Linear Perspective
The Pyramid of Sight and the Science of Representation
Perspective is the best guide to the art of Painting.
— Leonardo da Vinci, §40
Overview
Leonardo placed perspective at the very foundation of painting — the first thing a student should learn. His theory begins with the "Pyramid of Sight": every object sends its image to the eye through a pyramid of lines converging to a single point. This is both a physical description of how vision works and a practical method for constructing accurate images.
He distinguished between simple perspective (linear diminution) and complex perspective (involving color, atmosphere, and sharpness), and his treatment is both more rigorous and more practical than that of his predecessors, including Alberti.
The Elements of Perspective
§42–46 — Point, line, angle, surface, solid
42. All the problems of perspective are made clear by the five terms of mathematicians, which are: — the point, the line, the angle, the superficies and the solid. The point is unique of its kind. And the point has neither height, breadth, length, nor depth, whence it is to be regarded as indivisible and as having no dimensions in space.
44. OF THE NATURAL POINT.
The smallest natural point is larger than all mathematical points, and this is proved because the natural point has continuity, and any thing that is continuous is infinitely divisible; but the mathematical point is indivisible because it has no size.
Ash. I
Definition of Perspective
§50 — The Pyramid of Sight
50. Perspective is a rational demonstration by which experience confirms that every object sends its image to the eye by a pyramid of lines; and bodies of equal size will result in a pyramid of larger or smaller size, according to the difference in their distance, one from the other. By a pyramid of lines I mean those which start from the surface and edges of bodies, and, converging from a distance meet in a single point. A point is said to be that which, having no dimensions, cannot be divided, and this point placed in the eye receives all the points of the cone.
MS. A
"Every object sends its image to the eye by a pyramid of lines." This single sentence is the foundation of Western pictorial representation. Everything that follows — in Leonardo, in Descartes, in photography, in CGI — is an elaboration of this principle. -D
On Drawing Outline
§48–49 — The painter's instruction
48. OF DRAWING OUTLINE.
Consider with the greatest care the form of the outlines of every object, and the character of their undulations. And these undulations must be separately studied, as to whether the curves are composed of arched convexities or angular concavities.
49. The boundaries of bodies are the least of all things. The proposition is proved to be true, because the boundary of a thing is a surface, which is not part of the body contained within that surface; nor is it part of the air surrounding that body, but is the medium interposted between the air and the body. Wherefore O painter! do not surround your bodies with lines, and above all when representing objects smaller than nature; for not only will their external outlines become indistinct, but their parts will be invisible from distance.
Leonardo didn't invent linear perspective — Brunelleschi and Alberti got there first. But he understood it more deeply than anyone before him, and he integrated it into a complete theory of vision that includes atmospheric effects, color change over distance, and the physiology of the eye itself. His perspective is not just a drawing technique — it's a theory of perception. -D