ON METHODS OF INSTRUCTION IN ELEMENTARY GEOMETRY. 11 



explanation in ■whicli the notion of rotation is freely but judiciously used. The 

 Syllabus does not (like Euclid) limit the notion of an angle to ono less than 

 two right angles, but it does not explicitly recognize an angle greater than 

 four right angles. Possibly, considering the difficulties of expression which 

 the complete notion of an angle of unlimited magnitude involves, this limita- 

 tion at the outset is wise. The Committee note with approval the use of the 

 term conjugate for the two angles which, being contained by the same pair of 

 lines drawn from a point, together make up four right angles. 



They also approve the introduction of the term " identically equal " for 

 figures which, differing only in respect of position, can be made to coincide 

 •with one another, whilo the term "equal" is reserved for such as are equal 

 in area, but not necessarily in other respects. 



The Syllabus divides the Axioms (as, indeed, Euclid did) into General 

 Axioms (Euclid's Kouai eyvoiai), which find their fitting place in the Logical 

 Introduction, and specially Geometrical Axioms (Euclid's aJr/z/iara), which 

 arc nearly those of EucUd — that about the equality of right angles being 

 omitted, while that asserting that "two straight lines cannot enclose a space" 

 is extended so as to assert coincidence beyond as well as between the two 

 points which coincide. 



The Postulates are those of Euclid's ' Elements,' with a modification in the 

 third postulate, which admits of the direct transference of distances by the 

 compasses, as before remarked. 



The Theorems of Book I. are mainly those of Euclid I. 1-34, rearranged. 

 The guiding principle of the rearrangement appears to have been the nearness 

 or remoteness of the theorems from the possibility of proof by the direct appli- 

 cation of the fundamental principle of superposition, the free use of this 

 principle being indicated as desirable in many cases where Euclid j^refers to 

 keep it out of sight. 



The discussion of the cases of identical equality of two triangles is rendered 

 complete by the introduction of a theorem asserting the true conclusion from 

 the equality of two sides and a non-included angle in each, namely, that the 

 other non-included angles are either equal or supplementary, and that in the 

 former case only are the triangles identically equal. 



For the treatment of Parallels, Playfair's Axiom that "Two straight lines 

 that intersect one another cannot both be parallel to the same straight line," 

 has been substituted for Euchd's twelfth Axiom, and, in the opinion of the 

 Committee, judiciously. It may, in fact, be regarded as merely an improved 

 form of that axiom. 



5. Book II. Areas. 



This book contains in thirteen Theorems the various theorems contained in 

 Euclid between I. 35 and the end of Book II. Beyond noting the fact that it 

 brings together more completely than in Euclid those theorems which are 

 naturally related to one another, no comment is necessary which is not of the 

 nature of that detailed criticism which the Committee do not think it their 

 duty to offer. 



6. Book III. TJie Circle. 



In this Book the sequence of Theorems differs materially from that of 

 Euclid, those propositions being placed first which are fundamental in the 

 sense that they follow directly from supei-position. Other criticisms which 



