0.5

1 cm

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ENGINEER AND MACHINIST’S DRAWING-BOOK.

F G, touching the given circle. The operation is the

same whether the point B he within or without the

circle.

(Fig. 89.)

Problem XIX. — To draw tangents to two given

circles.

lsi Method.—Draw the straight line ABC through

the centres of the two given circles, from the centres A, B,

(Fig. 90.)

draw parallel radii AD, B E, in the same direction; join

D E and produce it to meet the centre line at C, and from

C draw tangents to one of the circles by Problem XIY.

Those tangents will touch both circles, as required.

2d Method.—Draw A B; and in the larger circle draw

any radius A H, on which set off H G equal to the radius

(Fig. 91.)

c

of the smaller circle; on A describe a circle with the

radius A G, and draw tangents B I, B K, to this circle

from the other centre B; from A and B draw perpen-

diculars to these tangents, and join CD, E F, for the

required tangents.

Note.—The second method is useful when the diameters

of the circles are nearly equal.

Problem XX.—Between two inclined lines to draiu

a series of circles touching these lines and touching each

other.

Bisect the inclination of the given lines A B, CD, by

the line N 0; this is the centre line of the circles to be

(Tig. 92.)

inscribed. From a point P in this line draw the perpen-

dicular P B to the line A B, and from P describe the

circle B D touching the given lines and cutting the centre

line at E ; from E draw E F perpendicular to the centre

(Fig. 93.)

line, cutting A B at F, and from F describe an arc E G,

cutting A B at G; draw G H parallel to B P, giving

H the centre of the second touching circle, described

with the radius H E or H G. By a similar process the

third circle I N is determined. And so on.

Inversely, the largest circle may be described first, and

the smaller ones in succession.

Note.—This problem is of frequent use in scroll work.

Problem XXI.—Between two inclined lines to draw a

circular segment to fill up the angle, and touching the

lines.

Let AB, D E be the inclined lines; bisect the inclination

by the line F C,

and draw the per-

pendicular A F D

to define the limit

within which the

circle is to be

drawn. Bisect the

angles A and D by

f a lines cutting at C,

and from C with radius C F, draw the arc H F G as

required.

Problem XXII.—To fill up the angle of a straight line

and a circle, with a circular arc of a given radius.

In the given circle A D draw a radius C B and produce

it, set off B E equal to the radius of the required arc, and

on the centre C with the

{Fig. 94.)

radius C E, draw the arc

E F. Draw G F parallel

to the given line H I, at

the distance G PI equal to

the radius of the required

arc, and cutting the arc

E F at F. Then F is the

required centre ; draw the

perpendicular FI, or F C cutting the circle at A, and

with the radius F A or FI describe the arc AI as

required.

Problem XXIII.—To fill up the angle of a straight

line and a circle, with a circular arc to join the circle

at a given point.

In the given circle draw the radius C A and produce it,

at A draw a tangent

mg. 95.) _ °

meeting the given

line at D, bisect the

angle A D E so

formed with a line

cutting the radius

C A at F ; and on

the centre F de-

scribe the arc AE

as required.

Problem XXIV.—To describe a circular arc joining

two circles, and to touch one of them at a given point.

Let A B and F G be the given circles, to be joined by

an arc touching one of them at F. Draw the radius E F,

and produce it both ways; set off F H equal to the radius

ENGINEER AND MACHINIST’S DRAWING-BOOK.

F G, touching the given circle. The operation is the

same whether the point B he within or without the

circle.

(Fig. 89.)

Problem XIX. — To draw tangents to two given

circles.

lsi Method.—Draw the straight line ABC through

the centres of the two given circles, from the centres A, B,

(Fig. 90.)

draw parallel radii AD, B E, in the same direction; join

D E and produce it to meet the centre line at C, and from

C draw tangents to one of the circles by Problem XIY.

Those tangents will touch both circles, as required.

2d Method.—Draw A B; and in the larger circle draw

any radius A H, on which set off H G equal to the radius

(Fig. 91.)

c

of the smaller circle; on A describe a circle with the

radius A G, and draw tangents B I, B K, to this circle

from the other centre B; from A and B draw perpen-

diculars to these tangents, and join CD, E F, for the

required tangents.

Note.—The second method is useful when the diameters

of the circles are nearly equal.

Problem XX.—Between two inclined lines to draiu

a series of circles touching these lines and touching each

other.

Bisect the inclination of the given lines A B, CD, by

the line N 0; this is the centre line of the circles to be

(Tig. 92.)

inscribed. From a point P in this line draw the perpen-

dicular P B to the line A B, and from P describe the

circle B D touching the given lines and cutting the centre

line at E ; from E draw E F perpendicular to the centre

(Fig. 93.)

line, cutting A B at F, and from F describe an arc E G,

cutting A B at G; draw G H parallel to B P, giving

H the centre of the second touching circle, described

with the radius H E or H G. By a similar process the

third circle I N is determined. And so on.

Inversely, the largest circle may be described first, and

the smaller ones in succession.

Note.—This problem is of frequent use in scroll work.

Problem XXI.—Between two inclined lines to draw a

circular segment to fill up the angle, and touching the

lines.

Let AB, D E be the inclined lines; bisect the inclination

by the line F C,

and draw the per-

pendicular A F D

to define the limit

within which the

circle is to be

drawn. Bisect the

angles A and D by

f a lines cutting at C,

and from C with radius C F, draw the arc H F G as

required.

Problem XXII.—To fill up the angle of a straight line

and a circle, with a circular arc of a given radius.

In the given circle A D draw a radius C B and produce

it, set off B E equal to the radius of the required arc, and

on the centre C with the

{Fig. 94.)

radius C E, draw the arc

E F. Draw G F parallel

to the given line H I, at

the distance G PI equal to

the radius of the required

arc, and cutting the arc

E F at F. Then F is the

required centre ; draw the

perpendicular FI, or F C cutting the circle at A, and

with the radius F A or FI describe the arc AI as

required.

Problem XXIII.—To fill up the angle of a straight

line and a circle, with a circular arc to join the circle

at a given point.

In the given circle draw the radius C A and produce it,

at A draw a tangent

mg. 95.) _ °

meeting the given

line at D, bisect the

angle A D E so

formed with a line

cutting the radius

C A at F ; and on

the centre F de-

scribe the arc AE

as required.

Problem XXIV.—To describe a circular arc joining

two circles, and to touch one of them at a given point.

Let A B and F G be the given circles, to be joined by

an arc touching one of them at F. Draw the radius E F,

and produce it both ways; set off F H equal to the radius