathematical Master, Mackie Academy, Stonehaven. 

 [Pp. xii + 243.] (London : G. Bell & Sons, 1919. Price $s.) 



This is the latest volume of " Bell's Mathematical Series for Schools and 

 Colleges," and fulfils its purpose of being a thorough introduction to the formal 

 study of analytical geometry, in which the many illustrative examples and suitable 

 exercises form a very important feature. After a short historical note and an 

 explanation of the idea of co-ordinates, the question of loci is discussed, and then 

 the equation of a straight line, the finding of the angle between two straight 

 lines, the question of gradients, and perpendiculars to a straight line. Then 

 circles and their properties are considered in the case where the origin is the 

 centre. After this, pencils of straight lines, the homogeneous equation of the 

 second degree, the general equation of a circle, tangents and normals of a circle, 

 conjugate points and poles and polars, and orthogonal and coaxial circles are 

 treated. 



The historical note is a brief reminder of Ahmes, Thales and a few other 

 ancient Greeks, Kepler, Desargues, and " a Frenchman named Descartes." It 

 is rather misleading to state (p. 2) that " Descartes was the first to perceive that 

 a point could be fixed on a plane by the help of two axes, and that the laws of 

 algebra could then be applied to the solution of geometrical problems." It seems 

 not quite satisfactory to treat geometrical curves as the result of motion under 



