0.5

1 cm

GEOMETRICAL CONSTRUCTIONS—DRAWING OF ELEMENTARY FORMS.

29

side b c applied to the square-blade, and to draw tire two

sides A C, B C. If the given side A B be upright {Fig. 125),

apply the long side a b to tlie square-blade, and draw as

before.

(Fig. 124.)

cV

a

A /

\

(Fig. 125.)

To draw a regular hexagon about a circle, with two of

its sides parallel to the lower edge of the board: with the

T square, draw the centre line A B {Fig. 126), and the upper

and lower sides, D E, F G, touching the circle, all off the

left edge of the board; and apply the set-square of 60°

touching the circle for the four remaining sides, as shown

in the figure.

(Fig. 127.)

** F K

When two of the sides are to be drawn parallel to the

end of the board, draw the perpendiculars A B, D E, F G

{Fig. 127), through the centre, and touching the circle for

the two sides; with the set-square of 60°, draw the four

remaining sides touching the circle, as in dotting.

When the hexagons are to be inscribed in the circle,

first draw the centre-line A B {Fig. 109), as a diameter, and

from the ends A, B, with the set-square, draw four sides

cutting the circle at D, E, F, G, and join D E, F G.

The triangles of 45° and 60° are useful in setting out

the centre-lines of wheels with 3, 4, 6, 8, &c. arms, by

drawing lines through the centre of the wheel. One illus-

tration will suffice:—to set out 12 spokes in a wheel. Draw

two diameters A B, CD, parallel to the two edges of the

board; in the quadrant A C, draw radii E a, E b, with

the long and the short sides of

the triangle against the square-

blade. These will divide the

quadrant equally; and the same

construction being employed for

the other quarters of the circle,

12 centre-lines, equally distant,

will be described. Should the

triangle be large enough to embrace the whole circle at

once, the opposite quadrants A C and B D may be divided

with the same setting of the triangle.

The foregoing illustrations of the use of the set-squares,

and the facilities they offer, will suffice, if well-studied, to

explain the nature of their application, and will lead the

way to many other abbreviations. In full-size drawings,

it is commonly impracticable to employ T squares; recourse

must then be had to the geometrical construction for each

problem, as already discussed.

(Fig. 12S.)

(Fig. 126)

A short method of dividing a line or surface into a

number of equal parts, is illustrated by {Fig. 129); and it is

convenient where an ordinary rule does not evenly mea-

sure the dimension. Suppose the width A C is to be

divided into seven equal parts, and that it measures 7|

inches; an ordinary inch rule, it is plain, does not afford

the subdivisions when applied directly; but if 14 inches of

length, or double the number of parts, be applied obliquely

across the space between the parallels A B, CD, so as to

measure it exactly, then point off 2 inch intervals on the

edge of the rule, and in this way seven equal subdivisions

will be effected, through which parallels may be drawn.

Or, in the same way, apply times 7, or 10^ inches,

obliquely across, as in the figure, and set off inch inter-

vals for the points of division. Again, suppose a breadth

of 9| inches is to be divided into 10 equal parts, a slight

inclination of the rule will give an oblique measure of 10

inches, and the divisions may then be marked off.

Simple Applications of Regular Figures.

Problem XLII.—To cover a surface with equilateral

triangles, hexagons, or lozenges.

Describe an equi-

(Fig. 130.) _ x

lateral triangle A B C,

and produce the sides

indefinitely. Set off,

from one angle A,

equal intervals at

a, b, a', b', &c., as re-

quired ; and through

these points draw

parallels to each of

the sides of the tri-

The area will be covered with triangles as re-

(Fig. 131.)

angle.

quired.

For hexagons, the

same operation is fol-

lowed as for tri-

angles ; and the

figures are com-

pleted in the obvious

manner shown in the

annexed figure. This

is the only case in

which regular poly-

gons, of more than

four sides, can be made to cover a surface without inter-

stices or waste area. It is thus that the cellular structure

of the honey-comb is designed.

29

side b c applied to the square-blade, and to draw tire two

sides A C, B C. If the given side A B be upright {Fig. 125),

apply the long side a b to tlie square-blade, and draw as

before.

(Fig. 124.)

cV

a

A /

\

(Fig. 125.)

To draw a regular hexagon about a circle, with two of

its sides parallel to the lower edge of the board: with the

T square, draw the centre line A B {Fig. 126), and the upper

and lower sides, D E, F G, touching the circle, all off the

left edge of the board; and apply the set-square of 60°

touching the circle for the four remaining sides, as shown

in the figure.

(Fig. 127.)

** F K

When two of the sides are to be drawn parallel to the

end of the board, draw the perpendiculars A B, D E, F G

{Fig. 127), through the centre, and touching the circle for

the two sides; with the set-square of 60°, draw the four

remaining sides touching the circle, as in dotting.

When the hexagons are to be inscribed in the circle,

first draw the centre-line A B {Fig. 109), as a diameter, and

from the ends A, B, with the set-square, draw four sides

cutting the circle at D, E, F, G, and join D E, F G.

The triangles of 45° and 60° are useful in setting out

the centre-lines of wheels with 3, 4, 6, 8, &c. arms, by

drawing lines through the centre of the wheel. One illus-

tration will suffice:—to set out 12 spokes in a wheel. Draw

two diameters A B, CD, parallel to the two edges of the

board; in the quadrant A C, draw radii E a, E b, with

the long and the short sides of

the triangle against the square-

blade. These will divide the

quadrant equally; and the same

construction being employed for

the other quarters of the circle,

12 centre-lines, equally distant,

will be described. Should the

triangle be large enough to embrace the whole circle at

once, the opposite quadrants A C and B D may be divided

with the same setting of the triangle.

The foregoing illustrations of the use of the set-squares,

and the facilities they offer, will suffice, if well-studied, to

explain the nature of their application, and will lead the

way to many other abbreviations. In full-size drawings,

it is commonly impracticable to employ T squares; recourse

must then be had to the geometrical construction for each

problem, as already discussed.

(Fig. 12S.)

(Fig. 126)

A short method of dividing a line or surface into a

number of equal parts, is illustrated by {Fig. 129); and it is

convenient where an ordinary rule does not evenly mea-

sure the dimension. Suppose the width A C is to be

divided into seven equal parts, and that it measures 7|

inches; an ordinary inch rule, it is plain, does not afford

the subdivisions when applied directly; but if 14 inches of

length, or double the number of parts, be applied obliquely

across the space between the parallels A B, CD, so as to

measure it exactly, then point off 2 inch intervals on the

edge of the rule, and in this way seven equal subdivisions

will be effected, through which parallels may be drawn.

Or, in the same way, apply times 7, or 10^ inches,

obliquely across, as in the figure, and set off inch inter-

vals for the points of division. Again, suppose a breadth

of 9| inches is to be divided into 10 equal parts, a slight

inclination of the rule will give an oblique measure of 10

inches, and the divisions may then be marked off.

Simple Applications of Regular Figures.

Problem XLII.—To cover a surface with equilateral

triangles, hexagons, or lozenges.

Describe an equi-

(Fig. 130.) _ x

lateral triangle A B C,

and produce the sides

indefinitely. Set off,

from one angle A,

equal intervals at

a, b, a', b', &c., as re-

quired ; and through

these points draw

parallels to each of

the sides of the tri-

The area will be covered with triangles as re-

(Fig. 131.)

angle.

quired.

For hexagons, the

same operation is fol-

lowed as for tri-

angles ; and the

figures are com-

pleted in the obvious

manner shown in the

annexed figure. This

is the only case in

which regular poly-

gons, of more than

four sides, can be made to cover a surface without inter-

stices or waste area. It is thus that the cellular structure

of the honey-comb is designed.