We have seen how Brook Taylor’s Euclidean conception of space without a coordinate system and without a privileged reference system led him to condemn the use in perspective theory of horizon lines. In the expanded preface to the second edition of Linear Perspective, he had said,
The Term of Horizontal Line, for instance, it apt to confine the Notions of a Learner to the Plane of the Horizon, and to make him imagine, that the Plane enjoys some particular Privileges, which make the Figures in it more easy and more convenient to be described, by the means of that Horizontal Line, than the Figures in any other Plane; as if all other Planes might not as conveniently be handled, by finding other Lines of the same nature belonging to them.
In his own work, of course, Taylor is having none of this:
But in this Book I make no difference between the Plane of the Horizon, and any other Plane whatsoever; for since Planes, as Planes, are alike in Geometry, it is most proper to consider them as so, and to explain their Properties in general, leaving the Artist himself to apply them in particular Cases, as Occasion requires.
Taylor had not felt a need to burden the reader with such lengthy explanations in the first edition, but he very subtly got his point across right at the beginning of his text. He opens Section One of the book, before he even gives his first formal definitions with an explanation of what exactly perspective is:
Perspective is the Art of drawing on a Plane the Appearances of any Figures, by the Rules of Geometry.
The eighteenth century contained a long-rumbling debate about the extent to which artists should be bound by the rules of geometry in laying out their paintings and how much geometric rules could be trumped by other optical considerations. Taylor the mathematician comes down strictly on the side of geometry. (Kirby, with less confidence in his mathematical ability, equivocates.)
In order to understand the Principles of this Art, we must consider, That a Picture painted in its utmost degree of Perfection, ought so to affect the Eye of the Beholder, that he should not be able to judge whether what he sees be only a few Colours laid artificially on a Cloth, or the very Objects there represented, seen thro’ the Frame of the Picture, as thro’ a Window.
This conception of painting as representing a view through a window goes back at least to Alberti. Taylor explains further that this effect is not solely due to following the rules of geometry, that the geometrical rules of perspective are necessary, but not sufficient:
To produce this Effect, it is plain the Light ought to come from the Picture to the Spectator’s Eye, in the very same manner, as it would do from the Objects themselves, if they really were where they seem to be; that is every Ray of Light ought to come from any Point of the Picture to the Spectator’s Eye with the same Colour, the same strength of Light and Shadow, and in the same Direction, as it would do from the corresponding Point of the real Object, if it were placed where it is imagined to be.
As will be his usual practice, Taylor here gives a general description of what is to be done, and then gives a specific simple example:
So that (Fig. 1.) if EF be a Picture, and abcd be the Representation of any Object on it, and ABCD be the real Object placed where it should seem to be to a Spectator’s Eye in O; then ought the Figure abcd to seem exactly to cover the Figure ABCD, and the Rays AO, BO, CO, &c. that go from any Points A, B, C< &c. of the original Objects to the Spectator’s Eye O, ought to cut the Picture in the corresponding Points a, b, c, &c. of the Representation.
Now this is an admirably clear, simple and straightforward description of how the method of perspective is supposed to work. The kicker is in Figure 1, the first figure in the book a reader would encounter, and one that clearly establishes Taylor’s priorities.
Not a Horizon Line to be seen.