The Circle, Radius, Diameter.

AD, BO= DO, and AO is common; therefore the angles AOB and AOD are equal, and, as they are adjacent, each of them must, by definition, art. 20, be a right angle, so that the two diagonals of a rhombus bisect each other at right angles.

CHAPTER VIII.

THE CIRCLE AND THE MEASURE OF ANGLES.

84. Definitions. The circumference of a circle is a curved 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.

85. Definitions. The radius of a circle is the straight line, as AB, AC, AD (fig. 43), drawn from the centre to the circumference.

The diameter of a circle is the straight line, as BD, drawn through the centre, and terminated each way by the circumference.

86. Corollary. Hence, all the radii of a circle are equal, and all its diameters are also equal, and double of the radius.

87. Theorem. Every diameter, as BD (fig. 43), bisects the circle and its circumference.

Demonstration. For if the figure BCD be folded over upon the part BED, they must coincide; otherwise there

Inscribed Lines.

would be points in the one or the other unequally distant from the centre.

88. Definition. A semicircumference is one half of the circumference, and a semicircle is one half of the circle itself.

89. Definition. An arc of a circle is any portion of its circumference, as BFE.

90. Definition. The chord of an arc is the straight line, as BE, which joins its extremities.

91. Theorem. Every chord is less than the diam

eter.

Demonstration. Thus BE (fig. 43) is less than DB. For, joining AE, we have BD BA+ AE, but BE< BA+AE, therefore BE < BD.

=

92. Definition. A straight line is said to be inscribed in a circle, when its extremities are in the circumference of the circle.

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

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

Demonstration. For, by arts. 38 and 41, only two equal straight lines can be drawn from the same point to the same straight line.

95. Theorem. In the same circle, or in equal circles, equal angles ACB, DCE (fig. 44), which have

Angles proportional to their Arcs.

their vertices at the centre, intercept upon the circumference equal arcs AB, DE.

Demonstration. 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.

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

Demonstration. 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 ACa, 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 ACa, a Cb, b Cc, &c., while the angle ACB is divided into the 4 equal angles ACa, &c.

The arcs AB and AD are, at the same time, divided into the equal parts Aa, 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.

Infinitely Small Quantities. Measure of Angles.

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 ACa, 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 ACa will be a common divisor of the given angles.

Now the angle ACa 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, certainly, be neglected, as "dust on the balance;" and the above demonstration is thus extended to the case of incommensurable angles.

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 less than any assignable quantity.

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 the arc as the measure af the angle.

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

Degree, Minute, Second, &c.; Quadrant.

101. Definitions. In order to compare together different arcs and angles, every circumference of a circle is supposed to be divided into 360 equal arcs called degrees, and marked thus (0). For instance, 60° is read 60 degrees.

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

Each minute is 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°.

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 not to any other.

102. 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′.

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

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