6. If two lines of incidence, having equal inclination, be the one in a thinner, the other in a thicker medium, the sine of the angle of their inclination will be a mean proportional between the two sines of their angles refracted.
For let the strait line A B (in [fig. 3]) have its inclination in the thinner medium, and be refracted in the thicker medium in B I; and let E B have as much inclination in the thicker medium, and be refracted in the thinner medium in B S; and let R S, the sine of the angle refracted, be drawn. I say, the strait lines R S, A F, and I K are in continual proportion. For it is, as the density of the thicker medium to the density of the thinner medium, so R S to A F. But it is also as the density of the same thicker medium to that of the same thinner medium, so A F to I K. Wherefore R S. A F :: A F. I K are proportionals; that is, R S, A F, and I K are in continual proportion, and A F is the mean proportional; which was to be proved.
If the angle of inclination be semirect, and the line of inclination be in the thicker medium, and the proportion of their densities be the same with that of the diagonal to the side of a square, and the separating superficies be plain, the refracted line will be in the separating superficies.
7. If the angle of inclination be semirect, and the line of inclination be in the thicker medium, and the proportion of the densities be as that of a diagonal to the side of its square, and the separating superficies be plain, the refracted line will be in that separating superficies.
For in the circle A C ([fig. 4]) let the angle of inclination A B C be an angle of 45 degrees. Let C B be produced to the circumference in D; and let C E, the sine of the angle E B C, be drawn, to which let B F be taken equal in the separating line B G. B C E F will therefore be a parallelogram, and F E and B C, that is F E and B G equal. Let A G be drawn, namely the diagonal of the square whose side is B G, and it will be, as A G to E F so B G to B F; and so, by supposition, the density of the medium, in which C is, to the density of the medium in which D is; and so also the sine of the angle refracted to the sine of the angle of inclination. Drawing therefore F D, and from D the line D H perpendicular to A B produced, D H will be the sine of the angle of inclination. And seeing the sine of the angle refracted is to the sine of the angle of inclination, as the density of the medium, in which is C, is to the density of the medium in which is D, that is, by supposition, as A G is to F E, that is as B G is to D H; and seeing D H is the sine of the angle of inclination, B G will therefore be the sine of the angle refracted. Wherefore B G will be the refracted line, and lye in the plain separating superficies; which was to be demonstrated.
Coroll. It is therefore manifest, that when the inclination is greater than 45 degrees, as also when it is less, provided the density be greater, it may happen that the refraction will not enter the thinner medium at all.
If a body be carried in a strait line upon another body, and do not penetrate it, but be reflected from it, the angle of reflection will be equal to the angle of incidence.
8. If a body fall in a strait line upon another body, and do not penetrate it, but be reflected from it, the angle of reflection will be equal to the angle of incidence.
Let there be a body at A (in [fig. 5]), which falling with strait motion in the line A C upon another body at C, passeth no further, but is reflected; and let the angle of incidence be any angle, as A C D. Let the strait line C E be drawn, making with D C produced the angle E C F equal to the angle A C D; and let A D be drawn perpendicular to the strait line D F. Also in the same strait line D F let C G be taken equal to C D; and let the perpendicular G E be raised, cutting C E in E. This being done, the triangles A C D and E C G will be equal and like. Let C H be drawn equal and parallel to the strait line A D; and let H C be produced indefinitely to I. Lastly let E A be drawn, which will pass through H, and be parallel and equal to G D. I say the motion from A to C, in the strait line of incidence A C, will be reflected in the strait line C E.
For the motion from A to C is made by two coefficient or concurrent motions, the one in A H parallel to D G, the other in A D perpendicular to the same D G; of which two motions that in A H works nothing upon the body A after it has been moved as far as C, because, by supposition, it doth not pass the strait line D G; whereas the endeavour in A D, that is in H C, worketh further towards I. But seeing it doth only press and not penetrate, there will be reaction in H, which causeth motion from C towards H; and in the meantime the motion in H E remains the same it was in A H; and therefore the body will now be moved by the concourse of two motions in C H and H E, which are equal to the two motions it had formerly in A H and H C. Wherefore it will be carried on in C E. The angle therefore of reflection will be E C G, equal, by construction, to the angle A C D; which was to be demonstrated.