8 REPORT— 1883. 
to geometry, and will then speak of Riemann, but I will first refer to 
another passage of the Logic. 
Speaking of the truths of arithmetic Mill says (p. 297) that even here 
there is one hypothetical element: ‘In all propositions concerning 
numbers a condition is implied without which none of them would be 
true, and that condition is an assumption which may be false. The con- 
dition is that 1=1: that all the numbers are numbers of the same or of 
equal units.’ Here at least the assumption may be absolutely true; one 
shilling = one shilling in purchasing power, although they may not be 
absolutely of the same weight and fineness: but it is hardly necessary ; 
one coin + one coin = two coins, even if the one be a shilling and the 
other a half-crown. In fact, whatever difficulty be raisable as to 
geometry, it seems to me that no similar difficulty applies to arithmetic ; 
mathematician or not, we have each of us, in its most abstract form, the 
idea of a number; we can each of us appreciate the truth of a pro- 
position in regard to numbers; and we cannot but see that a truth in 
regard to numbers is something different in kind from an experimental 
truth generalised from experience. Compare, for instance, the proposition 
that the sun, having already risen so many times, will rise to-morrow, and 
the next day, and the day after that, and so on; and the proposition 
that even and odd numbers succeed each other alternately ad infinitum: 
the latter at least seems to have the characters of universality and 
necessity. Or again, suppose a proposition observed to hold good for a 
long series of numbers, one thousand numbers, two thousand numbers, as 
the case may be: this is not only no proof, but it is absolutely no evidence, 
that the proposition is a true proposition, holding good for all numbers 
whatever ; there are in the Theory of Numbers very remarkable instances 
of propositions observed to hold good for very long series of numbers, 
which are nevertheless untrue. 
I pass in review certain mathematical theories. 
In arithmetic and algebra, or say in analysis, the numbers or magni- 
tudes which we represent by symbols are in the first instance ordinary 
(that is, positive) numbers or magnitudes. We have also in analysis and 
in analytical geometry negative magnitudes ; there has been in regard to 
these plenty of philosophical discussion, and I might refer to Kant’s 
paper, ‘ Ueber die negativen Grdéssen in die Weltweisheit’ (1763), but the 
notion of a negative magnitude has become quite a familiar one, and has 
extended itself into common phraseology. I may remark that it is used 
in a very refined manner in bookkeeping by double entry. 
But it is far otherwise with the notion which is really the funda- 
mental one (and I cannot too strongly emphasise the assertion) under- 
lying and pervading the whole of modern analysis and geometry, that of 
imaginary magnitude in analysis and of imaginary space (or space as a 
locus in quo of imaginary points and figures) in geometry: I use in each 
case the word imaginary as including real. This has not been, so far as 
