subtracted from the diameter of the required circle, must not be less that the distance between the centres of the given circles. 140. PROT. XLIX. Το άταν αν arc of a circle passing inrougi ihrer giver points (not in the same straight szne, nutnou nnding or using the centre of the circle. This is required to be done in drawing the parallels of latitude for mans and also in laying down railwaycurves; in both which cases as well as in some others, the centry is often inconveniently remote.] Let A. E. C be the three given points, of which B hes between the other two. Through A and C fix two pins, pegs, or naiis, in the plane surface on which the arc is to be drawn. Take two straight-edges" or "fiat-rulers", and lay them fiat or the surface with the edge of one resting or 4. and of the other or C: and bring them togetuer, until the same straight edges, which pass through 4 and C. intersect in the point B. When that is the case, festes the rulers tightly together at their junction, so that ghewerds, during the operation, the angle betweet tuent cannot rart. Then place a marker at B, in contact with the surface, and while this marker retains a fixed position with respect to the instrument, slide the Wive instrument on the pins at A and C. so as to bring the juntion of the straight edges, which was at B, first to Å end then to C, and the marker during this operation Win Uwe on the are required. For the curved line, whatever it be, certainly passes through A, B, and C: and, if AC be joined, AC subtelde the same angle at every point in the curved line, which is a well-known property of a segment of a circle (59. COR There is an instrument called a Bevel commonly used "the above purpose by those who have frequent occato draw such arcs. 147. PROP. L. An arc, or a segment, of a circle being given, to complete the circle of which it is a part. F D B C Let ABC be the given arc, or segment; if the former, join A and C, the extreme points, by the chord AC. In either case bisect AC in D by the straight line A DE at right angles to AC (101). Join AB, B being any point in the given arc, and bisect AB in like manner by the straight line FO at right angles to AB, meeting DE in O. Then with centre O, and radius OA, describe a circle, and it will be the circle required. For, it may easily be shewn, that OA = OC = OB, and B is any point in the given arc; .. the whole arc is a part of the circumference thus drawn. 148. PROP. LI. From a given circle to cut off a segment, which shall contain an angle equal to a given angle*. Let ABC be the given circle, A and B being any points in its circumference; through B draw BD touching the circle (135); and draw the chord BC such that DBC= the given angle (105). Then BAC is the required segment. For, joining AB, AC, 4 BAC in the alternate segment (63) DBC= the given angle. = COR. If the given angle be a B D right angle, the segment will be a semi-circle, which of course, may be cut off the given circle by drawing any diameter of the circle. If the given angle be acute, the segment will be greater than a semi-circle. And, if the given angle be obtuse, the segment will be less than a semi-circle (54). The learner must bear in mind the Definition of 'angle in a segment'; see (52 Cor.). 149. PROT LII. Upon a given straight line to construc, & segment of a circle which shali contain an angle equa. to 4. Free) angu. Let be the given straight line; and through the point draw the straight me A ( A touches the circle at A And the angle in the segment' ADB=1BAC (63) = tin giver angie. 1 the giver angie be a right angle, it will then only be necessary to describe a semi-circit on AB as a diameter, and that semi-circle will be the segment required. INSURIBED AND CIRCUMSCRIBED FIGURES, AND CONSTRUCTION OF POLYGONS. 150. DEFINITION 1. A RECTILINEAL FIGURE is sait. to be inscribed in another rectilineal figure, when all the angular points of the inscribed figure are upon the sides of the figure in which it is inscribed, each upon Thus, one triangle is inscribed in another triangle, not merely when the one is situated within the other, as in the first or second of the annexed diagrams, but when / * A exyment of a circle is defined (48) to be a portion of the circle nded by an are and its chord. The chord is sometimes called the of the segment, that is, it is a straight line upon which the segment sed to stand. also each side of the outer triangle has upon it the vertex of each one of the angles of the inner triangle, as in the 3rd diagram. DEF. 2. A Rectilineal Figure is said to be 'circumscribed' about another rectilineal figure, when all the sides of the circumscribed figure pass through the angular points of the figure about which it is circumscribed, each through each. Thus, in the preceding diagrams, the larger triangle is not 'circumscribed about the lesser in the 1st and 2nd, but only in the 3rd. DEF. 3. A Rectilineal Figure is said to be 'inscribed in a circle', when all the angular points of the inscribed figure are upon the circumference of the circle. DEF. 4. A Rectilineal Figure is said to be 'circumscribed about a given circle', when each side of the circumscribed figure touches the circle. DEF. 5. A Circle is said to be inscribed' in a rectilineal figure, when it is so drawn as to touch each side of the figure. DEF. 6. A Circle is said to be 'circumscribed' about a rectilineal figure, when its circumference passes through all the angular points of the figure. It is important for the learner to take good heed to these Definitions, because the words inscribed' and 'circumscribed' are allowed to have only the technical meanings here assigned to them, whereas the tendency is to give them a much wider meaning, which leads to serious error. For instance, the careless student would say, that the triangle in the annexed fig. (1), is inscribed in the circle, or the circle circumscribed about the triangle; but it is not so, according to the Definition, which requires that each angle of the inscribed figure have its vertex in the circumference of the circle, as shewn in fig. (2). (1) In fact, to call the circle in fig. (1) the circumscribing circle of the triangle would be to define nothing, because there are an infinite number of such circles, having the common property of passing through two of the vertices of the triangle; but when we speak of the circle circumscribing the triangle according to Definition, as shewn in fig. (2), we speak of a particular well-defined circle, for there is one such circle, and one only (134 Cor. 1). 151. PROP. LIII. In a given circle to inscribe a triangle similar, that is, equiangular, to a given triangle. Let ABC be the given circle; and DEF the given triangle. Draw the straight line GAH touching the circle in the point A; and from A draw the chords AB, AC, such that HAC: = < DEF, B D H That ABC is a triangle inscribed in the circle is plain, because it has the vertex of each of its angles on the circumference. Also, by (63), ▲ ABC = ▲ HAC = ▲ DEF; and ACB = 4 GAB = 2 DFE; .. the remaining ‹ BAC = LEDF (37); that is, the triangle ABC is equiangular, and similar, to the triangle DEF. Obs. Since the point A was taken arbitrarily any where in the circumference, there may be any number of such triangles, as ABC, inscribed in the circle. But they will all be equal and similar to one another, being different only in position. It is also to be noted, that in the above construction we have the solution of the following Problem:— "Out of a given circle to cut the greatest triangle similar to a given triangle'. 152. PROP. LIV. About a given circle to circumscribe a triangle similar, that is, equiangular, to a given triangle. Let ABC be the given circle; and DEF the given triangle. Produce EF both ways indefinitely to points G and H; find the centre O of the given circle, and draw |