5°S 
The British Association . 
[October, 
plane surface and a straight line may be regarded as special 
instances of surfaces and lines whose curvature is every- 
where uniform or constant. And it is perhaps not difficult 
to sec that, when the special notions of flatness and straight- 
ness are abandoned, many properties of geometrical figures 
which we are in the habit of regarding as fundamental will 
undergo profound modification. Thus a plane may be con- 
sidered as a special case of the sphere, — viz., the limit to 
which a sphere approaches when its radius is increased 
without limit. But even this consideration trenches upon 
an elementary proposition relating to one of the simplest of 
geometrical figures. In plane triangles the interior angles 
are together equal to two right angles ; but in triangles 
traced on the surface of a sphere this proposition does not 
hold good. To this other instances might be added. The 
principle of representing space of one kind by that of 
another, and figures belonging to one by their analogues in 
the other, is not only recognised as legitimate in pure 
mathematics, but has long ago found its application in 
cartography. In maps or charts, geographical positions, 
the contour of coasts, and other features, belonging in 
reality to the earth’s surface, are represented on the flat ; 
and to each mode of representation, or projection as it is 
called, there corresponds a special correlation between the 
spheroid and the plane. To this might perhaps be added 
the method of descriptive geometry, and all similar pro- 
cesses in use by engineers, both military and civil. 
With regard to pure and applied mechanics, it has often, 
said Mr. Spottiswoode, been asked whether modern research 
in the field of pure mathematics has not so completely out- 
stripped its physical applications as to be practically useless ; 
whether the analyst and the geometer might not now, and 
for a long time to come, fairly say, “ Hie artem remumque 
repono,” and turn his attention to mechanics and physics. 
That the Pure has outstripped the Applied is largely true ; 
but that the former is on that account useless is far from 
true. Its utility often crops up at unexpected points ; wit- 
ness the aids to classification of physical quantities, furnished 
by the ideas (of Scalar and Vector) involved in the Calculus 
of Quaternions ; or the advantages which have accrued to 
physical astronomy from Lagrange’s Equations, and from 
Hamilton’s Principle of Varying Action ; on the value of 
Complex Quantities, and the properties of general Integrals, 
and of general theorems on integration for the Theories of 
Electricity and Magnetism. 
These extensions of mathematical ideas would, however, 
