Syllabus of Plane Geometry

Abstract

THE readers of NATURE are so well acquainted with the genesis and growth of the Association whose syllabus has recently been given to the public, that we are relieved from all necessity of explaining what objects it has in view. The main result of its five years of labour is this Syllabus, and we shall here briefly exhibit some of its chief features. It is a double syllabus, being a syllabus of geometrical constructions and a syllabus of plane geometry. The former is very brief, and contains such constructions as can be made with the ruler and compasses only. This subject of constructive geometry has been tried in many schools of late and has been found generally to answer the end in view. Boys thus obtain some idea of the objects of pure geometry and of what is involved in the postulates of the science. The more important syllabus is prefaced by a Logical Introduction- not that the Association wishes “to imply by this that the study of geometry ought to be preceded by a study of the logical independence of associated theorems.” The opinion of the compilers is “that at first all the steps by which any theorem is demonstrated should be carefully gone through by the student, rather than that its truth should be inferred from the logical rules here laid down. At the same time they strongly recommend an early application of general logical principles.” The President, in one of his addresses, states that the object of this introduction is “to guide the teacher immediately, and the student ultimately.” It contains certain general axioms (as the whole is greater than its part), and taking as its typical theorem, if A is B then C is D, it explains what is meant by its contrapositive (if C is not D, A is not B), by its converse (if C is D, A is B), and by its obverse (if A is not B, C is not D). This last term we have heard strongly condemned; it was substituted (see Fifth Annual Report) for the more usual term opposite on the ground that, in logic, two opposite propositions cannot be true together. The terminology, however, to our mind, is a matter of no great consequence. For proving converse theorems frequent use is recommended in the work of a “ Rule of Identity“here given, i.e. if there is but one A and but one B, then if A is B, it necessarily follows that B is A. (De Morgan's illustration is given in Wilson's Geometry.)

Syllabus of Plane Geometry

(corresponding to Euclid, Books i.-vi.) Prepared by the Association for the Improvement of Geometrical Teaching. (London: Macmillan, 1875.)