Vertical line

“ Given a limited inclined plane and a vertical line having their highest point in common, it is required to find a point in the vertical line extended such that a body will fall from it and then traverse the inclined plane in the same time which is required to traverse the inclined plane alone starting from rest at the top of said plane. ”

“ 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. ”

“ From each point marked we draw a line parallel to the base till it reaches the scale. From the point where it touches the line AO, raise perpendicular as a, which gives the height required at that distance, and must be referred back to the figure itself.
Fig. 78.
XXVIII How to Determine the Heights of Figures on a Level Plane First Case.
This is but a repetition of the previous figure, excepting that we have substituted these schoolgirls for the vertical lines. If we wish to make some taller than the others, and some shorter, we can easily do so, as must be evident (see Fig. 79) . ”

“ To sum up, the eye, owing to the laws of its mechanism, follows a line much more easily in the central than in the peripheral parts of the field, and in the central parts it follows a vertical line more easily than an oblique, and an horizontal more easily than a vertical. ”

“ 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. ”

“ When the elevation is determined by sights the angle of elevation is the angle between the line of sight and the axis of the piece, when these lines are in the same vertical plane,—or the angle between the line of sight and a plane containing the axis of the piece and a horizontal line intersecting it at right angles, when they are not. The graduations of tangent scales and fixed breech-sights give this angle in degrees. The graduation of the pendulum hausse gives the angle correctly only when the line of sight is horizontal. ”

“ 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. ”

“ The circle being drawn, a line A, 179, is drawn through its centre, and from its intersection with the circle as at b, here a step on each side is marked as c, d, then lines c to d, and c and d to e, where A meets, the circle will describe a trigon. If the figure is to stand vertical, all that is necessary is to draw the line a vertical, as in Figure 180. A ready method of getting the dimension across corners, across the flats, or the length of a side of a given polygon, is by means of diagrams, such as shown in the following figures, which form excellent examples for practice. ”

“ We interpret vertical lines as moving upward, partly because the eye moves upward in scanning them, partly because we ourselves move in lines of this general direction in going from the bottom to the top of a building. Even when we are at the top of a building we apprehend its vertical lines as rising rather than as descending, because we ourselves had to rise in order to get there. ”

“ The GROUND-PLANE is the plane on which we stand; the base-line is an imaginary line passing through the middle of the feet as we stand square and erect; and the vertical plane is supposed to stand on the base-line and perpendicular to it. ”

“ How many pairs of vertical angles are formed by the intersection of two lines?
Read all the pairs of vertical angles formed by the lines l m, n p.
LESSON NINTH. KINDS OF ANGLES. RIGHT ANGLES. What do we call the angles a o c, c o b? (Diagram 9.)
Are they equal to each other?
Then they are called right angles.
A right angle is one of two adjacent angles that are equal to each other.
Are the adjacent angles c o b, b o d equal to each other?
Then what are they called?
Read the right angles below the line a b. On the left of c d.
Read three right angles whose vertices are at p.
24
Diagram 9. ”

“ And whenever you get long vertical lines in a composition, no matter whether it be a cathedral interior, a pine forest, or a row of scaffold poles, you will always have the particular feeling associated with rows of vertical lines in the abstract. And further, whenever you get the swinging lines of the volute, an impression of energy will be conveyed, no matter whether it be a breaking wave, rolling clouds, whirling dust, or only a mass of tangled hoop iron in a wheelwright's yard. ”

“ So accustomed is the eye to vertical lines in tree trunks, tall buildings, and thousands of other examples that the upward eye movement in viewing an object, having a predominance of vertical elements, seemingly adds to its height. ”

“ The angular difference at any place, between the direction of a plumb line (the vertical) and the perpendicular to the reference ellipsoid. This difference seldom exceeds 30". Often expressed in two components, meridian and prime vertical. Also called STATION ERROR.
deflection of the vertical correction. . The correction due to deflection of the vertical resulting from irregularities in the density and form of the earth. Deflection of the vertical affects the accuracy of sextant altitudes.
deflector. , n. An instrument for measuring the directive force acting on a magnetic compass. ”

“ To render the differences of altitude more apparent, these vertical distances are plotted to a much larger scale than the horizontal, as explained in a preceding Section. To erect the perpendiculars, the T and set square furnish the most convenient means. The detail points thus determined upon the perpendiculars represent [94] the points in the surface of the ground at which the levels were taken, and by joining these points we obtain the surface line. An example will clearly show the method pursued. Let it be required to lay down the section from the following level-book:—
Fig. ”

“ For vertical planes it will be a pair of generators of the cylinder. For other planes the vanishing curves will be ellipses having their centre at the eye The projections of vertical lines will be vertical lines on the cylinder. ”

“ To lay off the directions of these resultants, from any convenient point, , say, in the vertical through , as a center, describe an arc, , with a convenient radius, . With the same radius and as a center, describe the arc, , cutting the trial planes (produced if necessary) at , , ..., . Let be the point where the line of natural slope from cuts the arc, . On laying off the chords, , , ..., equal to the chords, , , ..., respectively, it will follow that , , ..., will make angles, , below the normals to the planes, , , ..., respectively. ”

“ Having the horizontal sections given, we may obtain a complete knowledge of the direction in which the ground slopes by drawing perpendicular to them any number of lines of greatest descent; the degree of declivity is expressed by purely conventional means. The means adopted for this purpose are of two kinds. One depends upon the principle of vertical illumination, in which the maximum quantity of light is reflected upwards to the eye by a horizontal surface, and a minimum by a surface inclined 45° to the horizon. ”

“ Should any part of a plan be a circle, it will when projected on to an oblique plane be an ellipse.
Thus, take any one round piece of wood, cut one end off square to its side, this end will be a true circle. Cut the other end to any angle, which will then be an ellipse, and when the piece is stood on end will be vertical over the circle, its plan. Fig. 1 shows this. And it will be seen, no matter what angle the oblique plane may be, the minor axis never changes, it is always the same length as on plan, as are all lines parallel to it ”

“ If a point lies in the horizontal plane, its elevation lies in the axis and the plan coincides with the point itself. If a point lies in the vertical plane, its plan lies in the axis and the elevation coincides with the point itself. ”

“ It will be seen that if a horizontal line is drawn from the foot of one of these figures, say G, to point O on the edge of the incline, then dropped vertically to o·, then again carried on to o·· where the other figure g is, we find it is the same height and also that the other vanishing scale is the same width at that distance, so that we can work from either one or the other. In the event of the rising ground being uneven we can make use of the scale on the level plane. ”

“ For working sections the horizontal scale adopted is usually three or four chains to the inch, and the vertical scale 30 or 40 feet to the inch. A working section should show the level of the ground, the level of the proposed work, and the height of embankment or depth of cutting at every point of the ground where the level has been taken, these quantities being found by calculation, not by measurement on the paper. ”

“ But there is nothing to prevent us from assuming that between the verticals and horizontals we have mentioned there lie an indefinite number of other similar lines, to which intermediary fractional numbers will be assigned. Henceforth every point of the plane can be regarded as defined by the intersection of some particular horizontal and some particular vertical. ”

“ 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. ”

“ Try it and see if it is not true that lines in one direction never indicate any other direction; the vertical can never be mistaken for the horizontal, or the reverse. For the representation of a level plain, make simple strokes in the horizontal direction with the side of the crayon, and to represent a vertical surface as a cliff, make a stroke in the vertical direction with the same broad side of the crayon. Oblique surfaces, as slopes, are to be drawn with oblique strokes, and curved surfaces like rounded hills, represented by continuous upward and downward strokes. (See Fig. 1.) ”

“ If the base and the difference of the base angles of a triangle be given, the rectangle contained by the perpendiculars from the vertex on two lines through the middle point of the base, parallel to the internal and external bisectors of the vertical angle, is constant. ”

“ A space that is arched looks wider than it actually is, for the eye unconsciously follows the lines of the arch, and a distance or width effect is the result. The same space treated with a straight line is quickly bridged. The same space treated with lines that come to an angle looks narrower, for the reason that the eye becomes focused by the apex of the angle, and a height effect, not a breadth effect, is the result. ”

“ Straighten one face with fore plane, jack plane, or jointer, and smooth with smoothing plane. This face, called the working face, becomes the basis from which all the other sides are squared.
Second. Plane one of the adjoining edges and make square with the working face. This edge, known as the joint edge, must be thoroughly tested throughout its entire length with the try-square, and must be square with the working face at every point.
Third. Set marking gauge at required width and with gauge block against the joint edge, gauge a fine line on working face. ”

“ That if we now make one of the two horizontal axes less and the other greater than the vertical axis, the instantaneous pole will separate from the axis of the instrument, and the axis will incline more and more till the spinning can no longer go on, on account of the obliquity. ”

“ Contour and surface enrichment should never appear to compete for attention in the same design.
Fobs
Plate 54 shows flat planes, the service of which suggests vertical inceptive axes. Figure 380 is noted as an exception to this vertical inceptive axis as it possesses a vertical primary mass but with radial inceptive axes. The interesting manner by which the dynamic leaves of the outer border transmit their movement to the inner border, which in turn leads toward the point of concentration, is worthy of attention. ”

“ 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. ”

“ Three straight lines not in the same plane meet at a point, and a plane cuts these straight lines at equal distances from the point of intersection: shew that the perpendicular from that point on the plane will meet it at the centre of the circle described about the triangle formed by the portion of the plane intercepted by the planes passing through the straight lines.
424. Give a geometrical construction for drawing a straight line which shall be equally inclined to three straight lines meeting at a point. ”