Joshua Kirby claimed in his Method of Perspective that he was making Brook Taylor’s work easier to understand for gentlemen and practitioners. Brook Taylor’s Linear Perspective was published in 1715, with a revised edition in 1719. His work is austere, rigorous and mathematically challenging. Perhaps as a result, it went largely unread, and, even after it was popularized by Kirby, Highmore, and others, it seems to have been more appreciated than read. In recent scholarship, Kirsti Andersen has studied Taylor’s work most deeply.
When Brook Taylor wrote Linear Perspective, few in Britain had gone before him, and certainly not attempting a coherent theoretical approach; in this respect Britain was far behind the Continent. The inadequacy of his predecessors caused Taylor to sweep away the old vocabulary and replace it with his own set of terms and concepts, including linear perspective. As he wrote in his Preface,
In this Treatise I have endeavour’d to render the Art of Perspective more general, and more easy, than has yet been done. In order to do this, I find it necessary to lay aside the common Terms of Art, which have hitherto been used, such as Horizontal Line, Points of Distance, &c. and to use new ones of my own; such as seem to be more significant of the Things they express, and more agreeable to the general Notion I have formed to my self of this Subject.
He succeeded in his aim of rendering the theory of perspective `more general’, but perhaps not `more easy’ to mere mortals. Like many mathematicians, his definition of `easy’ did not match that of the general populace. His constructions are simple, they use few ideas and few construction lines, but they require a great deal of mathematical maturity and do not lend themselves easily to actual practice.
The first clue to his generality lies in his getting rid of the horizon line. Taylor inhabited a Euclidean geometrical world, not a Cartesian one. To him,
Perspective is the Art of drawing on a Plane the Appearances of any Figures, by the Rules of Geometry.
Definition I. The Center of the Picture is that Point where a Line from the Spectator’s Eye cuts it …at Right Angles.
Taylor has three ingredients: a spectator, a picture plane, and the original plane of the objects to be represented. He sees no reason why the picture plane should be perpendicular to the ground plane, and hence, he has no need of horizons and horizontals. Here’s his illustration, showing the ‘leaning plane’.
I just love his approach, but then I don’t have to use it to paint. He introduces the terms vanishing point and vanishing line, and treats the one and two-dimensional cases as on equal footing in a manner which is wonderful to behold. Many of Taylor’s propositions take various pieces as given (in the Euclidean sense) and require finding the remaining points. The proofs unfold in a Euclidean manner, and there are practically no examples. He also, as you would expect, leans heavily on ratio theory. Here is Theorem 2:
Any Line in the Representation of a Figure parallel to the Picture, is to its Original Line, as the Principal Distance is to the Distance between the Spectator’s Eye, and the Plane of the Original Figure.
And here is Proposition 10:
Having given the Center and Distance of the Picture, and the Vanishing Line of a Plane, and the Vanishing Point of the Intersection of that Plane, with another Plane perpendicular to it; to find the Vanishing Line of that other Plane.
He covers the whole theory of perspective in 40 pages.
Andersen, K., 1992, Brook Taylor’s Work on Linear Perspective: A Study of Taylor’s Role in the History of Perspective Geometry. Including Facsimiles of Taylor’s Two Books on Perspective. New York: Springer.
Andersen, K., 2006. The Geometry of an Art. The History of the Mathematical Theory of Perspective from Alberti to Monge. New York: Springer.