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.

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 mustdecide 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?

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.

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:—

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.

Fig. 60

Here, for example, is a certain circular arc. It is perhaps best definedand 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.

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.

Fig. 62

Here is a curve which, so far as the arcs are concerned, of which it is composed, resembles the curve ofFig. 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 inFig. 62will 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.

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.

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.

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.

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.

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.

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.

62. Harmony of Direction means no change of direction.

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.

Fig. 68

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

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.

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.

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.

Fig. 72

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

Fig. 73

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

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.

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.

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.

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.

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.

Fig. 79

InFig. 79there 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.

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.

Fig. 81

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

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.

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.

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 thecenter, 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.

Fig. 84

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

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.

Fig. 86

In this case the balance in the line itself is just as good as it is inFig. 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.

Fig. 87

In this case of double inversion, also, we have balance. The balance is more distinctly felt than it was inFig. 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.

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.

Fig. 89

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

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.

Fig. 92

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

Fig. 92

In this line, which is composed wholly of inclinations to the right orleft, 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.

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:—

Fig. 94

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

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.

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 havesaid, onpage 24, the symmetrical framing must be sufficiently attractive to hold the eye steadily at the center, otherwise it does not serve its purpose.

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

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.

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.

Fig. 99

InFig. 99we 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.

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.

Fig. 101

In this case, by omitting the long vertical line I have diminished the amount of convergence downward. In that way I have given predominance to the upward movement. Instead of altogether omitting the long vertical line, I might have changed its tone from black to gray. That would cause an approximate instead of complete disappearance. It should be remembered that in all these cases the habit of reading comes in to facilitate the movements to the right. It is easier for the eye to move to the right than in any other direction, other things being equal. The movement back to the beginning of another line, which is of course inevitable in reading, produces comparatively little impression upon us, no more than the turning of the page. The habit of reading to the right happens to be our habit. The habit is not universal.

80. Reading repetitions and alternations to the right, always, I, for a long time, regarded such repetitions and alternations as rhythmical, until Professor Mowll raised the question whether it is necessary to read all alternations to the right when there is nothing in the alternations themselves to suggest a movement in one direction rather than another. Why not read them to the left as well as to the right? We at once decided that the movement in a Rhythm must be determined by the character of the Rhythm itself, not by any habit of reading, or any other habit, on our part. It was in that way that we came to regard repetitions and alternations as illustrations of Harmony rather than ofRhythm. Rhythm comes into the Harmony of a Repeated Relation when the relation is one which causes the eye to move in one direction rather than another, and when the movement is carried on from repetition to repetition, from measure to measure.

81. The repetition of a motive at equal intervals, when there is no movement in the motive, gives us no feeling of Rhythm.

Fig. 102

In this case, for example, we have a repetition in the line of a certain symmetrical shape. As there is no movement in the shape repeated, there is no Rhythm in the repetition. There is nothing to draw the eye in one direction rather than another. The attractions at one end of the line correspond with the attractions at the other.

82. The feeling of Rhythm may be induced by a regular diminution of measure or scale in the repetitions of the motive and in the intervals in which the repetitions take place.

Fig. 103

In this case the shape repeated is still symmetrical, but it is repeated with a gradual diminution of scale and of intervals, by which we are given a feeling of rhythmic movement. The change of scale and of intervals, to induce a sense of rhythmic motion, must be regular. To be regular the change must be in the terms of one or the other of the regular progressions; the arithmetical progression, which proceeds by acertain addition, or the geometrical, which proceeds by a certain multiplication. The question may arise in this case (Fig. 103)whether the movement of the Rhythm is to the right or to the left. I feel, myself, that the movement is to the right. In diminishing the scale of the motive and of the intervals we have, hardly at all, diminished the extent of the tone-contrast in a given angle of vision.See Fig. 41, p. 27, showing the increase of attractions from one visual angle to another. At the same time we come at the right end of the progression to two or more repetitions in the space of one. We have, therefore, established the attraction of a crowding together at the right end of the series. See the passage (p. 43) on the attractiveness of a line. The force of the crowding together of attractions is, I feel, sufficient to cause a movement to the right. It must be remembered, however, that the greater facility of reading to the right is added here to the pull of a greater crowding together of attractions in the same direction, so the movement of the Rhythm in that direction may not be very strong after all. If the direction of any Rhythm is doubtful, the Rhythm itself is doubtful.

83. The feeling of Rhythm may be induced, as I have said, by a gradual increase of the number of attractions from measure to measure, an increase of the extent of tone-contrast.

Fig. 104

Increasing the extent of tone-contrast and the number of attractions in the measures of the Rhythm inFig. 103, we are able to force the eye to follow the series in the direction contrary to the habit of reading, that is to say from right to left.

A decrease in the forces of attraction in connection with a decrease ofscale is familiar to us all in the phenomena of perspective. The gradual disappearance of objects in aerial perspective does away with the attraction of a greater crowding together of objects in the distance.

Fig. 105

In this case the diminution of scale has been given up and there is no longer any crowding together. There is no chance of this rhythm being read from left to right except by an effort of the will. The increase of attractions toward the left is much more than sufficient to counteract the habit of reading.

84. The force of a gradual coming together of attractions, inducing movement in the direction of such coming together, is noticeable in spiral shapes.

Fig. 106

In this case we have a series of straight lines with a constant and equal change of direction to the right, combined with a regular diminution of measures in the length of the lines, this in the terms of an arithmetical progression. The movement is in the direction of concentration and it is distinctly marked in its measures. The movement is therefore rhythmical.

Fig. 107

In this case we have a series of straight lines with a constant change of direction to the right; but in this case the changes of measure in the lines are in the terms of a geometrical progression. The direction is the same, the pull of concentration perhaps stronger.

Fig. 108

In this Rhythm there is an arithmetical gradation of measures in the changes of direction, both in the length of the legs and in the measureof the angles. The pull of concentration is, in this case, very much increased. It is evident that the legs may vary arithmetically and the angles geometrically; or the angles arithmetically and the legs geometrically.

85. If, in the place of the straight lines, which form the legs, in any of the examples given, are substituted lines which in themselves induce movement, the feeling of Rhythm may be still further increased, provided the directions of movement are consistent.

Fig. 109

In this case the movement is in the direction of increasing concentration and in the direction of the convergences.

If the movement of the convergences be contrary to the movement of concentration, there will be in the figure a contrary motion which may diminish or even entirely prevent the feeling of Rhythm. If the movement in one direction or the other predominates, we may still get the feeling of Rhythm, in spite of the drawback of the other and contrary movement.

Fig. 110

In this case the linear convergences substituted for the straight lines are contrary to the direction of increasing concentration. The movement is doubtful.

86. Corresponding rhythms, set in contrary motion, give us the feeling of Balance rather than of Rhythm. The balance in such cases is a balance of movements.

Fig. 111

This is an example of corresponding and opposed rhythms producing the feeling, not of Rhythm, but of Balance.

LINES IN DIFFERENT ATTITUDES

87. Any line or linear progression may be turned upon a center,and so made to take an indefinite number and variety of attitudes. It may be inverted upon an axis, and the inversion may be turned upon a center producing another series of attitudes which, except in the case of axial symmetry in the line, will be different from those of the first series.

Fig. 112

In this case the line changes its attitude.

Fig. 113

In this case I have inverted the line, and turning the inversion upon a center I get a different set of attitudes.

Fig. 114

In this case, which is a case of axial symmetry in the line, the inversion gives us no additional attitudes.

88. When any line or linear progression is repeated, without change of attitude, we have a Harmony of Attitudes.

Fig. 115

This is an illustration of Harmony of Attitudes. It is also an illustration of Interval-Harmony.

89. We have a Harmony of Attitudes, also, in the repetition of any relation of two or more attitudes, the relation of attitudes being repeated without change of attitude.

Fig. 116

We have here a Harmony of Attitudes due to the repetition of a certain relation of attitudes, without change of attitude.

90. When a line or linear progression is inverted upon any axis or center, and we see the original line and its inversion side by side, we have a Balance of Attitudes.

Fig. 117

The relation of attitudes I, II, of III, IV, and of I, II, III, IV, is that of Symmetrical Balance on a vertical axis. The relation of attitudes I, IV, and of II, III, is a relation of Balance but not of Symmetrical Balance. This is true, also, of the relation of I, III and of II, IV. Double inversions are never symmetrical, but they are illustrations of Balance. The balance of double inversions is central, not axial. These statements are all repetitions of statements previously made about positions.

91. It often happens that a line repeated in different attitudes gives us the sense of movement. It does this when the grouping of the repetitions suggests instability. The movement is rhythmical when it exhibits a regularity of changes in the attitudes and in the intervals of the changes.

Fig. 118

In this case we have a movement to the right, but no Rhythm, the intervals being irregular.

Fig. 119

In this case the changes of attitude and the intervals of the changes being regular, the movement becomes rhythmical. The direction of the rhythm is clearly down-to-the-right.

92. In the repetition of any line we have a Harmony, due to the repetition. If the line is repeated in the same attitude, we have a Harmony of Attitudes. If it is repeated in the same intervals, we have a Harmony of Intervals. We have Harmony, also, in the repetition of any relation of attitudes or of intervals.

We have not yet considered the arrangement or composition of two or more lines of different measures and of different shapes.

93. By the Composition of Lines I mean putting two or more lines together, in juxtaposition, in contact or interlacing. Our object in the composition of lines, so far as Pure Design is concerned, is to achieve Order, if possible Beauty, in the several modes of Harmony, Balance, and Rhythm.

94. We have Harmony in line-compositions when the lines which are put together correspond in all respects or in some respects, when they correspond in attitudes, and when there is a correspondence of distances or intervals.

Fig. 120

In this case the lines of the composition correspond in tone, measure, and shape, but not in attitude; and there is no correspondence in distances or intervals.

Fig. 121

In this case the attitudes correspond, as they did not inFig. 120. There is still no correspondence of intervals.

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