Not inju- 
rious to 
abstract 
science, 
(28.) 
806 
lesson more and more loudly in the ears of mankind. 
The era of Newton and Leibnitz was grandly distin- 
guished by the continually increasing applications of 
mathematics to physics, whereof Newton was the great 
teacher; the century 1750-1850, whilst profiting 
by the lessons of the past, has added almost a new 
one in the eminently practical character of its science, 
and in the no less scientific character of its practice. 
The result of these gradual modifications of human 
knowledge has not been in the slightest degree in- 
jurious to the real progress of the more abstract in- 
gredient of the mixed sciences. Did mathematics 
ever flourish more vigorously than under Newton? 
has pure physical science had greater triumphs than 
in the era of Volta, Watt, and Young? It was 
precisely because the new application of mathematics 
stimulated their growth, because abstract relations 
of quantity were vivified by concrete solutions of 
physical problems, that a new geometry arose. 
Dynamics could hardly be said to exist as a science 
without the invention of Fluxions as a language by 
which its conditions and results might be expressed ; 
and from that time onwards, the necessities of the 
natural philosopher have been the prime sources of 
inspiration to the geometer, while the subjects have 
become so blended that a mere discoverer in mathe- 
matics has become a singularity. It would be hardly 
possible to point out any mathematician of the 
highest class since Newton, or but a few of the second 
class, who have not contributed almost as much to 
physical science as they have to analysis. Of purely 
mathematical discoveries, the great majority have 
been called forth by the immediate necessity arising 
from some problem requiring solution in astronomy, 
mechanics, optics, or heat. Lagrange’s method of 
Variations of arbitrary Constants in Integration, the 
artifices for the computation of attractions by Laplace’s 
coefficients ; the introduction of the method of fac- 
torials by Kramp in his solution of the problem of 
refraction, and numberless improvements in the 
Theory of Definite Integrals by Fourier and his suc- 
cessors, sufficiently warrant the statement, and show 
how richly the physical sciences have repaid to the 
purely mathematical ones the debt which they origi- 
nally owed. One other conclusion may be drawn 
from these and parallel facts. It is that the com- 
binations arising out of external phenomena are more 
suggestive of the possible relations of number and 
quantity than is the most unlimited stretch of fancy 
and imagination; and I believe it will be conceded 
that, with few exceptions, theorems of the greatest 
value and beauty have been more frequently dis- 
covered during the attempt to solve some physical 
or at least geometrical problem, than in compre- 
hensive yet indefinite attempts to generalize the re- 
lations of abstract magnitude. 
These views are strikingly confirmed by the his- 
MATHEMATICAL AND PHYSICAL SCIENCE, 
(Diss. VI, 
torical fact of the paucity of pure mathematicians, 
and of distinct mathematical treatises of a strictly 
original character in an age distinguished by the 
diffusion of mathematical knowledge, and in countries 
(like France) most celebrated for its triumphs. There 
are not, perhaps, much more than half a dozen 
really great mathematicians of the last seventy years, 
who have not left treatises more numerous and more 
distinguished on physical science, treated mathe- 
matically, than on pure mathematics. Among the 
exceptions which more immediately occur, are Monge, 
Legendre, and Abel. And of distinct treatises, whilst 
we have the Mécanique Analytique, the Mécanique 
Céleste, the Théorie de la Chaleur, and numberless 
others, containing precious mathematical develop- 
ments, in connection with the applications which 
suggested them, the purely mathematical memoirs of 
the same period are to be sought chiefly in the form 
of detached essays, in the ponderous volumes of 
Academical Transactions. 
One point in the History of Mathematics has espe- 
cial interest for the English reader, and as such may 
be touched upon here with reference to the progress 
of science for the last three quarters of a century. 
The national pride of England in the triumphs of 
Newton impelled her ablest mathematicians to at- 
tempt to carry forward the synthetic methods which 
he had chiefly used, at least in his published works, 
to the more arduous and intricate questions of Me- 
chanics and Astronomy which presented themselves 
for solution in the course of the 18th century. Mac- 
laurin was almost the last Englishman of that period 
whose mathematical writings came into direct com- 
petition with the rising schools of Germany and 
France. The labours of Matthew Stewart, and 
Simpson were mostly geometrical ; those of Landen 
and Waring, though profound, created little general 
impression; and, gradually, the extent and difficulty 
of the foreign mathematics, increased by the use of the 
Leibnitzian notation of differentials which was ab- 
solutely unfamiliar in England, deterred almost every 
one even from perusing the writings of Clairaut and 
D'Alembert, Lagrange and Laplace. Of the conti- 
nental mathematicians, Euler was probably the best 
known, owing to the lucidity of his writings and 
their eminently practical tendencies. Some idea may 
be formed of the negation of mathematical talent in 
Britain during the later portion of the last century, 
when we find D’Alembert declaring, in 1769, that if 
an Englishman is to be elected one of the eight 
foreign associates of the Academy of Sciences, he 
will vote for Earl Stanhope as the best mathemati- 
cian there, as he believes, not having read any of 
his works! If the choice was to be free, he should 
prefer M. de Lagrange!!! A more cutting, though 
unintentional satire on the state of Mathematics in 
this country could not have been written. 
1 Letters of eminent persons, addressed to David Hume, edited by Mr Burton, p. 215. 
The pure 
Mathe- 
matics, 
(29.) 
Their pro- 
gress inthis 
country. 
