Then _abcd_ is the square required. COROLLARY I. It is obvious that any rectangle in a horizontal plane may be drawn by this problem, merely making _ab′_, on the measuring-line, Fig. 20., equal to the sight-magnitude of one of its sides, and _ac′_ the sight-magnitude of the other. COROLLARY II. Let _abcd_, Fig. 22., be any square drawn in perspective. Draw the diagonals _ad_ and _bc_, cutting each other in _C_. Then _C_ is the center of the square. Through _C_, draw _ef_ to the vanishing-point of _ab_, and _gh_ to the vanishing-point of _ac_, and these lines will bisect the sides of the square, so that _ag_ is the perspective representation of half the side _ab_; _ae_ is half _ac_; _ch_ is half _cd_; and _bf_ is half _bd_. [Illustration: Fig. 22.] COROLLARY III. Since _ABCD_, Fig. 20., is a square, _BAC_ is a right angle; and as _TV_ is parallel to _AB_, and _TV′_ to _AC_, _V′TV_ must be a right angle also. As the ground plan of most buildings is rectangular, it constantly happens in practice that their angles (as the corners of ordinary houses) throw the lines to the vanishing-points thus at right angles; and so that this law is observed, and _VTV′_ is kept a right angle, it does not matter in general practice whether the vanishing-points are thrown a little more or a little less to the right or left of _S_: but it matters much that the relation of the vanishing-points should be accurate. Their position with respect to _S_ merely causes the spectator to see a little more or less on one side or other of the house, which may be a matter of chance or choice; but their rectangular relation determines the rectangular shape of the building, which is an essential point. PROBLEM IX. TO DRAW A SQUARE PILLAR, GIVEN IN POSITION AND MAGNITUDE, ITS BASE AND TOP BEING IN HORIZONTAL PLANES. Let _AH_, Fig. 23., be the square pillar. Then, as it is given in position and magnitude, the position and magnitude of the square it stands upon must be given (that is, the line _AB_ or _AC_ in position), and the height of its side _AE_. [Illustration: Fig. 23.] [Illustration: Fig. 24.] Find the sight-magnitudes of _AB_ and _AE_. Draw the two sides _ab_, _ac_, of the square of the base, by Problem VIII., as in Fig. 24. From the points _a_, _b_, and _c_, raise vertical lines _ae_, _cf_, _bg_. Make _ae_ equal to the sight-magnitude of _AE_. Now because the top and base of the pillar are in horizontal planes, the square of its top, _FG_, is parallel to the square of its base, _BC_. Therefore the line _EF_ is parallel to _AC_, and _EG_ to _AB_. Therefore _EF_ has the same vanishing-point as _AC_, and _EG_ the same vanishing-point as _AB_. From _e_ draw _ef_ to the vanishing-point of _ac_, cutting _cf_ in _f_. Similarly draw _eg_ to the vanishing-point of _ab_, cutting _bg_ in _g_.