The Notebooks

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 point and diverge again beyond it to reproduce and re-magnify the real size of that image. But their impressions will appear reversed--as is shown in the first, above; where it is said that every image intersects as it enters the narrow openings made in a very thin substance. Read the marginal text on the other side. In proportion as the opening is smaller than the shaded body, so much less will the images transmitted through this opening intersect each other. The sides of images which pass through openings into a dark room intersect at a point which is nearer to the opening in proportion as the opening is narrower. To prove this let _a b_ be an object in light and shade which sends not its shadow but the image of its darkened form through the opening _d e_ which is as wide as this shaded body; and its sides _a b_, being straight lines (as has been proved) must intersect between the shaded object and the opening; but nearer to the opening in proportion as it is smaller than the object in shade. As is shown, on your right hand and your left hand, in the two diagrams _a_ _b_ _c_ _n_ _m_ _o_ where, the right opening _d_ _e_, being equal in width to the shaded object _a_ _b_, the intersection of the sides of the said shaded object occurs half way between the opening and the shaded object at the point _c_. But this cannot happen in the left hand figure, the opening _o_ being much smaller than the shaded object _n_ _m_. It is impossible that the images of objects should be seen between the objects and the openings through which the images of these bodies are admitted; and this is plain, because where the atmosphere is illuminated these images are not formed visibly. When the images are made double by mutually crossing each other they are invariably doubly as dark in tone. To prove this let _d_ _e_ _h_ be such a doubling which although it is only seen within the space between the bodies in _b_ and _i_ this will not hinder its being seen from _f_ _g_ or from _f_ _m_; being composed of the images _a_ _b_ _i_ _k_ which run together in _d_ _e_ _h_. [Footnote: 81. On the original diagram at the beginning of this chapter Leonardo has written "_azurro_" (blue) where in the facsimile I have marked _A_, and "_giallo_" (yellow) where _B_ stands.] [Footnote: 15--23. These lines stand between the diagrams I and III.] [Footnote: 24--53. These lines stand between the diagrams I and II.] [Footnote: 54--97 are written along the left side of diagram I.] 82. An experiment showing that though the pupil may not be moved from its position the objects seen by it may appear to move from their places. If you look at an object at some distance from you and which is below the eye, and fix both your eyes upon it and with one hand firmly hold the upper lid open while with the other you push up the under lid--still keeping your eyes fixed on the object gazed at--you will see that object double; one [image] remaining steady, and the other moving in a contrary direction to the pressure of your finger on the lower eyelid. How false the opinion is of those who say that this happens because the pupil of the eye is displaced from its position. How the above mentioned facts prove that the pupil acts upside down in seeing. [Footnote: 82. 14--17. The subject indicated by these two headings is fully discussed in the two chapters that follow them in the original; but it did not seem to me appropriate to include them here.] Demostration of perspective by means of a vertical glass plane (83-85). 83. OF THE PLANE OF GLASS. Perspective is nothing else than seeing place [or objects] behind a plane of glass, quite transparent, on the surface of which the objects behind that glass are to be drawn. These can be traced in pyramids to the point in the eye, and these pyramids are intersected on the glass plane. 84. Pictorial perspective can never make an object at the same distance, look of the same size as it appears to the eye. You see that the apex of the pyramid _f c d_ is as far from the object _c_ _d_ as the same point _f_ is from the object _a_ _b_; and yet _c_ _d_, which is the base made by the painter's point, is smaller than _a_ _b_ which is the base of the lines from the objects converging in the eye and refracted at _s_ _t_, the surface of the eye. This may be proved by experiment, by the lines of vision and then by the lines of the painter's plumbline by cutting the real lines of vision on one and the same plane and measuring on it one and the same object. 85. PERSPECTIVE. The vertical plane is a perpendicular line, imagined as in front of the central point where the apex of the pyramids converge. And this plane bears the same relation to this point as a plane of glass would, through which you might see the various objects and draw them on it. And the objects thus drawn would be smaller than the originals, in proportion as the distance between the glass and the eye was smaller than that between the glass and the objects. PERSPECTIVE. The different converging pyramids produced by the objects, will show, on the plane, the various sizes and remoteness of the objects causing them. PERSPECTIVE. All those horizontal planes of which the extremes are met by perpendicular lines forming right angles, if they are of equal width the more they rise to the level of eye the less this is seen, and the more the eye is above them the more will their real width be seen. PERSPECTIVE. The farther a spherical body is from the eye the more you will see of it. The angle of sight varies with the distance (86-88) 86. A simple and natural method; showing how objects appear to the eye without any other medium.