BOOK II.

THE CIRCLE, AND THE MEASUREMENT OF ANGLES.

DEFINITIONS.

1. The circumference of a circle is a curve line, all the points of which are equally distant from a point within, called the centre.

The circle is the space terminated by this curved line.

N. B. Sometimes, in common language, the circle is confounded with its circumference; but it will be always easy to recur to the correct expression, by recollecting that the circle is a surface, which

A

F

E

has length and breadth, while the circumference is but a line.

G

D

B

2. Every straight line, CA, CE, CD, etc., drawn from the centre to the circumference, is called a radius or semi-diameter; every line, as AB, which passes through the centre, and which is terminated on both sides, at the circumference, is called a diameter.

From the definition of a circle, it follows, that all the radii are equal; that all the diameters are equal also, and each double of the radius.

3. A portion of the circumference, such as FHG, is called an arc. The chord or subtense of the arc is the straight line FG which joins its two extremities.

4. A segment is the surface or portion of a circle comprised between the arc and its chord.

N. B. To the same chord, FG, correspond always two arcs, FHG, FEG, and, consequently, also two segments; but the smaller one is always meant unless the contrary is expressed.

5. A sector is the part of the circle included between an arc, DE, and the two radii, CD, CE, drawn to the extremities of that arc.

6. A straight line is said to be inscribed in a circle, when its extremities are in the circumference, as AB.

An inscribed angle is one, such as BAC, whose vertex is in the circumference, and which is formed by two chords.

An inscribed triangle is one which, like BAC, has its three vertices in the circumference.

B

And, in general, an inscribed figure is one of which all the angles have their vertices in the circumference. The circle is said at the same time to be circumscribed about this figure.

7. A secant is a line which meets the circumference in two points: AB is a secant.

8. A tangent is a line which has only one point in common with the circumference: CD is a tangent.

The point, M, is called the point of contact.

A

-B

C

9. In like manner, two circumferences are tangent to each other

when they have only one point in common.

10. A polygon is circumscribed about a circle when

all its sides are tangents to the circumference: in the same case we say that the circle is inscribed in the polygon.

PROPOSITION I.

THEOREM.

Every diameter, AB, divides the circle and its circumference into two equal parts.

For, if we apply the figure AEB to AFB, the common base, AB, retaining its position, the curve line AEB must fall exactly on the curve line AFB, otherwise, there would be, in the one or the other, points unequally distant from the centre, which is contrary to the definition of a circle.

F

D

B

C

E

PROPOSITION II.

THEOREM.

Every chord is less than the diameter.

For, if the radii AC, CD be drawn to the extremities of the chord AD, we shall have the A straight line AD < AC + CD, or AD < AB.

F

D

B

C

E

COR. Hence, the greatest straight line which can be inscribed in a circle is equal to its diameter.

PROPOSITION III.

THEOREM.

A straight line cannot meet a circumference in more than two points. For, if it met it in three, those three points would be equally distant from the centre; there would, therefore, be three equal straight lines drawn from the same point to the same straight line, which is impossible (Book I., Prop. XVII. Cor. 2).

PROPOSITION IV.

THEOREM.

In the same circle, or in equal circles, equal arcs are subtended by equal chords; and, conversely, equal chords subtend equal arcs.

D

M

If the radii AC and EO are equal, and the arc AMD equal to the arc ENG, then the chord AD will be equal to the chord EG. For, since the diameters AB, EF, are equal, the semicircle AMDB may be applied exactly to the semicircle ENGF, and the curve line AMDB will coincide entirely with the curve line ENGF. But the part AMD is equal to the part ENG by hypothesis;

N

E

F

C

B

K

therefore, the point D will fall on the point G; therefore, the chord AD is equal to the chord EG.

Conversely, supposing the radius AC = EO, if the chord AD = EG, then will the arc AMD be equal to the arc ENG. For, drawing the radii CD, OG, the two triangles ACD, EOG, having all their sides equal, each to each, viz.: AC = EO, CD = OG, and AD = EG, are themselves equal (Book I., Prop. XII.); therefore, the angle ACD = EOG. But, placing the semicircle ADB on its equal, EGF, since the angle_ACD = EOG, it is evident that the radius CD will fall on the radius OG, and the point D on the point G; therefore, the arc AMD is equal to the arc ENG.

PROPOSITION V.

THEOREM.

In the same circle, or in equal circles, a greater arc is subtended by a greater chord; and, conversely, the greater chord subtends the greater arc; the arcs being always supposed to be less than a semi-circumference.

than ACD; therefore (Book I., Prop. XI.), the third side, AH, is greater than the third side, AD; therefore, the chord which subtends the greater arc is the greater.

Conversely, if we suppose the chord AH greater than AD, we conclude from the same triangles that the angle ACH is greater than ACD, and, therefore, that the arc AH is greater than AD.

SCHOLIUM. The arcs here treated are less than the semicircumfer

ence.

If they were greater, the reverse property would exist in them, as the arc increased, the chord would diminish, and conversely; thus, the arc AKBĎ being greater than AKBH, the chord AD of the first is less than the chord AH of the second.

PROPOSITION VI.

THEOREM.

The diameter, GH, perpendicular to a chord, AB, divides this chord and also the arcs AGB, AHB, which it subtends, into two equal parts.

Let C be the centre of the circle, and D the point in which the diam、 eter HG meets the chord AB. The radii CA, CB, considered with regard to the perpendicular CD, are two equal oblique lines; hence, they lie equally distant from that perpendicular (Book I., Prob. XVII., Cor. 3); hence ADDB, or the chord AB is bisected at D.

H

Again, as we have shown that CG is perpendicular to AB at its middle point, every point of the perpendicular must be equidistant from A and B. Now, G is one of those points; therefore the chords GA and GB are equal; hence, the arcs GA and GB are also equal or the arc, AGB, is bisected at the point G. we may show that the arc AHB is bisected at the point H.

G

C

B

D

In like manner

SCHOLIUM. The centre, C, the middle point, D, of the chord AB, and the middle points, G and H, of the arcs subtended by this chord, are four points situated in the same line, perpendicular to the chord. Now, two points determine a straight line; therefore, every straight line which passes through two of the points mentioned, must necessarily pass through the third, and will be perpendicular to the chord.

It follows, also, that the perpendicular, drawn to a chord at its middle point, will pass through the centre and through the middle points of the arcs subtended by that chord. Also, the geometric locus of the middle points of a system of parallel chords is the diameter perpendicular to those chords.

PROPOSITION VII.

THEOREM.

Through three given points, A, B, C, not in a straight line, one circumference may always be made to pass, and but one.

We are to prove that there is one point, and only one, equally distant from the three points, A, B, C.

Draw AB, BC, and bisect those straight lines by the perpendiculars DE, FG. These lines, DE and FG, being perpendicular respectively to two lines, AB and BC, which meet, will meet each other in a point, O (Book I., Prop. XXV., Cor.).

And, moreover, since this point, O, lies

G

E

F

K

D

B