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:s 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 2 1/2 feet, the pillar's height will be 1 1/2 foot, and so on.

Fig. 53.

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, 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.