than the sum of their radii, they are external, one to the other:

2o. When this distance is equal to the sum of the radii, they are tangent, externally :

3°. When this distance is less than the sum, and greater than the difference of the radii, they intersect each other:

4°. When this distance is equal to the difference of their radii, one is tangent to the other, internally :

5°. When this distance is less than the difference of the radii, one is wholly within the other:

6°. When this distance is equal to zero, they have a common centre; or, they are concentric.

PROPOSITION XV. THEOREM.

In equal circles, radii making equal angles at the centre, intercept equal arcs of the circumference; conversely, radii which intercept equal arcs, make equal angles at the centre.

1o. In the equal circles ADB and EGF, let the angles ACD and EOG be equal: then the arcs AMD and ENG are equal.

B

M

F

For, draw the chords AD and EG; then the triangles ACD and EOG have two sides and their included angle, in the one, equal to two sides and their included angle, in the other, each to each. They are, therefore, equal in all respects; consequently, AD is equal to EG. But, since the chords AD and EG are equal, the arcs AMD and ENG are also equal (P. IV.); which was to be proved.

2o. Let the arcs AMD and ENG be equal: then the angles ACD and EOG are equal.

For, since the arcs AMD and ENG are equal, the chords AD and EG are equal (P. IV.); consequently, the triangles ACD and EOG have their sides equal, each to each; they are, therefore, equal in all respects:

Q Q

M

N

hence, the angle ACD is equal to the angle EOG; which was to be proved.

In equal circles, commensurable angles at the centre proportional to their intercepted arcs.

In the equal circles, whose centres are C and O, let the angles ACB and DOE be commensurable; that is, be exactly measured by a common unit: then are they proportional to the intercepted arcs AB and DE.

Let the angle M be a common unit; and suppose, for example, that this unit is contained 7 times in the angle ACB, and 4 times in the angle DOE. Then, suppose ACB be divided into 7 angles, by the radii Cm, Cn, Cp, &c. ; and DOE into 4 angles, by the radii Ox, Oy, and Oz, each equal to the unit M.

From the last proposition, the arcs Am, mn, &c., Dx, xy, &c., are equal to each other; and because there are 7 of these arcs in AB, and 4 in DE, we shall have,

If any other numbers than 7 and 4 had been used, the same proportion would have been found; which was to be proved.

Cor. If the intercepted arcs are commensurable, they are proportional to the corresponding angles at the centre, as may be shown by changing the order of the couplets in the above proportion.

In equal circles, incommensurable angles at the centre are proportional to their intercepted arcs.

In the equal circles, whose

centres are C and O, let

ACB and FOH be incommensurable : then are

they

proportional to the arcs AB

and FH.

DIOB

For, let the less angle FOH, be placed upon the greater angle ACB, so that it shall take the position ACD. Then,

Conceive the arc AB to be divided into equal parts, each less than DO: there will be at least one point of division between D and O; let I be that point; and draw Cl. Then the arcs AB, Al, will be commensurable, and we shall have (P. XVI.),

angle ACB

: angle ACI :: arc AB

: arc Al.

Comparing the two proportions, we see that the antecedents are the same in both: hence, the proportional (B. II., P. IV., C.); hence,

consequents are

angle ACD angle ACI ::
:

arc AO : arc Al.

But, AO is greater than Al: hence, if this proportion is true, the angle ACD must be greater than the angle ACI. On the contrary, it is less: hence, the fourth term of the assumed proportion can not be greater than AD.

In a similar manner, it may be shown that the fourth term can not be less than AD: hence, it must be equal to AD; therefore, we have,

angle ACB : angle ACD :: arc AB

which was to be proved.

: arc AD;

Cor. 1. The intercepted arcs are proportional to the corresponding angles at the centre, as may be shown by

changing the order of the couplets in the preceding proportion.

Cor. 2. In equal circles, angles at the centre are proportional to their intercepted arcs, and the reverse, whether they are commensurable or incommensurable.

Cor. 3. In equal circles, sectors are proportional to their angles, and also to their arcs.

Scholium. Since the intercepted arcs are proportional to the corresponding angles at the centre, the arcs may be taken as the measures of the angles. That is, if a circumference be described from the vertex of any angle, as a centre, and with a fixed radius, the arc intercepted between the sides of the angle may be taken as the measure of the angle. In Geometry, the right angle, which is measured by a quarter of a circumference, or a quadrant, is taken as a unit. If, therefore, any angle is measured by one half or two thirds of a quadrant, it is equal to one half or two thirds of a right angle.

An inscribed angle is measured by half of the arc included

between its sides.

There may be three cases: the centre of the circle

may lie on one of the sides of the

angle; it may lie within the angle; or, it may lie without the angle.

1o. Let EAD be an inscribed angle, one of whose sides AE passes through the centre: then it is measured by half of the arc DE.

E