DERIVATION OF THE 'UNIT CONE Referring now to the spherical cone in which the diameter of the base equals the altitude, it follows that it might be described as being equivalent to a relative aperture of f/ 1, and as every point of the surfaces about us in nature is a point of convergence of cones of light emitted from all the objects which confront it, it be comes necessary to evaluate these cones as a means of determining one of the elements of actinicity. A unit of measurement must then be sought and defined.
Since it is the custom to graduate lens dia phragm scales so that each number in the series indicates a speed half that of the next lower, the idea was suggested that by successively halving a spherical surface a small cone might be attained approximating in value one of the smallest practical values in the f system. This is evidently the case since an f / 1 cone has been seen to be approximately of a sphere. Now since f / 64 is the smallest practical relative aper ture used in photography, it is at once suggested to adopt the corresponding cone as the unit cone. Since f / 1 equals =6 of a sphere, and the cone values diminish inversely to the squares of the f values, f / 64 will equal (A x 4) = 65536 of a sphere. As mathematical accuracy is not demanded this fraction may conveniently be abbreviated to .‘ ,.4000 and may be written ab as has already been explained. By the adop tion of this particular number the geometrical progression with ratio 2 is preserved which much facilitates mental calculations. Practi cally all calculations in this treatise are based on this geometrical progression which has al ready been employed in the chapter on time.
Since it has been shown that the unit cone system and the f system of designating lens speeds are entirely consistent mutually, it is the present purpose to extend the latter as a basis for the calculation involved in many other problems in light. For example if a circular illuminated disk of any diameter, as a b, in the accompanying diagram, be thought of as sending light from each of its points to a point c whose distance from a b is equal to the diameter of the disk a b we may speak of the distance from a b to c as the focal length of the cone and in this case also the relative aperture or the solid angle of the cone at c would be f / 1 just as if a b were a lens transmitting light. Just as in the case of a lens it is the ratio of a b to the distance of c which is important and upon which the working speed depends, and not the value of either alone. It is thus seen that the f system so long used in de noting lens speeds is equally applicable to the ordinary conditions of light in nature. It should be remembered however that the f system is only a convenience in describing the form of cones, and that comparisons and computation are possi ble only on the basis of the solid angle of the cones, which is their basic, or working-quantity value. Hence the need of the cone unit valua tions on lens stops instead of the f and U. S. scales which confuse by not evaluating. In this case the cone value would be 4,096 or 4M units as will be seen from the table which shortly will follow.