272 
CORRESPONDENCE. 
On the second page, read " ended in the vague." On the third, 
undulating again for modulating ; Czermak for Azermath ; and, 
lastly, substitute for ^ig. 
In note, " my " should be " our." 
Very respectfully, 
Christopher Johnston, M.D. 
Professor Abbe on Dr. Pigott's Paper " On the Present 
Limits of Vision. " 
To the Editor of the ' Monthly Microscopical Journal.^ 
Jena University, October 9, 1876. 
Sir, — In the October issue of the ' Monthly Microscopical Journal ' 
I find a paper by Dr. Royston-Pigott, " On the Present Limits of 
Vision," from which I understand that Mr. Helmholtz and myself 
have " jiopularized " a doctrine of Lagrange about the limits of 
vision, as deduced from the wave-principle. In particular. Dr. 
Royston-Pigott, p. 180, quotes as "deduced from those of Lagrange" 
a formula which is the keystone of our theory, establishing the exact 
relation between the limits of visibility and the angular aperture of 
an optical system. 
I myself till now was not aware of this connection with Lagrange, 
nor will Mr. Helmholtz be, I am sure. I was of opinion that the 
said doctrine as a whole, and the quoted formula in particular, had 
been advanced by me for the first time in 1873 in Max Schultze's 
' Archiv,' and a short time afterwards — quite independently and from 
a difi'erent point of view — by Mr. Helmholtz in the ' Proceedings of 
the Berlin Academy ' and in Poggendorfi"s ' Annalen.' 
To my knowledge there is only a theorem by Lagrange of a 
purely geometrical character, which Mr. Helmholtz quotes and takes 
as a starting point in his deduction ; but this theorem has no relation 
at all either to the wave-principle or to the dijBfraction theory, and, 
besides, is essentially limited to the supposition of infinitely small 
apertures. For both reasons this theorem — either by itself or with 
all the other propositions in Lagrange's paper " Sur une Loi 
generale d'Optique " — is manifestly insufficient to form any conclu- 
sion aiming at the limits of vision ; for such a conclusion involves a 
complete theory of the difiraction effect in opticai systems, and at the 
same time a proposition about the convergence of pencils in aplanatic 
systems of finite (i. e. great) apertures. 
Now, from Dr. Royston-Pigott's decided assertions, I must infer, 
as many readers of the ' Monthly Microscopical Journal ' will do, 
that in some paper hitherto ignored by other writers Lagrange 
has already expressed a clear notion of the said problem in general, 
and, besides that, has treated the special questions upon which the 
solution essentially depends. 
