(20.) 
Pure ma- 
thematics 
—their 
progress 
and tech- 
nicality. 
804 
before him and since by Galileo, Newton, and their 
disciples. 
With regard to Pure Mathematics, and their pro- 
gress during the last seventy years, to the difficulty 
arising from the extent to which the review must have 
enlarged this Essay, and the enormous and dispropor- 
tionate labour it must have cost (a labour far greater 
to the present writer than to one by taste and habit 
more addicted to the study of merely abstract Ana- 
lysis), a conclusive argument against their systematic 
introduction into this historical sketch is to be found 
in the very nature of these modern improvements. 
All sciences—but especially the abstract sciences— 
tend to become more intensely technical the farther 
they are pursued. These especially are incapable of 
popular treatment, although in their applications to 
physical science they occasionally admit of it in a 
very remarkable manner. The progress of analysis 
cannot be even enunciated or expressed but in the 
language of analysis, and the History becomes almost 
a Treatise, or, if not a Treatise, something nearly as 
technical. It is partly for this reason that the his- 
tory of the Pure Mathematics has so seldom been 
even attempted to be written. Mathematicians have, 
since the time of D’Alembert, been noted for being 
more ready themselves to publish than to become 
acquainted with what others have done; and one con- 
sequence of this has been the formation of a mathe- 
matical literature, able, profound, and original, but 
cumbrous, fragmentary, and full of repetitions.? 
Besides, the seventeenth century had attained the 
vantage-ground of those grand and striking im- 
provements in methods to which no subsequent im- 
provements, however real and ingenious, can by 
possibility compare. We shall never have inventions 
comparable, in universality and importance, to the 
application of Algebra to Geometry, and the dis- 
covery of Fluxions. These also admit of being at 
least partly explained in language not obtrusively 
technical, and have been so explained by the facile pen 
of Playfair ; but all subsequent discoveries have been 
but enlargements and improvements on these pri- 
mary and distinguishing ones; and before the date 
at which our present discourse properly opens, even 
the larger generalizations of Newton’s fertile calcu- 
lus—the method, namely, of Variations, and the in- 
tegration of partial differential equations, had been 
MATHEMATICAL AND PHYSICAL SCIENCE. 
(Diss. VI. 
established, by the genius of Euler and Lagrange, on 
an impregnable basis. 
The intense, and praiseworthy, and successful 
labours of their followers have been, then, chiefly de- 
voted to the occupation of the fields of conquest thus 
summarily opened; or, rather, to storming, one by one, 
fortresses still unreduced, after the main resisting army 
had been first routed in the open field. To quit meta- 
phor, the efforts of mathematicians have for many 
years been chiefly applied to rendering possible the 
solution of problems involving quantities which ac- 
tually occurred in the course of the rapid simulta- 
neous advances of physical science. They are in a 
manner inseparable from the branches of physics in 
aid of which they were originally called forth, and 
will therefore be most properly noticed, however 
briefly, in the chapters of this Dissertation where 
their application is considered. Some farther obser- 
vations on this subject will be found in the imme- 
diately succeeding section of the present Essay, 
In reviewing the progress of science—physical 
science in particular—during the last seventy or 
eighty years, I have thought it advisable not to 
21.) 
(22.) 
Subdivi- 
sion of Phy- 
sico-Mathe- 
subdivide the subjects too minutely, and, following matical 
nearly the arrangements of Dr Whewell’s excellent Sciences. 
treatise, already quoted,* I shall treat, in suecessive 
chapters, of Analytical Mechanics including Physical 
Astronomy as their loftiest and most successful ap- 
plication; of Astronomy as a science of observation ; 
of Mechanics, with reference to the intimate consti- 
tution of matter, including Hydrodynamics, Acous- 
tics, and Civil Engineering ; of Optics, or Light; of 
Heat, including the Daltonian theory of the gases 
and chemical elements ; and, finally, of the large and 
comprehensive science of Electro-magnetism, includ- 
ing ordinary and Voltaic Electricity, Terrestrial Mag- 
netism, and Diamagnetism the discovery of Dr Fa- 
raday, 
The arrangement of the chapters is thus strictly 
Methodical ; but in the subdivision into sections, I 
have allowed the Biographical principle to pred 
(23.) 
rrange- 
; ment partly 
~ methodical, 
nate, thus giving as much as possible a historical partly bio- 
character to the whole, and endeavouring to intro- §r#phical. 
duce the reader to the intellectual acquaintance of 
the eminent men who are selected for notice on the 
principles which have been already detailed. In some 
1 Specimens of what a history of pure mathematics would be, and must be, are to be found in the able “ Re ” of Dr 
Peacock and Mr Leslie Hllis, in the Transactions of the British Association for 1833 and for 1846. A glance at these profound 
and very technical essays will show the impossibility of a popular mode of treatment, whilst the difficulty and labour of pro- 
ducing such summaries may be argued from their exceeding rarity in this or any other language. 
2 The celebrated Lagrange, in his later years, contrasting the mathematical works of his own generation with those which he 
studied when a youth, is said to have observed :—“ I pity the young mathematicians who have so many theories to wade through. 
If I were to begin, I would not study ; these large quartos frighten me too much.”—(Thomson’s Annals, vol. iv.) And it is stated 
that whilst his own most abstruse investigations were conducted in Paris, he kept the perusal of M. Gauss’s writings for the tran- 
quil retirement of the country,—a distinction intelligible enough between the intense effort of invention more than sustained by 
the vis viva of genius which prompts it, and the strain required to master the dead weight of reasoning imposed upon the mind 
by the discoveries of another. Dr Young, in his biography of Lagrange, observes upon the voluminous mathematical literature 
of his time, that “ unless something be done to check the useless accumulation of weighty materials, the fabric of science will sink 
in a few ages under its own insupportable bulk.” The fact is that a large proportion of the mathematical writings of even 
eminent authors are in a few years forgotten, or only casually consulted on some matter of history. 
3 History and Philosophy of the Inductive Sciences. 
