SYLLABUS OF GEOMETRICAL CONSTRUCTIONS [NOTE. The Association recommend that beginners, prior to or concur. rently with the study of Theoretical Geometry, should be exercised in simple Geometrical Drawing, in order to familiarise them with the conceptions, and give them some general notions of the nature of the results, of the Science of Geometry. With this object the following Syllabus of Constructions has been drawn up.] THE following constructions are to be made with the Ruler and Compasses only; the ruler being used for drawing and producing straight lines, the compasses for describing circles and for the transference of distances. 1. The bisection of an angle. 2. The bisection of a straight line. 3. The drawing of a perpendicular at a point in, and from a point outside, a given straight line, and the deter mination of the projection of a finite line on a given 4. The construction of an angle equal to a given angle; of 6. The construction of a triangle, having given (a) three sides; (B) two sides and contained angle; (7) two angles and side adjacent ; (6) two angles and side opposite. 7. The drawing of tangents to circles, under various conditions. 8. The inscription and circumscription of figures in and about circles; and of circles in and about figures. 7 and 8 may be deferred till the Straight Line and Triangles have been studied theoretically, but should in all cases precede the study of the Circle. The above constructions are to be taught generally, and illustrated by one or more of the following classes of problems :(a) The making of constructions involving various combinations of the above in accordance with (general, i.e., not numerical) conditions, and exhibiting some of the more remarkable results of Geometry, such as the circumstances under which more than two straight lines pass through a point, or more than two points lie on a straight line. (B) The making of the above constructions and combina- (8) The use of the protractor and scale of chords, and the INTRODUCTION [NOTE.-In the following Introduction are collected together certain general axioms which, though frequently used in Geometry, are not peculiar to that science, and also certain logical relations, the distinct apprehension of which is very desirable in connection with the demonstrations of the Propositions. They are brought together here for convenience of reference, but it is not intended to imply by this that the study of Geometry ought to be preceded by a study of the logical interdependence of associated theorems. The Association think 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.] 1. Propositions admitted without demonstration are called Axioms. 2. Of the Axioms used in Geometry those are termed General which are applicable to magnitudes of all kinds. The following is a list of certain general axioms frequently used : (a) The whole is greater than its part. (b) The whole is equal to the sum of its parts. (c) Things that are equal to the same thing are equal to one another. (d) If equals are added to equals the sums are equal. (e) If equals are taken from equals the remainders are equal. (ƒ) If equals are added to unequals the sums are unequal, the greater sum being that which is obtained from the greater magnitude. (g) If equals are taken from unequals the remainders are unequal, the greater remainder being that which is obtained from the greater magnitude. (h) The halves of equals are equal. 3. A Theorem is a proposition enunciating a fact whose truth is demonstrated from known propositions. These known propositions may themselves be Theorems or Axioms. 4. The enunciation of a Theorem consists of two parts-the hypothesis, or that which is assumed, and the conclusion, or that which is asserted to follow therefrom. in the typical Theorem, If A is B, then C is D, (i), Thus the hypothesis is that A is B, and the conclusion, that C is D. From this Theorem it necessarily follows that: If C is not D, then A is not B, (ii). Two such Theorems as (i) and (ii) are said to be contrapositive, each of the other. 5. Two Theorems are said to be converse, each of the other, when the hypothesis of each is the conclusion of the other. Thus, If C is D, then A is B, (iii) is the converse of the typical Theorem (i). |