BN, a common tangent to all these partial waves.  And if one would know how the wave AC has come progressively to BN, it is necessary only to draw in the same figure the straight lines ko parallel to BN, and all the lines Kl parallel to AC.  Thus one will see that the wave CA, from being a straight line, has become broken in all the positions LKO successively, and that it has again become a straight line at BN.  This being evident by what has already been demonstrated, there is no need to explain it further.

Now, in the same figure, if one draws EAF, which cuts the plane ab at right angles at the point A, since ad is perpendicular to the wave AC, it will be DA which will mark the ray of incident light, and an which was perpendicular to BN, the refracted ray:  since the rays are nothing else than the straight lines along which the portions of the waves advance.

Whence it is easy to recognize this chief property of refraction, namely that the Sine of the angle Dae has always the same ratio to the Sine of the angle Naf, whatever be the inclination of the ray DA:  and that this ratio is the same as that of the velocity of the waves in the transparent substance which is towards AE to their velocity in the transparent substance towards AF.  For, considering ab as the radius of a circle, the Sine of the angle BAC is bc, and the Sine of the angle ABN is an.  But the angle BAC is equal to Dae, since each of them added to CAE makes a right angle.  And the angle ABN is equal to Naf, since each of them with Ban makes a right angle.  Then also the Sine of the angle Dae is to the Sine of Naf as bc is to an.  But the ratio of bc to an was the same as that of the velocities of light in the substance which is towards AE and in that which is towards AF; therefore also the Sine of the angle Dae will be to the Sine of the angle Naf the same as the said velocities of light.

To see, consequently, what the refraction will be when the waves of light pass into a substance in which the movement travels more quickly than in that from which they emerge (let us again assume the ratio of 3 to 2), it is only necessary to repeat all the same construction and demonstration which we have just used, merely substituting everywhere 3/2 instead of 2/3.  And it will be found by the same reasoning, in this other figure, that when the piece C of the wave AC shall have reached the surface ab at B, all the portions of the wave AC will have advanced as far as BN, so that bc the perpendicular on AC is to an the perpendicular on BN as 2 to 3.  And there will finally be this same ratio of 2 to 3 between the Sine of the angle Bad and the Sine of the angle Fan.

Hence one sees the reciprocal relation of the refractions of the ray on entering and on leaving one and the same transparent body:  namely that if Na falling on the external surface ab is refracted into the direction ad, so the ray ad will be refracted on leaving the transparent body into the direction an.