Lateral line

“ Extremes meet, and so the attempt to avoid absolute horizontal lines has completely failed, because in their place we have a succession of vertical members placed side by side in such endless numbers that we really think more of the horizontal arrangement of these members, slight though they are, than we should of the simplest defined horizontal line. ”

“ If from any line A or B or C (Fig. 18) , &c., we drop perpendiculars from different points of those lines on to a horizontal plane, the intersections of those verticals with the plane will be on a line called the horizontal trace or projection of the original line. ”

“ Hence we are peculiarly inclined to think that a line which we perceive to be in such a plane is a vertical line. But to see a whole lot of lines at once, all ready to throw their images upon the vertical meridian, is a thing that has hardly ever happened to us, except when they all have been vertical lines. Hence when that happens we have a still stronger tendency to think that what we see before us is a group of vertical lines. ”

“ When the figure or object appears as dark upon a light ground, however, the outline should be within the silhouette, obviously, or its delicate boundary is lost.
Linear Expression of Movement
Another important attribute of line is its power of expressing or suggesting movement. By a law of inseparable association, undulating lines approaching the horizontal, or leading down to it, are connected with the sense of repose; whereas broken curves and rectangular lines always suggest action and unrest, or the resistance to force of some kind. ”

“ If a vertical line, drawn through the centre of the trunk, be taken as a median line in a simple design of the human figure, all subsequent patterns which are drawn will be grouped symmetrically about it. ”

“ Therefore a Pair of Lines, &c. Q. E. D.
(κ) . II. 3.
Through a given point, without a given Line, a Line may be drawn such that the two Lines are equally inclined to any transversal.
Take a second point, on the same side of the given Line and at the same distance from it; and join the 2 points.
Then the Line, so drawn, and the given Line, are equally inclined to any transversal. ”

“ 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. ”

“ His subjects—of whom the small Lines were men and the Points Women—were all alike confined in motion and eyesight to that single Straight Line, which was their World. It need scarcely be added that the whole of their horizon was limited to a Point; nor could any one ever see anything but a Point. Man, woman, child, thing—each was a Point to the eye of a Linelander. ”

“ For they each have the same direction as that Line, and therefore the same direction as the other.'
Min. I am willing to grant you, without any proof, that, if such Lines existed, they would have the same direction with regard to each other. The phrase 'they each have' is not remarkably good English. However, you may proceed.
Niemand reads.
P. 12. Ax. 9. 'An angle may be conceived as transferred from one position to another, the direction of its arms remaining the same.'
Min. Let us first consider the right arm by itself. ”

“ The complete intersection of the surface by the tangent plane is made up of the line counting once and of a residual curve of the order n-I. A quadric surface is a regulus in a twofold manner, for there are on the surface two systems of lines each of which is a regulus. ”

“ Perpendicular and horizontal straight lines are the more easy to examine, because their images fall directly between two opposing muscles. An oblique line is difficult to examine, and we instinctively turn the head, in order to bring it in the plane of rotation of one or the other set of muscles. In following a curved line with the eye, two muscles are used together, one contracting more rapidly than the other. A curve is therefore more difficult to observe, or run the eye over, than a straight line, and the difficulty increases with the subtileness of the curvature. ”

“ Follow with your eye any two adjacent lines, and you will see that where they are close to each other the surface has an abrupt change of level; where they are further apart the surface is nearly horizontal. Where the surface approaches the perpendicular, as on the sides, the dark line showing the separation of the strata is thin, because it has been cut through nearly at right angles. Where the surface is more horizontal the dark line is broader, because it has been cut through obliquely, the breadth varying steadily with the angle of inclination. ”

“ Now it is essential that the edge be ground so that its rate of approach to the back shall be absolutely uniform from hilt to point-plane, and, further, that the line of intersection of the edge- plane and the back-plane shall be equidistant throughout from the back and the edge. ”

“ 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. ”

“ 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. ”

“ In the same manner, whatever be the number of points on which the body may rest, its virtual base will be found by supposing straight lines drawn, joining the several points successively. When the line of direction falls within this base, the body will always stand firm, and otherwise not. ”

“ The accompanying charts comprise a series of projections formed on the following principle; A vertical line, one inch long, is divided by horizontal lines into eight parts, representing as many decades. The angle to the left of the vertical line is supposed to represent males, and that to the right, females; and the shading of either angle indicates the predomination of the sex it stands for. Figures attached to the ends of the horizontal lines indicate by thousandths the number of individuals in each decade of life. ”

“ 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. ”

“ If any line in a plane revolves about an external axis in the plane, the area of the curved surface generated by the revolution is equal to the product of the length of the line and the distance travelled by the centroid of the line. ”

“ 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. ”

“ If the angle of inclination be semirect, and the line of inclination be in the thicker medium, and the proportion of their densities be the same with that of the diagonal to the side of a square, and the separating superficies be plane, the refracted line will be in the separating superficies.—8. If a body be carried in a strait line upon another body, and do not penetrate the same, but be reflected from it, the angle of reflection will be equal to the angle of incidence.—9. The same happens in the generation of motion in the line of incidence. ”

“ The apical pole of the vertical main axis (principal or longitudinal axis) is different from the basal pole. The ventral pole of the horizontal sagittal axis (or dorso-ventral axis) is different from the dorsal pole. The right pole of the horizontal transverse axis (lateral or frontal axis) is equal to the left pole. ”

“ The truth is, supposing the brain to unite, instead of separate, the frontal line of the face and the axial line of the body, the bending of the continuous right line formed by them in the lowest vertebrate fishes into the two parallel lines in man, the one facio-abdominal and the other occipito-dorsal, is not by angles at all, but by curvatures, and the union of the parallels is by an arch over the head. The arc in each case is a greater or less part of a circle, according to the grade of intellectual and moral development. ”

“ All things considered, we may plant ourselves anew on the base-line of the old facial angle, assured that it is what its name signifies—fundamental, the centre of support and dependence between the transitional and variable lines, and presenting fixed extremities, constituting axial centres, in relation to which the movable lines are radii, forming with the baseline angles of all degrees between one and ninety in the development of animal life, from the lowest vertebrate form up to its highest and most perfect type. ”

“ The young philosopher was unable to answer this question, and his father, therefore, informed him that, if a perpendicular line were drawn from the centre of gravity of a body to the centre of the earth, such a line would be termed the line of direction; along which every body, not supported, endeavours to fall ”

“ 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. ”

“ All horizontal lines in nature that are parallel to {36} PEN DRAWING. By Jules Girardet. Showing a mantel a little below the eye. The student should practice drawing interiors with the purpose of learning the theory of perspective from every object drawn. The horizontal lines of the picture frame, for example, tip in the opposite direction to the mantel, because they are above the eye. Had the mantel been a few inches higher it would have been drawn as a perfectly horizontal line. {37} CRISPI AS CÆSAR IN HIS ROBE DE CHAMBRE. A political caricature from Don Chisciotte. ”

“ Accordingly, a body hanging by a line, which over a level surface would be drawn by the earth's attraction into a direction with reference to its point of suspension, the prolongation of the line of which would pass through the center of the earth—that is, would be perpendicular, or plumb—would be attracted and turned away from the perpendicular by a mass like a mountain in the neighborhood. ”

“ For orientation in three dimensions of space we must extend our ideas and consider position with reference to east and west and up and down. A man lying on his side with his head to the north and looking east, would not be similarly oriented to a man standing upright and looking east. To provide for the complete specification of how a body is oriented in space we must have in the body a plane of reference, and a line of reference in this plane, belonging to the body and moving with it. ”

“ Then, at the points at which the parallel lines cut the line called the horizon, the letter S is to be on the right and the letter V on the left, and from the extremity of the semicircle, at the point G, draw a line parallel to the axis, extending to the left-hand semicircle at the point H. This parallel line is called the Logotomus. ”

“ For the same reason, the horizontal line, in which innumerable objects intervene between the eye and the horizon, appears so much longer than the line of altitude of the meridian, that the vault of the sky does not seem a hemisphere, but a far smaller segment of a great sphere. ”

“ A perpendicular or vertical line is one perfectly upright, as a string, with a weight at the end of it, will hang from a nail, or from the hand.
In making horizontal lines, the pupil should make them parallel to the top or bottom of his slate or paper, and in making perpendiculars they should be parallel to the sides of the slate or paper. Parallel lines are lines running in the same direction equally distant from each other in every part ”

“ Given two horizontal planes cut by a vertical line, it is required to find a point on the upper part of the vertical line from which bodies may fall to the horizontal planes and there, having their motion deflected into a horizontal direction, will, during an interval equal to the time of fall, traverse distances which bear to each other any assigned ratio of a smaller quantity to a larger. ”

“ Then bisect the narrow transverse strip which is above the line e. f. in the line i. k. by the point m.; and indicate the breadth of the letter, to the right of the semicircle, by the point n. in the line g. h.; and afterwards draw from the point m. above the line e. f. in the direction of f. a short horizontal line as great as need be: then describe a semicircle which shall include this line, and the point n., and, at the top, the side a. b.; and through n. let pass a vertical line. These all combine to form, below, the concave of the curved limb, and above, its convex. ”

“ Let a point move from the origin in a straight line at an angle of 45° with the axis of X to the point () , thence vertically in a straight line to the point () , thence obliquely in a straight line to the point () , etc The broken line thus described (continued indefinitely forwards and backwards) is the limiting form of the curve as the number of terms increases indefinitely. ”