532 PHILOSOPHICAL TBANS ACTIONS. ' ^ANNO I7OO. 



Whence according to Cardan's rules, arises tliis theorem. From the cube of t 

 subtract the product of2tr into the excess of the secant of the same arch above 

 the radius ; the difference will be a lesser cube. And the sum of the same, adding 

 4trr, will be a greater cube. The sum of the sides of each cube, and of t, 

 will be equal to the tangent of the angle of incidence, half of which will be also 

 the tangent of the angle of refraction ; whence arises the ratio required. 



Of this lake the following example. In a drop of oil of turpentine, the 

 distance of the primary iris from the point opposite to the sun, is observed to 

 be 25° 40'. The ratio of refraction is required. 



t = tangent 12° 50' = 0,2278063 



8 = secant of the same = 1,0256197 



ttt = 0,01 182217 

 s — r into 2 tr = 0,011 67265 



difference, or lesser cube = O,000l4g52. ^0,0530773 



sum 0,02349482 

 4trr = 0,91122525 



greater cube 0,93472007.^0,9777486 



t = 0,2278063 



T = tangent incidence 51° 32' 1,2586322 

 -i-T = tangent refraction 52° ll' 0,6293161 



Finally, as v' tt + 4, is to ■/ tt + 1, so is r to s, so is 1 to 0,68026. And 

 this ratio approaches nearly to that, which we find by experiment obtains in 

 glass, and most other pellucid solids. But a diamond does not only exceed all 

 other diaphanous bodies in hardness and value, but alSo in this refractive 

 virtue; its ratio being nearly as 5 to 2, or more truly, as 100 to 41. But 

 of these perhaps more at large in a proper place. 



While I was employed in writing this, the very skilful geometrician, Mr. De 

 Moivre, at my request, took the pains to find a like equation for the ratio of 

 the secondary iris, when the diameter is given. By this the ratio may be de- 

 termined very accurately, but the equation being biquadratic, the calculation 

 cannot be performed with the same'ease. The equation is t^ + ^t t' — 2TTrr 

 — 4-r* = O. Here t is the tangent of the angle of refraction, t the tangent of 

 half the distance of the rainbow from the point opposite to the sun, and the 



