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.
Fig. 51.
Let A B, [Fig. 51.], be our sheet of paper, some 3 feet wide. Make S T equal to 1½ foot. Draw the line of sight through S. Produce T S, and make D S equal to 2 feet, therefore T D equal to 3½ feet. Draw D C, equal to 2 feet; C P, equal to 4 feet. Join T C (cutting the sight-line in Q) and T P.
Let fall the vertical Q P′, 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 [p70] ]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 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, 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 magnitude, 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 vertical 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.
