13 REPORT — 1876. 1 



miglit be offered on this part of the Syllabus are chiefly on points of detail on 

 which the Committee think it unnecessary here to enter. 



They would remark, however, with respect to the two modes of treatment 

 of tangents in the Syllabus, that they would not recommend the second 

 (depending on the notion of limits) in any case as a substitute for the first, 

 however desirable it may be that it should be freely used by way of illustra- 

 tion and as leading up to the methods of Higher Geometry. 



7. Eooks IV and V. Ratio and Proportion, and their application to Geometry 



A theory of Proportion which shall be at once perfectly rigorous and com- 

 plete is necessarily difficult. The Committee recognize with satisfaction that 

 the Syllabus does not attempt to attain simplicity by any sacrifice of rigour, 

 nor in Book IV. by any sacrifice of completeness. In Book IV. the theory is 

 essentially that of Euclid in his famous, though (at the present day) little 

 studied, Fifth Book : it is suggested, however, by an unusually full indication 

 in this part of the Syllabus of the forms of demonstration recommended, that 

 his theory may be presented in a form more easy to be grasped and applied 

 by the adoption of the late Prof, de Morgan's notation, in which magnitudes 

 are denoted by capital letters, instead of by straight lines, and their multiples 

 by prefixing to the capitals small letters denoting integral numbers, instead 

 of denoting them by longer lines. Opinions will probably differ as to the 

 wisdom of retaining Euclid's treatment in any shape*; but the Committee 

 doubt whether any rival theory, which is equaUij rigorous and equallij com- 

 plete, would be more generally accepted. 



It may, however, be thought that this complete theory is one which the 

 ordinary student can hardly be expected to master at an early stage of his 

 mathematical studies, even though he may be well prepared for the study of the 

 geometrical applications of the theory of Proportion. At the same time it is 

 undesirable that the study of Similarity of Figures &c. should be commenced 

 without some definite groundwork of demonstrated properties of Eatios and 

 Proportions. The Syllabus suggests a mode of meeting this difficulty by pre- 

 fixing to Book V. an indication of a method of treatment of the general 

 doctrine of proportion, in which greater simplicity is obtained, not by the 

 sacrifice of rigour, but by a certain sacrifice of completeness, in limiting the 

 magnitudes considered to such as are commensKrahle. 



The notion of Eatio may be regarded as an extension and generalization of 

 the notion of quantvpUcity, the simplest expression of which is contained in 

 the question, "How many times does a magnitude A contain another magni- 

 tude B ? " This question may be generalized so as to apply to any pair of 

 commensurable magnitudes in two ways — the question taking the shape either 

 "How many times does A contain some aliquot part of B?" or else " What 

 multiples of A and B are equal to one another?" The former leads to a 

 treatment of proportion such as is usually given with more or less exactness 

 in treatises on Arithmetic or Algebra, while the latter leads to a treatment 

 similar in principle to Euclid's, but simplified by its limitation to commensu- 

 rables. The Syllabus indicates a few of the more important general properties 

 of proportion which ought to be proved by one or other of these methods, but 

 leaves it open to the teacher to adopt whichever mode of treatment he may 

 prefer. 



In the Geometrical Applications of Proportion the Syllabus groups together 

 * Prof. Cayley is strongly of opinion that it^^ought to be retained. 



