BOOK III.

DEFINITIONS.

1. Every line which is not straight is called a curve line. 2. A circle is a space enclosed by a curve line, every point in which is equally distant from a point within the figure; which point is called the centre.

3. The boundary of a circle is called its circumference.

4. A radius is a line drawn from the centre to the circumference.

5. A diameter is a line which passes through the centre, and has its extremities in the circumference.

A diameter, therefore, is double the radius.

In the circle AEFBD, of which Cis the centre, CD is the radius, and AB the diameter.

6. An arc is any portion of the circumference.

7. The chord of an arc is the straight line joining its extremities. It is said to subtend the arc.

8. A segment of a circle is the portion included by an arc and its chord.

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G

B

The space EFGE included by the arc EFG, and the chord EG is a segment; so also is the space included by the same chord and the arc EADBG.

9. A sector of a circle is the portion included by two radii and the intercepted arc.

The space CBDC is a sector of the circle.

10. A tangent is a line which touches the circumference, that is, it has but one point in common with it, which point is called the point of contact.

11. One circle touches another when their circumferences have one point in common, and only one.

12. A line is inscribed in a circle when its extremities are in the circumference.

13. An angle is inscribed in a circle when its sides are inscribed.

14. A polygon is inscribed in a circle when its sides are inscribed, and under the same circumstances the circle is said to circumscribe the polygon.

Thus AB is an inscribed line, ABC an inscribed angle, and the figure ABCD is an inscribed quadrilateral.

15. A circle is inscribed in a polygon when its circumference touches each side, and the polygon is said to be circumscribed about the circle.

16. By an angle in a segment of a circle is to be understood, an angle whose vertex is in the arc, and whose

D

B

sides intercept the chord; and by an angle at the centre is meant one whose vertex is at the centre. In both cases the angles are said to be subtended by the chords or arcs which their sides include.

POSTULATE.

From any point as a centre with any radius, a circumference may be described.

A diameter divides a circle and its circumference into two equal parts; and, conversely, the line which divides the circle into two equal parts is a diameter. Let AB be a diameter of the circle AEBD, then the portions AEB, ADB, are equal both in surface and boundary.

Suppose the portion AEB were to be applied to the portion ADB, while the line AB still remains common to both, there must be an entire coincidence; for if any part of the boundary AEB were A to fall either within or without the boundary ADB, lines from the centre to the circumference could not all be equal. Therefore a diameter divides the circle and its circumference in two equal parts.

E

Conversely, the line dividing the circle into two equal parts is a diameter.

For, let AB divide the circle into two equal parts, then, if the centre is not in AB, let AF be drawn through it, which is,

therefore, a diameter, and, consequently, divides the circle into two equal parts; hence the portion AEF is equal to the portion AEFB, which is absurd.

Cor. The arc of a circle, whose chord is a diameter, is a semi-circumference, and the included segment is a semi-circle.

PROPOSITION II. THEOREM.

Any line inscribed in a circle lies wholly within the circle.

Let the line AB have its extremities in the circumference of a circle, whose centre is C; this line shall lie wholly within the circle.

For, to whatever point D, between the extremities of AB, a line CD from the centre be drawn, it must be shorter than CA or CB (Prop. XXII. Cor. 2. B. I.); AB, therefore, lies wholly within the circle.

Cor. Every point, moreover, in the production of AB is farther from the centre than the circumference.

In the same, or in equal circles, equal angles at the centre are subtended by equal arcs.

Let C be the centre of a circle, and let the angle ACB be equal to the angle ECD, then the arcs AB, ED, subtending these angles are equal.

Join AB, ED.

Then the triangles ACB, DCE, having two sides and the included angle in the one, equal to two sides and the included angle in the other, are equal; so that if ACB be applied to DCE, there shall be an entire coincidence, the point A coinciding with D, and B with E; the two extremities, therefore, of the arc AB thus coinciding with those of the arc DE; all the intermediate parts

must coincide, inasmuch as they are all equally distant from

the centre.

Cor. 1. It follows, moreover, that equal angles at the centre are subtended by equal chords.

Cor. 2. If the angle at the centre of a circle be bisected, both the arc and the chord which it subtends shall also be bisected.

Scholium.

The above reasoning obviously applies to the case of equal circles, as the one would entirely coincide with the other.

PROPOSITION IV. THEOREM. (Converse of Prop. III.) In the same circle equal arcs subtend equal angles at the centre.

For, let the arc AB be equal to the arc DE, then is the angle ACB equal to the angle DCE.

For, if the arc AB were to be applied to the arc DE, they would coincide; so that the extremities AB of the chord AB would coincide with those of the chord DE; these chords are, therefore, equal: hence the angle ACB is equal to the angle DCE (Prop. XXV. B. I.).

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B

E

Cor. 1. Equal chords subtend equal angles at the centre. Cor. 2. Therefore equal chords subtend equal arcs; and, conversely, equal arcs are subtended by equal chords.

Cor. 3. The angle at the centre, subtended by half a semicircumference, is a right angle; for the adjacent angles subtended by the two halves are equal.

Scholium.

Similar reasoning evidently applies to equal circles.

PROPOSITION V. THEOREM.

A perpendicular from the centre of a circle to any chord bisects it, and also the arc which it subtends. The perpendicular CDE, from the centre C to the chord AB, bisects AB.

For, if CA, CB be drawn, these lines will be equal; therefore their extremities A, B are equally distant from the perpendicular (Prop. XXII. Schol. B.ʻI.).

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E

It, moreover, follows (Pro. IX. Cor. 1. B. 1.) that the angle ACB is bisected by CDE; therefore, also, the arc AEB is bisected (Prop. III. Cor. 2.).

Cor. 1. Since the perpendicular, from the centre, joins the centre with the middle of the chord, or with the middle of the arc, it follows, conversely, that the line joining the centre, and middle of the chord, or the middle of the arc, must be perpendicular to the chord.

Cor. 2. And a perpendicular, through the middle of the chord, passes through the centre, and through the middle of the arc, bisecting the angle which it subtends at the centre.

PROPOSITION VI. THEOREM.

Equal chords are equidistant from the centre of the circle, and, conversely, equidistant chords are equal. In the circle ABED, let the chords AB, DE be equal, then the perpendiculars CF, CG, from the centre, shall likewise be equal.

For, since the chords are bisected in F and G, (Prop. V.) AF is equal to DG; therefore the right angled triangles AFC, DGC, having the hypothenuse and a side in each equal, are equal (Prop. XXII. Cor. 6. B.I.); therefore CF is equal to CG.

Conversely, if the distances CF, CG are equal, then, in the right angled triangles AFE, DGE, there will be the

hypothenuse AC, and a side CF in the one, equal to the hypothenuse DC and a side CG in the other; therefore AF is equal to DG: consequently AB, the double of AF, is equal to DE, the double of DG.

The longer the chord is, the nearer it is to the centre; and, conversely, the nearer any chord is to the centre, the longer it is.

Of the two chords AB, DE, let DE be the longer, then shall DE be nearer to the centre than AB; that is, the perpendicular CG shall exceed the perpendicular CF.

For, let CB, CE be joined.

Then the triangles BFC, EGC, being right angled, and having equal hypothenuses CB, CE, the squares of CF FB,