us, not the array of symbols and brackets which form the armoury of
the mathematician, or the dry results which are only the monuments of
his conquests, but the mathematician himself, with all his human
faculties directed by his professional sagacity to the pursuit,
apprehension, and exhibition of that ideal harmony which he feels to
be the root of all knowledge, the fountain of all pleasure, and the
condition of all action. The mathematician has, above all things, an
eye for symmetry; and Professor Sylvester has not only recognized the
symmetry formed by the combination of his own subject with those of
the former Presidents, but has pointed out the duties of his successor
in the following characteristic note:--

"Mr Spottiswoode favoured the Section, in his opening Address, with a
combined history of the progress of Mathematics and Physics; Dr.
Tyndall's address was virtually on the limits of Physical Philosophy;
the one here in print," says Prof. Sylvester, "is an attempted faint
adumbration of the nature of Mathematical Science in the abstract.
What is wanting (like a fourth sphere resting on three others in
contact) to build up the Ideal Pyramid is a discourse on the Relation
of the two branches (Mathematics and Physics) to, their action and
reaction upon, one another, a magnificent theme, with which it is to
be hoped that some future President of Section A will crown the
edifice and make the Tetralogy (symbolizable by _A+A'_, _A_, _A'_,
_AA'_) complete."

The theme thus distinctly laid down for his successor by our late
President is indeed a magnificent one, far too magnificent for any
efforts of mine to realize. I have endeavoured to follow Mr
Spottiswoode, as with far-reaching vision he distinguishes the systems
of science into which phenomena, our knowledge of which is still in