THE APEETOMETER 393 



It thus appears (1) that dry and immersion objectives having 

 different angular apertures, if of the same eqt< indent aperture, are 

 designated by the same term. Thus objectives of 60 in air, or 44 

 in water, or 38V 3 in oil, have identically the same aperture, and are 

 known by the same designation of '5 X.A. 



(2) The penetrating power of any objective is proportional to 



\r~r-, and its illuminating power to (X.A.) 2 . Therefore, if we 



double the X.A. we halve the penetrating power, and inci-ea.se the 

 illuminating power four times. 



In comparing the penetrating and illuminating powers of objec- 

 tives, however, care must be taken to avoid a popular error, by 

 making them between objectives of different foci. 



It cannot, for example, be said that a J-inch objective of '8 X.A. 

 has half the penetrating power of a Vincli of '4 X.A. Neither can 

 it be said that it has four times the illuminating power. What is 

 meant is that a ^-inch of - 8 X.A. has half the penetrating and four 

 times the illuminating power of a ^-inch objective of '4 X.A. 



But because penetrating and illuminating powers diminish as 

 the square of the foci, a Viiich objective of -(5 X.A. has four times 

 the illuminating and nearly four times the penetrating power of a 

 ^-iiich of (> X.A. ; but these conditions only hold when a full 

 illuminating cone is employed, in other words, when the back lens 

 of the objective, as seen when the eye-piece is removed, is full of 

 light. Thus if a small cone of illumination is used with the ^-inch 

 objective of '(5 X.A., its illuminating power would be much 

 diminished, while its penetrating power would be much increased. 



The old nomenclature, in use before numerical aperture was so 

 happily introduced, did not of course admit of comparisons of pene- 

 trating and illuminating powers by inspection ; which, however, is 

 a manifest advantage, Contributing to accuracy and precision in 

 important directions. 



(3) It may be well, for the sake of completeness, to repeat : here 

 that the resolving power of an objective is directly proportional to 

 its numerical aperture. If we double the X.A. we also double the 

 resolving power ; and this not simply with objectives of the same 

 foci, as in the case of penetrating and illuminating powers. Thus it 

 is not only true that a ^-inch objective of '6 X.A. resolves twice as 

 many lines to the inch as a ^-inch of '3 X.A., but so also does a 

 jr-inch of 1'4 X.A. resolve twice, and only twice, as manv as a J-inch 

 of -7 X\A. 



Within certain limits, then, the advantage lies with long foci of 

 wide angle, because we thus secure the greatest resolving power 

 with the greatest penetrating and illuminating powers. 



From what has here been shown, then, it becomes evident that 

 the employment of the microscope as an instrument of precision is 

 largely due to Abbe's work, and that the introduction of numerical 

 aperture, with its strictly accurate meaning, has been a practical 

 gain of untold value. But this has been greatly enriched by his 

 having introduced a thoroughly simple and useful apertometer. This 



1 Chapter I. 



