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treatise on Elementary Geometry, but simply as an edition of a part of the Syllabus with the demonstrations supplied and suitable exercises inserted.

Probably many teachers will find in this all that they require in a Text Book for their pupils, being satisfied to supply the needful illustrations, explanations, and developments of the subject in their oral teaching. Still there is, doubtless, room for other treatises embodying such illustrative and explanatory matter; but the Association is of opinion that such treatises should rather be the work of individual authors than of an Association.

Accordingly, should any author desire to publish a further treatise, based on the present work, the Council of the Association would be glad to authorise his free use of the work on terms to be arranged by communication with them through the publishers or the honorary secretaries.

SYLLABUS

OF

GEOMETRICAL CONSTRUCTIONS.

[NOTE.—The Association recommend that beginners, prior to or concurrently with the study of Theoretical Geometry, should be exercised in simple Geometrical Drawing, in order to familiarize 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

straight line. 4. The construction of an angle equal to a given angle; of

an angle equal to the sum of two given angles, etc. 5. The drawing of a line parallel to another under various

conditions—and hence the division of lines into aliquot parts, in given ratio, etc.

6. The construction of a triangle, having given
(a) three sides;
(B) two sides and contained angle;
() two angles and side adjacent;
() two angles and side opposite.
7. The drawing of tangents to circles, under various condi-

tions.
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 com

binations of the above in accordance with general (i.c., 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

tions of them to scale (but without the protractor). (v) The application of the above constructions to the in

direct measurement of distances. (8) The use of the protractor and scale of chords, and the

application of these to the laying off of angles, and the indirect measurement of angles.

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