Proof. Let AB be a tangent at A. Draw any secant as AM. The ▲ ACM is isosceles, and the ≤ CAM is less than 90°. As the secant rotates about A as a pivot, and M approaches A, the ACM will approach 0, and the Z CAM will approach 90°.

After M has passed through A, the angle CAM" will be greater than 90°, because CAK is less than 90°.

As the change has been a continuous one, the angle made with CA changing gradually from an acute to an obtuse angle must have passed through 90°. And furthermore, when the second point is on either side of A the angle is oblique, it must be 90° when the second point coincides with the pivotal point, and the rotating line has become a tangent.

45. THEOREM.

Q. E. D.

The perpendicular bisector of a chord of a circle will pass through the centre and will bisect the arcs subtended by the chord.

Section 20 furnishes the proof for the first part of the theorem.

P

For the establishing of the second part, the analysis suggests that we revolve one portion of the figure on PQ as an axis. PQ is the diameter of the circle that is the perpendicular bisector of AB.

When revolved, MA will coincide with MB, and the point A will fall at B. The points Q and P will remain stationary.

A

M

FIG. 64.

The two circumferences will coincide, for every point of each will be at the same distance from C.

Hence QA will coincide with QB, and PA will coincide with PB.

NOTE. The chord AB subtends the two arcs AQB and APB. But ordinarily, in speaking of the arc subtended by a chord, the lesser arc is the one understood.

Exercises.-1. Show that chords AQ and BQ would be equal to each other; and that chords AP and BP would also be equal to each other.

2. Show how to draw a tangent at a given point of a circumfer

ence.

3. Having given a circumference, show how to find the centre.

46. Recalling the matter in §§ 7, 11, and 21, and again observing the generation of an angle by the rotation of a line about one of its points as a pivot, we are prepared to develop another relation; viz. the

THEOREM. If a circumference be constructed with the vertex of an angle as its centre, the arc included between the lines forming the angle will be the same fractional part of the entire circumference that the angle is of 360°.

During a complete rotation every point in the line AB will generate a circumference. Let B represent one of these points.

If the angular magnitude about A be generated by a uniform rotation of the line AB, any point in the line AB will move at a uniform rate.

FIG. 65.

The angle and the circumference are generated with uniformity; they are begun at the same instant; and the 360° of rotation are completed at the instant the circumference is completed. At any stage of the proceeding, therefore, the angle generated will be the same fractional part of 360° that the arc generated by any point is of the entire circumference, or

COROLLARY. Two angles having their vertices at the centre of a circle will have the same ratio as the intercepted

Dividing the members of (1) by the members of (2), we get

NOTES.1. A corollary is a subsidiary theorem that follows from a principal one. In this work there are very few corollaries presented; it being preferred to establish the facts and relations as original exercises.

2. Because of the fact that when the vertex of an angle is at the centre of a circle, its sides intercept the same fractional part of the circumference that the angle is of 360°, we say that the angle is measured by the intercepted arc.

For purposes of numerical description, in speaking of a single angle when the angle itself is not represented in the drawing, an angle of 1o is applied as many times as it will be contained in the angle; then the one sixtieth part of one degree (or an angle of one minute) is applied to the remainder as many times as it will be contained in it; then to the second remainder is applied an angle of one second; and if a nearer approximation is desired, the decimal subdivisions of a second are applied to the succeeding remainders until the numerical description is as accurate as the circumstances demand.

In the same way, when numerical description is needed in order to represent an arc of a circumference, an arc which subtends an angle of 1o is applied as far as possible; to the first remainder is applied an arc that subtends an angle of 1', until there is a remainder less than 1'; to this is applied the arc that subtends 1", etc.

Ordinary surveying instruments describe an angle to within 30"; very accurate geodetic instruments to within 10"; ordinary astronomical instruments to within 3"; and the best to within".

47. Definitions.

If two chords intersect on the circumference of a circle, the angle they make is said to be an inscribed angle, and the intercepted arc is said to subtend the angle.

THEOREM.

An inscribed angle is measured by half of

the intercepted arc.

(a) If one of the chords be a diameter as AB, draw the auxiliary line CD.

The three cases are all the possible ones, and each having been demonstrated, the theorem is established.