The English works of Thomas Hobbes of Malmesbury, Volume 01 (of 11)

by Thomas Hobbes

4. If from a point, whatsoever the medium be, endeavour be propagated every way into all the parts of that medium; and to the same endeavour there be obliquely opposed another medium of a different nature, that is, either thinner or thicker; that endeavour will be so refracted, that the sine of the angle refracted, to the sine of the angle of inclination, will be as the density of the first medium to the density of the second medium, reciprocally taken.

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 medium in which is I K or B D, so the sine of the angle refracted to the sine of the angle of inclination. And by the same reason it may be shown, that as the density of the thinner medium is to the density of the thicker medium, so will K I the sine of the angle refracted be to A F the sine of the angle of inclination.

Secondly, let the body, which endeavoureth every way, be in the thicker medium at I. If, therefore, both the mediums were of the same density, the endeavour of the body in I B would tend directly to L; and the sine of the angle of inclination L M would be equal to I K or B H. But because the density of the medium, in which is I K, to the density of the medium, in which is L M, is as B H to B G, that is, to A F, the endeavour will be propagated further in the medium in which L M is, than in the medium in which I K is, in the proportion of density to density, that is, of M L to A F. Wherefore, B A being drawn, the angle refracted will be C B A, and its sine A F. But L M is the sine of the angle of inclination; and therefore again, as the density of one medium is to the density of the different medium, so reciprocally is the sine of the angle refracted to the sine of the angle of inclination; which was to be demonstrated.

In this demonstration, I have made the separating superficies B b plane by construction. But though it were concave or convex, the theorem would nevertheless be true. For the refraction being made in the point B of the plane separating superficies, if a crooked line, as P Q, be drawn, touching the separating line in the point B; neither the refracted line B I, nor the perpendicular B D, will be altered; and the refracted angle K B I, as also its sine K I, will be still the same they were.

The sine of the refracted angle in one inclination is to the sine of the refracted angle in another inclination, as the sine of the angle of that inclination is to the sine of the angle of this inclination.

5. The sine of the angle refracted in one inclination is to the sine of the angle refracted in another inclination, as the sine of the angle of that inclination to the sine of the angle of this inclination.

For seeing the sine of the refracted angle is to the sine of the angle of inclination, whatsoever that inclination be, as the density of one medium to the density of the other medium; the proportion of the sine of the refracted angle, to the sine of the angle of inclination, will be compounded of the proportions of density to density, and of the sine of the angle of one inclination to the sine of the angle of the other inclination. But the proportions of the densities in the same homogeneous body are supposed to be the same. Wherefore refracted angles in different inclinations are as the sines of the angles of those inclinations; which was to be demonstrated.

If two lines of incidence, having equal inclination, be one in a thinner the other in a thicker medium, the sine of the angle of inclination will be a mean proportional between the two sines of the refracted angles.