[30] As in algebraic science, much depends, in complicated perspective, on the student’s ready invention of expedients, and on his quick sight of the shortest way in which the solution may be accomplished, when there are several ways. [31] The greatest masters are also fond of parallel perspective, that is to say, of having one side of their buildings fronting them full, and therefore parallel to the picture plane, while the other side vanishes to the sight-point. This is almost always done in figure backgrounds, securing simple and balanced lines. 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. [Illustration: Fig. 51.] Let _AB_, Fig. 51., be our sheet of paper, some 3 feet wide. Make _ST_ equal to 1½ foot. Draw the line of sight through _S_. Produce _TS_, and make _DS_ equal to 2 feet, therefore _TD_ equal to 3½ feet. Draw _DC_, equal to 2 feet; _CP_, equal to 4 feet. Join _TC_ (cutting the sight-line in _Q_) and _TP_. Let fall the vertical _QP′_, 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 _TD_, 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, _QP_, 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 _OQP_, Fig. 52., then the image of its side, _QP_, being, like every other magnitude, limited on the glass _AB_ 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 _QP_ as _TS_ is to _TP_. But if we let fall a vertical _QD_ from _Q_, so as to get the vertical height of the pyramid, then it is true that _Q′S_ is to _QD_ as _TS_ is to _TD_. [Illustration: Fig. 52.]