262 
NATURE 
[ Fan. 6, 1870 
Others find its justification, its “ raison d’étre,” in its being 
either the torch-bearer leading the way, or the hand- 
maiden holding up the train of Physical Science; anda 
very clever writer in a recent magazine article, expresses 
his doubts whether it is, in itself, a more serious pursuit, or 
more worthy of interesting an intellectual human being, than 
the study of chess problems or Chinese puzzles.* What is it 
to us, they say, if the three angles of a triangle are equal 
to two right angles, or if every even number is, or may be, 
the sum of two primes,f or if every equation of an odd 
degree must have a real root? How dull, stale, flat and 
unprofitable are such and such like announcements! 
Much more interesting to read an account of a marriage 
in high life, or the details of an international boat-race. 
But this is like judging of architecture from being shown 
some of the brick and mortar, or even a quarried stone of 
a public building—or of painting from the colours mixed on 
the palette, or of music by listening to the thin and 
screechy sounds produced by a bow passed haphazard 
over the strings of a violin. The world of ideas which it 
discloses or illuminates, the contemplation of divine 
beauty and order which it induces, the harmonious con- 
nexion of its parts, the infinite hierarchy and absolute 
evidence of the truths with which mathematical science 
is concerned, these, and such like, are the surest grounds 
of its title to human regard, and would remain unimpaired 
were the plan of the universe unrolled like a map at our 
feet, and the mind of man qualified to take in the whole 
scheme of creation at a glance. 
In conformity with general usage, I have used the word 
mathematics in the plural ; but I think it would be desir- 
able that this form of word should be reserved for the 
applications of the science, and that we should use mathe- 
matic in the singular number to denote the science itself, 
in the same way as we speak of logic, rhetoric, or (own sister 
to algebrat) music. Time was when all the parts of the 
subject were dissevered, when algebra, geometry, and 
arithmetic cither lived apart or kept up cold relations 
of acquaintance confired to occasional calls upon one 
another; but that is now at an end; they are drawn 
together and are constantly becoming more and more 
intimately related and connected by a thousand fresh 
ties, and we may confidently look forward to a time 
when they shall form but one body with one soul. 
Geometry formerly was the chief borrower from arith- 
metic and algebra, but it has since repaid its obligations 
with overflowing usury ; and if I were asked to name, 
in cne word, the pole-star round which the mathematical 
firmament revolves, the central idea which pervades as a 
hidden spirit the whole corpus of mathematical doctrine, 
I shculd point to Continuity as contained in our notions 
* Ts it not the same disregard of principles, the same indifference to truth 
for its own sake, which prompts the question ‘‘ Where’s the good of it?” in 
reference to speculative science, and ‘** Where’s the harm of it?” in reference 
to white lies and pious frauds? In my own experience I have found that the 
very sume people who delight to put the first question are in the habit of 
acting upon the denial implied in the second. Adit im mores incurta. 
+ ‘This theorem still awaits proof; it is stated, I believe, in Euler's corre- 
spondence with Goldbach; I re-discovered it in ignorance of Euler’s having 
mentioned it, in connection with a theory of my own concerning cubic forms. 
‘The evidence in its fayour is zdwe¢évx of the undemonstrative or purely accu- 
mulative kind, and it may or may not turn out eventually to be true. As 
a most learned scholar who heard this address given at Exeter remarked to 
me not many days ago, it is certainly by no process of deduction that we 
make out that five times six is thirty. I mention this, because I know some, 
who agree, or did agree, with Professor Huxley’s published opinions about 
mathematics, are under the impression that the higher processes of mind in 
mathematics only concern ‘‘the aristocracy of mathematicians:” on the 
contrary, they lie at the very foundations of the subject. There are besides, 
and in abundance, mathematical processes which only by a forced interpre- 
tation can be brought under the head of demonstration, whether deductive or 
inductive, and really belong to a sort of artistic and constructive faculty, 
such for example as evaluating definite integrals, or making out the best way 
one can the number of distinct branches, and the general character of each 
branch of a curve from its algebraical equation. 
{I have elsewhere (in my Trilogy published in the ‘ Philosophical Trans- 
actions”) referred to the close connection between these two cultures, not 
merely as having Arithmetic for their common parent, but as similar in their 
habits and affections. I have called “‘ Music the Algebra of sense, Algebra 
the Music of the reason; Music the dream, Algebra the waking life—the 
soul of each the same !” 
of space, and say, It is this, it is this! Space is the 
Grand Continuum from which, as from an inexhaustible 
reservoir, all the fertilizing ideas of modern analysis are 
derived ; and as Brindley, the engineer, once allowed before 
a parliamentary committee that, in his opinion, rivers were 
made to feed navigable canals, I feel sometimes almost 
tempted to say that one principal reason for the existence 
of space, or at least one principal function which it 
discharges, is that of feeding mathematical invention. 
Everybody knows what a wonderful influence geometry 
has exercised in the hands of Cauchy, Puiseux, Riemann, 
and his followers Clebsch, Gordan, and others, over the 
very form and presentment of the modern calculus, and 
how it has come to pass that the tracing of curves, which 
was once to be regarded as a puerile amusement, or at best 
useful only to the architect or decorator, is now entitled to 
take rank as a high philosophical exercise, inasmuch as 
every new curve or surface, or other circumspection of 
space, is capable of being regarded as the synthesis and em- 
bodiment of some specific organised system of continuity.* 
The early study of Euclid made me a hater of geometry, 
which I hope may plead my excuse if I have shocked the 
opinions of any in this room (and I know there are some 
who rank Euclid as second in sacredness to the Bible 
alone, and as one of the advanced outposts of the British 
Constitution) by the tone in which I have previously 
alluded to it asa school-book; and yet, in spite of this 
repugnance, which had become a second nature in me 
whenever I went far enough into any mathematical 
question, I found I touched, at last, a geometrical bottom ; 
so it was, I may instance, in the purely arithmetrical theory 
of partitions ; so, again, in one of my more recent studies 
the purely algebraical question of the invariantive criteria 
of the nature of the roots of an equation of the fifth degree ; 
—the first inquiry landed me in a new theory of polyhedra, 
the latter found its perfect and only possible complete} 
solution in the construction of a surface of the ninth order 
and the sub-division of its infinite contents into three 
distinct natural regions. 
Having thus expressed myself at greater length 
* M. Camille Jordan's application of Dr. Salmon’s Eikosi-heptagram to 
Abelian functions is one of the most recent instances of this reyerse action 
of geometry on analysis. Mr. Crofton’s admirable apparatus of a reticu- 
lation with infinitely fine meshes rotated successively through indefinitely 
small angles, which he applies to obtaining whole families of definite 
integrals, is another equally striking example of the same phenomenon, 
+ Complete in the sense of x2zversal, more than perfect or complete in the 
ordinary sense. Two critcria are absolutely fixed ; but in addition to thesetwo 
an additional criterion or set of criteria must be introduced to make the system 
of conditions sufficient. ‘The number of such set may be either one or what- 
ever number we please, and into such one or into each of the set (if more 
than one) an indefinite number of arbitrary parameters (limited) may be 
introduced, Now the geometrical construction I arrive at contains implicitly 
the totality of all these infinitely varied forms of criteria, or sets of criteria, 
and without it, the existence and possibility of such variety in the shape of 
the solution could never have been anticipated or understood. My truly 
eminent friend M. Charles Hermite (Membre de l'Institut), with all the 
efforts of his extraordinary analytical power, and with the knowledge of my 
results to guide him, has only been able by the non-geometrical method to 
arrive at one form of solution consisting of a third criterion absolutely definite 
and destitute of a single variable parameter, As is well known, I haye made 
a very important use of a criterion of the same form as M. Hermite’s, but 
containing one arbitrary parameter (limited). The subject will be found re- 
sumed from the point where I left it, and pursued in considerable detail by 
Prof. Cayley, in one of his more recent memoirs on Quartics in the Philo- 
sophical Transactions. M. Hermite it was who first surprised Invariantists 
(’Eglise Invariantive, as we are sometimes styled) by an @ fr%or7 demon- 
stration that the nature of the roots or factors of quartics could in general 
be found by means of invariantive criteria. This was known to be possible up 
to the fourth order of binary quartics, and impossible for the fourth. M. 
Hermite showed that this negation which seemed to stop the way to further 
progress was an exceptional case; that whereas for the second, third, fifth, 
sixth, and all higher degrees the thing could be done, for the fourth alone it 
was impossible: as regards linear quartics, the question does not arise. I 
look upon this failure of a law for one term in the middle of an infinite 
progression as an unparalleled wzixacle of arithmetic, far more real and 
deeper seated than the one alluded to by Mr. Babbage in connection 
with the discontinuous action of a supposed machine in his ninth Bridgwater 
Treatise. > 
1 So I found, as a pure matter of observation, that allineation (alignemen?) 
in ornamental gardening—7.e. the method of putting trees in positions to form 
a very great number or the greatest number possible of straight rows, of which 
a few special cases only had been previously considered as detached porismatic 
problems, forms part of a great connected theory of the pluperfect points 
on a cubic curve, those points, of which the nine points of inflection and 
Pliicker’s twenty-seven points serve as the lowest instances, 
