382 
often did, with apparent readiness) his beloved mathe- 
matics to other claims. Another friend says: ‘‘ He was 
a man of rare powers, and as guileless as he was richly 
gifted.” 
Of some men it is said that they were never young, of 
others that they became old while their contemporaries 
were still lads; and it has been stated as a general law, 
in scientific thought at least, that the best and most 
original ideas have always been conceived before the 
age of thirty. But whatever may be the case in this 
respect with the generality of men, Henry Smith was as 
young and vigorous in intellect at the age of fifty-six, the 
limit to which he attained, as he was when he gained the 
first of his many University honours. It was his fresh- 
ness of mind, his vivid appreciation and intelligent enjoy- 
ment of everything going on, not only in science, but 
also in life, whether social or political, which made us 
forget that his years, like ours, were passing away, and 
that the number of them was finite. It was his genial 
presence, his sympathetic attention, his ready counsel, his 
sound judgment, his happy mode of dealing with both 
men and things, which make us already feel a loss which 
we cannot as yet fully appreciate, but which we can never 
hope again completely to replace. 
Of many Greek towns it is related that each has claimed for 
itself the honour of having been the birthplace of Homer ; 
in like manner, many branches of knowledge, and avoca- 
tions of life, might claim to have been the favourite 
pursuit of Henry Smith. But however proficient, or even 
prominent he may have been in other subjects, it was in 
mathematics that he mainly showed the originality of his 
genius, and that he has left any permanent record of work 
of the highest kind. 
Among the great works which it was long hoped that 
he would have accomplished was his treatise on the 
Theory of Numbers. This subject, which during the 
present generation has been so marvellously generalised 
as to undergo a complete transfiguration since it was pre- 
sented to us in the work of Barlow and in the ordinary 
educational books on Algebra, formed for many years a 
serious study on the part of Prof. Henry Smith. The 
papers in which the researches of mathematicians on this 
subject are recorded are scattered through the pages of 
various periodicals, so that it is not easy to realise the 
steps which each writer has contributed to the general 
progress, nor to assign to each his relative position. But 
this is not all, nor even the worst; it has been a 
prevailing custom, too prevalent we think, among mathe- 
maticians of late years, to publish results alone, without 
proof of their statements, and even without indication of 
the train of argument which led them to their: conclu- 
sions. This naturally entails on the part of the reader 
either a strong act of faith or a difficult and, as we hold, 
unnecessary effort. It need hardly be added that in endea- 
vouring to digest and present to his readers what had 
been done by others in his subject, Henry Smith adopted 
the latter course ; and, with a sagacity in which few could 
haye rivalled him, he assimilated all these fragments, and 
utilising the valuable among these désyecta membra, and 
rejecting the worthless, he brought them into harmony, 
and was in a fair way to produce from them a Structure 
intelligible in itself, and capable of forming a groundwork 
for further developments. But while our author was dis- 
NAL ORE 
[ Fed. 22, 1883 
cussing what had been already done, the very materials 
upon which he was engaged were growing apace, and his 
self-imposed task accumulated upon him. Of unfinished 
work, or of ‘“‘ragged ends” as he used to call them, he 
was as nearly intolerant as he could be of anything; 
and it is not clearly known whether he ever made up 
his mind to complete what he had undertaken up to a 
certain date or not. In any case what he had already 
long ago achieved in this matter must have been a 
gigantic work; and it remains only to hope that his 
manuscripts have been left in such a state that others may 
be able to wield the weapons which he had forged. 
The results of his preliminary studies were given in his six 
invaluable Reports on the Theory of Numbers, published 
in the volumes of the British Association for 1859 and 
following nearly consecutive years; and these alone are 
sufficient to show the extent of his reading and the firm 
grasp which he had of the subject. The following ex- 
tracts from the first and third of these Reports indicate 
both the wide range of the theory and the magnitude of 
the portion which still remains to be achieved :— 
“ There are two principal branches of the higher arith- 
metic: the Theory of Congruences and the Theory ot 
Homogeneous Forms. Ina general point of view these 
two theories are hardly more distinct from one another 
than are in algebra the two theories to which they re- 
spectively correspond, namely, the Theory of Equations 
and that of Homogeneous Functions; and it might, at 
first sight, appear as if there were not sufficient founda- 
tion for the distinction. But, in the present state of our 
knowledge, the methods applicable to, and the researches 
suggested by, these two problems, are sufficiently distinct 
to justify their separation from one another.”’ 
“Tt is hardly necessary to state that what has been 
done towards obtaining a complete solution of the Repre- 
sentation of Numbers by Forms, and the Transformation 
of Forms, is but very little compared with what remains 
to be done. Our knowledge of the algebra of homoge- 
neous forms (notwithstanding the accessions which it has 
received in recent times [1861]), is far too incomplete to 
enable us even to attempt a solution of them co-extensive 
with their general expression. And even if our algebra 
were so far advanced as to supply us with that knowledge 
of the invariants and other concomitants of homogeneous 
forms, which is an essential preliminary to an investiga- 
tion of their arithmetical properties, it is probable that 
this arithmetical investigation itself would present equal 
difficulties. The science, therefore, has as yet had to 
confine itself to the study of particular sorts of forms ; 
and of these (excepting linear forms, and forms contain- 
ing only one indeterminate) the only sort of which our 
“knowledge can be said to have any approach to complete- 
ness are the binary quadratic forms, the first in order of 
simplicity, as they doubtless are in importance.” 
Prof. Smith’s sphere of utility was, as indeed is pretty 
well known, not confined to his University, nor to science 
as such, but extended, among other directions, even to 
departments of the State. Passing over the Royal Com- 
missions on Scientific Education and on the University 
of Oxford, of both of which he was a leading member, 
mention must not be omitted of the Meteorological 
Council of which he was chairman. That body, nominated 
by the Royal Society, but appointed by the Government, | 
