ALGEBRAIC SYMBOLS. The advantage of these, in a work like the present, may be thus illustrated: Let / denote the length, b the breadth, and d the depth of an iron beam. If it be desired to express the product of the length and breadth, divided by the depth, it is done as follows: That is to say, multiplication is expressed by simply writing the letters which represent numbers one after the other; division, by drawing a line under the dividend, and writing the divisor below. The sum of the length and breadth, divided by the depth, would be expressed briefly thus: 2 + 6 The square of the length, multiplied by the cube of the breadth, thus: Ꭸ b The square root of the length, divided by the fourth root of the breadth, thus: ντ The square root of the difference of the length and breadth, divided by the depth, thus: d The square root of the quotient of the sum and difference of the length and breadth, thus: Any other letters-as a, b, c, &c.-may stand for the given dimensions. These explanations will serve to give the sense of the symbols which will be met with throughout the work. PRACTICAL GEOMETRY. A 1. From any given point, in a straight line, to erect a perpendicular; or, to make a line at right angles with a given line. On each side of the point A, from which the line is to be made, take equal distances, as A b, A c; and from b and c as centres, with any distance greater than b A or c A, describe arcs cutting each other at d; then will the line Ad be the perpendicular required. 2. When a perpendicular is to be made at or near e the end of a given line. With any convenient radius, and with any distance from the given line A b, describe a portion of a circle, as b A c, cutting the given point in A; draw, through the centre of the circle n, the line bnc; and a line from the point A, cutting the intersections at C, is the perpendicular required. 3. To do the same otherwise. From the given point A, with any conve nient radius, describe the arc de b; from d cut the arc in c, and from c cut the arc in b; also from c and b as centres, describe ares cutting each other in t; then will the line At be the perpendicular as required. Note. When the three sides of a triangle are in the proportion of 3. 4, and 5 equal parts, respectively, two of the sides form a right angle; and observe that in each of these or the preceding problems, the perpendiculars may be continued below the given lines, if necessarily required. d 4. To bisect any given angle. From the point A as a centre, with any radius less than the extent of the angle, describe an arc, as cd; and from c and d as centres, describe arcs cutting each other at b; then will the line Ab bisect the angle as required. A 5. To find the centre of a circle, or radius, that shall cut any three given points, not in a direct line. From the middle point b as a centre, with any radius, as bc, bd, describe a portion of a circle, r. as es d; and from and t as centres, with an equal radius, cut the portion of the circle in es and ds; draw lines through where the arcs cut each other; and the intersection of the lines at s is the centre of the circle as required. 6. To find the centre of a given circle. Bisect any chord in the circle, as A B, by a perpendicular, CD; bisect also the diameter ED in f; and the intersection of the lines at f is the centre of the circle required. 7. To find the length of any given arc of a circle. With the radius AC, equal to 4th the length of the chord of the arc A B, and from A as a centre, cut the are in e; also from B as a centre, with equal radius, cut the chord in b; draw the line Cb; and twice the length of the line is the length of the arc nearly. 8. Through any given point, to draw a tangent to a circle. Let the given point be at A; draw the line A C, on which describe the semicircle ADC; draw the line ADB, cutting the circumference in D, which is the tangent as required. C f B B 9. To draw from or to the circumference of a circle lines tending towards the centre, when the centre is inaccessible. Divide the whole or any given portion of the circumference into the desired number of equal parts; then, with any radius less than the distance of two divisions, describe arcs cutting each other, as A 1, B 1, C2, D 2, &c.; draw the B D A F lines C 1, B 2, D 3, &c., which lead to the centre, as required. To draw the end lines. As Ar, Fr, from C describe the arc r, and with the radius C 1, from A or F as centres, cut the former arcs at r, or r, and the lines. Ar, Fr, will tend to the centre as required. 10. To describe an arc, or segment of a circle, of large radii. Of any suitable material, construct a triangle, as A B C; make A B, BC, each equal in length to the chord of the arc DE, and height, twice that of the arc Bb. At each end of the chord D E fix a pin, and at B, in the triangle, fix a tracer (as a pencil), move the triangle along the pins as guides; and the tracer will describe the arc required. 11. Or otherwise. Draw the chord A CB; also draw the line HD I, parallel with the chord, and equal to the height of the segment; bisect the chord H 2 3 D D 3 2 n 1 3 3 2 A 1 2 3 C3 2 1 B in C, and erect the perpendicular CD; join A Î, DB; draw A H perpendicular to A D, and BI perpendicular to BD; erect also the perpendiculars An, B n; divide AB and H I into any number of equal parts; draw the lines 11, 22, 33, &c.; likewise divide the lines An, B n, each into half the number of equal parts; draw lines to D from each division in the lines An, Bn, and, through where they intersect the former lines, describe a curve, which will be the arc or segment required. 12. To describe an ellipse, having the two diameters given. On the intersection of the two diameters as a centre, with a radius equal to the difference of the semidiameters, describe the arc a b; and from b as a centre, with half the chord be a, describe the arc ed; from o, as a centre, with the distance o d, cut the diameters in dr, dt; draw the lines r, s, s, and t, s, s; then from r and t describe the arcs s, 8, 8, 8; also from d and d, describe the smaller ares 8, 8, 8, which will complete the ellipse as required. ь 13. To describe an elliptic arch, the width and rise of span being given. Biseet with a line at right angles the chord or span A B; erect the perpendicular A q, and draw the line D equal and parallel to A C; bisect A C and Aq in r and n; make Cl equal to CD, and draw the line lrq; draw also the line n s D; bisect &D with a line at right angles, and meeting the line CD in g; draw the line gq, make CP equal to Ck, and draw the line g Pi; then from g as a centre, with the radius g D, describe the arc s Di; and from k and P as centres, with the ra dius A k, describe the arcs A s and B i, which completes the arch as required. Or, 14. Bisect the chord AB, and fix at right angles any straight guide, as be; prepare, of any suitable material, a rod or staff, equal to half the chord's length, as def; from the end of the staff, equal to the height of the arch, fix a pin e, and at the extremity a tracer f; move the staff, keeping its end to the guide and the fixed pin to the chord, and the tracer will describe one half the arc required. 15. To describe a parabola, the dimensions being given. Let A B equal the length, and CD the breadth of the required parabola; divide CA, CB into any number of equal parts; also divide the perpen diculars Aa and Bb 3 2 a D into the same num- 1 ber of equal parts; A. then from a and b 3 C draw lines meeting each division on the line A CB; and a curve line drawn through each intersection will form the parabola required. 16. To obtain by measurement the length of any direct line, though intercepted by some material object. Suppose the distance between A and B is required, but the right line is intercepted by the object C. On the point d, with any convenient radius, describe the arc c c, make the arc twice the radius in length, through which draw the line dee; and on e describe another |