Some Remarks on the Ajpertometer. By Prof. E. Ahhe, 29 



mating at a glance the fifths of an interval. The possible error of 

 this estimation will hardly exceed half the unit of the second decimal, 

 since nobody with a moderate amount of care would read • 03 if the 

 point in question is decidedly nearer to ' 02 than to • 03. The 

 error of measurement will be limited, therefore, to about ^ per cent. 

 of the unit, corresponding to half a degree, or 30' in angle, as far 

 as the actual reading is concerned. 



An exactness of reading to this extent is evidently more than 

 suflScient. An unavoidable amount of uncertainty resulting from 

 the nature of the object, and many other sources of slight error, will 

 always limit the real exactness of observation beyond 1 per cent, of 

 the unit, different observers and different methods of equal reliability 

 being supposed. In low powers slight deviations in the length of 

 tube ; in high powers slight alterations of the cover-adjustment, will 

 admit of much greater difference than the error of reading will 

 introduce. It should be observed that in high-angled objectives 

 the aperture has not the same value for different colours, owing to 

 the difference of focal length (or of amplification), even in objectives, 

 which are perfectly achromatic in the ordinary sense. In the case 

 of very large angles, the aperture, angular or numerical, will be 

 greater for the blue rays than for the red, generally by more than 

 1 per cent. Last, not least, there is no possible interest, either 

 practical or scientific, appertaining to single degrees, or half-degrees, 

 of apertm-e angles ; for no microscopist in the world will be able to 

 make out any difference in the performance of objectives, as long as 

 the numerical apertures do not differ by several per cent., other 

 circumstances being equal. 



For these reasons I consider all attempts at very accurate 

 measurements of this kind to be useless. 



There has been made another objection, from quite a different 

 point of view, by Mr. Wenham,* who declares the apertometer to 

 measure the angle of field, instead of, or in addition to, the angle of 

 aperture. I hope Mr. Wenham will abandon this objection after 

 having considered the dioptrical proposition spoken of early in this 

 paper, and its bearing on the subject. By reducing to a pin-holo 

 the critical diaphragm in apertometric observation he will be able 

 to confine the entering pencils to one-hundredth of the ordinary 

 field of vision of the object-glass. But then he will at once 

 perceive that a greater or less diameter of the entrance-area has 

 no influence at all on the outcome of the observation, not even 

 if this area should be much greater than the field of vision in the 

 ordinary microscopic use of the objective, provided the diaphragm 

 be in its correct position. Too large a field taken as entrance-area 

 in telescopic vision will deteriorate definition and for this reason 



* ' Am. Quar. Micr. Journ.,' i. (1879) p. 280. 



