On Geometrical Optics. 233 



Waves march always at right angles to their surfaces ; a 

 change in the form of the surface alters the direction of 

 march. The wave-surface is to be considered instead of 

 the " ray." The curvature of the surface therefore becomes 

 the all-important consideration. All that any lens or mirror 

 or any system of lenses or mirrors can do to a wave of light 

 is to imprint a curvature upon the surface of the wave. If 

 the wave is initially a plane wave, then the curvature imprinted 

 upon it by the lens or mirror will result in making it either 

 march toward a point (a real focus) or inarch as from a point 

 (a virtual focus) . If the wave possesses an initial curvature, 

 then all that the lens or mirror can do is to imprint another 

 curvature upon its surface, the resultant curvature being 

 simply the algebraic sum of the initial and the impressed 

 curvatures. As will be seen, in the new method the essential 

 thing to know about a lens or mirror is the curvature which 

 it can imprint on a plane wave : this is, indeed, nothing else 

 than what the opticians call its " power ;" the focal power 

 being inversely proportional to the so-called focal length. 

 Another, but less vital point in the method, is the abandon- 

 ment of the use of the so-called index of refraction in favour 

 of a quantity reciprocally related to it, and here denominated 

 the velocity-constant. The use of the index of refraction 

 dates from a time anterior to the discovery that refraction 

 was a mere consequence of the difference of velocity of the 

 waves in different media. The index of refraction is a mere 

 ratio between the sines (or originally the cosecants) of the 

 observed angles of incidence 'and refraction. The uselessness 

 of clinging to it as a foundation for lens formulae is shown by 

 the simple fact that, in order to accomplish the very first 

 stage of reasoning in the orthodox way of establishing the 

 formulae, we abandon the sines and write simply the cor- 

 responding angles, as Kepler did before the law of Snell was 

 discovered. The elementary formulae of lenses are, in fact, 

 where Kepler left them. It is now common knowledge 

 that the speed of light, on which refraction depends, is 

 less in optically dense media than in air. The speed of 

 light in air is not materially different from one thousand 

 million feet per second, or thirty thousand million centi- 

 metres per second. If we take the speed of light in air 

 as unity, then the numeric expressing the speed in denser 

 media, such as glass or water, will be a quantity less than 

 unity, and will differ for light of different wave-lengths. It 

 is here preferred to take the speed of light in air, rather than 

 in vacuo, as unity, because lenses and optical instruments 

 in general are used in the air. The numeric expressing the 



