General remarks on perspective (40-41).

40.

ON PAINTING.

Perspective is the best guide to the art of Painting.

[Footnote:  40.  Compare 53, 2.]

41.

The art of perspective is of such a nature as to make what is flat appear in relief and what is in relief flat.

The elements of perspective—­Of the Point (42-46).

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.  The line is of three kinds, straight, curved and sinuous and it has neither breadth, height, nor depth.  Hence it is indivisible, excepting in its length, and its ends are two points.  The angle is the junction of two lines in a point.

43.

A point is not part of a line.

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.

[Footnote:  This definition was inserted by Leonardo on a MS. copy on parchment of the well-known "Trattato d’Architettura civile e militare" &c. by FRANCESCO DI GIORGIO; opposite a passage where the author says:  ’In prima he da sapere che punto e quella parie della quale he nulla—­Linia he luncheza senza apieza; &c.]

45.

1, The superficies is a limitation of the body. 2, and the limitation of a body is no part of that body. 4, and the limitation of one body is that which begins another. 3, that which is not part of any body is nothing.  Nothing is that which fills no space.

If one single point placed in a circle may be the starting point of an infinite number of lines, and the termination of an infinite number of lines, there must be an infinite number of points separable from this point, and these when reunited become one again; whence it follows that the part may be equal to the whole.

46.

The point, being indivisible, occupies no space.  That which occupies no space is nothing.  The limiting surface of one thing is the beginning of another. 2.  That which is no part of any body is called nothing. 1.  That which has no limitations, has no form.  The limitations of two conterminous bodies are interchangeably the surface of each.  All the surfaces of a body are not parts of that body.

Of the line (47-48).

47.

DEFINITION OF THE NATURE OF THE LINE.

The line has in itself neither matter nor substance and may rather be called an imaginary idea than a real object; and this being its nature it occupies no space.  Therefore an infinite number of lines may be conceived of as intersecting each other at a point, which has no dimensions and is only of the thickness (if thickness it may be called) of one single line.