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.