14. Lay down to scale and calculate the areas of fields of the following, dimensions,— Ist. Left o C 525 0 586 1253 B B 200 400 600 846 C 50 85 64 O Ans. Area 3 acres 2 roods 32 perches. 2nd. A field of four sides, ABCD. CIRCLES AND SECANTS. 193. We have thus far considered the chief properties of the circumference and the chief methods of describing it; we will now examine the relations between circles and straight lines. Any straight line which cuts a circle is termed a secant. The perpendicular let fall from the centre upon a chord is less than the radius of the circle. 194. Join the centre O to the extremities A and B of the chord (fig. 168); then A OB O, since they are radii of the same circle, and therefore A O B is an isosceles triangle. If a line be drawn from the centre of the circle to the middle point of the chord, it will be perpendicular to the chord (§ 73), and therefore less than the oblique line A O from the same point ($35.) The line OC is the symmetrical axis of the triangle A O B, and if it be produced to cut the arc A B in the point D, the line O D is evidently also the symmetrical axis of the arc A B. Since only one perpendicular can be drawn to the straight line A B, there is only one symmetrical axis, and therefore the perpendicular drawn from the centre of a circle to a chord bisects the chord and the arc it subtends. Every diameter is a symmetrical axis of the circumference (§ 59). 195. The circle is the only curve in which every diameter is a line of symmetry, for the demonstration of this property depends upon the equality of the lines which join the centre to the extremities of the chord; that is, of the radii (§ 43). Hence the circumference is the only curve whose parts coincide when superposed. This property is applied in many ways; one of the most curious being the construction of the spirit-level. This instrument (fig. 169) consists of a glass tube closed at both ends, slightly curved, and nearly filled with liquid, only a small bubble of air being left in it. The tube is mounted in brass, so that when the instrument is placed upon a horizontal plane, the bubble occupies a determined position, which necessarily changes as soon as one of the extremities is raised, since it tends always to place itself in the most elevated part of the Fig. 169. tube. In levels constructed so that the position of the bubble determines the slope of the plane, the upper edge of the tube must be an arc of a circle. From a given centre to describe a circle cutting a straight line in a given point. 196. With the compass take the distance from the given point O (fig. 168) to the point A, then on the straight line with O as a centre, describea circumference. To draw a circle of known radius cutting a straight line in two given points. 197. The centre of the required circle will be at the given distance from the two given points A and B, and must therefore be the point of intersection of the two circumferences described from A and B as centres with the given radius (fig. 170). Suppose these circumferences intersect in O and O'. The circle described either from O or O' as centre, with the given radius, will pass through the two points A and B. In order that the problem may be possible, the two arcs must cut one another, and therefore the given radius must be greater than the half of A B. Fig. 170. B It is evident that the centre of the circle must be upon the perpendicular erected at the middle of A B (§ 194), hence if a circle be described from one of the points A and B with the given radius, intersecting this perpendicular in the points OO', the required centre must be one of these points. To draw a circle passing through three given points. 198. Let A B C be the three given points (fig. 171). Join the points A and B; the centre of the circle passing through these points must lie in D E, a perpendicular at the middle point D of A B. Join also B and C ; the centre of the circle which passes through these points must lie in FG, a perpendicular at the middle point of B C. Hence, the centre of a circle passing through the three points A B C must be at O, where these perpendiculars cut each other. Describe then X Fig. 171. a circle with O as centre and OA as radius; it will pass through the points B and C, and will be the circle required. As the two perpendiculars in which the centre of the required circle must lie can intersect in one point O only, it follows that there can be no other circle which will pass through the three points А В С. From this it follows that two circumferences cut one another, at most, in two points; since without coinciding they cannot have three points in common. To find the centre of a circle already described. 199. Draw two chords, A B, BC (fig. 171); at the middle of these chords erect perpendiculars; their point of intersection will be the centre of a circle passing through the three points, and we know there can be but one circle fulfilling these conditions. 4 To divide an arc into two equal parts. A Fig. 172. 200. Draw a perpendicular from the centre of the circle (fig. 172), upon the chord of the given arc, cutting the arc in D; the point D bisects the arc (§ 194). The line bisecting an angle is the symmetrical axis of the bent line forming the two sides of the angle. 201. The bisecting line of an angle is that which divides it into two equal parts, and it is evident that the two sides coincide if the figure be folded about the bisecting line. If an arc be described with the apex as centre, the line also bisects this arc. To divide an angle into two, four, eight, etc., equal parts. 202. The division of an angle into two equal parts is effected by describing from the apex of the angle an arc of any radius (fig. 173) and bisecting this arc. Fig. 173. Sometimes the apex of the angle is not known. The bisecting line may then be obtained by considering that each point in a line bisecting an angle is equidistant from the two sides of the angle. This evidently results from the symmetry of the figure with respect to the bisecting line. Now all points which are at the same distance from a straight line are upon a parallel to this straight line. To find a point then in the line bisecting the angle between the two lines A B, CD (fig. 174), draw a parallel, A' B', to one of sides, A B; at any point erect a perpendicular, |