MICROSCOPY. 441 
the secret of his peculiar belief in this case may be fully studied 
out, notwithstanding the unscientific method which he has chosen, 
in this instance, of appealing from principles to facts. 
The principle involved in this discussion has long been under- 
stood. An objective varies in working focal length, and in angular 
aperture, according to the medium through which it works; and 
this variation has a definite ratio to the refractive indices of the 
media compared. By a simple and undisputed mathematical 
computation, the sine of the semi-aperture in air is to the sine of 
its semi-aperture in another medium, as the index of refraction of 
that medium is to the index of air: or, as the index of air is unity, 
the sine of its semi-aperture in any medium is equal to the sine of 
its semi-aperture in air divided by the index of the other medium. 
This theoretical ratio is easily verified by experiment, as instanced 
by Mr. Brakey in the case under consideration, where an angle of 
145° in air should give a fraction over 91° in water and 79° in 
balsam so thin that its index was an arithmetical mean between 
that of balsam and that of turpentine, while in hard balsam hav- 
ing an index of 1.549 its aperture would have fallen to 76°. As 
the angle in air approaches the extreme limit of 170° or upwards, 
the balsam angle rises so slowly that the above 79° would scarcely 
reach 83°, the extreme angle for pure balsam being necessarily 
still smaller. This reasoning assumes only that the extreme ray 
above the front combination, capable of entering into the image 
when the objective is worked dry, is the extreme also when ad- 
justed for immersion work. 
Mr Tolles has aa declined either to accept or to con- 
trovert this well known theory, preferring simply to offer proof of 
his ability to excel this limit, without reconciling such result with 
the mathematical doctrine. Whether he utilizes rays beyond the 
extreme ray dry, or whether he measures rays not capable of form- 
ing a (good) image he does not state, and we can only conjecture. 
His ear ly publications seem to claim *‘ collecting” power for more 
extreme rays; but his letter to the March number of the Monthly 
Mic. Jour. practically disclaims this doctrine, and hints at a higher 
refracting power than crown glass has, in the front lens, as the 
Secret of his excessive angle. Curiously this letter happens to be 
published in the same number with Mr. Brakey’s explanation that 
the result is independent of the quality of the first lens, its index 
of refraction occurring twice in the computation in such positions 
as to cancel itself. 
