This is self evident only in the middle diagram; but it is equally true of the side figures if we conceive of the lines 4 f, x n v m, y l k v, and 4 e, as prolonged beyond the semicircle of the horizon.] and is in a straight line with the centre of the primary shadow, with the centre of the body casting it and of the derivative light and with the centre of the window and, finally, with the centre of that portion of the source of light which is the celestial hemisphere, y h is the centre of the derived shade, l h of the primary shadow, l of the body throwing it, l k of the derived light, v is the centre of the window, e is the final centre of the original light afforded by that portion of the hemisphere of the sky which illuminates the solid body.

[Footnote:  Compare the diagram on Pl.  IV, No. 3.  In the original this drawing is placed between lines 3 and 22; the rest, from line 4 to line 21, is written on the left hand margin.]

174.

THE FARTHER THE DERIVED SHADOW IS PROLONGED THE LIGHTER IT BECOMES.

You will find that the proportion of the diameter of the derived shadow to that of the primary shadow will be the same as that between the darkness of the primary shadow and that of the derived shadow.

[Footnote 6:  Compare No. 177.] Let a b be the diameter of the primary shadow and c d that of the derived shadow, I say that a b going, as you see, three times into d c, the shadow d c will be three times as light as the shadow a b. [Footnote 8:  Compare No. 177.]

If the size of the illuminating body is larger than that of the illuminated body an intersection of shadow will occur, beyond which the shadows will run off in two opposite directions as if they were caused by two separate lights.

On the relative intensity of derived shadows (175-179).

175.

ON PAINTING.

The derived shadow is stronger in proportion as it is nearer to its place of origin.

176.

HOW SHADOWS FADE AWAY AT LONG DISTANCES.

Shadows fade and are lost at long distances because the larger quantity of illuminated air which lies between the eye and the object seen tints the shadow with its own colour.

177.

a b will be darker than c d in proportion as c d is broader than a b.

[Footnote:  In the original MS. the word lume (light) is written at the apex of the pyramid.]

178.

It can be proved why the shadow o p c h is darker in proportion as it is nearer to the line p h and is lighter in proportion as it is nearer to the line o c.  Let the light a b, be a window, and let the dark wall in which this window is, be b s, that is, one of the sides of the wall.

Then we may say that the line p h is darker than any other part of the space o p c h, because this line faces the whole surface in shadow of [Footnote:  In the original the diagram is placed between lines 27 and 28.] the wall b s.  The line o c is lighter than the other part of this space o p c h, because this line faces the luminous space a b.