That is to say in the object t and the object e the proportion of their distances from the eye a is quintuple.  I remove each from its place and set it farther from the eye by one of the 5 parts into which the proposition is divided.  Hence it happens that the nearest to the eye has doubled the distance and according to the last proposition but one of this, is diminished by the half of its whole size; and the body e, by the same motion, is diminished 1/5 of its whole size.  Therefore, by that same last proposition but one, that which is said in this last proposition is true; and this I say of the motions of the celestial bodies which are more distant by 3500 miles when setting than when overhead, and yet do not increase or diminish in any sensible degree.

871.

a b is the aperture through which the sun passes, and if you could measure the size of the solar rays at n m, you could accurately trace the real lines of the convergence of the solar rays, the mirror being at a b, and then show the reflected rays at equal angles to n m; but, as you want to have them at n m, take them at the. inner side of the aperture at cd, where they maybe measured at the spot where the solar rays fall.  Then place your mirror at the distance a b, making the rays d b, c a fall and then be reflected at equal angles towards c d; and this is the best method, but you must use this mirror always in the same month, and the same day, and hour and instant, and this will be better than at no fixed time because when the sun is at a certain distance it produces a certain pyramid of rays.

872.

a, the side of the body in light and shade b, faces the whole portion of the hemisphere bed e f, and does not face any part of the darkness of the earth.  And the same occurs at the point o; therefore the space a o is throughout of one and the same brightness, and s faces only four degrees of the hemisphere d e f g h, and also the whole of the earth s h, which will render it darker; and how much must be demonstrated by calculation. [Footnote:  This passage, which has perhaps a doubtful right to its place in this connection, stands in the Manuscript between those given in Vol.  I as No. 117 and No. 427.]

873.

THE REASON OF THE INCREASED SIZE OF THE SUN IN THE WEST.

Some mathematicians explain that the sun looks larger as it sets, because the eye always sees it through a denser atmosphere, alleging that objects seen through mist or through water appear larger.  To these I reply:  No; because objects seen through a mist are similar in colour to those at a distance; but not being similarly diminished they appear larger.  Again, nothing increases in size in smooth water; and the proof of this may be seen by throwing a light on a board placed half under water.  But the reason why the sun looks larger is that every luminous body appears larger in proportion as it is more remote. [Footnote:  Lines 5 and 6 are thus rendered by M. RAVAISSON in his edition of MS. A. “De meme, aucune chose ne croit dans l’eau plane, et tu en feras l’experience en calquant un ais sous l’eau.”—­Compare the diagrams in Vol.  I, p. 114.]