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Suppose a painter sees a path through a field sown with poppies and paints it: at one end of the chain of events is the field of poppies, at the other a canvas with pigment disposed on its surface. We can recognize that the latter represents the former, so I shall suppose that despite the differences between a field of poppies and a piece of canvas, despite the transformation that the artist has effected in what he saw to make it take the form of a picture, something has remained unaltered and on this something recognition depends. The elements that go to make up the unaltered aspect of the transformation I shall call invariants.
The artist is not the only person involved in looking at a picture; recognition of what the picture represents could not occur if the observer were to rely exclusively on his sense of smell. The wider his experience of art the more likely he would be to interpret the painting correctly.
In many pictures the effectiveness of the representation would depend on perspective. A peculiar feature of this domain is that a completely circular pond, for example, might be represented by an ellipse, or a path with borders running parallel to each other might be represented by two lines that meet. Indeed the representation of pond or path would be less adequate if it were a circle or parallel lines. Accordingly we assume that in ellipse and intersecting lines, circular pond and parallel borders, is some quality that is invariant under artistic creation.
Suppose now that we view a stretch of railway line that is straight as far as the eye can see. The two lines of the track will be seen to converge. We know that if we were to test the convergence by walking up the line this convergence would not be confirmed; but, if we
were to walk far enough and to look back the way we had come, the convergence would appear to lie behind us and to be confirmed by our sense of sight; the two parallel lines meet in a point. Where then is this point?
One theory would expla