This section is from the "The elements of drawing & the elements of perspective" book, by J. M. Dent & Sons. Also see Amazon: The Elements Of Drawing & The Elements Of Perspective.
P' Q': P Q:: S T:D T
P dash Q dash is to p q as S T to d t
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 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.
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. 1: 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 how to base their drawings on the measurement and place of the object, I have known practised students represent a parish church, certainly in true perspective, but with a nave about two miles and a half long.
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 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.Measurements assumed are generally very different from any likely to occur in practice. Thus, in Fig. 3., the distance d s would be probably in practice half a mile or a mile, and the distance t s, 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 g h given, Fig. 1.;
The Sight-point S given;
The Station-point t given; and
The three distances of the object1, direct, lateral, and vertical, with its dimensions, given.
The size of the picture, conjecturally limited by the dotted circle, is to be determined after wards at our pleasure.On these conditions I proceed at once to construction.
More accurately, the three distances of any point, either in the object itself, or indicative of its distance.