Geometrical Optics. 235 



But, for small apertures, AM is small compared with 2r, and 

 may be neglected in the denominator, whence 



MA ilj-1. 



Twice* the sagitta represents numerically the curvature. 

 The error is less than 1 per cent, when the semi-angle sub- 

 tended at the centre is 10°; less than two per cent, when it 

 is 15°; less than five per cent, when it is 25°. 



If the method of reckoning curvatures by means of the 

 sagitta required justification, that is afforded by the fact that 

 the practical method of measuring the curvatures of lenses 

 and mirrors by the spherometer consists essentially in applying 

 a micrometer-screw to measure the sagitta of the arc subtended 

 by a fixed chord, the diameter of the contact circle drawn 

 through the three feet of the instrument. In this case, as 

 indeed in all cases where accuracy, not approximation, is 

 desired, the basis for calculation of the correction exists in the 

 actual size of the diameter of the contact circle, which is a 

 fixed parameter for all measurements made with the instru- 

 ment. 



The sign of the curvature remains to be defined. In the 

 case of actual waves of light, the sign adopted will be + for 

 the curvature of waves which are converging upon a real 

 focus; — for those which are diverging either from a luminous 

 source or from a virtual focus. This agrees with the practice 

 of the ophthalmists and of the opticians, who always describe 

 a converging lens as positive. A positive lens is one which 

 imprints a positive curvature upon a plane wave which 

 traverses it. 



The unit of curvature, whether of the wave-surface itself 

 or of the surface of any mirror or lens, will be taken so as 

 to accord with modern ophthalmic and optical practice as the 

 dioptrie ; that is to say, the curvature of a circle of one 

 metre radius will be taken as unity. The dioptrie, originally 

 proposed by Monoyer as the unit of focal power of a lens, 

 was formally adopted in 1875 by the International Medical 

 Congress at Brussels, and its great convenience has led to its 

 universal adoption for the enumeration of the focal powers of 

 lenses. That lens which has a focal length of 1 metre is said 

 to have a focal power of one dioptrie. In other words, such 



* Though the sagitta is numerically half the curvature, since all the 

 formulas of first approximation are homogeneous and of the first degree as 

 regards sagittee and curvatures, the numerical factor .§ disappears in pass- 

 ing from sagittse to curvatures, or vice versa. 



