Read The Notebooks of Leonardo Da Vinci for Free Online Page B

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 � quella parie della
quale he nulla—Linia he luncheza senza �pieza; &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.
HOW WE MAY CONCLUDE THAT A SUPERFICIES TERMINATES IN A POINT?
    An angular surface is reduced to a point where it terminates in an
angle. Or, if the sides of that angle are produced in a straight
line, then—beyond that angle—another surface is generated,
smaller, or equal to, or larger than the first.
    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 nature of the outline.
    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, as is proved in its
place. But the lateral boundaries of these bodies is the line
forming the boundary of the surface, which line is of invisible
thickness. 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.
    50.
    Definition of Perspective.
    [Drawing is based upon perspective, which is nothing else than a
thorough knowledge of the function of the eye. And this function
simply consists in receiving in a pyramid the forms and colours of
all the objects placed before it. I say in a pyramid, because there
is no object so small that it will not be larger than the