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 _ST_ 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 _ST_ 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 _ST_ 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 _ST_ 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 _AB_ (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, _CD_, into an equal number of parts, bearing to each other the same proportions as the parts of _AB_, and arranged in the same order. Draw the two lines parallel to each other, as in the figure. Join _AC_ and _BD_, and produce the lines _AC_, _BD_, till they meet in _P_. Join _aP_, _bP_, _cP_, cutting _cD_ in _f_, _g_, _h_. Then the line _CD_ is divided as required, in _f_, _g_, _h_. In the figure the lines _AB_ and _CD_ are accidentally perpendicular to _AP_. There is no need for their being so. [Illustration: Fig. 53.] Now, to return to our first problem.