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by entire numbers, the arcs AB, DE, will be to each other as the angles ACB, DCE.
121. Scholium. Reciprocally, if the arcs AB, DE, are to each other as two entire numbers, the angles ACB, DCE, will be to each other as the same numbers, and we shall have always ACB: DCE::AB: DE; for the partial arcs Am, mn, &c., Dx, xy, &c., being equal, the partial angles AC m, m Cn, &c., DC x, x Cy, &c., are also equal.
122. Whatever may be the ratio of two angles ACB, ACD, (fig. 63), these two angles will always be to each other as the Fig. 63. arcs AB, AD, intercepted between their sides, and described from their vertices, as centres, with equal radii.
Demonstration. Let us suppose the less angle placed in the greater; if the proposition enunciated be not true, the angle ACB will be to the angle ACD as the arc AB is to an arc greater or less than AD. Let this arc be supposed to be greater, and let it be represented by AO; we shall have,
angle ACB: angle ACD :: arc AB : arc AO.
Let us now imagine the arc AB to be divided into equal parts, of which each shall be less than DO; there will be at least one point of division between D and O; let I be this point, and join CI; the arcs AB, AI, will be to each other as two entire numbers, and we shall have, by the preceding theorem,
angle ACB angle ACI :: arc AB : arc AI.
Comparing these two proportions together, and observing, that the antecedents are the same, we conclude that the consequents are proportional (111),* namely,
angle ACD: angle ACI :: arc AO: arc AI.
But the arc AO is greater than the arc AI; it is necessary, then, in order that this proportion may take place, that the angle ACD should be greater than the angle ACI; but it is less; it is therefore impossible, that the angle ACB should be to the angle ACD, as the arc AB is to an arc greater than AD.
By a process of reasoning altogether similar, it may be shown, that the fourth term of the proportion cannot be less than AD;
* The reference by Roman numerals is to the Introduction. GEOM.
therefore it is exactly AD, and we have the proportion
angle ACB angle ACD :: arc AB : arc AD.
123. Corollary. Since the angle at the centre of a circle and the arc intercepted between its sides have such a connexion, that, when one increases or diminishes in any ratio whatever, the other increases or diminishes in the same ratio, we are authorized to establish one of these magnitudes as the measure of the other; thus we shall, in future, take the arc AB as the measure of the angle ACB. The only thing to be observed in the comparison of angles with each other is, that the arcs, which are used to measure them, must be described with equal radii. This is to be understood in the preceding propositions.
124. Scholium. It may seem more natural to measure a quantity by another quantity of the same kind, and upon this principle it would be convenient to refer all angles to the right angle; and thus, the right angle being the unit of measure, the acute angle would be expressed by a number comprehended between O and 1, and an obtuse angle by a number between 1 and 2. But this manner of expressing angles would not be the most convenient in practice. It has been found much more simple to measure them by arcs of a circle, on account of the facility of making arcs equal to given arcs, and for many other reasons. Besides, if the measure of angles by the arcs of a circle be in some degree indirect, it is not the less easy to obtain, by means of them, the direct and absolute measure; for, if we compare the arc, which is used as the measure of an angle, with the fourth part of the circumference, we have the ratio of the given angle to a right angle, which is the absolute measure.
125. Scholium II. All that has been demonstrated in the three preceding propositions, for the comparison of angles with arcs, is equally applicable to the purpose of comparing sectors with arcs; for sectors are equal, when their arcs are equal, and in general they are proportional to the angles; hence two sectors ACB, ACD, taken in the same circle, or in equal circles, ≈re to each other as the arcs AB, AD, the bases of these sectors.
It will be perceived, therefore, that the arcs of a circle, which are used as a measure of angles, will also serve as the measure of different sectors of the same circle or of equal circles.
126. The inscribed angle BAD (fig. 64, 65), has for its Fig. 64 measure the half of the arc BD comprehended between its sides.
Demonstration. Let us suppose, in the first place, that the centre of the circle is situated in the angle BAD (fig. 64); we Fig. 64 draw the diameter AE, and the radii CB, CD. The angle BCE, being the exterior angle of the triangle ABC, is equal to the sum of the two opposite interior angles, CAB, ABC. But, the triangle BAC being isosceles, the angle CAB ABC; hence the angle BCE is double of BAC. The angle BCE, having its vertex at the centre, has for its measure the arc BE; therefore the angle BAC has for its measure the half of BE. For a similar reason, the angle CAD has for its measure the half of ED; therefore BAC+ CAD, or BAD, has for its measure the half BE+ED, or the half of BD.
Let us suppose, in the second place, that the centre C (fig. 65), Fig. 65. is situated without the angle BAD; then, the diameter AE being drawn, the angle BAE will have for its measure the half of BE, and the angle DAE the half of DE; hence their difference BAD will have for its measure the half of BE minus the half of ED, or the half of BD.
Therefore every inscribed angle has for its measure the half of he arc comprehended between its sides.
127. Corollary 1. All the angles BAC, BDC (fig. 66), &c., Fig. 66. inscribed in the same segment, are equal; for they have each for their measure the half of the same arc BOC.
128. Corollary II. Every angle BAD (fig. 67), inscribed in Fig. 67. a semicircle, is a right angle; for it has for its measure the half of the semicircumference BOD, or the fourth of the circumfer
To demonstrate the same thing in another way, draw the ra dius AC; the triangle BAC is isosceles, and the angle
the triangle CAD is also isosceles, and the angle CAD=ADC; hence BAC+ CAD, or BAD≈ ABD + ADB. But, if the two angles B and D of the triangle ABD are together equal to the third BAD, the three angles of the triangle will be equal to twice the angle BAD; they are also equal to two right angles; therefore the angle BAD is a right angle.
129. Corollary 1. Every angle BAC (fig. 66), inscribed in a segment greater than a semicircle, is an acute angle; for it has for its measure the half of the arc BOC less than a semicircumference.
And every angle BOC, inscribed in a segment less than a semicircle, is an obtuse angle; for it has for its measure the half of the arc BAC greater than a semicircumference.
130. Corollary iv. The opposite angles A and C (fig. 68) of an inscribed quadrilateral ABCD are together equal to two right angles; for the angle BAD has for its measure the half of the arc BCD, and the angle BCD has for its measure the half of the arc BAD; hence the two angles BAD, BCD, taken together, have for their measure the half of the circumference; therefore their sum is equal to two right angles.
131. The angle BAC (fig. 69), formed by a tangent and a chord, has for its measure the half of the arc AMDC, compre
hended between its sides.
Demonstration. At the point of contact A draw the diameter AD; the angle BAD is a right angle (110), and has for its measure the half of the semicircumference AMD; the angle DAC has for its measure the half of DC; therefore BAD + DAC, or BAC, has for its measure the half of AMD plus the half of DC, or the half of the whole arc AMDC.
It may be demonstrated, in like manner, that CAE has for its measure the half of the AC, comprehended between its sides.
Problems relating to the two first Sections.
132. To divide a given straight line AB (fig. 70) into two equal parts.
Solution. From the points A and B, as centres, and with a radius greater than the half of AB, describe two arcs cutting each other in D; the point D will be equally distant from the points A and B; find, in like manner, either above or below the line AB, a second point E equally distant from the points A and
B; through the two points D and E draw the line DE; this line will divide the line AB into two equal parts in the point C.
For, the two points D and E being each equally distant from the extremities A and B, they must both be in the perpendicular which passes through the middle of AB. But through two given points only one straight line can be drawn; therefore the line DE will be this perpendicular, which divides the line AB into two equal parts in the point C.
133. From a given point A (fig. 71), in the line BC, to erect a Fig. 71. perpendicular to this line.
Solution. Take the points B and C, at equal distances from A; and from B and C, as centres, with a radius greater than BA, describe two arcs cutting each other in D; draw AD, which will be the perpendicular required.
For, the point D, being equally distant from B and C, must be in a perpendicular to the middle of BC (55); therefore AD is this perpendicular.
134. Scholium. The same construction will serve to make a right angle BAD at a given point A in a given line BC.
135. From a given point A (fig. 72), without the straight line Fig. 72 BD, to let fall a perpendicular upon this line.
Solution. From A, as a centre, with a radius sufficiently great, describe an arc cutting the line BD in two points B and D ; then find a point E equally distant from the points B and D (132), and draw AE, which will be the perpendicular required.
For the two points A and E are each equally distant from the points B and D; therefore the line AE is perpendicular to the middle of BD.
136. At a given point A (fig. 73), in the line AB, to make an Fig. 73. angle equal to a given angle K.
Solution. From the vertex K, as a centre, with any radius, describe an arc IL meeting the sides of the angle, and from the point A, as a centre, with the same radius, describe an indefinite