The refractive index is the ratio between the sine of the angle of incidence on a lens surface and that of the angle of refraction in passing the surface. Fig. 38 shows the relation of the incident and refracted rays in passing from air into the glass lens surface _L_, and the sines of the angles which determine n, the conventional symbol for the index of refraction. Here _i_ is the angle of incidence and _r_ the angle of refraction i.e. n = _s_/_s′_. The indices of refraction are usually given for specific colors representing certain lines in the spectrum, commonly _A_¹, the potassium line in the extreme red, _C_ the red line due to hydrogen, _D_ the sodium line, _F_ the blue hydrogen line and _G′_ the blue-violet line hydrogen line, and are distinguished as n_{_c_}, n_{_d_}, n_{_f_}, etc. The standard dispersion (dn) for visual rays is given as between _C_ and _F_, while the standard refractivity is taken for _D_, in the bright yellow part of the spectrum. (Note. For the convenience of those who are rusty on their trigonometry, Fig. 39 shows the simpler trigonometric functions of an angle. Thus the sine of the angle _A_ is, numerically, the length of the radius divided into the length of the line dropped from the end of the radius to the horizontal base line, i.e. _bc_/_Ob_, the tangent is _da_/_Ob_, and the cosine _Oc_/_Ob_.) Ordinarily the index of refraction of the crown was taken as about 3/2, that of the flint as about 8/5. As time has gone on and especially since the new glasses from the Jena works were introduced about 35 years ago, one cannot define crowns and flints in any such simple fashion, for there are crowns of high index and flints of low dispersion. [Illustration: FIG.39.—The Simple Trigonometric Functions of an Angle.] The following table gives the optical data and chemical analyses of a few typical optical glasses. The list includes common crowns and flints, a typical baryta crown and light flint, and a telescope crown and flint for the better achromatization of objectives, as developed at the Jena works. The thing most conspicuous here as distinguishing crowns from flints is that the latter have greater relative dispersion in the blue, the former in the red end of the spectrum, as shown by the bracketed ratios. This as we shall see is of serious consequence in making achromatic objectives. In general, too, the values of ν for flints are much lower than for crowns, and the indices of refraction themselves commonly higher. As we have just seen, glass comes to the optician in blocks or discs, for miscellaneous use the former, three or four inches square and an inch think, more or less; for telescope making the latter. The discs are commonly some ten percent greater in diameter than the finished objective for which they are intended, and in thickness from 1/8 to 1/10 the diameter. They are commonly well annealed and given a preliminary polish on both sides to facilitate close inspection.