Arrange the three distances of the object on your paper, as in Fig. 4.[10] Join _CT_, cutting _GH_ in _Q_. From _Q_ let fall the vertical line _QP′_. Join _PT_, cutting _QP_ in _P′_. _P′_ is the point required. If the point _P_ is _above_ the eye of the observer instead of below it, _CP_ is to be measured upwards from _C_, and _QP′_ drawn upwards from _Q_. The construction will be as in Fig. 5. [Illustration: Fig. 5.] And if the point _P_ is to the right instead of the left of the observer, _DC_ is to be measured to the right instead of the left. The figures 4. and 5., looked at in a mirror, will show the construction of each, on that supposition. Now read very carefully the examples and notes to this problem in Appendix I. (page 69). I have put them in the Appendix in order to keep the sequence of following problems more clearly traceable here in the text; but you must read the first Appendix before going on. [8] More accurately, “To fix on the plane of the picture the apparent position of a point given in actual position.” In the headings of all the following problems the words “on the plane of the picture” are to be understood after the words “to draw.” The plane of the picture means a surface extended indefinitely in the direction of the picture. [9] The sentence within brackets will not be repeated in succeeding statements of problems. It is always to be understood. [10] In order to be able to do this, you must assume the distances to be small; as in the case of some object on the table: how large distances are to be treated you will see presently; the mathematical principle, being the same for all, is best illustrated first on a small scale. Suppose, for instance, _P_ to be the corner of a book on the table, seven inches below the eye, five inches to the left of it, and a foot and a half in advance of it, and that you mean to hold your finished drawing at six inches from the eye; then _TS_ will be six inches, _TD_ a foot and a half, _DC_ five inches, and _CP_ seven. PROBLEM II. TO DRAW A RIGHT LINE BETWEEN TWO GIVEN POINTS. [Illustration: Fig. 6.] Let _AB_, Fig. 6., be the given right line, joining the given points _A_ and _B_. Let the direct, lateral, and vertical distances of the point _A_ be _TD_, _DC_, and _CA_. Let the direct, lateral, and vertical distances of the point _B_ be _TD′_, _DC′_, and _C′B_. Then, by Problem I., the position of the point _A_ on the plane of the picture is _a_. And similarly, the position of the point _B_ on the plane of the picture is _b_. Join _ab_. Then _ab_ is the line required. COROLLARY I. If the line _AB_ is in a plane parallel to that of the picture, one end of the line _AB_ must be at the same direct distance from the eye of the observer as the other. Therefore, in that case, _DT_ is equal to _D′T_.