0.5

1 cm

GEOMETRICAL CONSTRUCTIONS—DRAWING OF ELEMENTARY FORMS.

23

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

tangent required.

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-

cessible.

Problem XVII.—To draw tangents to a circle, from a

point without it.

{Fig. 87.)

2d Method.-

(Fig. 88.)

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

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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

tangent required.

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-

cessible.

Problem XVII.—To draw tangents to a circle, from a

point without it.

{Fig. 87.)

2d Method.-

(Fig. 88.)

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