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BOOK III. The Circle.

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Section s. Elementary Properties
Section 2. Chords
Section 3.

Angles in Segments
Section 4. A. Tangents (treated directly).
Section 4. B. Tangents (treated by the method of limits)
Section 5. Two Circles
Section 6. Problems
Section 7. The Circle in connection with Areas

30 31 32 33 35

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BOOK IV. Fundamental Propositions of Proportion.


Section 1.
Section 2.

Of Ratio and Proportion
Fundamental Geometrical Propositions


BOOK V. Proportion.

50 53

Section 1. Similar Figures
Section 2. Areas
Section 3. Loci and Problems

56 58





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.

The bisection of an angle.

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 determination 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, &c. 5. The drawing of a line parallel to another under

various conditions-and hence the division of lines

into aliquot parts, in given ratio, &c.
6. The construction of a triangle, having given

(a) three sides;
(B) two sides and contained angle;
(y) two angles and side adjacent;
(8) 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 in Geometry. 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.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 combi

nations of them to scale (but without the protractor). (y) The application of the above constructions to the

indirect 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.





[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 connexion 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.
Of the Axioms used in Geometry those are termed
General which are applicable to magnitudes of all
kinds: the following is a list of the general axioms

more frequently used.
(a) The whole is greater than its part.
(6) The whole is equal to the sum of its parts.

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