intersection of the lines at o is the centre of the circle required.

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.

A

A

D

B

E

B

With the radius A C, equal to th the length of the chord of the arc A B, and from A as a 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 AD, 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

c1, 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 centre, with half the chord bca, describe the arc c d; from o,

as a centre, with the distance o d, cut the diameters in d, r, d, t; draw the lines r s, 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.

Bisect with a line at right angles the chord or span A B, erect the perpendicular a 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 7 r q;

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 g Pi; 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 A s and Bi, which completes the arch as required. Or,

13. Bisect the chord A в, 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 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

A

B

P

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 в b into the same number of equal parts; then from a and b draw lines meeting each division on the line a C 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

B

convenient radius, describe the arc c c, make the arc twice the radius in length, through which draw the line d ce, 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 r, j, which will be a right line, and de or er, will equal the distance between d, r, by which the distance between A and B is obtained as required.

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

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 Dd at any angle to the line a B; bisect the line D d in c, through which draw the line в b, making cb equal to в c; join a c, and draw db a meeting A c, produced in a. Then ba,

h

equal to A B, is the distance a required.

17. Or otherwise,

Produce A B to any point D, and bisect B D in C; through D draw Da, making any angle with DA, and take D C, D b, equal to D C and D B respectively; join в c, c b, and a b. Through E, the intersection of в c, c b, draw D E F meeting a b in

a

F

B