Straight line

“ You think yourself the perfection of existence, while you are in reality the most imperfect and imbecile. You profess to see, whereas you can see nothing but a Point! You plume yourself on inferring the existence of a Straight Line; but I can see Straight Lines and infer the existence of Angles, Triangles, Squares, Pentagons, Hexagons, and even Circles. ”

“ If a straight line, falling on two other straight lines, make the exterior angle equal to the interior and opposite angle on the same side of the line, or make the interior angles on the same side together equal to two right angles, the two straight lines shall be parallel to one another. ”

“ The due mixture of the curved line with the straight, the growth of one from the other, the repetition of straight lines in suitable positions, all seem to have come to them by intuition which seldom erred. ”

“ If a straight line stand at right angles to each of two straight lines at the point of their intersection, it shall also he at right angles to the plane which passes through them, that is, to the plane in which they are. ”

“ But when a straight line, standing on another straight line, makes the adjacent angles equal to one another, each of the angles is called a right angle, and the straight line which stands on the other is called a perpendicular to it. ”

“ It must travel continually in the direction of a straight line, and at the same time continually out of the direction of a straight line. It must move in a straight direction and out of a straight direction at once. ”

“ Fig. 34.
Well, as curves are more beautiful than straight lines, it is necessary to a good composition that its continuities of object, mass, or colour should be, if possible, in curves, rather than straight lines or angular ones. Perhaps one of the simplest and prettiest examples of a graceful continuity of this kind is in the line traced at any moment by the corks of a net as it [Pg 370] is being drawn: nearly every person is more or less attracted by the beauty of the dotted line. ”

“ Two straight lines meeting in opposing directions give an impression of abruptness, severity, or even violence; the difference of movement being emphasized. If a third line is added, as in the sketches below, the opposition is softened and an effect of unity and completeness produced.
This combination typifies beauty itself which has been defined as consisting of elements of difference harmonized by elements of unity.
A very common example of Transition is the bracket, No. 15. The straight line is modified into curves and may be elaborated with great complexity of modelling. ”

“ Who can decide between rectilinear and curvilinear geometry?—between the theory of straight lines and the theory of curved lines? If, in His work, the mysterious Maker, who attains His ends with miraculous directness, only makes use of the straight line to divide it at a right angle and obtain a curve, man himself cannot rely on it: the bullet a man wishes to send in a straight line follows a curve, and when you want to hit a point in space with certainty you propel the ball on its cruel parabola. ”

“ Further, supposing the straight lines FG, HI, etc., to be [Pg 51] drawn from an infinitude of points in the straight line AB and to terminate on the line DE (which is straight or may be considered as such) , let the different penetrabilities at the different heights in the air between A and B be represented by all these lines ”

“ The problem of drawing straight lines at random, under various limitations of direction or intersection, is familiar enough, but I do not know that any one has suggested the drawing of a line whose shape as well as position shall be of a purely random character. For simplicity we suppose the line to be confined to a plane. ”

“ Just in the same way straightness mingled with curvature, that is to say, the close approximation of part of any curve to a straight line, gives to such curve all its spring, power, and nobleness: and even perfect straightness, limiting curves, or opposing them, is often pleasurable: yet, in the abstract, straightness is always ugly, and curvature always beautiful. ”

“ Is there any motion in a straight line? A straight line infinitely projected becomes a circle, it returns to the starting point. You must end where you begin; and as you began in God, you must go back to God. ”

“ Naturally man tended to lose his sense of scale and relation. A straight line, or a combination of straight lines, may have still a sort of artistic unity, but what can be done in art with a series of negative symbols? Even if the negative were continuous, the artist might express at least a negation; ”

“ LINE UNITY is that principle which appeals to the eye by the relation of lines in the design to the space decorated. Curved lines for curved forms and straight lines in the designs decorating straight lined forms will do much toward making all parts consistent. A line not too curved and with a blending of straight lines with the curved direction produces a line of character and strength for design rendering. ”

“ They are not ideally straight, and the only possible imperfection affecting all parts of all straight lines equally must be curvature. The ideal straight line, that is to say, the shortest distance, between two points must be curved, and of course equally curved, for any inequality would lengthen the distance. ”

“ Straight lines exist which have the property that any one of them may be produced both ways without limit, that through any two points in space such a line may be drawn, and that any two of them coincide throughout their indefinite extensions as soon as two points in the one coincide with two points in the other. ”

“ What applies to distance applies equally to the straight line. There is nothing in the actual world having exactly the properties that straight lines were supposed to have; the nearest approach is the track of a light ray. Straight lines have to be replaced by geodesics, which are defined by what they do at each point, not all at once, like Euclidean straight lines. ”

“ If for example, when my Father, the Triangle, approaches me, he happens to present his side to me instead of his angle, then, until I have asked him to rotate, or until I have edged my eye round him, I am for the moment doubtful whether he may not be a Straight Line, or, in other words, a Woman. ”

“ The seventeenth proposition of the sixth book is the problem “To divide a straight line, whether internally or externally, so that the rectangle under its segments shall be equivalent to a given rectangle.” ”

“ Fig. 6.
As outlines are composed of profiles, the same laws govern. That the curved line is the line of beauty stands out most evidently in the study of antique designs. Vertical lines and horizontal lines are the lines of support and strength, and must always have proper consideration; but in pure ornament the office of straight lines seems to be confined to connecting curves and to emphasizing their contrasts. ”

“ Through a given point, a whole pencil of straight lines may be drawn, none of which shall cut a given line, though infinitely produced, and limited, at the two extremes, by lines that cut the given line at infinite distances in the opposite directions. ”

“ Nie. We have that. At p. 20 we give you 'Axiom IV. Through two points always one, and only one, Line can be drawn.' And at p. 18 we at last distinguish 'Line' and 'curve.' 'A straight Line will in future be called a Line simply. All other Lines will be called curved Lines, or curves.'
Min. Better late than never: though it makes wild work of your former theory—in which you got the notion of 'Line' from a bent wire, and of 'curve' from the path of a moving point. Now for the Definition of 'angle.'
Nie. ”

“ It follows that a natural object is associated also with a straight line. Three points , and on a rigid body thus lie in a straight line when, the points and being given, is chosen such that the sum of the distances and is as short as possible. ”

“ If we should give to a number of adults the order to draw a straight line upon the blackboard, each person would draw a long line proceeding in a different direction, some beginning from one side, some from another, and almost all would succeed in making the line straight. ”