332 Transactions of the Society. 
chromatics, the goal to aim at is • 26 N.A. for every 100 diameters 
of amplification. 
We see, therefore, that Carpenter has in this table hit upon the 
ideal for the most critical work with low powers. It is easy to see 
how he arrived at it. These are the lenses he worked with, the plane 
mirror and daylight would yield a 3/4 cone, and the cone would be 
focused on the object. He therefore unconsciously attained a critical 
image, and working largely with these lenses, he came to the only 
conclusion he could with justice arrive at, viz. that a large aperture 
gave the best results. 
As his powers advance his images become more and more uncritical, 
and consequently we see the ratios of aperture to power becoming 
more and more fantastic. Taking the 1 /4 for instance, 50° to 80°, 
or *42 N.A. to *64 N.A., with powers of 200 to 300, if he had 
been a consistent reasoner, his 1 -in. objective ought to have had 15° of 
aperture, or a trifle less than that possessed by the earliest achromatic 
inches made in this country. 
In my own mind there is not the slightest doubt but that a 1/4 
should be capable of bearing a power of 400 diams. ; therefore, in 
accordance with the ideal stated above, viz. '26 N.A. per 100 diams., 
its aperture ought to be 1 * 04 N.A. It is nevertheless highly 
impolitic to make the 1/4 an immersion, therefore ‘95 N.A. is the 
aperture it ought to have, because that is the highest aperture that 
can he advantageously put into a dry lens. 
With regard to this table of apertures, Dr. Carpenter says, “He 
has the satisfaction of finding that his opinions on this point, which 
are based on long experience in the microscopic study of a wider range 
of animal and vegetable objects than has fallen within the purview of 
most of his contemporaries, are in accordance with the conclusions 
drawn by Professor Abbe from his profound investigations into the 
theory of microscopic vision.” It will be as well to turn to the 
reference given (this Journal, 1882, pp. 300, 460), and see what are 
the ratios of aperture to power that Abbe gives: — 1-in., power 
40-80, aperture 18°; 1 /2-in., power 80-160, aperture 35°; 1/4-in., 
aperture 60° : we see therefore that it is only in the 1/4-in. that they 
agree ; in respect of Carpenter’s favourite objective, the inch, they dis- 
agree very materially. 
Abbe, although his conclusions are fallacious, is at least consistent 
with regard to the apertures of his 1-in. and 1/4- in., which Carpenter 
is not : this entirely cuts away the ground for Dr. Carpenter’s 
satisfaction. 
The views on penetrating powers which find acceptance among 
microscopists at the present time are those laid down by Prof. Abbe 
in this Journal. 
He divides the penetrating power into two parts, one he calls 
“ the accommodation depth,” the other “ the focal depth,” and he 
states that “ the visual depth ” is the sum of these two quantities. 
