490 
ARTICLE XIX. 
Abstract of some of the principal Propositions of Gauss’s 
Dioptrie Researches*. 
1. THE demonstrations here offered of some of the principal 
propositions of the Dioptric Researches of Gauss, though less 
elegant and perhaps less rigorous than those given by the author, 
are rather shorter, and more in accordance with the methods pur- 
sued by the optical writers of this country. 
The author begins by observing that the consideration of the 
path of a ray making a very small angle with the axis of the 
lenses through which it is refracted affords results of great ele- 
gance, which, though they may appear to have been exhausted 
by the labours of Cotes, Euler, Lagrange, and Mobius, still leave 
much to be desired. An essential defect in the propositions 
enunciated by those mathematicians is, that the thickness of the 
lens is neglected, an omission that impresses upon the investiga- 
tions a character of inaccuracy that greatly diminishes their value. 
The idea of the axis and of the focus of a lens is perfectly clear. 
Not so that of the focal length, which most writers define as the 
distance of the focus from the middle point of the lens, having 
previously assumed, either tacitly or expressly, that the thickness 
of the lens is indefinitely small, therefore in practice admit an 
inaccuracy of the order of the thickness of the lens. When an 
attempt is made to define the focal length more accurately, it is 
stated to be either the distance of the focus from the nearest sur- 
face of the lens, or its distance from the centre of the lens, or 
else its distance from a point half-way between the two surfaces 
of the lens. Different from all of these is that value of the foc 
length which must be assumed in order that the linear magnitude 
of the image of an indefinitely distant object may correspond 
to its angular magnitude. This last is in fact the only correc 
value of the focal length. 
The author then proceeds to say that he does not consider 1 
superfluous labour to devote a few pages to this perfectly elemen 
tary investigation, principally in order to show that, in the elegan 
* Communicated by W. H. Miller, Esq., Professor of Mineralogy in th 
University of Cambridge. 
