ADDRESS. 25 
simple case, that of a body not acted upon by any forces, is a very 
interesting one in the mathematical point of view. 
I may mention a few other instances where a practical or physical 
question has connected itself with the development of mathematical 
theory. I have spoken of two map-projections—the stereographic, 
dating from Ptolemy; and Mercator’s projection, invented by Edward 
Wright about the year 1600: each of these, as a particular case of the 
orthomorphic projection, belongs to the theory of the geometrical repre- 
sentation of an imaginary variable. I have spoken also of perspective, 
and of the representation of solid figures employed in Monge’s descriptive 
geometry. Monge, it is well known, is the author of the geometrical 
‘theory of the curvature of surfaces and of curves of curvature: he was 
led to this theory by a problem of earthwork; from a given area, covered 
with earth of uniform thickness, to carry the earth and distribute it over 
an equal given area, with the least amount of cartage. For the solution 
‘of the corresponding problem in solid geometry he had to consider the 
intersecting normals of a surface, and so arrived at the curves of curvature. 
‘(See his ‘Mémoire sur les Deéblais et les Remblais,’ Mem. de l’Acad., 
1781.) The normals of a surface are, again, a particular case of a doubly 
infinite system of: lines, and are so connected with the modern theories of 
congruences and complexes. 
The undulatory theory of light led to Fresnel’s wave-surface, a 
surface of the fourth order, by far the most interesting one which had 
then presented itself. A geometrical property of this surface, that of 
having tangent planes each touching it along a plane curve (in fact, a 
circle), gave to Sir W. R. Hamilton the theory of conical refraction. 
‘The wave-surface is now regarded in geometry as a particular case of 
Kummer’s quartic surface, with sixteen conical points and sixteen sin- 
gular tangent planes. . 
My imperfect acquaintance as well with the mathematics as the 
physics prevents me from speaking of the benefits which the theory of 
Partial Differential Equations has received from the hydrodynamical 
‘theory of vortex motion, and from the great physical theories of heat, 
electricity, magnetism, and energy. 
It is difficult to give an idea of the vast extent of modern mathematics. 
This word ‘extent’ is not the right one: I mean extent crowded with 
beautiful detail—not an extent of mere uniformity such as an objectless 
plain, but of a tract of beautiful country seen at first in the distance, 
but which will bear to be rambled through and studied in every detail of 
hillside and valley, stream, rock, wood, and flower. But, as for anything 
else, so for a mathematical theory—beauty can be perceived, but not 
explained. As for mere extent, I can perhaps best illustrate this by 
‘speaking of the dates at which some of the great extensions have been 
‘made in several branches of mathematical science. 
