points, A, B, C, in the circumference, well apart; with one
radius describe arcs from these three points, cutting each
other; and draw the two lines D E, F G through their in-
tersections, according to Problem I. for bisecting an arc.
The point 0, where they cut, is the centre of the circle or arc.
Problem XIY.—To describe a circle passing through
three given points.
Join the given points A, B, C (Fig. 82), and proceed as
in last problem to find the centre O, from which the circle
may be described.
Note.—This problem is of various utility: in striking
out the circular arches of bridges upon centering, when
the span and rise are given ; describing shallow pans, or
dished covers of vessels; or finding the diameter of a fly-
wheel, or any other object of large diameter, when only
a part of the circumference is accessible.
Problem XY.—To describe a circle passing through
three given points, when the centre is not available.
1st Method.—From the extreme points A, B, as centres,
describe arcs A H, B G. Through the third point C draw
A E and B F, cutting the arcs. Divide A F and B E into
any number of equal parts, and set off a series of equal
parts of the same length on the upper portions of the arcs
beyond the points E, F. Draw straight lines B L, B M,
&c. to the divisions in A F; and A I, A K, &c. to the
divisions in EG; the successive intersections N, O, &c.
of these lines, are points in the circle required, between
the given points A and C, which may be filled in accord-
ingly. Similarly, the remaining part of the curve B C,
may be described.
2d Method.—Let A, D, B, be the given points; draw
A B, A D, D B, and e f parallel to A B. Divide D A into
a number of equal parts, 1, 2, 3, &c,, and from D describe
arcs through these points to meet e f. Divide the arc A e
into the same number of equal parts, and draw straight
lines from D to the points of division. The intersections
of these lines successively with the arcs 1, 2, 3, &c., are
points in the circle, which may be filled in as before.
Note.—The second method is not perfectly true, but
sufficiently so for arcs less than one-fourth of a circle.
When the middle point is equally distant from the
extremes, the vertical C D is the rise of the arc; and this
problem is serviceable for setting out circular arcs of large
radius, as for bridges of very great span, where the centre
is unavailable; and for the outlines of bridge-beams,
steam-engine beams, connecting-rods, and the like.
Problem XVI.—To draw a tangent to a circle, from a
given point in the circumference.
1 si Method.—^Through the given point A (Fig. 85), draw
the radial line A C, and the perpendicular F G for the
2d Method.—From A (Fig. 86) set off equal segments
A B, AD; join B D, and draw A E parallel to it, for the
tangent. This method is useful when the centre is inac-
Problem XVII.—To draw tangents to a circle, from a
point without it.
1st Method. — Draw
A C from the given point
A to the centre of the
circle, bisect it at D, from
D describe an arc through
C cutting the circle at
E, F. Draw A E, A F,
for the required tangents.
-From A with the radius A C, describe an
arc BCD; and from C,
with a radius equal to the
diameter of the given circle
E F, cut the arc at B, D ;
join B C, CD, cutting the
circle at E, F, and draw
A E, A F, for tangents.
Note.—When a tangent
is already drawn, the precise
point of contact may be
found by drawing a per-
pendicular to it from the centre.
Problem XVIII.—To describe a circle from a given
point to touch a given circle.
D E being the given circle, and B the point, draw B C
to the centre C, and produce it, if necessary, to cut the
circle at A, and with B A as radius describe the circle