18 REPORT—1883. 
the plane of the picture of the line joining the two positions of the 
eye); the figure in space is thus represented on the plane by two figures, 
which are such that the lines joining corresponding points of the two 
figures pass always through the fixed point. And such two figures com- 
pletely replace the figure in space ; we can by means of them perform 
on the plane any constructions which could be performed on the figure in 
space, and employ them in the demonstration of properties relating to 
such figure. A curious extension has recently been made: two figures 
in space such that the lines joining corresponding points pass through a 
fixed point have been regarded by the Italian geometer Veronése as repre- 
sentations of a figure in four-dimensional space, and have been used for 
the demonstration of properties of such figure. 
I referred to the connection of Mathematics with the notions of 
space and time, but I have hardly spoken of time. It is, I believe, usually 
considered that the notion of number is derived from that of time; thus 
Whewell in the work referred to, p. xx, says number is a modification of 
the conception of repetition, which belongs to that of time. I cannot 
recognise that this is so: it seems to me that we have (independently, I 
should say, of space or time, and in any case not more depending on time 
than on space) the notion of plurality ; we think of, say, the lettersa, b, c, 
&c., and thence in the case of a finite set—for instance a, b, c, d, e—we 
arrive at the notion of number; coordinating them one by one with any 
other set of things, or, suppose, with the words first, second, &c., we find 
that the last of them goes with the word fifth, and we say that the number 
of things is = five: the notion of cardinal number would thus appear to 
be derived from that of ordinal number. 
Questions of combination and arrangement present themselves, and 
it might be possible from the mere notion of plurality to develope a 
branch of mathematical science; this, however, would apparently be of a 
very limited extent, and it is difficult not to introduce into it the notion 
of number; in fact, in the case of a finite set of things, to avoid asking the 
question, How many? If we do this, we have a large enough subject, 
including the partition of numbers, which Sylvester has called Tactic. 
From the notion thus arrived at of an integer number, we pass to that 
of a fractional number, and we see how by means of these the ratio of 
any two concrete magnitudes of the same kind can be expressed, not 
with absolute accuracy, but with any degree of accuracy we please: for 
instance, a length is so many feet, tenths of a foot, hundredths, thousandths, 
&c.; subdivide as you please, non constat that the length can be expressed 
accurately, we have in fact incommensurables; as to the part which these 
‘play in the Theory of Numbers, I shall have to speak presently: for the 
moment I am only concerned with them in so far as they show that 
we cannot from the notion of number pass to that which is required 
in analysis, the notion of an abstract (real and positive) magnitude 
susceptible of continuous variation, The difficulty is got over by a 
