Tag Archives: vanishing point

More on Brook Taylor’s “Linear Perspective”

Brook Taylor’s book, Linear Perspective (1715) is short, only 40 pages, but dense. For later comparison with Kirby’s exposition, I want to bring out a few more features of Taylor’s text.

Taylor divides his book into five sections:

Section I: Containing an Explanation of those things that are necessary to be understood in order to the Practice of Perspective.

    This section contains his 10 definitions, along with several ‘Corollaries’, and the first four propositions.

Section II: Propositions relating to the General Practice of Perspective.

    The longest section, this one includes Propositions 5-18, with a set of examples between Propositions 16 and 17.

Section III: Of finding the Shadows of given Figures

    A short section with three propositions on shadows.

Section IV: Of finding the Representations of the Reflections of Figures on polish’d Planes.

    Taylor dispenses with reflections in three quick propositions.

Section V: Of the Inverse Practice of Perspective and of the manner of Examining Pictures already drawn.

    The mathematician comes to the fore with the study of inverse problems. The first proposition is, “Having given the Representation of a Line divided into two Parts in a given Proportion; to find its Vanishing Point.” He then tackles similar problems for triangles, parallelograms, trapezia, and “right-angled Parallelepipedons”.

Definitions

For a flavor of Taylor’s style and terminology, here are the first few definitions, along with the figure they reference.

DEF. I. The Center of the Picture is that Point where a Line from the Spectator’s Eye cuts it (or its Plane continued beyond the Frame, if need be) at Right Angles.

If the Plane CD be the Picture, and O the Spectator’s Eye, then a Perpendicular let fall on the Picture from O, will cut it in its Center P.

DEF. II. The Distance of the Picture, or principal Distance, is the Distance between the Center of the Picture and the Spectator’s Eye

    In the same Figure PO is the Distance of the Picture.

DEF. III. The Intersection of an Original Line is that Point where it cuts the Picture.

    If IK be an Original Line cutting the Picture in C, then is C the Intersection of the Line IK.

DEF. IV. The intersection of an Original Plane, is that Line wherein it cuts the Picture.

    AB is an Original Plane cutting the Picture in the Line CQ, which therefore is its Intersection.

DEF. V. The Vanishing Point of an Original Line, is that Point where a Line passing thro’ the Spectator’s Eye, parallel to the Original Line, cuts the Picture.

    Such is the Point V, the Line OV being parallel to the Original Line IK.

COROL. I. Hence it is plain, that Original Lines, which are parallel to each other, have the same Vanishing Point. For one Line passing thro’ the Spectator’s Eye, parallel to them all, produces the Vanishing Point of ’em all, by this Definition.

COROL. 2. Those Lines that are parallel to the Picture have no Vanishing Points. Because the Lines which should produce the Vanishing Points, are in this Case also parallel to the Picture, and therefore can never cut it.

COROL. 3. The Lines that generate the Vanishing Points of two Original Lines, make the same Angle at the Spectator’s Eye, as the Original Lines do with each other.

DEF. VI. The Vanishing Line of an Original Plane, is that Line wherein the Picture is cut by a Plane passing thro’ the Spectator’s Eye parallel to the Original Plane.

    Such is the Line VS, the Plane EF, being parallel to the Original Plane AB.

COROL. I. Hence Original Planes, that are parallel, have the same Vanishing Line. For one Plane passing thro’ the Spectator’s Eye, parallel to them all, produces that Vanishing Line.

COROL. 2. All the Vanishing Points of Lines in parallel Planes, are in the Vanishing Line of those Planes. For the Lines that produce those Vanishing Points, (by Def. 5.) are all in the Plane that produces that Vanishing Line, (by this Def.)

COROL. 3. The Planes which produce the Vanishing Lines of two Original Planes, being parallel to the Original Planes, and passing both thro’ the Spectator’s Eye, (by this Def.) have their common Intersection passing thro’ the Spectator’s Eye, parallel to the Intersection of the Original Planes, and are inclined to each other in the same Angle as the Original Planes are. And hence,

COROL. 4. The Vanishing Point of the common Intersection of two Planes, is the Intersection of the Vanishing Lines of those Planes.

COROL. 5. The Vanishing Line, and Intersection of the same Original Plane, are parallel to each other. Because they are generated by parallel Planes. (By this Def. and Def. 4.)

I think this is all admirably clear, and shows how he treats lines and planes in a similar fashion. It also shows how his concepts of vanishing point and vanishing line are central to the way he is going to proceed.

Related Posts

Brook Taylor’s Linear Perspective

Who was Brook Taylor?

Method of Perspective

Related Works

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.

Brook Taylor’s Linear Perspective

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.

Related Posts

Who was Brook Taylor?

Method of Perspective

Related Works

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.