"attack or reply to. 
operations. 
Det. 30, 1869] 
A PLEA FOR THE MATHEMATICIAN 
[At the request of the Editor, Professor Sylvester has furnished 
the following abridgment of his opening address to the 
*Mathematical and Physical Section of the British Association 
at Exeter, with some supplemental notes appended in the 
course of its passing through the press.—ED. ] 
ie 
T is said of a great party leader and orator in the 
House of Lords that, when lately requested to make 
a speech at some religious or charitable meeting, he 
declined to do so on the ground that he could not speak 
unless he saw an adversary before him—somebody to 
In obedience to a somewhat similar 
combative instinct, I set to myself the task of considering 
certain recent utterances of a most distinguished member 
of this Association, one whom I no less respect for his 
- honesty and public spirit than I admire him for his genius 
and eloquence, but from whose opinions on a subject 
which he has not studied I feel constrained to differ. 
Gothe has said— 
“ Verstandige Leute kannst du irren sehn 
= In Sachen niimlich, die sie nicht verstehn.” 
Understanding people you may see erring—in those things, 
to wit, which they do not understand, 
I have'no doubt that had my distinguished friend, the 
probable President-elect of the next Meeting of the 
Association, applied his uncommon powers of reasoning, 
induction, comparison, observation, and invention to the 
study of mathematical science, he would have become as 
great a mathematician as he is now a biologist ; indeed 
he has given public evidence of his ability to grapple 
with the practical side of certain mathematical questions ; 
but he has not made a study of mathematical science as 
such: and the eminence of his position and the weight 
justly attaching to his name, render it only the more im- 
perative that any assertions proceeding from such a 
quarter, which may appear to be erroneous, or so expressed 
as to be conducive to error, should not remain unchal- 
lenged or be passed over in silence. 
He says “mathematical training “is almost purely 
deductive. The mathematician starts with a few simple 
propositions, the proof of which is so obvious that they 
are called self-evident, and the rest of his work consists 
of subtle deductions from them. The teaching of 
languages, at any rate as ordinarily practised, is of the 
same general nature—authority and tradition furnish the 
data, and the mental operations are deductive.” It would 
seem from this that, according to Prof. Huxley, the 
business of the mathematical student is from a limited 
number of propositions (bottled up and labelled ready for 
future use) to declare any required result by a process of 
the same general nature as a student of language employs 
in declining and conjugating his nouns and verbs: that 
to make out a mathematical proposition and to construe 
or parse a sentence are equivalent or identical mental 
Such an opinion scarcely seems to need 
serious refutation. The passage is taken from an article 
in Macmitlan’s Magazine for June last, entitled “ Scientific 
Education—Notes of an After-dinner Speech,” and I 
cannot but think would have been couched in more guarded 
terms by my distinguished friend had his speech been made 
_ before dinner instead of after. 
The notion that mathematical truth rests on the narrow 
basis of a limited number of elementary propositions, from 
which all others are to be derived by a process of logical 
inference and verbal deduction, has been stated still more 
strongly and explicitly by the same eminent writer in an 
article of even date with the preceding, in the Fortnightly 
Review, where we are told that “ Mathematics is that 
study which knows nothing of observation, nothing of 
experiment, nothing of induction, nothing of causation.” 
I think no statement could have been made more 
NATURE 
237 
opposite to the facts of the case: that mathematical 
analysis is constantly invoking the aid of new principles, 
new ideas, and new methods, not capable of being defined 
by any form of words, but springing direct from the 
inherent powers and activity of the human mind, and 
from continually renewed introspection of that inner world 
of thought of which the phenomena are as varied and 
require as close attention to discern as those of the outer 
physical world—to which the inner one in each individual 
man may, | think, be conceived to stand in somewhat the 
same general relation of correspondence as a shadow to 
the object from which it is projected, or as the hollow palm 
of one hand to the close fist which it grasps of the other: 
that it is unceasingly calling forth the faculties of obser- 
vation and comparison, that one of its principal weapons 
is induction, that it has frequent recourse to experimental 
trial and verification, and that it affords a boundless scope 
for the exercise of the highest efforts of imagination and 
invention.* 
Lagrange, than whom no greater authority could be 
quoted, has expressed emphatically his belief in the 
importance to the mathematician of the faculty of obser- 
vation :+ Gauss has called mathematics a science of the 
* The annexed instance of Mathematical Euristic is, I think, from its 
intrinsic interest, worthy of being put on record. The so-called canonical 
representation of a binary quartic of the eighth degree I found to be a 
quartic multiplied by itself, together with a sum of powers of its linear 
factors, just as for the fourth degree it was known to be a quadric into itself, 
together with a sum of powers of its factors: but for a sextic a cubic multi- 
plied into itself, with a tail of powers as before, was not found to answer. 
To find the true representation was like looking out into universal space for 
a planet desiderated according to Bode’s or any other empirical law. I found 
my desideratum as follows: I invented a catena of morphological processes 
which, applied to a quadric or to a quartic, causes each to reproduce itself: 
I then considered the two quadrics and two quartics to be noumenally dis- 
tinguishable (one as an auto-morphic derivative of the other), although phe- 
nomenally identical. ‘Ihe same catena of processes applied to the cubic gave 
no longer an identical but a distinct derivative, and the product of the two 
I regarded as the analogue of the before-mentioned square of the quadric or 
of the quartic. This product of a cubic by its derivative so obtained together 
with a sum of powers of linear factors of the original cubic, I found by 
actual trial to my great satisfaction satisfied the conditions of canonicity, and 
it was thus I was led up to the desired representation which will be found 
reproduced in one of Prof. Cayley’s memoirs on Quantics and in Dr. Salmon’s 
lectures on Modern Algebra. Here certainly induction, observation, inven- 
tion, and experimental verification all played their part in contributing to the 
solution of the problem. I discovered and developed the whole theory of 
canonical binary forms for odd degrees, and, as far as yet made out, for even 
degrees too, at one evening sitting, with a decanter of port wine to sustain 
nature’s flagging energies, in a back office in Lincoln’s-Inn-Fields. The work 
was done, and well done, but at the usual cost of racking thought—a brain 
on fire, and feet feeling, or feelingless, as if plunged in an ice pail. That 
night we slept no more. The canonisant of the quartic (its cubic covariant) 
was the first thing to offer itself in the inquiry. I had but to think the 
words ‘‘ Resultant of Quintic and its Canonisant,” and the octodecadic skew 
invariant would have fallen spontaneously into my lap. By quite another 
mode of consideration M. Hermite subsequently was led to the discovery of 
this, the key to the innermost sanctuary of Invariants—so hard is it in Euristic 
to see what lies immediately before one’s eyes. The disappointment weighed 
deeply, far too deeply, on my mind, and caused me to relinquish for long years 
a cherished field of meditation: but the whirligig of time brings about its 
revenges. Ten years later this same canonisant gave me the upper hand 
of my honoured predecessor and guide, M. Hermite, in the inquiry (referred to 
at the end of this address) concerning the invariantive criteria of the con- 
stitution of a quintic with regard to the real and imaginary. By its aid I 
discovered the essential character of the famous amphigenous surface of the 
ninth order, and its bicuspidal universal section of the fourth order (otherwise 
termed the Bicorn), as may be seen in the third part of my Triology, printed 
in the Philosophical Transactions. 
t+ Iwas under the conviction that a passage to that effect from Lagrange had 
been cited to me some years ago by M. Hermite of the Institute of France ; 
on applying to him on the subject, I received the following reply :— 
“Relativement a l’opinion que suivant vous j’aurais attribuee & Lagrange, 
je m’empresse de vous informer qu'il ne faut aucunement, & ma connaissance, 
l'en rendre responsable. Nous nous sommes entretenus du réle de la 
Jaculté d’observation dans les études que nous avons poursuivies de concert 
pendant bien des années, et c'est alors, sans doute, que je vous aurai conté 
une anecdote que je tiens de M. Chevreul. M. Chevreul, allant a l'Institut 
dans la voiture de Lagrange, a été vivement frappé du sentiment de plaisir 
avec lequel ce grand géometre lui faisait voir, dans un travail manuscrit, la 
beauté extérieure et artistique, si je peux dire, des nombreuses formules qui 
y, figuraient. Ce sentiment nous l’avons tous éprouvé en faisant, avec 
sincérité, abstraction de l'idée analytique dont les formules sont l’expression 
écrite. Il y a J&A, n’est-il point vrai, un imperceptible lien qui rattache au 
monde de I’art le monde abstrait de l’algébre et de l’analyse, et j’oserai méme 
vous dire que je crois a des sympathies réelles, qui vous font trouver un 
charme, dans les notations d’un auteur, et dans les répulsions qui éloignent 
d’un autre, par l’apparence seule des formules.” 
I am, however, none the less persuaded that on one or more than one 
occasion, M. Hermite, speaking of Lagrange, expressed to me, if not as I 
supposed on Lagrange’s, then certainly on his own high authority, ‘‘ that the 
faculty of observation was no less necessary for the successful cultivation of 
the pure mathematical than of the natural sciences.” 
