It is true that in drawing landscapes from nature the sizes and distances of the objects cannot be accurately known. When, however, we know how to draw them rightly, if their size were given, we have only to _assume a rational approximation_ to their size, and the resulting drawing will be true enough for all intents and purposes. It does not in the least matter that we represent a distant cottage as eighteen feet long, when it is in reality only seventeen; but it matters much that we do not represent it as eighty feet long, as we easily might if we had not been accustomed to draw from measurement. Therefore, in all the following problems the measurement of the object is given. The student must observe, however, that in order to bring the diagrams into convenient compass, the measurements assumed are generally very different from any likely to occur in practice. Thus, in Fig. 3., the distance _DS_ would be probably in practice half a mile or a mile, and the distance _TS_, from the eye of the observer to the paper, only two or three feet. The mathematical law is however precisely the same, whatever the proportions; and I use such proportions as are best calculated to make the diagram clear. Now, therefore, the conditions of a perspective problem are the following: The Sight-line _GH_ given, Fig. 1.; The Sight-point _S_ given; The Station-point _T_ given; and The three distances of the object,[7] direct, lateral, and vertical, with its dimensions, given. The size of the picture, conjecturally limited by the dotted circle, is to be determined afterwards at our pleasure. On these conditions I proceed at once to construction. [3] If the glass were not upright, but sloping, the objects might still be drawn through it, but their perspective would then be different. Perspective, as commonly taught, is always calculated for a vertical plane of picture. [4] Supposing it to have no thickness; otherwise the images would be distorted by refraction. [5] I say “height” instead of “magnitude,” for a reason stated in Appendix I., to which you will soon be referred. Read on here at present. [6] _P_ and _Q_ being points indicative of the place of the tower’s base and top. In this figure both are above the sight-line; if the tower were below the spectator both would be below it, and therefore measured below _D_. [7] More accurately, “the three distances of any point, either in the object itself, or indicative of its distance.” PROBLEM I. TO FIX THE POSITION OF A GIVEN POINT.[8] Let _P_, Fig. 4., be the given point. [Illustration: Fig. 4.] Let its direct distance be _DT_; its lateral distance to the left, _DC_; and vertical distance _beneath_ the eye of the observer, _CP_. [Let _GH_ be the Sight-line, _S_ the Sight-point, and _T_ the Station-point.][9] It is required to fix on the plane of the picture the position of the point P.