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Authors: Leonardo Da Vinci
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be proved by the 2nd of this which
shows: all the rays which convey the images of objects through the
air are straight lines. Hence, if the images of very large bodies
have to pass through very small holes, and beyond these holes
recover their large size, the lines must necessarily intersect.
[Footnote: 77. 2. In the first of the three diagrams Leonardo had
drawn only one of the two margins, et m .]
78.
Necessity has provided that all the images of objects in front of
the eye shall intersect in two places. One of these intersections is
in the pupil, the other in the crystalline lens; and if this were
not the case the eye could not see so great a number of objects as
it does. This can be proved, since all the lines which intersect do
so in a point. Because nothing is seen of objects excepting their
surface; and their edges are lines, in contradistinction to the
definition of a surface. And each minute part of a line is equal to
a point; for smallest is said of that than which nothing can be
smaller, and this definition is equivalent to the definition of the
point. Hence it is possible for the whole circumference of a circle
to transmit its image to the point of intersection, as is shown in
the 4th of this which shows: all the smallest parts of the images
cross each other without interfering with each other. These
demonstrations are to illustrate the eye. No image, even of the
smallest object, enters the eye without being turned upside down;
but as it penetrates into the crystalline lens it is once more
reversed and thus the image is restored to the same position within
the eye as that of the object outside the eye.
79.
OF THE CENTRAL LINE OF THE EYE.
Only one line of the image, of all those that reach the visual
virtue, has no intersection; and this has no sensible dimensions
because it is a mathematical line which originates from a
mathematical point, which has no dimensions.
According to my adversary, necessity requires that the central line
of every image that enters by small and narrow openings into a dark
chamber shall be turned upside down, together with the images of the
bodies that surround it.
80.
AS TO WHETHER THE CENTRAL LINE OF THE IMAGE CAN BE INTERSECTED, OR
NOT, WITHIN THE OPENING.
It is impossible that the line should intersect itself; that is,
that its right should cross over to its left side, and so, its left
side become its right side. Because such an intersection demands two
lines, one from each side; for there can be no motion from right to
left or from left to right in itself without such extension and
thickness as admit of such motion. And if there is extension it is
no longer a line but a surface, and we are investigating the
properties of a line, and not of a surface. And as the line, having
no centre of thickness cannot be divided, we must conclude that the
line can have no sides to intersect each other. This is proved by
the movement of the line a f to a b and of the line e b to e
f , which are the sides of the surface a f e b . But if you move
the line a b and the line e f , with the frontends a e , to the
spot c , you will have moved the opposite ends f b towards each
other at the point d . And from the two lines you will have drawn
the straight line c d which cuts the middle of the intersection of
these two lines at the point n without any intersection. For, you
imagine these two lines as having breadth, it is evident that by
this motion the first will entirely cover the other—being equal
with it—without any intersection, in the position c d . And this
is sufficient to prove our proposition.
81.
HOW THE INNUMERABLE RAYS FROM INNUMERABLE IMAGES CAN CONVERGE TO A
POINT.
Just as all lines can meet at a point without interfering with each
other—being without breadth or thickness—in the same way all the
images of surfaces can meet there; and as each given point faces the
object opposite to it and each object faces an opposite point, the
converging rays of the image can pass through the
shows: all the rays which convey the images of objects through the
air are straight lines. Hence, if the images of very large bodies
have to pass through very small holes, and beyond these holes
recover their large size, the lines must necessarily intersect.
[Footnote: 77. 2. In the first of the three diagrams Leonardo had
drawn only one of the two margins, et m .]
78.
Necessity has provided that all the images of objects in front of
the eye shall intersect in two places. One of these intersections is
in the pupil, the other in the crystalline lens; and if this were
not the case the eye could not see so great a number of objects as
it does. This can be proved, since all the lines which intersect do
so in a point. Because nothing is seen of objects excepting their
surface; and their edges are lines, in contradistinction to the
definition of a surface. And each minute part of a line is equal to
a point; for smallest is said of that than which nothing can be
smaller, and this definition is equivalent to the definition of the
point. Hence it is possible for the whole circumference of a circle
to transmit its image to the point of intersection, as is shown in
the 4th of this which shows: all the smallest parts of the images
cross each other without interfering with each other. These
demonstrations are to illustrate the eye. No image, even of the
smallest object, enters the eye without being turned upside down;
but as it penetrates into the crystalline lens it is once more
reversed and thus the image is restored to the same position within
the eye as that of the object outside the eye.
79.
OF THE CENTRAL LINE OF THE EYE.
Only one line of the image, of all those that reach the visual
virtue, has no intersection; and this has no sensible dimensions
because it is a mathematical line which originates from a
mathematical point, which has no dimensions.
According to my adversary, necessity requires that the central line
of every image that enters by small and narrow openings into a dark
chamber shall be turned upside down, together with the images of the
bodies that surround it.
80.
AS TO WHETHER THE CENTRAL LINE OF THE IMAGE CAN BE INTERSECTED, OR
NOT, WITHIN THE OPENING.
It is impossible that the line should intersect itself; that is,
that its right should cross over to its left side, and so, its left
side become its right side. Because such an intersection demands two
lines, one from each side; for there can be no motion from right to
left or from left to right in itself without such extension and
thickness as admit of such motion. And if there is extension it is
no longer a line but a surface, and we are investigating the
properties of a line, and not of a surface. And as the line, having
no centre of thickness cannot be divided, we must conclude that the
line can have no sides to intersect each other. This is proved by
the movement of the line a f to a b and of the line e b to e
f , which are the sides of the surface a f e b . But if you move
the line a b and the line e f , with the frontends a e , to the
spot c , you will have moved the opposite ends f b towards each
other at the point d . And from the two lines you will have drawn
the straight line c d which cuts the middle of the intersection of
these two lines at the point n without any intersection. For, you
imagine these two lines as having breadth, it is evident that by
this motion the first will entirely cover the other—being equal
with it—without any intersection, in the position c d . And this
is sufficient to prove our proposition.
81.
HOW THE INNUMERABLE RAYS FROM INNUMERABLE IMAGES CAN CONVERGE TO A
POINT.
Just as all lines can meet at a point without interfering with each
other—being without breadth or thickness—in the same way all the
images of surfaces can meet there; and as each given point faces the
object opposite to it and each object faces an opposite point, the
converging rays of the image can pass through the
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