First, let a body be in the thinner medium in A (fig. 3), and let it be understood to have endeavour every way, and consequently, that its endeavour proceed in the lines A B and A _b_; to which let B _b_ the superficies of the thicker medium be obliquely opposed in B and _b_, so that A B and A _b_ be equal; and let the strait line B _b_ be produced both ways. From the points B and _b_, let the perpendiculars B C and _b c_ be drawn; and upon the centres B and _b_, and at the equal distances B A and _b_ A, let the circles A C and A _c_ be described, cutting B C and _b c_ in C and _c_, and the same C B and _c b_ produced in D and _d_, as also A B and A _b_ produced in E and _e_. Then from the point A to the strait lines B C and _b c_ let the perpendiculars A F and A _f_ be drawn. A F therefore will be the sine of the angle of inclination of the strait line A B, and A _f_ the sine of the angle of inclination of the strait line A _h_, which two inclinations are by construction made equal. I say, as the density of the medium in which are B C and _b c_ is to the density of the medium in which are B D and _b d_, so is the sine of the angle refracted, to the sine of the angle of inclination. Let the strait line F G be drawn parallel to the strait line A B, meeting with the strait line _b_ B produced in G. Seeing therefore A F and B G are also parallels, they will be equal; and consequently, the endeavour in A F is propagated in the same time, in which the endeavour in B G would be propagated if the medium were of the same density. But because B G is in a thicker medium, that is, in a medium which resists the endeavour more than the medium in which A F is, the endeavour will be propagated less in B G than in A F, according to the proportion which the density of the medium, in which A F is, hath to the density of the medium in which B G is. Let therefore the density of the medium, in which B G is, be to the density of the medium, in which A F is, as B G is to B H; and let the measure of the time be the radius of the circle. Let H I be drawn parallel to B D, meeting with the circumference in I; and from the point I let I K be drawn perpendicular to B D; which being done, B H and I K will be equal; and I K will be to A F, as the density of the medium in which is A F is to the density of the medium in which is I K. Seeing therefore in the time A B, which is the radius of the circle, the endeavour is propagated in A F in the thinner medium, it will be propagated in the same time, that is, in the time B I in the thicker medium from K to I. Therefore, B I is the refracted line of the line of incidence A B; and I K is the sine of the angle refracted; and A F the sine of the angle of inclination. Wherefore, seeing I K is to A F, as the density of the medium in which is A F to the density of the medium in which is I K; it will be as the density of the medium in which is A F or B C to the density of the