Fig. 52.
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 [p71] ]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 [p72] ]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