REVIEWS 539 



In discussing the teaching of proportion, it is unnecessary to deal with how 

 far incommensurable magnitudes should be included. Every teacher who is not 

 confined by an antiquated syllabus must include in his course of geometrical 

 teaching the ratio of commensurables. Ratio and proportion are important 

 notions which every educated citizen should possess; in the course of geometry 

 and in the geometrical text-book they should be presented in the clearest form. 

 In the book, here reviewed, it is a little disconcerting to find that the definition 

 of ratio is first introduced in the midst of a subsection which is headed Measure- 

 ment of Angles, while in spite of its position, the earliest illustrations of ratio are 

 chosen not from angular, but from linear magnitudes. The words in which the 

 authors carefully weigh their final decision must be given, in order that those who 

 are not familiar with the modern geometries may have an opportunity of judging 

 how far Euclid has been improved upon by his modern rivals. Here is the 

 sentence which summarises our authors' views upon a most important and difficult 

 conception. " The numerical measure of one quantity divided by the numerical 

 measure of a second quantity of the same kind, provided the same unit has been 

 used in each case, is called the ratio of the first quantity to the second." The 

 statement is so comprehensive and is so carefully worded that one regrets that it 

 fails so completely to be a definition. It reminds us of a statement made by a 

 prominent champion of geometrical reform who once wrote that the test of the 

 utility of a proposition was that it could be used. Here we have a more com- 

 plicated fallacy, for, omitting unnecessary verbiage, and avoiding the undefined 

 words " division " and " measure," the only way in which the definition can be 

 stated is in the following terms : the ratio which the ratio of one quantity to a 

 certain quantity, called the unit, bears to the ratio of a second quantity to the 

 same unit, is called the ratio of the one quantity to the second quantity. The 

 word to be defined must be used no less than three times before the definition is 

 complete. If teachers of geometry upon the modern lines will turn to the 

 definitions given to their pupils in the text-books which they use, they may discover 

 one of the reasons why their scholars are finding the new geometry a difficult 

 subject. Once more it is the desire to make the subject practical which has led 

 the authors into such serious error. They might with advantage reflect upon the 

 danger of mixing ; we all know that there is an excellent drink called water and 

 one perhaps equally excellent, but certainly less accessible, called Chateau Lafitte, 

 and it is generally acknowledged that one whose taste is unspoilt can enjoy either 

 of these beverages ; but what can be the state of the man's palate who deliberately 

 mixes the two ? The most charitable view is that he hardly appreciates the ex- 

 cellence of either when taken separately. 



Too little has been said of the special features of the book. It has many 

 good points : it covers both plane and solid, and it abounds in instances of apt 

 illustration. Teachers who desire a text-book upon the so-called modern lines 

 will find much that will please them between its covers. 



C. 



Analytic Geometry and Principles of Algebra. By Alexander Ziwet 

 and Louis Allen Hopkins. [Pp. viii + 369.] (New York : The Mac- 

 millan Co., 191 3. Price 7s. net.) 



THIS book is an attempt to unite— more completely than is usually done — elementary 

 analytical geometry of two and three dimensions with the portions of algebra which 

 are needed for the discussion of the straight line and conic in two dimensions and 



