808 REPORT—1905. 
yary from year to year, but this variation is due to the predilection of the indi- 
vidual Presidents ; the types of address are but few in number, Sometimes, 
indeed, we have had addresses that cannot be ranged under any comprehensive 
type. Thus one year we had an account of a particular school of long-sustained 
consecutive research ; another year the President made a constructive (and perhaps 
defiant) defence of the merits of a group of subjects that were of special interest to 
himself, But there is one type of address which recurs with iterated frequency ; 
it is constituted by a general account of recent progress in discovery, or by a 
survey of modern advances in some one or other of the branches of science to 
which the multiple activities of our Section are devoted. No modern President 
has attempted a general survey of recent progress in all the branches of our group 
of sciences; such an attempt will probably be deferred until the Council discovers 
a President who, endowed with the omniscience of a Whewell, and graced with 
the tongue of men and of angels, shall once again unify our discussions. 
On the basis of this practice, it would have been not unreasonable on my part 
to have selected some topic from the vast range of pure mathematics, and to have 
expounded some body of recent investigations. There certainly is no lack of 
topics; our own day is peculiarly active in many directions, Thus, even if we 
leave on one side the general progress that has been made in many of the large 
branches of mathematics during recent years, it is easy to hint at numerous sub- 
jects which could occupy the address of a mathematical President. He might, 
for instance, devote his attention to modern views of continuity, whether of 
quantity or of space; he might be heterodox or orthodox as to the so-called 
laws of motion; he might expound his notions as to the nature and properties of 
analytic functionality ; a discussion of the hypotheses upon which a consistent 
system of geometry can be framed could be made as monumental as his ambition 
might choose; he could revel in an account of the most recent philosophical 
analysis of the foundations of mathematics, even of logic itself, in which all 
axioms must either be proved or be compounded of notions that defy resolution 
by the human intellect at the present day. Such discussions are bound to be 
excessively technical unless they are expressed in unmathematical phraseology ; 
when they are so expressed, and in so far as such expression is possible, they 
become very long and they can be very thin. Moreover, had I chosen any topic of 
this character, it would have been the merest natural justice to have given early 
utterance of the sibyllic warning to the uninitiated ; I must also have bidden the 
initiated that, as they come, they should summon all the courage of their souls. 
So I abstain from making such an experiment upon an unwarned audience; yet 
it is with reluctance that I have avoided subjects in the range which to me is of 
peculiar interest. 
On the other hand, I must ask your indulgence for not conforming to average 
practice and expectation, My desireis to mark the present occasion by an address 
of unspecialised type which, while it is bound to be mainly mathematical in 
tenor, and while it will contain no new information, may do little more than 
recall some facts that are known, and will comment briefly upon obvious 
tendencies. Let me beg you to believe that it is no straining after novelty which 
has dictated my choice; such an ambition has a hateful facility of being fatal 
both to the performer and to the purpose. It is the strangeness of our circum- 
stances, both in place and time, that has suggested my subject. With an adven- 
turous audacity that quite overshadows the spirit of any of its past enterprises, the 
British Association for the Advancement of Science has travelled south of the 
Equator and, in accepting your hospitality, proposes to traverse much of South 
Africa. The prophet of old declared that ‘many shall run to and fro, and know- 
ledge shall be increased’ ; if the second part of the prophecy is not fulfilled, it will 
not be for the want of our efforts to fulfil the first part. Andif the place and the 
range of this peripatetic demonstration of our annual corporate activity are 
unusual, the occasion chosen for this enterprise recalls memories that are funda- 
mental in relation to our subject. It is a modern fashion to observe centenaries, 
In this Section we are in the unusual position of being able to observe three 
scientific centenaries in one and the same year. Accordingly I propose to refer to 
