P′ Q′ ∶ P Q ∷ S T ∶ D T[eqn i]

or in words:

In which formula, recollect that P′ Q′ is the height of the appearance of the object on the picture; P Q the height of the object itself; S the Sight-point;

T the Station-point; D a point at the direct distance of the object; though the object is [p8] ]seldom placed actually on the line T S produced, and may be far to the right or left of it, the formula is still the same.

For let S, [Fig. 3.], be the Sight-point, and A B the glass—here seen looking down on its upper edge, not sideways;—then if the tower (represented now, as on a map, by the dark square), instead of being at D on the line S T produced, be at E, to the right (or left) of the spectator, still the apparent height of the tower on A B will be as S′ T to E T, which is the same ratio as that of S T to D T.

Fig. 3.

Now in many perspective problems, the position of an object is more conveniently expressed by the two measurements D T and D E, than by the single oblique measurement E T.

I shall call D T the “direct distance” of the object at E, and D E its “lateral distance.” It is rather a license to call D T its “direct” distance, for E T is the more direct of the two; but there is no other term which would not cause confusion.

Lastly, in order to complete our knowledge of the position of an object, the vertical height of some point in it, above or below the eye, must be given; that is to say, either D P or D Q in [Fig. 2.][Footnote 6] ]: this I shall call the “vertical distance” of the point given. In all perspective problems these three distances, and the dimensions of the object, must be stated, otherwise the problem is imperfectly given. It ought not to be required of us merely to draw a room or a church in perspective; but to draw this room from this corner, and that church on that spot, in perspective. For want of knowing [p9] ]how to base their drawings on the measurement and place of the object, I have known practiced students represent a parish church, certainly in true perspective, but with a nave about two miles and a half long.