Part 3

54. It is to be observed that single inversions in any direction, for example the relation of attitudes I and II, II and III, III and IV, IV and I, in Fig. 53, shows an opposition and Balance of Attitudes upon the axis of inversion. The relation of positions I and II and III and IV, the relation of the two groups on the left to the two groups on the right, illustrates the idea of Symmetry of Attitudes, the axis of balance being vertical. By Symmetry I mean, in all cases, right and left balance on a vertical axis. All double inversions, the relation of positions I and III, and II and IV, in Fig. 53, are Attitude-Balances, not on axes, but on centers. The balance of these double inversions is not symmetrical in the sense in which I use the word symmetry, nor is it axial. It is central.

THE ORDER OF RHYTHM IN ATTITUDES

55. When movement is suggested by any series of attitudes and the movement is regulated by equal or regularly progressive intervals, we have a Rhythm of Attitudes.

[Illustration: Fig. 57]

In this case the changes of attitude suggest a falling movement to the right and down. In the regular progression of this movement through marked intervals we have the effect of Rhythm, in spite of the fact that the relation of positions repeated has axial balance. The intervals in this case correspond, producing Interval-Harmony. The force of this Rhythm might be increased if the relation of positions repeated suggested a movement in the same direction. We should have Rhythm, of course, in the repetition of any such unstable attitude-rhythms at equal or lawfully varying intervals.

LINES

DEFINITION OF LINES

56. Taking any dot and drawing it out in any direction, or in a series or sequence of directions, it becomes a line. The line may be drawn in any tone, in any value, color or color-intensity. In order that the line may be seen, the tone of it must differ from the ground-tone upon which it is drawn. The line being distinctly visible, the question of tone need not be raised at this point of our discussion. We will study the line, first, as a line, not as an effect of light.

The line may be drawn long or short, broad or narrow. As the line increases in breadth, however, it becomes an area. We will disregard for the present all consideration of width-measures in the line and confine our attention to the possible changes of direction in it, and to possible changes in its length.

We can draw the line in one direction from beginning to end, in which case it will be straight. If, in drawing the line, we change its direction, we can do this abruptly, in which case the line becomes angular, or we can do it gradually, in which case it becomes curved. Lines may be straight, angular, or curved. They may have two of these characteristics or all three of them. The shapes of lines are of infinite variety.

CHANGES OF DIRECTION IN LINES

_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.

DIFFERENCES OF SCALE IN LINES

60. Having drawn a line of a certain shape, either straight or angular or curved, or partly angular, partly curved, we may change the measure of the line, in its length, without changing its shape. That is to say, we may draw the line longer or shorter, keeping all changes of direction, such as they are, in the same positions, relatively. In that way the same shape may be drawn larger or smaller. That is what we mean when we speak of a change of scale or of measure which is not a change of shape.

DIFFERENCES OF ATTRACTION IN LINES

61. A line attracts attention in the measure of the tone-contrast which it makes with the ground-tone upon which it is drawn. It attracts attention, also, according to its length, which is an extension of the tone-contrast. It attracts more attention the longer it is, provided it lies, all of it, well within the field of vision. It attracts attention also in the measure of its concentration.

[Illustration: Fig. 63]

Line “a” would attract less attention than it does if the tone-contrast, black on a ground of white paper, were diminished, if the line were gray, not black. In line “b” there is twice the extension of tone-contrast there is in “a.” For that reason “b” is more attractive. If, however, “a” were black and “b” were gray, “a” might be more attractive than “b,” because of the greater tone-contrast.

[Illustration: Fig. 64]

In this illustration the curved line is more attractive than the straight line because it is more concentrated, therefore more definite. The extent of tone-contrast is the same, the lines being of the same length.

[Illustration: Fig. 65]

In this line there is no doubt as to the greater attraction of the twisted end, on account of the greater concentration it exhibits. The extent of tone-contrast is the same at both ends. The force of attraction in the twisted end of the line would be diminished if the twisted end were made gray instead of black. The pull of concentration at one end might, conceivably, be perfectly neutralized by the pull of a greater tone-contrast at the other.

[Illustration: Fig. 66]

In “b” we have a greater extension of tone-contrast in a given space. The space becomes more attractive in consequence.

This might not be the case, however, if the greater extension of tone-contrast in one case were neutralized by an increase of tone-contrast in the other.

THE ORDER OF HARMONY IN LINES

62. Harmony of Direction means no change of direction.

[Illustration: Fig. 67]

In this case we have a Harmony of Direction in the line, because it does not change its direction.

63. Harmony of Angles. We may have Harmony in the repetition of a certain relation of directions, as in an angle.

[Illustration: Fig. 68]

The angle up 45° and down 45° is here repeated seven times.

[Illustration: Fig. 69]

In this case we have a great many angles in the line, but they are all right angles, so we have a Harmony of Angles.

[Illustration: Fig. 70]

In this case we have Harmony in the repetition of a certain relation of angles, that is to say, in the repetition of a certain form of angularity.

64. Equality of lengths or measures between the angles of a line means a Harmony of Measures.

[Illustration: Fig. 71]

In this case, for example, we have no Harmony of Angles, but a Harmony of Measures in the legs of the angles, as they are called.

65. We have a Harmony of Curvature in a line when it is composed wholly of arcs of the same radius and the same angle.

[Illustration: Fig. 72]

This is a case of Harmony of Curvature. There is no change of direction here, in the sequence of corresponding arcs.

[Illustration: Fig. 73]

Here, again, we have a Harmony of Curvature. In this case, however, there is a regular alternation of directions in the sequence of corresponding arcs. In this regular alternation, which is the repetition of a certain relation of directions, there is a Harmony of Directions.

[Illustration: Fig. 74]

In this case the changes of direction are abrupt (angular) as well as gradual. There is no regular alternation, but the harmony of corresponding arcs repeated will be appreciated, nevertheless.

66. Arcs produced by the same radius are in harmony to that extent, having the radius in common.

[Illustration: Fig. 75]

This is an example of a harmony of arcs produced by radii of the same length. The arcs vary in length.

67. Arcs of the same angle-measure produced by different radii are in Harmony to the extent that they have an angle-measure in common.

[Illustration: Fig. 76]

This is an example.

Arcs having the same length but varying in both radius and angle may be felt to be in Measure-Harmony. It is doubtful, however, whether lines of the same length but of very different curvatures will be felt to correspond. If the correspondence of lengths is not felt, visually, it has no interest or value from the point of view of Pure Design.

68. Any line may be continued in a repetition or repetitions of its shape, whatever the shape is, producing what I call a Linear Progression. In the repetitions we have Shape-Harmony.

[Illustration: Fig. 77]

This is an example of Linear Progression. The character of the progression is determined by the shape-motive which is repeated in it.

69. The repetition of a certain shape-motive in a line is not, necessarily, a repetition in the same measure or scale. A repetition of the same shape in the same measure means Measure and Shape-Harmony in the progression. A repetition of the same shape in different measures means Shape-Harmony without Measure-Harmony.

[Illustration: Fig. 78]

Here we have the repetition of a certain shape in a line, in a progression of measures. That gives us Shape-Harmony and a Harmony of Proportions, without Measure-Harmony.

70. In the repetition of a certain shape-motive in the line, the line may change its direction abruptly or gradually, continuously or alternately, producing a Linear Progression with changes of direction.

[Illustration: Fig. 79]

In Fig. 79 there is a certain change of direction as we pass from one repetition to the next. In the repetition of the same change of direction, of the same angle of divergence, we have Harmony. If the angles of divergence varied we should have no such Harmony, though we might have Harmony in the repetition of a certain relation of divergences. Any repetition of a certain change or changes of direction in a linear progression gives a Harmony of Directions in the progression.

[Illustration: Fig. 80]

In this case there is a regular alternation of directions in the repeats. The repeats are drawn first to the right, then up, and the relation of these two directions is then repeated.

71. By inverting the motive of any progression, in single or in double inversion, and repeating the motive together with its inversion, we are able to vary the character of the progression indefinitely.

[Illustration: Fig. 81]

In this case we have a single inversion of the motive and a repetition of the motive with its inversion. Compare this progression with the one in Fig. 77, where the same motive is repeated without inversion.

[Illustration: Fig. 82]

Here we have the same motive with a double inversion, the motive with its double inversion being repeated. The inversion gives us Shape-Harmony without Harmony of Attitudes. We have Harmony, however, in a repetition of the relation of two attitudes. These double inversions are more interesting from the point of view of Balance than of Harmony.

THE ORDER OF BALANCE IN LINES

72. We have Balance in a line when one half of it is the single or double inversion of the other half; that is, when there is an equal opposition and consequent equilibrium of attractions in the line. When the axis of the inversion is vertical the balance is symmetrical.

[Illustration: Fig. 83]

There is Balance in this line because half of it is the single inversion of the other half. The balance is symmetrical because the axis is vertical. The balance, although symmetrical, is not likely to be appreciated, however, because the eye is sure to move along a line upon which there is no better reason for not moving than is found in slight terminal contrasts. The eye is not held at the center when there is nothing to hold the eye on the center. Mark the center in any way and the eye will go to it at once. A mark or accent may be put at the center, or accents, corresponding and equal, may be put at equal distances from the center in opposite directions. The eye will then be held at the center by the force of equal and opposite attractions.

[Illustration: Fig. 84]

In this case the eye is held at the balance-center of the line by a change of character at that point.

[Illustration: Fig. 85]

In this case the changes of character are at equal distances, in opposite directions, from the center. The center is marked by a break. The axis being vertical, the balance is a symmetrical one.

73. The appreciation of Balance in a line depends very much upon the attitude in which it is drawn.

[Illustration: Fig. 86]

In this case the balance in the line itself is just as good as it is in Fig. 85; but the axis of the balance being diagonal, the balance is less distinctly felt. The balance is unsatisfactory because the attitude of the line is one which suggests a falling down to the left. It is the instability of the line which is felt, more than the balance in it.

[Illustration: Fig. 87]

In this case of double inversion, also, we have balance. The balance is more distinctly felt than it was in Fig. 86. The attitude is one of stability. This balance is neither axial nor symmetrical, but central.

74. A line balances, in a sense, when its inclinations are balanced.

[Illustration: Fig. 88]

This line may be said to be in balance, as it has no inclinations, either to the right or to the left, to suggest instability. The verticals and the horizontals, being stable, look after themselves perfectly well.

[Illustration: Fig. 89]

This line has two unbalanced inclinations to the left. It is, therefore, less satisfactory than the line in Fig. 88, from the point of view of Balance.

[Illustration: Fig. 90]

The two inclinations in this line counteract one another. One inclination toward the left is balanced by a corresponding inclination toward the right.

[Illustration: Fig. 91]

In this case, also, there is no inclination toward the left which is not balanced by a corresponding inclination toward the right.

[Illustration: Fig. 92]

In this line, which is composed wholly of inclinations to the right or left, every inclination is balanced, and the line is, therefore, orderly in the sense of Balance; more so, certainly, than it would be if the inclinations were not counteracted. This is the problem of balancing the directions or inclinations of a line.

75. A line having no balance or symmetry in itself may become balanced. The line may be regarded as if it were a series of dots close together. The line is then a relation of positions indicated by dots. It is a composition of attractions corresponding and equal. It is only necessary, then, to find what I have called the center of equilibrium, the balance-center of the attractions, and to indicate that center by a symmetrical inclosure. The line will then become balanced.

[Illustration: Fig. 93]

Here is a line. To find the center of its attractions it may be considered as if it were a line of dots, like this:—

[Illustration: Fig. 94]

The principle according to which we find the balance-center is stated on page 23. The balance-center being found, it must be indicated unmistakably. This may be done by means of any symmetrical inclosure which will draw the eye to the center and hold it there.

[Illustration: Fig. 95]

In this case the balance-center is indicated by a rectangular inclosure. This rectangle is not, however, in harmony of character with the line inclosed by it, which is curved.

[Illustration: Fig. 96]

In this case the balance-center is indicated by a circle, which, being a curve, is in harmony of character with the inclosed line, which is also a curve. I shall call this Occult Balance to distinguish it from the unmistakable Balance of Symmetry and other comparatively obvious forms of Balance, including the balance of double inversions. As I have said, on page 24, the symmetrical framing must be sufficiently attractive to hold the eye steadily at the center, otherwise it does not serve its purpose.

THE ORDER OF RHYTHM IN LINES

76. The eye, not being held on a vertical axis or on a balance-center, readily follows any suggestion of movement.

[Illustration: Fig. 97]

In this case there is no intimation of any vertical axis or balance-center. The figure is consequently unstable. There is a sense of movement to the right. This is due, not only to the inclinations to the right, but to the convergences in that direction.

[Illustration: Fig. 98]

In this case the movement is unmistakably to the left. In such cases we have movement, but no Rhythm.

77. Rhythm requires, not only movement, but the order of regular and marked intervals.

[Illustration: Fig. 99]

In Fig. 99 we have a line, a linear progression, which gives us the feeling of movement, unmistakably. The movement, which in the motive itself is not rhythmical, becomes rhythmical in its repetition at regular, and in this case equal, intervals. The intervals are marked by the repetitions.

78. It is a question of some interest to decide how many repetitions are required in a Rhythm. In answer to this question I should say three as a rule. A single repetition shows us only one interval, and we do not know whether the succeeding intervals are to be equal or progressive, arithmetically progressive or geometrically progressive. The rhythm is not defined until this question is decided, as it will be by two more repetitions. The measures of the rhythm might take the form of a repeated relation of measures; a repetition, for example, of the measures two, seven, four. In that case the relation of the three measures would have to be repeated at least three times before the character of the rhythm could be appreciated.

79. Any contrariety of movement in the motive is extended, of course, to its repetitions.

[Illustration: Fig. 100]

In this case, for example, there are convergences and, consequently, movements both up and down. This contrariety of movements is felt through the whole series of repetitions. Other things being equal, I believe the eye moves up more readily than down, so that convergences downward have less effect upon us than corresponding convergences upward.

[Illustration: Fig. 101]