describe a circle, and from B, with a radius = BC, describe another circle. 52. To divide a straight line AB into 2, 4, 8, 16, &c., equal parts, (fig. 47.) From A and B, as centres, with equal radii, which must be greater than AB, describe two circles; the circumferences of these circles will intersect at 2 points, C and D, one on each side of AB. Join C and D by a straight line. The line AB will be bisected at the point where CD crosses it. It is not necessary to describe entire circumferences; 2 intersecting arcs on each side of AB will be sufficient. If the line is very long, or the compasses very small, we can take any 2 points in the line for centres, so they are at equal distances from the ends of the line. By a similar process bisect each half of AB, and the whole line will be divided into 4 equal parts, (fig. 48.) Bisect each of these 4 parts, and the whole line will be divided into 8 equal parts. 53. To draw a curved line like a steel spring, (fig. 49.) Draw a straight line, and take in it any point A. From A, as a centre, with a small radius AB, describe a semi-circumference BC; then from B, as a centre, with radius BC, describe a second semi-circumference below the line AB; then from A, with radius AD, describe a third semi-circumference above the line AB; again from B, with a radius BE, describe a fourth semi-circumference, EF, below the line, and so on. There will thus be formed from connected semi-circumferences a connected curved line. From the construction it is evident that AB= AC, BC= BD, and AD=AE, consequently BD=CE; BE=BF, consequently CE DF, thus BC BD-CE=DF. = = = 54. To draw a curved line winding like a snailshell, (fig. 50.) Draw a straight line and take in it a point A. On each side of this point measure off in the line several small equal parts, for example 3, viz., AD, DH, HB, AC, CG, GL. From A, as a centre, with a radius AB, describe a circumference. Then, on each side of the line AB alternately, describe semi-circumferences, viz., from C, with radius CB; from D, with radius DE; from G, with radius GF. The required curved line will thus be formed. From the diagram it appears that LE contains 2 of the above equal parts; BF has 4; EI has 6. 55. To draw a serpentine line, (fig. 51.) Divide a straight line into any number of equal parts, for example, 12. From the 1st, 3d, 5th, 7th, 9th, and 11th points of division, as centres, with a radius equal to one part, describe semi-circumferences, alternately above and below the straight line. A curved line drawn in this manner resembles that made by a snake in motion, and is therefore called a serpentine line. 56. To draw a line which curves like the waves, (fig. 52.) Draw 3 parallel straight lines at equal distances from one another. We can do this with a ruler, for if it is well made the two edges will be parallel; draw lines along both edges, and we shall have 2 of the parallels; move the ruler and place one edge exactly upon one of the lines already drawn, then draw a line along the other edge, and we shall have 3 parallels. Divide these parallels into any number of equal parts, for example, 10. Then from the points of division 1, 3, 5, 7, 9, alternately on one and the other of the exterior parallels, with a radius equal to the distance between the first division point and the end of the middle parallel, describe arcs. These will be alternately above and below the middle parallel, and will form a continuous curved line, called a waving line. 57. To draw an Ellipse, (fig. 53.) Divide a straight line AD into 3 equal parts at the points B and C. From B, with radius BC, describe a circle; and from C, with radius CB, describe another circle; the circumferences of these circles will intersect at 2 points, E and F. From E draw the diameters EG and EH; and from F draw the diameters FI and FK. From E, as a centre, with radius EG, describe the arc GH, and from F, with radius FI, describe the arc IK. Thus from the 4 exterior arcs IK, KH,' HG, and GI, is formed a connected figure called an ellipse. 58. To draw an Oval, (fig. 54.) Describe a circumference and divide it into 4 equal parts; join the opposite division points by the diameters BC and DE; draw CD and BD, and produce them beyond D. From C, with radius CB, describe the arc BF; and from B, with radius BC, describe the arc CG; and from D, with radius DF, describe the arc FG. The arcs BC, CG, GF, and FB form, together, a figure shaped like an egg. II. ANGLES. 59. Before proceeding to the construction of angles, it will be well to make ourselves acquainted with the relation between angles and arcs. We have before seen that as the side AC (fig. 39) departs from AB, the angle which it makes with AB is constantly increasing. Now let us suppose that each point in AC describes at the same time an arc of a circle. It is evident that such arcs bear a certain fixed relation to the angle and to one another, since all are made by one and the same motion of the side AC, and begin, increase, and end simultaneously. These arcs differ in actual length; but each is the same fractional part of a whole circumference; if we suppose each circumference to be divided into the same number of equal parts, these arcs will contain an equal number of such parts. It is usual to divide a circumference into 360 equal parts called degrees, and marked thus (°); for example, 40° is read forty degrees. As all circumferences, whether of great or of small circles, are divided into 360°, it follows that a degree is not a fixed quantity, but varies for every different circumference. It merely expresses the magnitude of an arc as compared with the whole circumference of which it is a part, and not with any other circumference. Each degree is divided into 60 equal parts called minutes, and marked ('). Each minute is divided into 60 equal parts called seconds, marked ("). The division is sometimes carried to thirds and fourths, marked (''') (''''). We are thus furnished with a very convenient method of measuring angles. As the magnitude of an angle has no reference to the length of its sides, but to their mutual inclination, or the opening between them, either of the arcs described from its vertex as a centre, and intercepted, (taken between,) by its sides, may be taken as the measure of the angle, for they all contain the same number of degrees; which number of degrees denotes the size of the angle. 60. If the side AC be moved entirely round the point A, it will have made 4 R. A.; and at the same time each point in it will have described an entire circumference; thus a circumference, or 360°, is the measure of 4 right angles; and therefore a quarter of a circumference, or quadrant, as it is called, is the measure of 1 right angle; that is, the sides of every right angle 360° = 90° in the circumference 4 will intercept an arc of of a circle described from its vertex as a centre with any radius. We say therefore that a right angle is an angle of 90°; half a right angle is an angle of 45°; one third of a right angle is an angle of 30°; two thirds of a right angle is an angle of 60°, &c. 61. Having made ourselves acquainted with the principle upon which the mensuration of angles depends, we will now examine the protractor, (fig. 55,) which is an instrument used in plotting, that is, drawing upon paper angles whose magnitude is known; or for measuring angles already drawn upon paper. It is a semicircle of wood, metal, or horn, accurately divided into 180°. For the convenience of reckoning both ways, the degrees are numbered from the left towards the right, and from the right towards the left. The division lines are all drawn from a point in the middle of the diameter, |