I.

PRACTICE AND OBSERVATIONS ON THE PRECED

ING 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 4 feet beneath the eye, 2 feet to our right and 3 feet in direct distance from us. We intend to make a drawing of the pattern which shall be seen properly when held 1 foot from the eye. It is required to fix the

position of the corner of the piece of pattern.

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 3 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 mag

nitude, 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 ver tical 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 AB, and false for all lines which are inclined to it at an angle.

Hence generally. Let P Q (Fig. 2. in Introduction, p. 11) 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 "sightmagnitude" 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 opera

tions, to measure the other dimensions of objects by the sightmagnitude 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 "sightmagnitude" of any line, I mean, always, find the magnitude which bears to that line the proportion of s T to DT; or, in simpler terms, reduce the line to the scale which you have fixe 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 sr equal to 2 feet, the pillar's height will be 1 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