Then the construction will be as in Fig. 7.; and the student will find experimentally that _ab_ is now parallel to _AB_.[11] [Illustration: Fig. 7.] And that _ab_ is to _AB_ as _TS_ is to _TD_. Therefore, to draw any line in a plane parallel to that of the picture, we have only to fix the position of one of its extremities, _a_ or _b_, and then to draw from _a_ or _b_ a line parallel to the given line, bearing the proportion to it that _TS_ bears to _TD_. COROLLARY II. If the line _AB_ is in a horizontal plane, the vertical distance of one of its extremities must be the same as that of the other. Therefore, in that case, _AC_ equals _BC′_ (Fig. 6.). And the construction is as in Fig. 8. [Illustration: Fig. 8.] In Fig. 8. produce _ab_ to the sight-line, cutting the sight-line in _V_; the point _V_, thus determined, is called the VANISHING-POINT of the line _AB_. Join _TV_. Then the student will find experimentally that _TV_ is parallel to _AB_.[12] COROLLARY III. If the line _AB_ produced would pass through some point beneath or above the station-point, _CD_ is to _DT_ as _C′D′_ is to _D′T_; in which case the point _c_ coincides with the point _c′_, and the line _ab_ is vertical. Therefore every vertical line in a picture is, or may be, the perspective representation of a horizontal one which, produced, would pass beneath the feet or above the head of the spectator.[13] [11] For by the construction _AT_ ∶ _aT_ ∷ _BT_ ∶ _bT_; and therefore the two triangles _ABT_, _abT_, (having a common angle _ATB_,) are similar. [12] The demonstration is in Appendix II. Article I. [13] The reflection in water of any luminous point or isolated object (such as the sun or moon) is therefore, in perspective, a vertical line; since such reflection, if produced, would pass under the feet of the spectator. Many artists (Claude among the rest) knowing something of optics, but nothing of perspective, have been led occasionally to draw such reflections towards a point at the center of the base of the picture. PROBLEM III. TO FIND THE VANISHING-POINT OF A GIVEN HORIZONTAL LINE. [Illustration: Fig. 9.] Let _AB_, Fig. 9., be the given line. From _T_, the station-point, draw _TV_ parallel to _AB_, cutting the sight-line in _V_. _V_ is the Vanishing-point required.[14] COROLLARY I. As, if the point _b_ is first found, _V_ may be determined by it, so, if the point _V_ is first found, _b_ may be determined by it. For let _AB_, Fig. 10., be the given line, constructed upon the paper as in Fig. 8.; and let it be required to draw the line _ab_ without using the point _C′_. [Illustration: Fig. 10.] Find the position of the point _A_ in _a_. (Problem I.) Find the vanishing-point of _AB_ in _V_. (Problem III.) Join _aV_. Join _BT_, cutting _aV_ in _b_. Then _ab_ is the line required.[15] COROLLARY II.