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PRESIDENTIAL ADDRESS TO THE 
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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 and which are nevertheless untrue. 
I pass in review certain mathematical theories. 
In arithmetic and algebra, or say in analysis, the numbers or magnitudes 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, TJeber die negativen Grossen 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 fundamental one (and 
I cannot too strongly emphasise the assertion) underlying 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 I am aware, a subject of philosophical discussion or enquiry. As 
regards the older metaphysical writers this would be quite accounted for by saying 
that they knew nothing, and were not bound to know anything, about it; but at 
present, and, considering the prominent position which the notion occupies—say even 
that the conclusion were that the notion belongs to mere technical mathematics, or 
has reference to nonentities in regard to which no science is possible, still it seems to 
me that (as a subject of philosophical discussion) the notion ought not to be thus 
ignored; it should at least be shown that there is a right to ignore it. 
Although in logical order I should perhaps now speak of the notion just referred 
to, it will be convenient to speak first of some other quasi-geometrical notions; those 
of more-than-three-dimensional space, and of non-Euclidian two- and three-dimensional 
space, and also of the generalised notion of distance. It is in connexion with these 
that Riemann considered that our notion of space is founded on experience, or rather 
that it is only by experience that we know that our space is Euclidian space. 
It is well known that Euclid’s twelfth axiom, even in Playfair’s form of it, has 
been considered as needing demonstration; and that Lobatschewsky constructed a 
perfectly consistent theory, wherein this axiom was assumed not to hold good, or say 
a system of non-Euclidian plane geometry. There is a like system of non-Euclidian