Appendix
I.
PRACTICE AND OBSERVATIONS.
II.
DEMONSTRATIONS.
I Practice and Observations on the Preceding Problems
Problem I.
An example will be necessary to make this problem clear to the general student.
The nearest corner of a piece of pattern on the carpet is 4½ feet beneath the eye, 2 feet to our right and 3½ feet in direct distance from us. We intend to make a drawing of the pattern which shall be seen properly when held 1½ foot from the eye. It is required to fix the position of the corner of the piece of pattern.
Let A B, Fig. 51., be our sheet of paper, some 3 feet wide. Make S T equal to 1½ foot. Draw the line of sight through S. Produce T S, and make D S equal to 2 feet, therefore T D equal to 3½ feet. Draw D C, equal to 2 feet; C P, equal to 4 feet. Join T C (cutting the sight-line in Q) and T P.
Let fall the vertical Q P′, then P′ is the point required.
If the lines, as in the figure, fall outside of your sheet of paper, in order to draw them, it is necessary to attach other sheets of paper to its edges. This is inconvenient, but must be done at first that you may see your way clearly; and sometimes afterwards, though there are expedients for doing without such extension in fast sketching.
It is evident, however, that no extension of surface could be of any use to us, if the distance T D, instead of being 3½ feet, were 100 feet, or a mile, as it might easily be in a landscape.
It is necessary, therefore, to obtain some other means of construction; to do which we must examine the principle of the problem.
In the analysis of Fig. 2., in the introductory remarks, I used the word “height” only of the tower, Q P, because it was only to its vertical height that the law deduced from the figure could be applied. For suppose it had been a pyramid, as O Q P, Fig. 52., then the image of its side, Q P, being, like every other magnitude, limited on the glass A B by the lines coming from its extremities, would appear only of the length Q′ S; and it is not true that Q′ S is to Q P as T S is to T P. But if we let fall a vertical Q D from Q, so as to get the vertical height of the pyramid, then it is true that Q′ S is to Q D as T S is to T D.
Supposing this figure represented, not a pyramid, but a triangle on the ground, and that Q D and Q P are horizontal lines, expressing lateral distance from the line T D, still the rule would be false for Q P and true for Q D. And, similarly, it is true for all lines which are parallel, like Q D, to the plane of the picture A B, and false for all lines which are inclined to it at an angle.
Hence generally. Let P Q (Fig. 2. in Introduction, p. 6) be any magnitude parallel to the plane of the picture; and P′ Q′ its image on the picture.
Then always the formula is true which you learned in the Introduction: P′ Q′ is to P Q as S T is to D T.
Now the magnitude P dash Q dash in this formula I call the “SIGHT-MAGNITUDE” of the line P Q. The student must fix this term, and the meaning of it, well in his mind. The “sight-magnitude” of a line is the magnitude which bears to the real line the same proportion that the distance of the picture bears to the distance of the object. Thus, if a tower be a hundred feet high, and a hundred yards off; and the picture, or piece of glass, is one yard from the spectator, between him and the tower; the distance of picture being then to distance of tower as 1 to 100, the sight-magnitude of the tower’s height will be as 1 to 100; that is to say, one foot. If the tower is two hundred yards distant, the sight-magnitude of its height will be half a foot, and so on.
But farther. It is constantly necessary, in perspective operations, to measure the other dimensions of objects by the sight-magnitude of their vertical lines. Thus, if the tower, which is a hundred feet high, is square, and twenty-five feet broad on each side; if the sight-magnitude of the height is one foot, the measurement of the side, reduced to the same scale, will be the hundredth part of twenty-five feet, or three inches: and, accordingly, I use in this treatise the term “sight-magnitude” indiscriminately for all lines reduced in the same proportion as the vertical lines of the object. If I tell you to find the “sight-magnitude” of any line, I mean, always, find the magnitude which bears to that line the proportion of S T to D T; or, in simpler terms, reduce the line to the scale which you have fixed by the first determination of the length S T.
Therefore, you must learn to draw quickly to scale before you do anything else; for all the measurements of your object must be reduced to the scale fixed by S T before you can use them in your diagram. If the object is fifty feet from you, and your paper one foot, all the lines of the object must be reduced to a scale of one fiftieth before you can use them; if the object is two thousand feet from you, and your paper one foot, all your lines must be reduced to the scale of one two-thousandth before you can use them, and so on. Only in ultimate practice, the reduction never need be tiresome, for, in the case of large distances, accuracy is never required. If a building is three or four miles distant, a hairbreadth of accidental variation in a touch makes a difference of ten or twenty feet in height or breadth, if estimated by accurate perspective law. Hence it is never attempted to apply measurements with precision at such distances. Measurements are only required within distances of, at the most, two or three hundred feet. Thus it may be necessary to represent a cathedral nave precisely as seen from a spot seventy feet in front of a given pillar; but we shall hardly be required to draw a cathedral three miles distant precisely as seen from seventy feet in advance of a given milestone. Of course, if such a thing be required, it can be done; only the reductions are somewhat long and complicated: in ordinary cases it is easy to assume the distance S T so as to get at the reduced dimensions in a moment. Thus, let the pillar of the nave, in the case supposed, be 42 feet high, and we are required to stand 70 feet from it: assume S T to be equal to 5 feet. Then, as 5 is to 70 so will the sight-magnitude required be to 42; that is to say, the sight-magnitude of the pillar’s height will be 3 feet. If we make S T equal to 2½ feet, the pillar’s height will be 1½ foot, and so on.
And for fine divisions into irregular parts which cannot be measured, the ninth and tenth problems of the sixth book of Euclid will serve you: the following construction is, however, I think, more practically convenient:—
The line A B (Fig. 53.) is divided by given points, a, b, c, into a given number of irregularly unequal parts; it is required to divide any other line, C D, into an equal number of parts, bearing to each other the same proportions as the parts of A B, and arranged in the same order.
Draw the two lines parallel to each other, as in the figure.
Join A C and B D, and produce the lines A C, B D, till they meet in P.
Join a P, b P, c P, cutting c D in f, g, h.
Then the line C D is divided as required, in f, g, h.
In the figure the lines A B and C D are accidentally perpendicular to A P. There is no need for their being so.
Now, to return to our first problem.
The construction given in the figure is only the quickest mathematical way of obtaining, on the picture, the sight-magnitudes of D C and P C, which are both magnitudes parallel with the picture plane. But if these magnitudes are too great to be thus put on the paper, you have only to obtain the reduction by scale. Thus, if T S be one foot, T D eighty feet, D C forty feet, and C P ninety feet, the distance Q S must be made equal to one eightieth of D C, or half a foot; and the distance Q P′, one eightieth of C P, or one eightieth of ninety feet; that is to say, nine eighths of a foot, or thirteen and a half inches. The lines C T and P T are thus practically useless, it being only necessary to measure Q S and Q P, on your paper, of the due sight-magnitudes. But the mathematical construction, given in Problem I., is the basis of all succeeding problems, and, if it is once thoroughly understood and practiced (it can only be thoroughly understood by practice), all the other problems will follow easily.
Lastly. Observe that any perspective operation whatever may be performed with reduced dimensions of every line employed, so as to bring it conveniently within the limits of your paper. When the required figure is thus constructed on a small scale, you have only to enlarge it accurately in the same proportion in which you reduced the lines of construction, and you will have the figure constructed in perspective on the scale required for use.
Problem IX
The drawing of most buildings occurring in ordinary practice will resolve itself into applications of this problem. In general, any house, or block of houses, presents itself under the main conditions assumed here in Fig. 54. There will be an angle or corner somewhere near the spectator, as A B; and the level of the eye will usually be above the base of the building, of which, therefore, the horizontal upper lines will slope down to the vanishing-points, and the base lines rise to them. The following practical directions will, however, meet nearly all cases:—
Let A B, Fig. 54., be any important vertical line in the block of buildings; if it is the side of a street, you may fix upon such a line at the division between two houses. If its real height, distance, etc., are given, you will proceed with the accurate construction of the problem; but usually you will neither know, nor care, exactly how high the building is, or how far off. In such case draw the line A B, as nearly as you can guess, about the part of the picture it ought to occupy, and on such a scale as you choose. Divide it into any convenient number of equal parts, according to the height you presume it to be. If you suppose it to be twenty feet high, you may divide it into twenty parts, and let each part stand for a foot; if thirty feet high, you may divide it into ten parts, and let each part stand for three feet; if seventy feet high, into fourteen parts, and let each part stand for five feet; and so on, avoiding thus very minute divisions till you come to details. Then observe how high your eye reaches upon this vertical line; suppose, for instance, that it is thirty feet high and divided into ten parts, and you are standing so as to raise your head to about six feet above its base, then the sight-line may be drawn, as in the figure, through the second division from the ground. If you are standing above the house, draw the sight-line above B; if below the house, below A; at such height or depth as you suppose may be accurate (a yard or two more or less matters little at ordinary distances, while at great distances perspective rules become nearly useless, the eye serving you better than the necessarily imperfect calculation). Then fix your sight-point and station-point, the latter with proper reference to the scale of the line A B. As you cannot, in all probability, ascertain the exact direction of the line A V or B V, draw the slope B V as it appears to you, cutting the sight-line in V. Thus having fixed one vanishing-point, the other, and the dividing-points, must be accurately found by rule; for, as before stated, whether your entire group of points (vanishing and dividing) falls a little more or less to the right or left of S does not signify, but the relation of the points to each other does signify. Then draw the measuring-line B G, either through A or B, choosing always the steeper slope of the two; divide the measuring-line into parts of the same length as those used on A B, and let them stand for the same magnitudes. Thus, suppose there are two rows of windows in the house front, each window six feet high by three wide, and separated by intervals of three feet, both between window and window and between tier and tier; each of the divisions here standing for three feet, the lines drawn from B G to the dividing-point D fix the lateral dimensions, and the divisions on A B the vertical ones. For other magnitudes it would be necessary to subdivide the parts on the measuring-line, or on A B, as required. The lines which regulate the inner sides or returns of the windows (a, b, c, etc.) of course are drawn to the vanishing-point of B F (the other side of the house), if F B V represents a right angle; if not, their own vanishing-point must be found separately for these returns. But see Practice on Problem XI.
Interior angles, such as E B C, Fig. 55. (suppose the corner of a room), are to be treated in the same way, each side of the room having its measurements separately carried to it from the measuring-line. It may sometimes happen in such cases that we have to carry the measurement up from the corner B, and that the sight-magnitudes are given us from the length of the line A B. For instance, suppose the room is eighteen feet high, and therefore A B is eighteen feet; and we have to lay off lengths of six feet on the top of the room wall, B C. Find D, the dividing-point of B C. Draw a measuring-line, B F, from B; and another, g C, anywhere above. On B F lay off B G equal to one third of A B, or six feet; and draw from D, through G and B, the lines G g, B b, to the upper measuring-line. Then g b is six feet on that measuring-line. Make b c, c h, etc., equal to b g; and draw c e, h f, etc., to D, cutting B C in e and f, which mark the required lengths of six feet each at the top of the wall.
Problem X.
This is one of the most important foundational problems in perspective, and it is necessary that the student should entirely familiarize himself with its conditions.
In order to do so, he must first observe these general relations of magnitude in any pyramid on a square base.
Let A G H′, Fig. 56., be any pyramid on a square base.
The best terms in which its magnitude can be given, are the length of one side of its base, A H, and its vertical altitude (C D in Fig. 25.); for, knowing these, we know all the other magnitudes. But these are not the terms in which its size will be usually ascertainable. Generally, we shall have given us, and be able to ascertain by measurement, one side of its base A H, and either A G the length of one of the lines of its angles, or B G (or B′ G) the length of a line drawn from its vertex, G, to the middle of the side of its base. In measuring a real pyramid, A G will usually be the line most easily found; but in many architectural problems B G is given, or is most easily ascertainable.
Observe therefore this general construction.
Let A B D E, Fig. 57., be the square base of any pyramid.
Draw its diagonals, A E, B D, cutting each other in its center, C.
Bisect any side, A B, in F.
From F erect vertical F G.
Produce F B to H, and make F H equal to A C.
Now if the vertical altitude of the pyramid (C D in Fig. 25.) be given, make F G equal to this vertical altitude.
Then G B and G H are the true magnitudes of G B and G H in Fig. 56.
If G B is given, and not the vertical altitude, with center B, and distance G B, describe circle cutting F G in G, and F G is the vertical altitude.
If G H is given, describe the circle from H, with distance G H, and it will similarly cut F G in G.
It is especially necessary for the student to examine this construction thoroughly, because in many complicated forms of ornaments, capitals of columns, etc., the lines B G and G H become the limits or bases of curves, which are elongated on the longer (or angle) profile G H, and shortened on the shorter (or lateral) profile B G. We will take a simple instance, but must previously note another construction.
It is often necessary, when pyramids are the roots of some ornamental form, to divide them horizontally at a given vertical height. The shortest way of doing so is in general the following.
Let A E C, Fig. 58., be any pyramid on a square base A B C, and A D C the square pillar used in its construction.
Of the diagonals, F E, D E, choose the shortest (in this case D E), and produce it to cut the sight-line in V.
Therefore V is the vanishing-point of D E.
Divide D B, as may be required, into the sight-magnitudes of the given vertical heights at which the pyramid is to be divided.
From the points of division, 1, 2, 3, etc., draw to the vanishing-point V. The lines so drawn cut the angle line of the pyramid, B E, at the required elevations. Thus, in the figure, it is required to draw a horizontal black band on the pyramid at three fifths of its height, and in breadth one twentieth of its height. The line B D is divided into five parts, of which three are counted from B upwards. Then the line drawn to V marks the base of the black band. Then one fourth of one of the five parts is measured, which similarly gives the breadth of the band. The terminal lines of the band are then drawn on the sides of the pyramid parallel to A B (or to its vanishing-point if it has one), and to the vanishing-point of B C.
Erect the vertical D B and join G B and D G (G being the apex of pyramid).
Find the vanishing-point of D G, and use D B for division, carrying the measurements to the line G B.
In Fig. 59., if we join A D and D C, A D C is the vertical profile of the whole pyramid, and B D C of the half pyramid, corresponding to F G B in Fig. 57.
We may now proceed to an architectural example.
Let A H, Fig. 60., be the vertical profile of the capital of a pillar, A B the semi-diameter of its head or abacus, and F D the semi-diameter of its shaft.
Let the shaft be circular, and the abacus square, down to the level E.
Join B D, E F, and produce them to meet in G.
Therefore E C G is the semi-profile of a reversed pyramid containing the capital.
Draw F m, F n, to the vanishing-points of A H and A K. Then F n and F m are horizontal lines across the pyramid at the level F, forming at that level two sides of a square.
Complete the square, and within it inscribe a circle, as in Fig. 62., which is left unlettered that its construction may be clear. At the extremities of this draw vertical lines, which will be the sides of the shaft in its right place. It will be found to be somewhat smaller in diameter than the entire shaft in Fig. 60., because at the center of the square it is more distant than the nearest edge of the square abacus. The curves of the capital may then be drawn approximately by the eye. They are not quite accurate in Fig. 62., there being a subtlety in their junction with the shaft which could not be shown on so small a scale without confusing the student; the curve on the left springing from a point a little way round the circle behind the shaft, and that on the right from a point on this side of the circle a little way within the edge of the shaft. But for their more accurate construction see Notes on Problem XIV.
Problem XI
It is seldom that any complicated curve, except occasionally a spiral, needs to be drawn in perspective; but the student will do well to practice for some time any fantastic shapes which he can find drawn on flat surfaces, as on wall-papers, carpets, etc., in order to accustom himself to the strange and great changes which perspective causes in them.
The curves most required in architectural drawing, after the circle, are those of pointed arches; in which, however, all that will be generally needed is to fix the apex, and two points in the sides. Thus if we have to draw a range of pointed arches, such as A P B, Fig. 63., draw the measured arch to its sight-magnitude first neatly in a rectangle, A B C D; then draw the diagonals A D and B C; where they cut the curve draw a horizontal line (as at the level E in the figure), and carry it along the range to the vanishing-point, fixing the points where the arches cut their diagonals all along. If the arch is cusped, a line should be drawn, at F to mark the height of the cusps, and verticals raised at G and H, to determine the interval between them. Any other points may be similarly determined, but these will usually be enough. Figure 63. shows the perspective construction of a square niche of good Veronese Gothic, with an uncusped arch of similar size and curve beyond.
In Fig. 64. the more distant arch only is lettered, as the construction of the nearest explains itself more clearly to the eye without letters. The more distant arch shows the general construction for all arches seen underneath, as of bridges, cathedral aisles, etc. The rectangle A B C D is first drawn to contain the outside arch; then the depth of the arch, A a, is determined by the measuring-line, and the rectangle, a b c d, drawn for the inner arch.
A a, B b, etc., go to one vanishing-point; A B, a b, etc., to the opposite one.
In the nearer arch another narrow rectangle is drawn to determine the cusp. The parts which would actually come into sight are slightly shaded.
Problem XIV
Several exercises will be required on this important problem.
I. It is required to draw a circular flat-bottomed dish narrower at the bottom than the top; the vertical depth being given, and the diameter at the top and bottom.
Let a b, Fig. 65., be the diameter of the bottom, a c the diameter of the top, and a d its vertical depth.
Take A D in position equal to a c.
On A D draw the square A B C D, and inscribe in it a circle.
Therefore, the circle so inscribed has the diameter of the top of the dish.
From A and D let fall verticals, A E, D H, each equal to a d.
Join E H, and describe square E F G H, which accordingly will be equal to the square A B C D, and be at the depth a d beneath it.
Within the square E F G H describe a square I K, whose diameter shall be equal to a b.
Describe a circle within the square I K. Therefore the circle so inscribed has its diameter equal to a b; and it is in the center of the square E F G H, which is vertically beneath the square A B C D.
Therefore the circle in the square I K represents the bottom of the dish.
Now the two circles thus drawn will either intersect one another, or they will not.
If they intersect one another, as in the figure, and they are below the eye, part of the bottom of the dish is seen within it.
To avoid confusion, let us take then two intersecting circles without the inclosing squares, as in Fig. 66.
Draw right lines, a b, c d, touching both circles externally. Then the parts of these lines which connect the circles are the sides of the dish. They are drawn in Fig. 65. without any prolongations, but the best way to construct them is as in Fig. 66.
If the circles do not intersect each other, the smaller must either be within the larger or not within it.
If within the larger, the whole of the bottom of the dish is seen from above, Fig. 67. a.
If the smaller circle is not within the larger, none of the bottom is seen inside the dish, b.
If the circles are above instead of beneath the eye, the bottom of the dish is seen beneath it, c.
If one circle is above and another beneath the eye, neither the bottom nor top of the dish is seen, d. Unless the object be very large, the circles in this case will have little apparent curvature.
II. The preceding problem is simple, because the lines of the profile of the object (a b and c d, Fig. 66.) are straight. But if these lines of profile are curved, the problem becomes much more complex: once mastered, however, it leaves no farther difficulty in perspective.
Let it be required to draw a flattish circular cup or vase, with a given curve of profile.
The basis of construction is given in Fig. 68., half of it only being drawn, in order that the eye may seize its lines easily.
Two squares (of the required size) are first drawn, one above the other, with a given vertical interval, A C, between them, and each is divided into eight parts by its diameters and diagonals. In these squares two circles are drawn; which are, therefore, of equal size, and one above the other. Two smaller circles, also of equal size, are drawn within these larger circles in the construction of the present problem; more may be necessary in some, none at all in others.
It will be seen that the portions of the diagonals and diameters of squares which are cut off between the circles represent radiating planes, occupying the position of the spokes of a wheel.
Now let the line A E B, Fig. 69., be the profile of the vase or cup to be drawn.
Inclose it in the rectangle C D, and if any portion of it is not curved, as A E, cut off the curved portion by the vertical line E F, so as to include it in the smaller rectangle F D.
The planes which occupy the position of the wheel spokes will then each represent a rectangle of the size of F D. The construction is shown by the dotted lines in Fig. 69.; c being the center of the uppermost circle.
Within each of the smaller rectangles between the circles, draw the curve E B in perspective, as in Fig. 69.
Draw the curve x y, touching and inclosing the curves in the rectangles, and meeting the upper circle at y.32
Then x y is the contour of the surface of the cup, and the upper circle is its lip.
If the line x y is long, it may be necessary to draw other rectangles between the eight principal ones; and, if the curve of profile A B is complex or retorted, there may be several lines corresponding to X Y, inclosing the successive waves of the profile; and the outer curve will then be an undulating or broken one.
III. All branched ornamentation, forms of flowers, capitals of columns, machicolations of round towers, and other such arrangements of radiating curve, are resolvable by this problem, using more or fewer interior circles according to the conditions of the curves. Fig. 70. is an example of the construction of a circular group of eight trefoils with curved stems. One outer or limiting circle is drawn within the square E D C F, and the extremities of the trefoils touch it at the extremities of its diagonals and diameters. A smaller circle is at the vertical distance B C below the larger, and A is the angle of the square within which the smaller circle is drawn; but the square is not given, to avoid confusion. The stems of the trefoils form drooping curves, arranged on the diagonals and diameters of the smaller circle, which are dotted. But no perspective laws will do work of this intricate kind so well as the hand and eye of a painter.
IV. There is one common construction, however, in which, singularly, the hand and eye of the painter almost always fail, and that is the fillet of any ordinary capital or base of a circular pillar (or any similar form). It is rarely necessary in practice to draw such minor details in perspective; yet the perspective laws which regulate them should be understood, else the eye does not see their contours rightly until it is very highly cultivated.
Fig. 71. will show the law with sufficient clearness; it represents the perspective construction of a fillet whose profile is a semicircle, such as F H in Fig. 60., seen above the eye. Only half the pillar with half the fillet is drawn, to avoid confusion.
P Q the thickness of the fillet, sight-magnitude at the shaft’s center.
Round P a horizontal semicircle is drawn on the diameter of the shaft a b.
Round Q another horizontal semicircle is drawn on diameter c d.
These two semicircles are the upper and lower edges of the fillet.
Then diagonals and diameters are drawn as in Fig. 68., and, at their extremities, semicircles in perspective, as in Fig. 69.
The letters A, B, C, D, and E, indicate the upper and exterior angles of the rectangles in which these semicircles are to be drawn; but the inner vertical line is not dotted in the rectangle at C, as it would have confused itself with other lines.
Then the visible contour of the fillet is the line which incloses and touches33 all the semicircles. It disappears behind the shaft at the point H, but I have drawn it through to the opposite extremity of the diameter at d.
Turned upside down the figure shows the construction of a basic fillet.
The capital of a Greek Doric pillar should be drawn frequently for exercise on this fourteenth problem, the curve of its echinus being exquisitely subtle, while the general contour is simple.
Problem XVI
It is often possible to shorten other perspective operations considerably, by finding the vanishing-points of the inclined lines of the object. Thus, in drawing the gabled roof in Fig. 43., if the gable A Y C be drawn in perspective, and the vanishing-point of A Y determined, it is not necessary to draw the two sides of the rectangle, A′ D′ and D′ B′, in order to determine the point Y′; but merely to draw Y Y′ to the vanishing-point of A A′ and A′ Y′ to the vanishing-point of A Y, meeting in Y′, the point required.
Again, if there be a series of gables, or other figures produced by parallel inclined lines, and retiring to the point V, as in Fig. 72.,34 it is not necessary to draw each separately, but merely to determine their breadths on the line A V, and draw the slopes of each to their vanishing-points, as shown in Fig. 72. Or if the gables are equal in height, and a line be drawn from Y to V, the construction resolves itself into a zigzag drawn alternately to P and Q, between the lines Y V and A V.
The student must be very cautious, in finding the vanishing-points of inclined lines, to notice their relations to the horizontals beneath them, else he may easily mistake the horizontal to which they belong.
Thus, let A B C D, Fig. 73., be a rectangular inclined plane, and let it be required to find the vanishing-point of its diagonal B D.
Find V, the vanishing-point of A D and B C.
Draw A E to the opposite vanishing-point, so that D A E may represent a right angle.
Let fall from B the vertical B E, cutting A E in E.
Join E D, and produce it to cut the sight-line in V′.
So that if we now let fall the vertical V′ P from V′, and produce B D to cut V′ P in P, the point P will be the vanishing-point of B D, and of all lines parallel to it.35
Problem XVIII
Before examining the last three problems it is necessary that you should understand accurately what is meant by the position of an inclined plane.
Cut a piece of strong white pasteboard into any irregular shape, and dip it in a sloped position into water. However you hold it, the edge of the water, of course, will always draw a horizontal line across its surface. The direction of this horizontal line is the direction of the inclined plane. (In beds of rock geologists call it their “strike.”)
Next, draw a semicircle on the piece of pasteboard; draw its diameter, A B, Fig. 74., and a vertical line from its center, C D; and draw some other lines, C E, C F, etc., from the center to any points in the circumference.
Now dip the piece of pasteboard again into water, and, holding it at any inclination and in any direction you choose, bring the surface of the water to the line A B. Then the line C D will be the most steeply inclined of all the lines drawn to the circumference of the circle; G C and H C will be less steep; and E C and F C less steep still. The nearer the lines to C D, the steeper they will be; and the nearer to A B, the more nearly horizontal.
To fix the position of an inclined plane, therefore, is to determine the direction of any two lines in the plane, A B and C D, of which one shall be horizontal and the other at right angles to it. Then any lines drawn in the inclined plane, parallel to A B, will be horizontal; and lines drawn parallel to C D will be as steep as C D, and are spoken of in the text as the “steepest lines” in the plane.
But farther, whatever the direction of a plane may be, if it be extended indefinitely, it will be terminated, to the eye of the observer, by a boundary line, which, in a horizontal plane, is horizontal (coinciding nearly with the visible horizon);—in a vertical plane, is vertical;—and, in an inclined plane, is inclined.
This line is properly, in each case, called the “sight-line” of such plane; but it is only properly called the “horizon” in the case of a horizontal plane: and I have preferred using always the term “sight-line,” not only because more comprehensive, but more accurate; for though the curvature of the earth’s surface is so slight that practically its visible limit always coincides with the sight-line of a horizontal plane, it does not mathematically coincide with it, and the two lines ought not to be considered as theoretically identical, though they are so in practice.
It is evident that all vanishing-points of lines in any plane must be found on its sight-line, and, therefore, that the sight-line of any plane may be found by joining any two of such vanishing-points. Hence the construction of Problem XVIII.
Notes
Then by construction (Problem X.) B D and A F are both of the vertical height of the pyramid.
If it happens that the vanishing-points of the diagonals are awkwardly placed for use, bisect the nearest base line of the pyramid in B, as in Fig. 59.
Construct this pyramid, with the square of the abacus, in the required perspective, as in Fig. 61.; making A E equal to A E in Fig. 60., and A K, the side of the square, equal to twice A B in Fig. 60. Make E G equal to C G, and E D equal to C D. Draw D F to the vanishing-point of the diagonal D V (the figure is too small to include this vanishing-point), and F is the level of the point F in Fig. 60., on the side of the pyramid.
Draw the rectangle A C B D in position, and upon it construct two squares, as they are constructed on the rectangle A C D in Fig. 68.; and complete the construction of Fig. 68., making the radius of its large outer circles equal to A D, and of its small inner circles equal to A E.
32 This point coincides in the figure with the extremity of the horizontal diameter, but only accidentally.
33 The engraving is a little inaccurate; the inclosing line should touch the dotted semicircles at A and B. The student should draw it on a large scale.
Then, since the point E is vertically under the point B, the horizontal line E D is vertically under the inclined line B D.
34 The diagram is inaccurately cut. Y V should be a right line.
35 The student may perhaps understand this construction better by completing the rectangle A D F E, drawing D F to the vanishing-point of A E, and E F to V. The whole figure, B F, may then be conceived as representing half the gable roof of a house, A F the rectangle of its base, and A C the rectangle of its sloping side.
In nearly all picturesque buildings, especially on the Continent, the slopes of gables are much varied (frequently unequal on the two sides), and the vanishing-points of their inclined lines become very important, if accuracy is required in the intersections of tiling, sides of dormer windows, etc.
Obviously, also, irregular triangles and polygons in vertical planes may be more easily constructed by finding the vanishing-points of their sides, than by the construction given in the corollary to Problem IX.; and if such triangles or polygons have others concentrically inscribed within them, as often in Byzantine mosaics, etc., the use of the vanishing-points will become essential.
When, therefore, the line A B is horizontal (or marks the water surface), its direction is the direction of the inclined plane, and the inclination of the line D C is the inclination of the inclined plane. In beds of rock geologists call the inclination of the line D C their “dip.”