4.28 REPORT—1904, 
It is commonly reckoned that the first rude beginnings of Geometry date from 
the Egyptians. I am inclined to think that in one sense the matter is to be 
placed much further back, and that the dawn of geometric ideas is to be traced 
among the prehistoric races who carved rough but thoroughly artistic outlines of 
animals on their weapons. I do not know whether the matter has attracted 
serious speculation, but I have myself been led to wonder how men first arrived 
at the notion of an outline drawing. The primitive sketches referred to imme- 
diately convey to the experienced mind the idea of a reindeer or the like; but 
in reality the representation is purely conventional, and is expressed in a lan- 
guage which has to be learned. For nothing could be more unlike the actual 
reindeer than the few scratches drawn on the surface of a bone; and it is of 
course familiar to ourselves that it is only after a time, and by an insensible 
process of education, that very young children come to understand the meaning 
of an outline. Whoever he was, the man who first projected the world into two 
dimensions, and proceeded to fence off that part of it which was reindeer from 
that which was not, was certainly under the influence of a geometrical idea, and 
had his feet in the path which was to culminate in the refined idealisations of the 
Greeks, As to the manner in which these latter were developed, the only indica- 
tion of tradition is that some propositions were arrived at first in a more empirical 
or intuitional, and afterwards in a more intellectual way. So long as points had 
size, lines had breadth, and surfaces thickness, there could be no question of exact 
relations between the various elements of a figure, any more than is the case with 
the realities which they represent. But the Greek mind loved definiteness, and 
discovered that if we agree to speak of lines as if they had no breadth, and so 
on, exact statements became possible. If any one scientific invention can claim pre- 
eminence over all others, I should be inclined myself to erect a monument to the 
unknown inventor of the mathematical point, as the supreme type of that process 
of abstraction which has been a necessary condition of scientific work from the very 
beginning. 
It is possible, however, to uphold the importance of the part which Abstract 
Geometry has played, and must still play, in the evolution of scientific concep- 
tions, without committing ourselves to a defence, on all points, of the traditional 
presentment. The consistency and completeness of the usual system of defini- 
tions, axioms, and postulates has often been questioned; and quite recently a 
- more thorough-going analysis of the logical elements of the subject than has ever 
before been attempted has been made by Hilbert. The matter is a subtle one, 
and a general agreement on such points is as yet hardly possible. The basis for 
such an agreement may perhaps ultimately be found in a more explicit recognition 
of the empirical source of the fundamental conceptions. This would tend, at all 
events, to mitigate the rigour of the demands which are sometimes made for 
logical perfection. 
Even more important in some respects are the questions which have arisen in 
connection with the applications of Geometry to purposes of graphical representa- 
tion. It is not necessary to dwell on the great assistance which this method has 
rendered in such subjects as Physics and Engineering. The pure mathematician, 
for his part, will freely testify to the influence which it has exercised in the 
development of most branches of Analysis; for example, we owe to it all the 
leading ideas of the Calculus. Modern analysts have discovered, however, that 
Geometry may be a snare as well as a guide. In the mere act of drawing a curve 
to represent an analytical function we make unconsciously a host of assumptions 
which are difficult not merely to prove, but even to formulate precisely. It is now 
sought to establish the whole fabric of mathematical analysis on a strictly arith- 
metical basis. To those who were trained in an earlier school, the results so far are 
in appearance somewhat forbidding. If the shade of one of the great analysts of a 
century ago could revisit the glimpses of the moon, his feelings would, I think, be 
akin to those of the traveller to some medieval town, who finds the buildings he 
came to see obscured by scaffolding, and is told that the ancient monuments are 
all in process of repair. It is to be hoped that a good deal of this obstruction is 
only temporary, that most of the scaffolding will eventually be cleared away, and 
