OF THE OPINION OF SOME THAT A TRIANGLE CASTS NO SHADOW ON A PLANE SURFACE. Certain mathematicians have maintained that a triangle, of which the base is turned to the light, casts no shadow on a plane; and this they prove by saying [5] that no spherical body smaller than the light can reach the middle with the shadow. The lines of radiant light are straight lines [6]; therefore, suppose the light to be _g h_ and the triangle _l m n_, and let the plane be _i k_; they say the light _g_ falls on the side of the triangle _l n_, and the portion of the plane _i q_. Thus again _h_ like _g_ falls on the side _l m_, and then on _m n_ and the plane _p k_; and if the whole plane thus faces the lights _g h_, it is evident that the triangle has no shadow; and that which has no shadow can cast none. This, in this case appears credible. But if the triangle _n p g_ were not illuminated by the two lights _g_ and _h_, but by _i p_ and _g_ and _k_ neither side is lighted by more than one single light: that is _i p_ is invisible to _h g_ and _k_ will never be lighted by _g_; hence _p q_ will be twice as light as the two visible portions that are in shadow. [Footnote: 5--6. This passage is so obscure that it would be rash to offer an explanation. Several words seem to have been omitted.] On the relative depth of cast shadows (200-202). 200. A spot is most in the shade when a large number of darkened rays fall upon it. The spot which receives the rays at the widest angle and by darkened rays will be most in the dark; a will be twice as dark as b, because it originates from twice as large a base at an equal distance. A spot is most illuminated when a large number of luminous rays fall upon it. d is the beginning of the shadow _d f_, and tinges _c_ but _a_ little; _d e_ is half of the shadow _d f_ and gives a deeper tone where it is cast at _b_ than at _f_. And the whole shaded space _e_ gives its tone to the spot _a_. [Footnote: The diagram here referred to is on Pl. XLI, No. 2.] 201. _A n_ will be darker than _c r_ in proportion to the number of times that _a b_ goes into _c d_. 202. The shadow cast by an object on a plane will be smaller in proportion as that object is lighted by feebler rays. Let _d e_ be the object and _d c_ the plane surface; the number of times that _d e_ will go into _f g_ gives the proportion of light at _f h_ to _d c_. The ray of light will be weaker in proportion to its distance from the hole through which it falls. FIFTH BOOK ON LIGHT AND SHADE. Principles of reflection (203. 204). 203. OF THE WAY IN WHICH THE SHADOWS CAST BY OBJECTS OUGHT TO BE DEFINED. If the object is the mountain here figured, and the light is at the point _a_, I say that from _b d_ and also from _c f_ there will be no light but from reflected rays. And this results from the fact that rays of light can only act in straight lines; and the same is the case with the secondary or reflected rays. 204.