To these considerations it may be added, that as the angles BCH and FCA are each of them solely due to the refraction at C, as their common cause, they must be equal to each other, and BCA being common to both, the remaining angle ACH = the remaining angle BCF. We naturally begin by the consideration of the lowest ray FC, whose path being traced gives the direction of the two refracting sides BC and AC, leaving only the direction of the reflecting side BA to be determined. I shall not now explain the reason for neglecting entirely the consideration of the reflecting side at present, as I could not do so without anticipating what must be more fully discussed in the sequel; but I may content myself with stating, that as the positions of BC and AC depend upon the direction of the incident ray FC, and on the refractive index of the glass, this part of the investigation may be carried on apart from any interference with the reflecting side. As we know the relation existing between the angles of incidence and refraction, we might determine the relative positions of the sides AC and BC, by means of successive corrections obtained by protraction, tracing the paths of the rays from the horizontal directions backwards through the zones to the focus. This method, however, depends entirely upon accurate protraction, and is therefore unsatisfactory as a final determination, or if employed for any other purpose than that of affording a rough approximation to the value of the angle, a knowledge of which may occasionally save trouble in the employment of more exact means of determination. I have not, however, on any occasion employed this process, as I found that a little practice enabled me to make my first estimation very near the truth. I shall therefore at once proceed to give a view of the reasoning employed in the investigation. [Illustration: Fig. 72.] Referring to fig. 72, which shews the _first_ and _second_ zone of the upper series, we have FL Tan LCF = --; CL and if we make the known angle, SCF = α OCF = ξ = the complement of BCF = the angle of incidence for FC. DCΟ = γ = angle of refraction. LCF = θ = (HCF - 90°) = (2 α - 90°) SCD = (α + γ - ξ) And _m_ = the index of refraction for crown glass, we obtain the means of determining the angles γ and ξ in two equations, which are based upon the relation between the angles of incidence and refraction, and on the interdependence of the various angles about C. These primary equations are: sin ξ = _m_ . sin γ and [66]γ = 2 ξ - θ (making 2 α - 90° = θ)