- The Excesses of the Sines of Refraction of several sorts of Rays above their common Sine of Incidence when the Refractions are made out of divers denser Mediums immediately into one and the same rarer Medium, suppose of Air, are to one another in a given Proportion.
- The Proportion of the Sine of Incidence to the Sine of Refraction of one and the same sort of Rays out of one Medium into another, is composed of the Proportion of the Sine of Incidence to the Sine of Refraction out of the first Medium into any third Medium, and of the Proportion of the Sine of Incidence to the Sine of Refraction out of that third Medium into the second Medium.
By the first Theorem the Refractions of the Rays of every sort made out of any Medium into Air are known by having the Refraction of the Rays of any one sort. As for instance, if the Refractions of the Rays of every sort out of Rain-water into Air be desired, let the common Sine of Incidence out of Glass into Air be subducted from the Sines of Refraction, and the Excesses will be 27, 27-1/8, 27-1/5, 27-1/3, 27-1/2, 27-2/3, 27-7/9, 28. Suppose now that the Sine of Incidence of the least refrangible Rays be to their Sine of Refraction out of Rain-water into Air as 3 to 4, and say as 1 the difference of those Sines is to 3 the Sine of Incidence, so is 27 the least of the Excesses above-mentioned to a fourth Number 81; and 81 will be the common Sine of Incidence out of Rain-water into Air, to which Sine if you add all the above-mentioned Excesses, you will have the desired Sines of the Refractions 108, 108-1/8, 108-1/5, 108-1/3, 108-1/2, 108-2/3, 108-7/9, 109.
By the latter Theorem the Refraction out of one Medium into another is gathered as often as you have the Refractions out of them both into any third Medium. As if the Sine of Incidence of any Ray out of Glass into Air be to its Sine of Refraction, as 20 to 31, and the Sine of