As one point and the direction of a straight line, determine its position (53)—A straight line drawn perpendicularly through the middle of a chord, bisects the arc, and passes through the centre of the circle.

110. This enables us to find the centre of a circle to which any arc belongs (fig. 53). We have only to as- Fig. 53. sume two chords, AD, DB, in the arc, and draw perpendiculars through the middle of each; the point of meeting of these two lines is the centre of the circle.

The operation would be the same if we were required to draw a circular curve through the three given points A, D, B. If, however, these three points were in the same straight line, it is manifest that the perpendiculars would be parallel (23); they would therefore never meet; consequently--No three points in the same straight line, can be in the circumference of a circle.

Can you make two or more circumferences pass through three given points not in the same straight line? Can you make a circumference pass through any four points not in the same straight line?

111. As the tangent has but one point in common with the circumference, namely, the point of contact (fig. 54), every other point must be without, and conse- Fig. 54. quently farther from the centre of the circle; and if radius be drawn to this point, it will be less than any other straight line drawn from the centre to the tangent; it must consequently be perpendicular to the tangent (51); therefore-To draw a tangent to any point in a circumference, we have only to draw a straight line perpendicular to radius at that point.

As many circles, therefore, as have the point A, a point in the circumference, and BD, a common tangent, must have their centres in the same straight line CA, produced if necessary. In this case the circumferences are said to be tangent to each other.

When the circles have their centres on opposite sides of the point of contact, they are said to be tangent externally. When they have their centres on the same side, they are said to be tangent internally.

Can a straight line pass through the point of contact between a circumference and a rectilinear tangent, which shall cut neither of them?

Can you draw a circular line between a right-line tangent and the circumference ?

Fig. 55.

Can you draw a circular line between two circular curves tangent to each other, and having their centres on the same side of the point of contact?

112. If we draw the secant, MN (fig. 55) parallel to the tangent, it will be perpendicular to the radius CA; and as the interior portion of the secant, will be a chord, the arc which it subtends will be bisected by this radius (109) at the point of contact. So if another parallel secant OP be drawn, the arc which this cuts off will also be bisected by the same radius; so that we shall have the arc AO equal to the arc AP; and AM equal to AN; therefore MO and NP are equal. Hence--Parallels intercept equal arcs in the circumference.

113. We have seen (106) that equal angles at the centre of a circle, are subtended by equal arcs; if, therefore, ve find a common measure of the two arcs AB, A'B', (Ag. 56) drawn with equal radii; and divide each of the Fig. 56. arcs by this common measure; and to the several points of division draw radii; the two angles ACB, A'C'B', will be divided into equal angles. If the arc ab is a common measure of the arcs, and the angle a cb, a common measure of the two angles; as there will be in each of the sectors as many of the small arcs as angles; we A'C'B' A'B' a b

shall have

ACD

ac b

AB

ab

and

a c b

; as the
denominators are the same in both proportions the
numerators must be in proportion; which gives (79)
A'C'B' A'B'
-; that is, in circles drawn with equal radii,

ACB AB
the angles at the centre are to each other in the ratio of
the arcs which subtend them, if these arcs are commen-
surable.

we may

Suppose the arcs have no common measure ; take a measure of one of them, A'B' for instance (fig. 57), so small that by applying it to AB as many times as Fig. 57. it is contained in AB, the remainder b B which we will

designate by m, will be less than any assignable arc; and the angle b CB, which it subtends, less than any assignable angle, which indefinitely small angle we designate by n. The arcs A'B' and A b are commensurable, ACB n AB-m

and therefore give the proportion A'C'B'

A'B'

Now suppose that, to AB produced we apply this small measure of A'B', once more than before, it will extend to b'; and suppose also that the excess of this arc A b' over AB, is less than any assignable arc, which excess we designate by m', and the corresponding indefinitely small ACB+n'

angle by n'; we shall have

A'C'B'

AB+m'

A'B'

It is manifest, therefore, from these proportions, that, one of the arcs, with its angle, remaining the same, if any magnitude however small be subtracted from the other arc, a corresponding magnitude must be subtracted from its angle, to preserve the proportion; and if any magnitude however small be added to this arc, a corresponding magnitude must be added to its angle to preserve the proportion; from which it follows that if nothing be either added to or subtracted from the one, nothing must be either added to or subtracted from the other.

shall therefore have the proportion,

We

AB

ACB
A'C'B' A'B'.

even when the arcs are incommensurable. We therefore say--If from the vertices of two angles, arcs are described with the same radius, the portions of these arcs embraced by the sides, will be in the ratio of the angles.

114. We see from the preceding article, what is to be understood when it is said that—An angle has for its measure the circular are comprehended between its sides, the vertex being at the centre of the circle. By this we mean that the angle is the same part of four right-angles, as the arc is of an entire circumference or 360°. When therefore, we speak of an angle of 30°, we mean an angle which would embrace between its sides, 30 parts out of 360, into which any circumference described from the vertex of this angle as a centre, should be divided.*

115. Suppose that the vertex of the angle, instead of being at the centre of the circle, is at some point between the centre and the circumferance, as at C (fig. 58). If Fig. 58.

This is according to the ancient division of the circumference, mentioned in article 26; a different system has been adopted in France. This system divides the circumference into 400 degrees; each degree into 100 minutes, and each minute into 100 seconds.

we draw through the centre, the two lines a n and bm, parallel to the two lines AN and BM, they will make an angle at the centre equal to the angle to be measured (24); and the arc m n will be as much greater than MN as the arc a b is less than AB; so that the sum of the arcs intercepted by the lines passing through the centre, is equal to the sum of the arcs intercepted by the lines which meet at C; therefore the arc ab is equal to half the sum of the two arcs AB and MN. But a b measures the angle a Ob equal to ACB. Whence we have this truth--An angle whose vertex is in the interior of a circle, made by two chords, has for its measure half the sum of the intercepted arcs.

116. Suppose the vertex of the angle to be in the circumference (in which case the angle is called an inscribed angle). By drawing parallels to the chords through Fig. 59. the centre (fig. 59) we shall have the two arcs C n and Cm, equal to A a and B b; therefore the arc m n equals AaBb; consequently the arc a b equals A a + B b ; that is, the arc a b is half the arc AB; but a b measures the angle whose vertex is at O, equal to the inscribed angle.

When both the chords which form the angle are on the same side of the centre, as the angle ACB'; this angle may be considered as the difference of the two angles BCB', BCA; BCB' has for its measure half the arc BB'; BCA has for its measure half of AB; consequently, the remaining angle ACB must have for its measure half the remaining arc AB'; and thereforeAny inscribed angle has for its measure half the arc eu braced by its sides.

117. This result includes the case of the angle made by the chord and tangent; for if AC be made to revolve about the point C till it come into the position A/C as the angle ACB increases, the arc AB increases, half of it still measuring the angle; and this reasoning holds till AC come infinitely near to the position of the tangent. We infer, therefore, that the error, if there be one, must be infinitely small.

But to ascertain whether there is any error in this result, draw through the centre parallels to the chord and Fig. 60. tangent (fig. 60); the angle ACB will be equal to a c b, measured by the arc a b. On account of the parallels, the arcs C a and C n are equal; and we have the arc mn

equal to C a added to B b; but the arc m n is equal to ab; therefore the arc a b is equal to a C added to B b, that is, equal to half the arc C a B. Hence we have rigorously demonstrated that―The angle made by a tangent and chord, has for its measure half the included arc.

118. If the meeting of the two lines is beyond the circumference (fig. 61), draw M b parallel to CB; the arc Fig. 61 Ab will be equal to the difference between the two arcs AB and MN; and the angle AM b, will be equal to the angle C (24); but the angle AM b has for its measure half the arc A b (116); therefore--The angle made by the two secants meeting without the circumference, has for its measure half the difference of the two arcs embraced by the sides.

But

119. This includes the case of the angle made by two tangents; for drawing the chord A b (fig. 62) from one Fig. 62. point of contact parallel to the other tangent; the arcs AB and B b, will be equal (112), and the arc A b will be the difference between the two arcs A b B and AB. half this arc Ab measures the angle DAb (116), and conquently the angle ACB, equal to DAb. Hence we say-The angle made by two tangents, is measured by half the difference of the arcs between their points of contact.

120. From the reasoning in article 116, it follows that --All inscribed angles subtended by the same arc, are equal (fig. 63), as each has for its measure half this arc. Fig. 63 121. If the arc embraced by the angle is a semi-circumference, that is, if the two sides of the inscribed angle are in the extremities of a diameter (fig. 64), the Fig. 61. semi-are, which measures the angle, will be an arc of 90°; therefore (27)—An angle inscribed in a semicircle is a right-angle.

122. Two angles, inscribed in opposite segments of the same circle, as ACB, ADB (fig. 63), will have, each Fig. 63. for its measure, half the opposite arc; therefore the two will have for their measure, half the circumference; consequently-The sum of two angles inscribed in opposite segments of the same circle, is equal to two rightangles.

123. As every tangent to the circumference of a circle, is perpendicular to the radius whose extremity is the point of contact (111); if we were required to draw a tangent to the circumference BFD (fig. 65) from the point A Fig. 65. without the circle; we should draw a straight line from