divisions, describe arcs cutting each other, as A 1

B 1, C 2, D 2, &c.; draw the lines C 1, B 2, D 3, &c., which lead to the centre as required.

To draw the end lines.

B D

As A r, 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 A r, F r, 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, B C, each equal in length to the chord of the arc D E, and height, twice that of the arc

B b. 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.

H

3

2

3

2

Draw the chord A C B; also draw the line H D I, parallel with the 2 3 D 3 2 n 1 chord, and equal to the height of the segment; bisect the chord in C, and erect the perpendicular C D; join A D, D B; draw A H perpendicular to A D, and BI perpendicular to B D, erect also the perpendiculars A ̊n, B n; divide A B and H I into any

A. 1 2 3

C 3 2

number of equal parts; draw the lines 1 1, 22, 33, &c.; likewise divide the lines A n, B n, each into half the number of equal parts; draw lines to D from each division in the lines A n, B n, 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 semi-diameters, describe the arc a b; and from bas a centre, with half the chord b c a, describe the arc cd; from o, as a centre, with the distance o d,

cut the diameters in d r, dt; draw the lines r, s, s, and t, s, s; then from r and t describe the arcs s, s, 3, 5; also from d and d, describe the smaller arcs s, s, s, s, which will complete the ellipse as required.

13. To describe an elliptic arch, the width and rise of span being given.

Bisect with a line at right angles the chord or span A B; erect the perpendicular An q, and draw the line q D equal and parallel to A C; bisect A C and A q in r and n; make C equal to C D, and draw the line lrq; draw also the line n s D; bisect s D with a line at right angles, and meeting the line C D in

A

S

D

P

g; draw the line g q, 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 radius A k, describe the arcs A s and Bi which completes the arch as required. Or,

14. Bisect the chord A B, and fix at right angles any straight guide, as b c; prepare, of any suitable material, a rod or staff, equal to half the chord's length, as a de f; 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 C A, C B into any num

ber 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 A CB; and a curve line drawn through each intersection will form the parabola required.

16. To obtain by measurement the length of any line, though intercepted by some material object. Suppose the

distance be

tween A and B A

is required, but the right line is intercepted by the object C. On the point d, with any con

direct

B

venient radius, describe the arc c c, make the arc twice the radius in length, through which draw the line d c e, and on e describe another arc equal in length to once the radius, as eff; draw the line e fr equal to e fd; on r describe the arc jj, in length twice the radius; continue the line through rj, which will be a right line, and de, or er, equal the distance between dr, by which the distance between A and B is obtained as required.

17. A round piece of timber being given, out of which to cut a beam of strongest section.

Divide into three equal parts any diameter in the circle, as A d, e C; from d or e, erect a perpendicular meeting the circumference of the circle, as d B; draw A B and B C, also A D equal to B C, and D C equal to A B, and the rectangle will be a section of the beam as required.

D

18. To measure the distance between two objects, both being inaccessible.

From any point C draw a any line Cc, and bisect it in D; take any point E in the prolongation of A C, and draw the line E e, making De equal to DE; in like manner take any point F in the prolongation of B C, and make Df equal to F D. Produce A D and ec till they meet in a, and also B D and fc till they meet in b; then ab equal A B, or the distance between the objects as required.

E

19. To ascertain the distance, geometrically, of any inaccessible object on an equal plane.

d

Let it be required to find the distance between A and B, A being inaccessible; produce the line in the direction of A B to any point, as D; draw the line D d at any angle to the line AB; bisect the line D d, through which draw the line B b, making cb b equal to B c; draw the line dba; also through c, in the direction c A, draw the line ac A, intersecting the line dba; then b a equal B A, the distance required.

20. Otherwise.

Prolong A B to any point D, making B C equal to C D; draw the line D a at any angle with D A, and the line Cb similar to B c; draw also the line DE F, which intersects the line Da; then ab a equal B A, or the distance required.

21. To find the proper position for an eccentric, in relation to the crank in a steam engine, the angle of eccentric rod, and travel of the valve, being given.

Draw the right line A B, as the situation of the crank at commencement of the stroke; draw also the line C d, as the proper given angle of eccentric rod with the crank; then from C as centre, describe a circle equal to the travel

of the valve; draw the line e ƒ at right angles to the line