Angles proportional to their Arcs.

93. Corollary. Hence the greatest straight line which can be inscribed in a circle is equal to its di

ameter.

94. Theorem. A straight line cannot meet the circumference of a circle in more than two points.

Proof. For, by §§ 38 and 41, only two equal straight lines can be drawn from the same point to the same straight line; whereas, if a straight line could meet the circumference ABD (fig. 45) in the three points, ABD, three equal straight lines CA, CB, CD, would be drawn from the point C to this line.

95. Theorem. In the same circle, or in equal circles, equal angles ACB, DCE (fig. 44), which have their vertices at the centre, intercept upon the circumference equal arcs AB, DE.

Proof. Since the angles DCE and ACB are equal, one of them may be placed upon the other; and since their sides are equal, the point D will fall upon A, and the point E upon B. The arcs AB and DE must therefore coincide, or else there would be points in one or the other unequally distant from the centre.

96. Theorem. Reciprocally if the arcs AB, DE (fig. 44) are equal, the angles ACB and DCE must be equal.

Proof. For if the line CE be drawn, so as to make an angle DCE equal to ACB, it must pass through the extremity E of the arc DE, which is equal to AB.

97. Theorem. Two angles, as ACB, ACD (fig. 45), are to each other as the arcs AB, AD intercepted

Infinitely Small Quantities.

between their sides, and described from their vertices as centres, with equal radii.

Proof. Suppose the less angle placed in the greater, and suppose the angles to be to each other, for example, as 7 to 4; or, which amounts to the same, suppose the angle AC a, which is their common measure, to be contained 7 times in ACD, and 4 times in ACB; so that the angle ACD may be divided into the 7 equal angles AC a, a Cb, b Cc, &c., while the angle ACB is divided into the 4 equal angles AC a, &c.

The arcs AB and AD are, at the same time, divided into the equal parts A a, ab, bc, &c., of which AD contains 7 and AB 4; and therefore these arcs must be to each other as 7 to 4, that is, as the angles ACD and ACB.

98. Scholium. The preceding demonstration does not strictly include the case in which the two angles are incommensurable, that is, in which they have no common divisor. The divisor AC a, instead of being contained an exact number of times in the given angles ACB, ACD, is, in this case, contained in one or each of them a certain number of times plus a remainder less than the divisor. So that if these remainders be neglected, the angle AC a will be a common divisor of the given angles.

Now the angle AC a may be taken as small as we please; and therefore the remainders, which are neglected, may be as small as we please; less, then, than any assignable quantity, less than any conceivable quantity, that is, less than any possible quantity within the limits of human knowledge. Such quantities can, undoubtedly, be neglected, without any error; and the above demonstration is thus extended to the case of incommensurable angles.

Measure of Angles. Degree, Minute, Second, &c.; Quadrant.

99. The principle, involved in the reasoning just given, is general in its application; and may be stated as follows, using the term infinitely small quantity to denote a quantity which may be taken at pleasure, as small as we please, so that it may be supposed equal to nothing whenever we please.

Axiom. Infinitely small quantities may be neglected.

100. Corollary. Since the angle at the centre of a circle is proportional to the arc included between its sides, either of these quantities may be assumed as the measure of the other; and we shall, accordingly, adopt, as the measure of the angle, the arc described from its vertex as a centre and included between its sides.

But when different angles are compared with each other, the arcs, which measure them, must be described with equal radii.

101. Definitions. In order to compare together different arcs and angles, every circumference of a circle may be supposed to be divided into 360 equal arcs. called degrees, and marked thus (°). For instance, 60° is read 60 degrees.

Each degree may be divided into 60 equal parts called minutes, and marked thus (').

Each minute may be divided into 60 equal parts called seconds, and marked thus (").

When extreme minuteness is required, the division is sometimes extended to thirds and fourths, &c., marked thus ('''), (''''), &c.

A quadrant is a fourth part of a circumference, and contains 90°. This is called the sexagesimal division of the circle; another which is called the centesimal di

Inscribed Angle and Triangle.

vision has been introduced by the French geometers. They divide the quadrant into 100 degrees, the degree into 100 minutes, &c; so that by this method of division, the whole notation is decimal.

102. Scholium. As all circumferences, whether great or small, are divided into the same number of parts, it follows that a degree which is thus made the unit of arcs, is not a fixed value, but varies for every different circle. It merely expresses the ratio of an arc, namely, to the whole circumference of which it is a part, and nol to any other.

360

103. Corollary. The angle may be designated by the degrees and minutes of the arc which measures it; thus the angle which is measured by the arc of 17° 28′ may be called the angle of 17° 28'.

104. Corollary. The right angle is tnen an angle of 90°, and is measured by the, quadrant.

of

105. Corollary. The angle which is measured by the arc of one degree, that is, the angle of 1° is then a right angle, and has a fixed value, altogether independent, in its magnitude, of the radius of the arc by which it is measured.

The same is the case with an angle of any other value, so that the arcs AP, A'D', AD", &c. (fig. 46), of the same number of degrees, all measure the same angle C, the vertex of which is at their common centre.

106. Definitions. An inscribed angle is one, whose vertex is in the circumference of a circle, and which is formed by two chords, as BAC (fig. 47).

An inscribed triangle is a triangle whose three angles have their vertices in the circumference of the circle.

Inscribed Angle.

And, in general, an inscribed figure is one, all whose angles have their vertices in the circumference of the circle. In this case the circle is said to be circumscribed about the figure.

107. The inscribed angle BAC (figs. 47, 48, 49) has for its measure the half of the arc BC comprehended between its sides.

Proof. 1. If one of the sides is a diameter, as AC (fig. 47), O being the centre of the circle,

Join BO. Then the triangle AOB is isosceles, for the radii AO, BO are equal. Therefore the angles OAB and OBA are equal, and the exterior angle BOC being equal to their sum, by § 71, is equal to the double of either of them, as BAC. BAC is, therefore, half of BOC and has half its measure, or half of BC.

2. If the centre O falls within the angle, as in (fig. 48,)

Draw the diameter AOD; and, by the above, BAD has for its measure half of BD, and DAC half of DC; so that BAD DAC or BAC has for its measure half of BD+DC, or half of BC.

3. If the centre O falls without the angle, as in (fig. 49,)

Draw the diameter AOD; and BAD · DAC, or BAC has for its measure half of BD — DC, or half of BC.

108. Corollary. All the angles BAC, BDC (fig. 50), &c., inscribed in the same segment are equal.

Proof. For they have each for their measure the half of the same arc BEC.

109. Corollary. Every angle BAD (fig. 51) inscribed in a semicircle is a right angle.