The lines which regulate the inner sides or returns of the windows (_a_, _b_, _c_, etc.) of course are drawn to the vanishing-point of _BF_ (the other side of the house), if _FBV_ represents a right angle; if not, their own vanishing-point must be found separately for these returns. But see Practice on Problem XI. [Illustration: Fig. 55.] Interior angles, such as _EBC_, Fig. 55. (suppose the corner of a room), are to be treated in the same way, each side of the room having its measurements separately carried to it from the measuring-line. It may sometimes happen in such cases that we have to carry the measurement _up_ from the corner _B_, and that the sight-magnitudes are given us from the length of the line _AB_. For instance, suppose the room is eighteen feet high, and therefore _AB_ is eighteen feet; and we have to lay off lengths of six feet on the top of the room wall, _BC_. Find _D_, the dividing-point of _BC_. Draw a measuring-line, _BF_, from _B_; and another, _gC_, anywhere above. On _BF_ lay off _BG_ equal to one third of _AB_, or six feet; and draw from _D_, through _G_ and _B_, the lines _Gg_, _Bb_, to the upper measuring-line. Then _gb_ is six feet on that measuring-line. Make _bc_, _ch_, etc., equal to _bg_; and draw _ce_, _hf_, etc., to _D_, cutting _BC_ in _e_ and _f_, which mark the required lengths of six feet each at the top of the wall. PROBLEM X. This is one of the most important foundational problems in perspective, and it is necessary that the student should entirely familiarize himself with its conditions. In order to do so, he must first observe these general relations of magnitude in any pyramid on a square base. Let _AGH′_, Fig. 56., be any pyramid on a square base. [Illustration: Fig. 56.] The best terms in which its magnitude can be given, are the length of one side of its base, _AH_, and its vertical altitude (_CD_ in Fig. 25.); for, knowing these, we know all the other magnitudes. But these are not the terms in which its size will be usually ascertainable. Generally, we shall have given us, and be able to ascertain by measurement, one side of its base _AH_, and either _AG_ the length of one of the lines of its angles, or _BG_ (or _B′G_) the length of a line drawn from its vertex, _G_, to the middle of the side of its base. In measuring a real pyramid, _AG_ will usually be the line most easily found; but in many architectural problems _BG_ is given, or is most easily ascertainable. Observe therefore this general construction. [Illustration: Fig. 57.] Let _ABDE_, Fig. 57., be the square base of any pyramid. Draw its diagonals, _AE_, _BD_, cutting each other in its center, _C_. Bisect any side, _AB_, in _F_. From _F_ erect vertical _FG_. Produce _FB_ to _H_, and make _FH_ equal to _AC_. Now if the vertical altitude of the pyramid (_CD_ in Fig. 25.) be given, make _FG_ equal to this vertical altitude. Join _GB_ and _GH_.