0.5

1 cm

GEOMETRICAL CONSTRUCTIONS—DRAWING OF ELEMENTARY FORMS.

25

A C of tlie other circle, join C H and bisect it with the

perpendicular L I cutting E F at I. On the centre I

with radius I F, describe

the arc F A as required.

Note.—The processes

comprised in the fore-

going four problems are

of very general utility in

drawing, for rounding the

angles of framework, and

the other elements of ma-

chinery with precision.

SECTION III.

Problems on Circles and Rectilineal Figures.

Problem XXIY.—To construct a triangle upon a given

straight line or base, the length of the two sides being given.

First, an equilateral triangle (Fig. 97). On the ends

A, B, of the given line, with A B as radius, describe arcs

cutting at C, and draw AC, B C; then A B C is the

triangle required.

(Fig. 97.) (Fig. 9S.)

B

A Ii

C —---

Second, when the sides are unequal (Fig. 98). Let

A D be the base, and B and C the two sides. On either

end, as A, of the base-line, with the line B as radius,

describe an arc; and on D, with C as radius, cut the arc at

E. Draw A E, E D, then A E D is the triangle as required.

Note.—This construction is used also to find the position

of a point when its distances are given from two other

given points, whether joined by a line or not. The

parallel motion of Watt is a familiar case.

Problem XXY.—To construct a square or a rectangle

upon a given straight line.

First, a square (Fig. 99). Let A B be the given line;

on A and B as centres, with the radius A B, describe arcs

cutting at C ; on C, with the

same radius, describe arcs cut-

ting the others at D and E; and

on D and E, cut these at F, G.

Draw A F, B G, cutting the

arcs at H, I; and join H I, to

form the square as required.

Second, a rectangle (Fig. 100).

To the base E F, draw perpendiculars E IT, F G, equal

to the sides, and join G PI to complete the rectangle.

When the centre lines of the square or rectangle are

given, the figure may be described as follows:—

Let A B and C D (Fig. 101) be the centre lines, perpen-

dicular to each other, and E the middle point of the figure;

set off E F, E G, equal each to the half length of the rect-

(Fig. 100.) (Fig. 101.)

c

angle, and E H, E J, each equal to half the height. On

the centres H, J, with a radius equal to the half length,

describe arcs on both sides; and on F, G, with a radius

of half the height, cut these arcs at K, L, M, N. Join the

four intersections so formed, to complete the rectangle.

Problem XXYI.—To construct a parallelogram, of

vjhich the sides and one of the angles are given.

Let A and B be the lengths of the two sides, and C the

angle; draw a straight line, and set off D E equal to A;

from D draw D F equal

to B, and forming an angle

with D E equal to C; from

E with D F as radius,

describe an arc, and from

F with D E as radius, cut

the arc at G, and drawF G

and E G, to complete the parallelogram. Or, the remaining

sides may be drawn parallel to D E and D F, cutting at

G, and the figure thus completed.

Note.—This problem is an extension of the 24th, for

describing a triangle, and it is, like that problem, em-

ployed in the parallel motion ; also in many other obvious

cases.

Problem XXYII.—To describe a circle about a tri-

(Fig. 103.)

angle.

(Fig. 103.)

Bisect two of the sides A B,

A C, of the triangle, at E, F;

from these points draw perpen-

diculars cutting at K. From the

centre K, with K A as radius,

describe the circle A B C as re-

quired.

Note.—The construction for

this problem is the same as that

for Problem XIY.

Problem XXYIII.—To inscribe a circle in a triangle.

Bisect two of the angles A, C, of the triangle A B C, by

lines cutting at D; from D draw

a perpendicular D E to any side,

as A C; and with D E as radius,

from the centre D, describe the

circle required.

When the triangle is equila-

teral, the centre of the circle is more easily found by bisect-

ing two of the sides, and drawing perpendiculars, as in the

previous problem. Or, draw a perpendicular from one of

the angles to the opposite side, and from the side set off

one-third of the length of the perpendicular.

D

25

A C of tlie other circle, join C H and bisect it with the

perpendicular L I cutting E F at I. On the centre I

with radius I F, describe

the arc F A as required.

Note.—The processes

comprised in the fore-

going four problems are

of very general utility in

drawing, for rounding the

angles of framework, and

the other elements of ma-

chinery with precision.

SECTION III.

Problems on Circles and Rectilineal Figures.

Problem XXIY.—To construct a triangle upon a given

straight line or base, the length of the two sides being given.

First, an equilateral triangle (Fig. 97). On the ends

A, B, of the given line, with A B as radius, describe arcs

cutting at C, and draw AC, B C; then A B C is the

triangle required.

(Fig. 97.) (Fig. 9S.)

B

A Ii

C —---

Second, when the sides are unequal (Fig. 98). Let

A D be the base, and B and C the two sides. On either

end, as A, of the base-line, with the line B as radius,

describe an arc; and on D, with C as radius, cut the arc at

E. Draw A E, E D, then A E D is the triangle as required.

Note.—This construction is used also to find the position

of a point when its distances are given from two other

given points, whether joined by a line or not. The

parallel motion of Watt is a familiar case.

Problem XXY.—To construct a square or a rectangle

upon a given straight line.

First, a square (Fig. 99). Let A B be the given line;

on A and B as centres, with the radius A B, describe arcs

cutting at C ; on C, with the

same radius, describe arcs cut-

ting the others at D and E; and

on D and E, cut these at F, G.

Draw A F, B G, cutting the

arcs at H, I; and join H I, to

form the square as required.

Second, a rectangle (Fig. 100).

To the base E F, draw perpendiculars E IT, F G, equal

to the sides, and join G PI to complete the rectangle.

When the centre lines of the square or rectangle are

given, the figure may be described as follows:—

Let A B and C D (Fig. 101) be the centre lines, perpen-

dicular to each other, and E the middle point of the figure;

set off E F, E G, equal each to the half length of the rect-

(Fig. 100.) (Fig. 101.)

c

angle, and E H, E J, each equal to half the height. On

the centres H, J, with a radius equal to the half length,

describe arcs on both sides; and on F, G, with a radius

of half the height, cut these arcs at K, L, M, N. Join the

four intersections so formed, to complete the rectangle.

Problem XXYI.—To construct a parallelogram, of

vjhich the sides and one of the angles are given.

Let A and B be the lengths of the two sides, and C the

angle; draw a straight line, and set off D E equal to A;

from D draw D F equal

to B, and forming an angle

with D E equal to C; from

E with D F as radius,

describe an arc, and from

F with D E as radius, cut

the arc at G, and drawF G

and E G, to complete the parallelogram. Or, the remaining

sides may be drawn parallel to D E and D F, cutting at

G, and the figure thus completed.

Note.—This problem is an extension of the 24th, for

describing a triangle, and it is, like that problem, em-

ployed in the parallel motion ; also in many other obvious

cases.

Problem XXYII.—To describe a circle about a tri-

(Fig. 103.)

angle.

(Fig. 103.)

Bisect two of the sides A B,

A C, of the triangle, at E, F;

from these points draw perpen-

diculars cutting at K. From the

centre K, with K A as radius,

describe the circle A B C as re-

quired.

Note.—The construction for

this problem is the same as that

for Problem XIY.

Problem XXYIII.—To inscribe a circle in a triangle.

Bisect two of the angles A, C, of the triangle A B C, by

lines cutting at D; from D draw

a perpendicular D E to any side,

as A C; and with D E as radius,

from the centre D, describe the

circle required.

When the triangle is equila-

teral, the centre of the circle is more easily found by bisect-

ing two of the sides, and drawing perpendiculars, as in the

previous problem. Or, draw a perpendicular from one of

the angles to the opposite side, and from the side set off

one-third of the length of the perpendicular.

D