3. To do the same otherwise. From the given point a, with any convenient radius, describe the arc dc b; with the same radius from d, cut the arc in c, and from c, cut the arc in b; also from c and b as centres, with the same or any other radius, describe arcs cutting each other in t; then will the line at be the perpendicular required. d A 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 the preceding problems, the perpendiculars may be continued below the given lines, if necessarily required. 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 c d; and from c and d as centres, describe arcs catting each other at b; then will the line ab bisect the angle as required. d 5. To find the centre of a circle 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, as csd; and from and t as centres, with an equal radius, cut the portion of the circle in C, S and d, v; draw lines through where the arcs cut each other, and the c b d intersection of the lines at o is the centre of the circle required. 6. To find the centre of a given circle. Bisect any chord in the circle, as A B, by a perpendicular c D; bisect also the diameter E D in ƒ, 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 A c, equal to th the length of the chord of the arc A A B, and from A as a C A D centre, cut the arc in c; also from в as a centre, with equal radius, cut the chord in b; draw the line c b, 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 A DC; join A D, which, produced if necessary, is the tangent required. A D 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, c 2, D 2, &c.; draw the lines B D A F c 1, B2, D3, &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. On a given straight line, to describe an arc of a circle, the altitude being given. Let D E be the given straight line, в b the given altitude: join D B, B E, and of any suitable material construct a triangle, as a B C, making each of the sides A B, B C, equal at least to the chord D E, and the angle contained by them equal to D B E; at each end of the chord D E, fix a pin, and at в a tracer, as a pencil; move the triangle along the pins as guides, and the tracer will describe the arc required. 11. 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 semi-diameters, describe the arc a b, and from b as a S centre, with half the chord bca, describe the arc cd; from o, a as a centre, with the distance o d, cut the diameters in d, r, d, t; draw the lines rs, rs, ts, ts, then from r and t describe the arcs ss, ss; also from d and d describe the smaller arcs s s, s s, which will complete the ellipse as required. 12. To describe an elliptic arch, the width and rise of span being given. draw also the line n s D; bisect s D with a line at right angles, and meeting the line C D produced in g; draw the line g q, make C P equal to c k, and draw the line gPi; then from g as a centre, with the radius g D, describe the arc s D i, and from k and P as centres, with the radius A k, describe the arcs As and Bi, which completes the arch as required. Or, 13. Bisect the chord A в, and fix at right angles any straight guide, as bc; 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 ƒ; move the staff, keeping its end to the guide and the fixed pin to the chord, and the tracer will describe one-half the arch required. 14. To describe a parabola, the dimensions being given. Let A B equal the length, and C D the breadth of the required parabola: divide CA, C B into any number of equal parts, also divide the perpendiculars A a and b b into the same number of equal parts; then from a and b draw lines meeting each division on the line AC B, and a curve line drawn through each intersection will form the parabola required. 15. To obtain by measurement the length of any direct line, though intercepted by some material object. Suppose the distance between A and B is re- A quired, but the right line is intercepted by the object c. On the point d, with any d e B |