_Angles_
57. Regarding the line which is drawn as a way or path upon which we
move and proceed, we must decide, if we change our direction, whether
we will turn to the right or to the left, and whether we will turn
abruptly or gradually. If we change our direction abruptly we must
decide how much of a change of direction we will make. Is it to be a
turn of 30° or 60° or 90° or 135°? How much of a turn shall it be?
[Illustration: Fig. 58]
The above illustrations are easy to understand and require no
explanation. An abrupt change of 180° means, of course, returning upon
the line just drawn.
_Curves_
58. In turning, not abruptly but gradually, changing the direction at
every point, that is to say in making a curve, the question is, how
much of a turn to make in a given distance, through how many degrees
of the circle to turn in one inch (1″), in half an inch (½″), in two
inches (2″). In estimating the relation of arcs, as distances, to
angles of curvature, the angles of the arcs, the reader will find it
convenient to refer to what I may call an Arc-Meter. The principle of
this meter is shown in the following diagram:—
[Illustration: Fig. 59]
If we wish to turn 30° in ½″, we take the angle of 30° and look within
it for an arc of ½″. The arc of the right length and the right angle
being found, it can be drawn free-hand or mechanically, by tracing
or by the dividers. Using this meter, we are able to draw any curve
or combination of curves, approximately; and we are able to describe
and define a line, in its curvatures, so accurately that it can be
produced according to the definition. Owing, however, to the difficulty
of measuring the length of circular arcs accurately, we may find it
simpler to define the circular arc by the length of its radius and the
angle through which the radius passes when the arc is drawn.
[Illustration: Fig. 60]
Here, for example, is a certain circular arc. It is perhaps best
defined and described as the arc of a half inch radius and an angle
of ninety degrees, or in writing, more briefly, rad. ½″ 90°. Regarding
every curved line either as a circular arc or made up of a series of
circular arcs, the curve may be defined and described by naming the arc
or arcs of which it is composed, in the order in which they are to be
drawn, and the attitude of the curve may be determined by starting from
a certain tangent drawn in a certain direction. The direction of the
tangent being given, the first arc takes the direction of the tangent,
turning to the right of it or to the left.
[Illustration: Fig. 61]
Here is a curve which is composed of four circular arcs to be drawn in
the following order:—
Tangent up-right 45°, arc right radius 1″ 60°, arc left radius ⅓″ 90°,
arc right radius ¾″ 180°.
Two arcs will often come together at an angle. The definition of the
angle must be given in that case. It is, of course, the angle made by
tangents of the arcs. Describing the first arc and the direction (right
or left so many degrees) which the tangent of the second arc takes
from the tangent of the first arc; then describing the second arc and
stating whether it turns from its tangent to the right or to the left,
we shall be able to describe, not only our curves, but any angles which
may occur in them.
[Illustration: Fig. 62]
Here is a curve which, so far as the arcs are concerned, of which it is
composed, resembles the curve of Fig. 61; but in this case the third
arc makes an angle with the second. That angle has to be defined.
Drawing the tangents, it appears to be a right angle. The definition of
the line given in Fig. 62 will read as follows:—
Tangent down right 45°, arc left radius 1″ 60°, arc right radius ⅓″
90°, tangent left 90°, arc left ¾″ 180°.
59. In this way, regarding all curves as circular arcs or composed of
circular arcs, we shall be able to define any line we see, or any line
which we wish to produce, so far as changes of direction are concerned.
For the purposes of this discussion, I shall consider all curves as
composed of circular arcs.
There are many curves, of course, which are not circular in character,
nor composed, strictly speaking, of circular arcs. The Spirals are in
no part circular. Elliptical curves are in no part circular. All curves
may, nevertheless, be approximately drawn as compositions of circular
arcs. The approximation to curves which are not circular may be easily
carried beyond any power of discrimination which we have in the sense
of vision. The method of curve-definition, which I have described,
though it may not be strictly mathematical, will be found satisfactory
for all purposes of Pure Design. It is very important that we should be
able to analyze our lines upon a single general principle; to discover
whether they are illustrations of Order. We must know whether any given
line, being orderly, is orderly in the sense of Harmony, Balance, or
Rhythm. It is equally important, if we wish to produce an orderly as
distinguished from a disorderly line, that we should have some general
principle to follow in doing it, that we should be able to produce
forms of Harmony or Balance or Rhythm in a line, if we wish to do so.