720 REPORT—1890. 
and rewards the pure mathematician, and upon which his best work is most 
profitably spent. I do not wish to under-estimate the importance of such a 
subject as Finite Differences, in which a number of distinct problems are treated 
with more or less success by interesting methods specially adapted to their solution. 
Nor would I willingly undervalue the interest of those branches of mathematics 
which we owe to the mathematical necessities of physical inquiry. But it always 
appears to me that there is a certain perfection, and also a certain luxuriance and 
exuberance, in the pure sciences which have resulted from the unaided, and I might 
almost say inspired, genius of the greatest mathematicians which is conspicuously 
absent from most of the investigations which have had their origin in the attempt 
to forge the weapons required for research in the less abstract sciences. To 
illustrate my meaning, I may take as an example of a subject of the latter class 
the theory of Bessel’s functions. The object of mathematicians in this case has 
been to investigate the properties of functions which have already presented them- 
selves in Astronomy and Physics. Formule for their calculation by means of 
series, continued fractions, definite integrals, &c., have been obtained in profusion, 
numerous theorems of various kinds and applicable to different purposes have been 
discovered, extensions and developments have been made in all directions, and 
finally the large body of interesting analysis thus accumulated has been classified 
and systematised. But, valuable and suggestive as are many of the results and 
processes, such a collection of facts and investigations is necessarily fragmentary. 
We do not find the easy flow or homogeneity of form which is characteristic of a 
mathematical theory properly so called. In such a theory, as for example the 
theory of double periodicity (elliptic functions), the subject develops itself 
naturally as it proceeds; one group of results leads spontaneously to another ; new 
and unexpected prospects open of themselves; ideas the most novel and striking, 
which penetrate the mind with a charm of their own, spring directly out of the 
subject itself. We are surprised by the wonderful connections with other subjects 
which unexpectedly start into existence, and by the widely different methods of 
arriving at the same truths; in fact, as our knowledge progresses, we continually 
find that results which seemed to lie far away in the interior of the subject—so 
remote and concealed that, at first sight, we might think that no other path except 
the one actually pursued could have reached them—are actually close to its edge 
when approached from another side, or viewed from another standpoint. We 
notice, too, that any great theory gives rise to its own special analysis or algebra, 
frequently connecting together into one whole what were hitherto merely isolated 
and apparently independent analytical results, and affording a reason for their 
existence, and also—what is often even more interesting—a reason for the non- 
occurrence of others which analogy might have led us to expect. I do not pretend 
that there are not many branches of mathematics which partake of both these 
characters, nor do I suppose that the description I have given of a mathematical 
theory is at all peculiar to pure mathematics. Much of it is common to all scientific 
research in a fruitful field, though, possibly, we may not find elsewhere such pro- 
fusion of ideas or perfection of form. 
I have been tempted to speak at such length on the objects and aims of the 
mathematician by the feeling that they are not infrequently misunderstood by the 
workers in the less abstract sciences. I do not think that mathematical formule 
or processes, merely as such, are much more interesting to the pure than to the 
applied mathematician. The one studies number, quantity, and position, the other 
deals with matter and motion ; and in both cases the investigations are carried on 
by means of the same symbolic language. 
The order in which the subjects which form an ordinary mathematical course 
are presented to the student is regulated by the fact that portions of the elements 
of the pure sciences are required for the explanation and development of any exact 
science ; for example, a knowledge of the elements of trigonometry, analytical geo- 
metry, and differential and integral calculus, must necessarily precede any adequate 
treatment of mechanics, light, or electricity. The majority of students, after 
mastering a sufficient amount of pure mathematics to enable them to pass on 
to the physical subjects, continue to devote their attention to the latter, and 
