9. OUR author adds but one remark more upon refraction, which is, that if refraction be performed in the manner he has supposed from the light’s being pressed by the refracting power perpendicularly toward the surface of the refracting body, and consequently be made to move swifter in the body than before its incidence; whether this power act equally at all distances or otherwise, provided only its power in the same body at the same distances remain without variation the same in one inclination of the incident rays as well as another; he observes that the refracting powers in different bodies will be in the duplicate proportion of the tangents of the lead angles, which the refracted light can make with the surfaces of the refracting bodies[316]. This observation may be explained thus. When the light passes into any refracting substance, it has been shewn above that the sine of incidence bears a constant proportion to the sine of refraction. Suppose the light to pass to the refracting body A B C D (in fig. 134) in the line E F, and to fall upon it at the point F, and then to proceed within the body in the line F G. Let H I be drawn through F perpendicular to the surface A B, and any circle K L M N be described to the center F. Then from the points O and P where this circle cuts the incident and refracted ray, the perpendiculars O Q, P R being drawn, the proportion of O Q to P R will remain the same in all the different obliquities, in which the same ray of light can fall on the surface A B. Now O Q is less than F L the semidiameter of the circle K L M N, but the more the ray E F is inclined down toward the surface A B, the greater will O Q be, and will approach nearer to the magnitude of F L. But the proportion of O Q to P R remaining always the same, when O Q, is largest, P R will also be greatest; so that the more the incident ray E F is inclined toward the surface A B, the more the ray F G after refraction will be inclined toward the same. Now if the line F S T be so drawn, that S V being perpendicular to F I shall be to F L the semidiameter of the circle in the constant proportion of P R to O Q; then the angle under N F T is that which I meant by the least of all that can be made by the refracted ray with this surface, for the ray after refraction would proceed in this line, if it were to come to the point F lying on the very surface A B; for if the incident ray came to the point F in any line between A F and F H, the ray after refraction would proceed forward in some line between F T and F I. Here if N W be drawn perpendicular to F N, this line N W in the circle K L M N is called the tangent of the angle under N F S. Thus much being premised, the sense of the forementioned proposition is this. Let there be two refracting substances (in fig. 135) A B C D, and E F G H. Take a point, as I, in the surface A B, and to the center I with any semidiameter describe the circle K L M. In like manner on the surface E F take