726 REPORT—1890. 
greatest and most perfect of all the mathematical theories, should have been so 
little cultivated in this country, and that no portion of it should ever have been 
included in an ordinary course of mathematical study. It may be said to date from 
the year 1801, when Gauss published his ‘ Disquisitiones Arithmetice,’ so that it is 
nearly thirty years older than the Theory of Elliptic Functions, to which we 
may assion the date 1829, the year in which Jacobi’s ‘Fundamenta Nova’ appeared. 
But the latter theory has already found a congenial home among us, while the 
former is nowhere systematically studied, and is still without a text-book. The 
chapters in books upon Algebra which bear the title ‘Theory of Numbers’ give a 
misleading idea of the nature of the subject, the results there given being mainly 
introductory lemmas of the simplest kind. The theory has nothing to do with arith- 
metic in the ordinary sense of the word, or numerical tables, or the representation of 
numbers by figures in the decimal system or otherwise. All its results are actual 
truths of the most fundamental kind, which must exist in rerwm naturd. Its principal 
branches are the theory of forms and the so-called complex theories. Such a pro- 
position as that every prime number, which when divided by 4 leavesremainder 1, 
can always be expressed as the sum of two squares, and that this can be done in 
one way only, affords a good example of a very simple result in the theory of forms. 
It is entirely independent of any method of representing numbers, and merely asserts 
that if we have 5, 13, 17, 29,.&c., things—let us say marbles, to fix the ideas—we 
. can always succeed in so arranging them as to form them into two squares, and 
that for each number we can do this in but one way. Simple as such a theorem is 
to enunciate and comprehend, the demonstration is far from easy. This is charac- 
teristic of the whole subject ; simple propositions, which we can easily discover by 
trial, and of the universal truth of which we can feel but little doubt, require for 
their demonstration a refined and intricate analysis, founded upon the most difficult 
and imaginative conceptions which mathematics has as yet attained to in its 
struggles to grapple with the actual problems of the worlds of thought and matter. 
The theory of quantity consists of two distinct branches—one relating to discrete 
quantity and the other to continuous quantity. To the latter branch belong 
algebra and all the ordinary subjects of pure mathematics; the former bears the 
name of the theory of numbers. Its truths are of the most absolute kind, involving 
only the notions of number and arrangement; in fact, if we imagine all the exact 
sciences ranged in order, it naturally takes its place at one end of the series. Different 
sciences appeal to different intellects with very different force, but there are some 
minds over which the absolute character of the fundamental truths that belong to 
this theory and the absolute precision of its methods exercise the strongest fascina- 
tion, and excite an interest which neither the truths of geometry nor the most 
important discoveries depending upon the constitution of matter are capable of 
producing. 
Many of the greatest masters of the mathematical sciences were first attracted to 
mathematical inquiry by problems relating to numbers, and no one can glance at the 
periodicals of the present day which contain questions for solution without noticing 
how singular a charm such problems still continue to exert. This interest in numbers 
seems implanted in the human mind, and it is a pity that it should not have freer 
scope in this country. The methods of the theory of numbers are peculiar to itself, 
and are not readily acquired by a student whose mind has for years been familiarised 
with the very different treatment which is appropriate to the theory of continuous 
magnitude ; it is therefore extremely desirable that some portion of the theory 
should be included in the ordinary course of mathematical instruction at our 
Universities. From the moment that Gauss, in his wonderful treatise of 1801, laid 
down the true lines of the theory, it entered upon a new day, and no one is likely 
to be able to do useful work in any part of the subject who is unacquainted with 
the principles and conceptions with which he endowed it. 
Undoubtedly the subject is a difficult and intricate one even in its elementary 
parts, but there can be but little doubt that when the processes which are now 
only read by specialists on their way to the border become more generally known 
and studied, they will be found to admit of great simplification, It is in fact a 
territory where there is quite as much scope for the mathematician in simplifying 
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