PART SECOND. SECTION SECOND. CONSTRUCTION I. LINES. 47. Hitherto we have drawn figures on the board, and on the slate, by the eye, without regard to strict accuracy; we will now make use of instruments, and draw them with more care. To draw straight lines we make use of a ruler. That we may be accustomed to draw them in all positions of the ruler, we will take points in various parts of the paper, (or slate,) and connect them by straight lines, keeping the paper (or slate) always in the same position. 48. To draw circular lines we make use of a pair of compasses. These must be opened to a certain distance, the point of one leg fixed tight in the paper, and the point of the other leg moved on the surface of the paper about the fixed point, until the curved line thus made shall return into itself, and form a complete figure. This curved line is the circumference of a circle; the point of the fixed leg is the centre of the circle. The size of the circle depends upon the greater or less distance between the points of the compasses. The radius of the circle is exactly equal to this distance: hence, to express the size of a circle, we give the length of its radius; thus we say a circle of 1 inch, 1 foot, or 1000 feet radius. Many circles may be described, that is drawn, about the same centre; and their circumferences will remain in all parts at the same distance one from another. Two such circles, besides having a common centre, have the surface of the smaller in common, and the difference between their surfaces is an annular surface or ring. 49. Two circles in a plane may have various relative positions. They may have a common centre, or they may not. In the latter case the circles may lie entirely apart, or their circumferences may meet at 1 point on the outside, or may intersect at 2 points, so that they shall have in common a surface enclosed by 2 arcs. One circle may be entirely within the other, without having a common centre; of this there may be two cases, viz., the circumferences may be entirely separate, or they may touch at 1 point. We will draw several figures and examine them. (Fig. 46.) In 1 the distance between the centres of the circles == 0. In 2 the distance is greater than the sum of the 2 radii, by so much of the straight line joining the 2 centres as lies between the 2 circumferences. In 3 the distance between the 2 centres is equal to the sum of the 2 radii. In 4 the distance is less than the sum of the 2 radii by so much of the straight line joining the 2 centres as ies between the intersecting arcs. In 5 the distance is less than the difference of the 2 radii by so much of the radius of the greater circle passing through the centre of the lesser, as is contained between the 2 circumferences. In 6 the distance is exactly equal to the difference of the radii. 50. Suppose it is required to draw 10 straight lines, of which the 1st shall be 1 inch long, the 2d 2 inches long, and the 3d 3 inches long, and so on, the 10th being 10 inches long. Draw a straight line of any length; open the compasses so that the points shall be 1 inch apart, or, as it is more concisely expressed, take 1 inch between the points of the compasses, and apply them to the line. To get the 2d line, apply the compasses twice continuously to the line already drawn; or take 2 inches between the points and apply them once. Proceed in this manner; for the 10th line apply an opening of 1 inch 10 times continuously. If it is required to draw a straight line which shall be 2, 3, 4 or more times as long as another line; then draw a line of any length; take this length between the points of the compasses, and apply it as often as is required to a line of indefinite length. How shall we cut from a long line a part equal to a shorter one? Take the length of the shorter one between the points of the compasses; apply them to the longer line, placing one point at the end of the line. The remainder of the long line is the difference between the lines. 51. Suppose it is required to describe from the ends of the line AB, as centres, 2 circles which shall have the following relations one to the other. 1. Having nothing in common, (fig. 46. 2.) Fix one leg of the compasses at A, and with an opening less than AB, for example AC, describe a circle. = Then fix one leg of the compasses at B, and with an opening less than BC describe a circle. We have the 2 circles required. 2. The surfaces having no part in common, but the circumferences touching at 1 point, (fig. 46. 3.) From A, as a centre, with an opening of the compasses, or, (to speak more technically,) a radius, less than AB, for example AC, describe a circle; from B, as a centre, with a radius BC, describe another circle; we have the 2 circles required, which will touch each other externally. = 3. The circumferences touching at 2 points, (fig. 46. 4.) From A, as a centre, with a radius less than AB, describe a circle; from B, as a centre, with a radius greater than BC, but less than BA, describe another circle; we have the circles required. Suppose the line BA to be continued to E, it is evident that the radius of the circle described from B may be taken either equal to or greater than BA. If it be taken BE=BA+AE BA+AC, (since AE and AC are equal, being radii of the same circle,) then the first circle will meet the 2d at one point only, viz. at E. 4. Having the surface of one in common, without the circumferences touching each other, (fig. 46. 5.) Produce AB to C. From A, as a centre, with a radius AC, describe a circle; from B, with a radius less than BC, for example BD, describe another circle. We have the circles required. = 5. Having the surface of one in common, and the circumferences touching at 1 point, (fig. 46. 6.) Produce AB to any point C. From A, with a radius — AC, |