of Edinburgh, Session 1870-71. 413 
stration to an audience as a whole not mathematical. In no part 
of Physics have I found this so marked as in the most elementary 
portions of geometrical optics. Such a formula as 
when interpreted directly as signifying that “the sum of the 
reciprocals of the distances of the object and image from the sur- 
face of a concave spherical mirror, is equal to double the reciprocal 
of the radius of the mirror,” if understood at. all, is understood as a 
sort of memoria technica which enables the student to make calcu- 
lations; but unless he have some knowledge of mathematics it 
suggests absolutely no higher meaning. If, however, we give to 
the various terms of the formula their meanings in terms of the 
divergence of the incident and reflected beams, and of the normals 
to the reflecting surface, even the non-mathematical student easily 
understands the relation signified. I am indebted to Mr Sang for 
a reference to Lloyd On Light and Vision , 1831, in which this 
mode of presenting the subject is introduced, but I think the term 
“vergency” there used is hardly so convenient as the more com- 
monly employed word divergence. Our fundamental optical fact 
is that to produce the most distinct vision rays must diverge as if 
from a point about ten inches from the eye. No one has any diffi- 
culty in understanding this. As my object has been merely to men- 
tion to the Society what I have found to be a method (however 
trivial in itself, yet) of really considerable importance in teaching, 
I need do no more than give one simple example of its application, 
and that only to direct pencils of such small divergence that spheri- 
cal aberration may be neglected, A perfectly obvious set of modi- 
fications is introduced when we treat of oblique pencils, and pencils 
of large divergence, but students capable of understanding these 
do not require the adoption of such elementary methods of ex- 
planation. 
Take, then, the case of light refracted at a concave spherical sur- 
face, bounding a substance denser than air. If the incident and 
refracted rays make (small) angles a and /3 with the axis of the 
surface, and if y be the angle between the normal at the point of 
VOL, vii. 3 K 
