10 REPORT— 1876. , 



2. Logical Introduction, 



The Syllabus is further prefaced with an introduction, in which are collected 

 together and formulated the most important logical relations of the several 

 propositions logically associated with a given proposition, namely its converse, its 

 obverse (sometimes called its opiiosite), and its contrapositive. It is distinctly 

 stated, in a note prefixed to this introduction, that it is not intended that a 

 study of the abstract logical j-elations contained in it should precede the study 

 of Geometry, but that the introduction should be referred to from time to 

 time as instances of the applications of its principles arise, until the student 

 obtains such a grasp of the principles and rules as to be able to apply them 

 without difficulty. With this understanding the Committee regard the pro- 

 posed logical introduction as a valuable feature of the Syllabus. 



3. Separation of Theorems and Problems — Loci. 



Throughout the Syllabus, the Problems, instead of being interspersed among 

 the Theorems, are collected together in separate sections at the end of each 

 Eook. This may be regarded as equivalent to the assertion of the principle 

 that, while Problems are from their very nature dependent f()r the form, and 

 even the possibility, of their solution on the ai'bitrary limitation of the instru- 

 ments allowed to be used. Theorems being truths involving no arbitrary 

 element ought to be exhibited in a form and sequence independent of such 

 limitations. In other words, constructions may be rightly assumed in the 

 demonstrations of theorems, whether or not they have been shown previously 

 to be capable of being effected by ruler and compasses, provided only they 

 can be seen from the nature of the case, or be proved, to be possible. For 

 instance, the existence of the third part of an angle being regarded as 

 axiomatic, the impossibility of trisecting an angle with ruler and compasses 

 only ought to form no obstacle to the ])roof of a theorem for which the trisec- 

 tion of an angle is required. It should be remembered that the acceptance 

 of the principle here asserted by no means necessitates in ieacAinr/ that separa- 

 tion of Theorems from Problems which seems desirable in a syllabus. It is 

 probable that most teachers would prefer to introduce problems, not as a 

 separate section of geometry, but rather in connexion with the theorems with 

 Avhich they are essentially related. The S3'llabu8 in this respect leaves com- 

 plete freedom to the teacher. 



The early introduction of the notion of a Locus and its use in the solution 

 of problems by the intersection of Loci the Committee regard with favour ; and 

 they observe with satisfaction that the Syllabus rightly insists on the demon- 

 stration of two theorems (a theorem, and either its converse or its obverse) as 

 necessary for the complete establishment of a locus, a point which is too often 

 neglected in the investigation of loci, 



4. Book I. The Straight Line. 



The Definitions are substantially those of Euclid. An attempt to give a 

 real definition of a straight line (Euclid's is only verbal) is to be commended, 

 though the wording is difficult, and would for a beginner require detailed 

 and familiar explanation. 



The definition of an an|le is another of the elementary difficulties of Geo- 

 metry. The Syllabus in a note asserts that " an angle is a simple concept in- 

 capable of definition, properly so called," but enters into a somewhat detailed 



