BOOK II. THE CIRCLE. CONTENTS. INTRODUCTION. Definitions, and obvious Properties of the Circle SECTION I. PROPERTIES OF CENTRE. THEOREM I. Rotatory Properties of the Circle. Equal arcs of a circle subtend equal angles at the centre, and have equal chords; and conversely, equal angles at the centre cut off equal arcs and have equal chords; and equal chords in a circle cut off equal arcs, and subtend' equal angles at the centre. THEOREM 2. Symmetry of the Circle with respect to its Diameter. THEOREM 3. One circle, and only one circle, can be drawn to pass through three given points which are not in the same straight line THEOREM 4. Equal chords of a circle are equally distant from the centre, and conversely; and of two unequal chords the greater is nearer to the centre than the less, and conversely Exercises PAGE I 3 5 7 8 9 SECTION II. ANGLES IN SEGMENTS OF THE CIRCLE. THEOREM 5. The angle subtended at any point in the circum- ference by any arc of a circle is half of the angle subtended COR. I. Angles in the same segment of a circle are equal to one COR. 2. If a circle is divided into any two segments by a chord, THEOREM 7. The radius to the point of contact is at right angles THEOREM 8. If from the point of contact of a straight line and a circle a chord of the circle be drawn, the angles made by 18 THEOREM 9. From any point within or without a circle except the centre, two and only two normals can be drawn, one of which is the shortest, and the other the longest line that can be drawn from that point to the circumference: and as a point moves along the circumference from the extremity of the shortest to the extremity of the longest normal, its distance from the fixed point continually increases THEOREM IO. Intersection of Circles. The line that joins the centres of two intersecting circles, or that line produced, bisects at right angles their common chord PROBLEM 3. To cut from any circle a segment which shall be PROBLEM 4. On a given straight line to describe a segment of a 29 THEOREM 12. If from the centre of a circle radii are drawn to make equal angles with one another consecutively all round, then if their extremities are joined consecutively, a regular polygon will be inscribed in the circle, and if at their extre- mities, tangents are drawn, a regular polygon will be cir- THEOREM 13. In a regular polygon the bisectors of the angles PROBLEM 8. To construct a regular polygon of four, eight, six- PROBLEM 9. To construct regular polygons of three, six, THEOREM. The area of a circle is equal to half the rectangle BOOK III. PROPORTION. CONTENTS. INTRODUCTION. Measures. PROBLEM 1. To find the greatest common measure of two magnitudes, if they have a common measure THEOREM I. To prove that the side and diagonal of a square are incommensurable RATIO, continuity, incommensurables, compound ratio . THEOREM 2. If A and B be two fixed points in a straight line of indefinite length, and Pa moveable point in that line, then the ratio of PA to PB may have any value, from o to infinity, and there are two and only two positions of P such that PA: PB any given ratio. PROPORTION THEOREM 3. If A, B, C, D be four magnitudes such that B and D always contain the same aliquot part of A and C respectively the same number of times, however great the number of parts into which A and C are divided, then A : B :: C: D. FIVE COROLLARIES PAGE 45 46 48 49 53 54 55 56 b |