CONTENTS

BOOK I. The Circle.

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Section 1. Elementary Properties

Section II. Angles at the Centre and Sectors

Section III. Chords

Section IV. Angles in Segments

Section V. Tangents

Section VI. Two Circles

Section VII. Inscribed and Circumscribed Figures

Section VIII. The Circle in Connection with Areas

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Introductory Remarks

PART I. Commensurable Magnitudes only.

Section 1. Of Ratio and Proportion

Section II Fundamental Geometrical Propositions

Definitions

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PART II. Magnitudes without respect to Commensurability.

Section 1. Of Ratio and Proportion

Section II. Fundamental Geometrical Propositions

Definitions

THE ELEMENTS OF PLANE GEOMETRY.

BOOK III.

THE CIRCLE.

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SECTION I.

ELEMENTARY PROPERTIES.

DEF. 1. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another. This point is called the centre of the circle.

DEF. 2. A radius of a circle is a straight line drawn from the centre to the circumference,

DEF. 3.

A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.

THEOR. I. The distance of a point from the centre of a circle is less than, equal to, or greater than the radius, according as the point is within, on, or without the circum. ference.

then shall OP be less than, equal to, or greater than the radius, according as P is within, on, or without the circumference.

The straight line passing through O and P will meet the circumference in two points Q, Q', and in no other points, since there are only two points on the line whose distances from O are equal to the radius.

If P is between Q and Q', P is within the circumference, and OP is less than OQ, that is, less than the radius.

If P coincide with or Q', P is on the circumference, and OP is equal to OQ or OQ”, that is, equal to the radius.

If P is on OQ or OQ produced, P is without the circumference, and OP is greater than OQ or OQ', that is, greater than the radius.

Q.E.D.

COR. A point is within, on, or without the circumference of a circle, according as its distance from the centre is less than, equal to, or greater than the radius.