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TRANSACTIONS OF THE SECTIONS, 
Section AA-MATHEMATICAL AND PHYSICAL SCIENCE. 
PRESIDENT OF THE SECTION— 
Professor HENRICI, PH.D., F.R.S., President of the London Mathematical Society. 
THURSDAY, SEPTEMBER 20. 
The PresipeEntT delivered the following Address :— 
On reading through the addresses delivered by my predecessors in this office, I was 
struck by the fact that in nearly every case the speaker began with a lamentation 
over his unfitness for the work before him, and those seemed to me to be the more 
eloquent on these points who showed by their address that they least needed an 
excuse. The amount of excuse given appears in fact to be directly proportional 
to the gifts of the speaker, and hence inversely proportional to the need of such 
an excuse. 
Under these circumstances I cannot express my sense of my own unfitness for 
this post better than by saying nothing about it. I must, however, beg your 
indulgence for my shortcomings, both as regards my address and my manner of 
conducting the general business of this section, 
As the Presidential chair is occupied by one of the most illustrious of mathe- 
maticians, it would be presumptuous for me to attempt to give an account of the 
recent progress of mathematics. I propose only to speak for a short time on that 
part of mathematics which has always been most attractive to myself—that is, pure 
geometry as apart from algebra, but I shall confine myself to some considerations 
relating to the teaching of geometry in this country. Pure geometry seems to me 
to be of the greatest educational value, and almost indispensable in many appli- 
cations; but it has scarcely ever been introduced at Cambridge, the centre of 
mathematics and mathematical education in England. 
The number of geometrical methods now in use is astonishingly great. These 
differ on the one hand according to the nature of the result aimed at, but on the 
other according to the amount of algebra employed, and to the relation in which 
this algebra stands to the pure ‘ Anschauung.’ I use the word <Anschauung 
because I know of no English equivalent; the German word has the philosophic 
meaning rendered by intuition, and retains its original concrete meaning of looking 
at a thing, and might perhaps be translated: intuition by inspection. It is the 
inspection of figures which is of the greatest importance in geometry. It is 
hereby of little consequence whether the figures are seen by the physical eye or 
only mentally ; because the conception of that space in which we perceive every- 
thing and without which we can perceive nothing, which therefore is, according to 
Kant, a form of our Anschauung, is built up in our mind through many genera- 
tions in conformity with sensual impressions. 
It would be of interest, if time permitted, to follow up the gradual develop- 
ment and extension of geometry into the wider science of algebra, from the first 
introduction of the latter in the theory of proportion to the present state, where 
there exists really no essential difference between the two, where geometry 
is only one manifestation of algebra, but so complete a one, that at least within its 
number of dimensions it again contains algebra. 
In some of the methods just referred to, no algebra is used at all, whilst others 
may be distinguished according to the nature of the algebra used, whether 
