rn 
Fan. 6, 1870| 
A PEEA FOR THE MATHEMATICIAN 
Il. 
] MIGHT go on, were it necessary, piling instance upon 
instance to prove the paramount importance of the 
faculty of observation to the process of mathematical dis- 
covery.* Were it not unbecoming to dilate on one’s personal 
experience, I could tell a story of almost romantic interest 
about my own latest researches in a field where Geometry, 
Algebra, and the Theory of Numbers melt in a surprising 
manner into one another, like sunset tints or the colours 
of the dying dolphin, “ the last still loveliest ” (a sketch of 
which has just appeared in the Proceedings of the London 
Mathematical Society),f which would very strikingly illus- 
trate how much observation, divination, induction, experi- 
mental trial, and verification—causation, too (if that means, 
as, if it mean anything, I suppose it must, mounting from 
phenomena to their reasons or causes of being)—have to do 
with the work of the mathematician. In the face of these 
facts, which every analyst can vouch for out of his own know- 
ledge and personal experience, how can it be maintained, 
in the words of Professor Huxley (who, in this instance, is 
speaking of the sciences as they are in themselves and 
without any reference to scholastic discipline), that Mathe- 
matics “is that study which knows nothing of observation, 
nothing of induction, nothing of experiment, nothing of 
causation ”?+ 
I, of course, am not so absurd as to contend that the 
habit of observation of external nature || will be best or at 
all cultivated by the study of mathematics, leastways as 
that study is at present conducted ; and no one can desire 
more earnestly than myself to see natural and experimen- 
* Newton’s Rule (subsequently and for the first time deduced to demon- 
stration in No. 2 of the London Mathematical Society’s Proceedings) was to 
all appearance, and according to the more received opinion, obtained in- 
ductively by its author. So also my reduction of Euler’s problem of the 
Virgins (or rather one slightly more general than this) to the form of a 
question (or, to speak more exactly, a set of questions) in simple partitions 
was (strangely enough) first obtained by myself inductively, the result 
communicated to Prof. Cayley, and proved subsequently by each of us in- 
dependently, and by perfectly distinct methods. 
+ Under the title of ‘Outline Trace of the Theory of Reducible 
Cyclodes.” 
t Induction and analogy are the special characteristics of modern mathe- 
matics, in which theorems have given place to theories and no truth is regarded 
otherwise than as a link in an infinite chain. ‘‘Omne exit in infinitum” 
is their favourite motto and accepted axiom. No mathematician now-a-days 
sets any store on the discovery of isolated theorems, except as affording hints 
of an unsuspected new sphere of thought, like meteorites detached from some 
undiscovered planetary orb of speculation. The form, as well as matter, of 
mathematical science, as must be the case in any true living organic science, 
is in a constant state of flux and the position of its centre of gravity is liable 
to continual change. At different periods in its history, defined with more 
or less accuracy, as the science ‘of number or quantity, or extension or 
operation or arrangement, it appears, at present, to be passing through a phase 
in which the development of the notion of continuity plays the leading part. 
In exemplification of the generalising tendency of modern mathematies, 
take so simple a fact as that of two straight lines or two planes being in- 
capable of including ‘“‘a space. When analysed this statement will be 
found to resolve itself into the assertion that if two out of the four triads 
that can be formed with four points lie respectively 7 divectumz, the same must 
be true of the remaining two triads; and that if two of the five tetrads 
that can be formed with five points lie respectively 7 A/ano, the remaining 
three tetrads (subject to a certain obvious exception) must each do the same. 
This at least is one way of arriying at the notion of an unlimited rectilinear 
and planar scheme of points. The two statements above made, translated 
into the language of determinants, immediately suggest as their generalised 
expression my great ‘* Homaloidal Law,” which affirms that the vanishing of 
a certain specifiable number of minor determinants of a given order of any 
matrix (z.c. rectangular array of quantities) implies the simultaneous evanes- 
cence of all the rest of that order. I made (ter aZia) a beautiful application 
of this law (which is, I believe, recorded in Mr. Spottiswoode’s valuable 
treatise on Determinants, but where besides I know not) to the establishment 
of the well-known relations, wrung out with so much difficulty by Euler, 
between the cosines of the nine angles which two sets of rectangular ones in 
space make with one another. Thisis done by contriving a matrix such 
that the six known equations connecting the nine cosines taken both ways 
in sets of threes shall be expressed by the evanescence of six of its minors ; 
the simultaneous evanescence of the remaining minors given by the 
Homaloidal Law will then be found to express the Eulerian relations in 
question, which are thus obtained bya simple process of inspection and 
reading off, without any labour whatever. The fact that such a law, containing 
in a latent form so much refined algebra, and capable of such interesting im- 
mediate applications, should present itself to the observation merely as the 
extended expression of the ground of the possibility of our most elementary 
and seemingly intuitive conceptions concerning the right line and plane, has 
often filled me with amazement to think of. 
|| As the prerogative of Natural Science is to cultivate a taste for observa- 
tion, so that of Mathematics is, almost from the starting point, to stimulate 
the faculty of invention, 
NGA TLE 
261 
tal science introduced into our schools as a primary and 
indispensable branch of education : I think that that study 
and mathematical culture should go on hand in hand to- 
gether, and that they would greatly influence each other 
for their mutual good. I should rejoice to see mathematics 
taught with that life and animation which the presence 
and example of her young and buoyant sister could not 
fail to impart ; short roads preferred tolong ones ; Euclid 
honourably shelved or buried “deeper than did ever 
plummet sound” out of the schoolboy’s reach ; morpho- 
logy introduced into the elements of Algebra ; projection, 
correlation, and motion accepted as aids to geometry ; 
the mind of the student quickened and elevated and his 
faith awakened by early initiation into the ruling ideas 
of polarity, continuity, infinity, and familiarisation with 
the doctrine of the imaginary and inconceivable. 
It is this living interest in the subject which is so wanting 
in our traditional and medizeval modes of teaching. In 
France, Germany, and Italy, everywhere where I have 
been on the Continent, mind acts direct on mind in a 
manner unknown to the frozen formality of our academic 
institutions ; schools of thought and centres of real 
intellectual co-operation exist ; the relation of master and 
pupil is acknowledged as a spiritual and a lifelong tie 
connecting successive generations of great thinkers in an 
unbroken chain, just as we read, in the catalogue of our 
French Exhibition, or of the Salon at Paris, of this man 
or that being the pupil of one great painter or sculptor and 
the master of another. When followed out in this spirit, 
there is no study in the world which brings into more 
harmonious action all the faculties of the mind than the 
one of which I stand here as the humble representative 
and advocate. There is none other which prepares so 
many agreeable surprises for its followers, more wonderful 
than the transformation scene of a pantomime, or, like 
this, seems to raise them, by successive steps of initiation, 
to higher and higher states of conscious intellectual being. 
This accounts, I believe, in part for the extraordinary 
longevity of all the greatest masters of the Analytical art, the 
Dii Majores of the mathematical Pantheon. Leibnitz lived 
to the age of 70; Euler to 76 ; Lagrange to 77; Laplace 
to 78; Gauss to 78; Plato, the supposed inventor of the 
conic sections, who made mathematics his study and 
delight, who called them the handles or aids to philosophy, 
the medicine of the soul, and is said never to have let a 
day go by without inventing some new theorems, lived to 
82; Newton, the crown and glory of his race, to 85 ; 
Archimedes, the nearest akin, probably, to Newton in 
genius, to 75, and might have lived on to be roo, for aught 
we can guess to the contrary, when he was slain by the 
impatient and ill-mannered sergeant sent to bring him 
before the Roman General, in the full vigour of his 
faculties, and in the very act of working out a problem ; 
Pythagoras, in whose school, I believe, the word mathe- 
matician (used, however, in a somewhat wider than its 
present sense) originated, the second founder of geometry, 
the inventor of the matchless theorem which goes by his 
name, the precognizer of undoubtedly the miscalled Coper- 
nican theory, the discoverer of the regular solids and the 
musical canon (who stands at the very apex of this 
pyramid of fame), if we may accept the tradition, after 
spending 22 years studying in Egypt and 12 in Babylon, 
opened school when 56 or 57 years old in Magna Grecia, 
married a young wife when past 60, and died, carrying on 
his work with energy unspent to the last, at the age of 
99. The mathematician lives long and lives young; “the 
wings of his soul do not early drop off, nor do its pores 
become clogged with the earthy particles blown from the 
dusty highways of vulgar life.” 
Some people have been found to regard all mathematics, 
after the 47th proposition of Euclid, as a sort of morbid 
secretion, to be compared only with the pearl said to be 
generated in the diseased oyster, or, as I have heard it 
described, “une excroissance maladive de l’esprit humain, 
