any 
Cuar. I., § 2.] 
MATHEMATICS—PHYSICS—MECHANICAL ARTS. 
807 
(80.) The commencement of a better era originated, early spicuously introduced into the doctrine of the con- 
At Gam- 4 g ore See ; 
brid a in the present century, with Woodhouse at Cambridge, duction of Heat in consequence of the abrupt varia- 
? 
Edinburgh, and Playfair in Edinburgh, by both of whom the con- 
and Dublin. tinental methods were introduced into the studies of 
I Pete 
(31.) 
Improve- 
ments in 
their respective universities ; whilst Ivory, a native of 
Scotland, was the first to challenge, by his writings, a 
place in the list of great living mathematicians. The 
systematic form of the Mécanique Céleste rendered the 
subject more accessible than were the countless me- 
moirs by men of the highest name, which then filled 
the Transactions of Paris, Turin, Berlin, and St Peters- 
burg. But the notation of differentials, which could 
alone break down the barrier between the British and 
foreign mathematicians, was first introduced at Cam- 
bridge by the efforts of Sir John Herschel and Dean 
Peacock about 1816, soon after which the transla- 
tion of Lacroix’s Differential Calculus, which they 
superintended, came into use as an university text- 
book. From this time the works of foreign mathe- 
maticians began to be more generally read, particu- 
larly the writings of Laplace and Poisson; and 
within ten or a dozen years subsequently, a few 
active and undaunted men, chiefly of the Cambridge 
school, such as Mr Airy and Sir John Lubbock, 
grappled with the outstanding difficulties of physical 
astronomy, whilst a larger number applied them- 
selves to the most difficult parts of pure analysis, 
and acquired great dexterity in its use in the solu- 
tion of geometrical and mechanical problems. Such, 
for example, were Mr Babbage, Mr De Morgan, Mr 
Murphy, and Mr Green ; and at Dublin Sir William 
R. Hamilton and Mr MacCullagh, whose names will 
occur in other parts of this Dissertation. 
No new calculus or great general method in ana- 
lysis has resulted from these persevering labours, 
tion of physical circumstances at the boundary of 
the conducting body, enters largely into the specula- 
tions of mathematicians of the present century; and 
the doctrine of definite integrals so intimately con- 
nected with it has received a proportional extension. 
Next, analytical geometry has acquired a very great Analytical 
enlargement and by attention principally to symmetry 
in the arrangement of the results, solutions other- 
wise the most intricate are obtained with facility and 
directness. Of this we shall find examples in our 
history of the Undulatory Theory of Light. 
geometry. 
Lastly, The Calcu- 
notwithstanding the pre-eminently practical charac- yates Oye- 
ter of the mathematics of the last age, speculative 
geometers and analysts have found time to discuss 
the metaphysics of their respective sciences, both as 
regards the foundations of the Differential Calculus 
and as to the use of imaginary and other symbols in 
Algebra. An almost new branch of abstract science 
(though faintly foreshadowed by Leibnitz) has come 
into existence—the separation of symbols of opera- 
tion from symbols of quantity, and the treatment of the 
former like ordinary algebraic magnitudes. In some 
cases remarkable simplicity is thus introduced into 
the solution of problems, although perhaps few ma- 
thematicians would choose to depend implicitly upon 
the method in untried cases. Sir John Herschel and 
the late Mr Gregory? were amongst the most active in- 
troducers of this new algebra, but few of the more 
eminent living British or foreign mathematicians 
have failed to contribute their share to this more 
metaphysical department of analysis. 
I shall now attempt to consider more particu- 
tions. 
(32.) 
Connection 
larly the reciprocal relations of pure physical science 
Integra. Whether of British or foreign mathematicians, but 
and the mechanical arts. 
tion, an increased facility and power in applying the ex- eeenyerel 
_ Science and 
The dis- the arts. 
o 
a] 
1 
ous funce 
~ 
isting resources of mathematics to the solution of 
large classes of problems previously intractable, or 
resolved only indirectly or by approximation. The 
Integral Calculus, in particular, affords an almost 
boundless field for research, and each branch of 
science in succession—not only Physical Astronomy, 
but Optics, Heat. Electricity, and Civil Engineering 
—has offered problems of great importance, which 
awaited only the skill of the pure mathematician to re- 
solve in a practical and finite form.' Every year, and 
every civilized community, contribute to these real 
“improvements. The principle of discontinuity, con- 
This is evidently a very intimate one. 
coveries of pure physies (such as Astronomy, Acous- 
tics, Magnetism), are the results of either observation 
or experiment, and they, consist in generalizations, 
by means of which a multitude of facts are reduced 
under one simple expression of a more general fact 
or principle. But instruments often very compli- 
cated are necessary for observation and for experi- 
ment ; as telescopes in astronomy, organs in acoustics, 
properly magnetized and suspended steel bars in 
magnetism. Art is required to construct these. The 
highest possible degree of science, and the utmost 
1 For example the Lucasian Professor at Cambridge, Mr Stokes, has effected two previously impracticable integrations, one 
occurring in the theory of the rainbow, the other in that of railway girder bridges, 
2 Mr Duncan Gregory, a promising mathematician who died 23d February 1844, at the early age of 30, was the youngest Mr Duncan 
- son of Dr James Gregory, the late distinguished Professor of Medicine at Edinburgh. His name deserves a passing record, not only Gregory. 
from the influence he exercised on the progress of the English mathematics of his time, but as having revived the dormant charac- 
ter for this peculiar kind of talent, so long connected with the family of Gregory. He was, in fact, the lineal descendant of the 
inventor of the reflecting telescope. 
Mr Gregory was the first editor of the Cambridge Mathematical Journal, and author of an 
excellent book of Examples in the Differential and Integral Calculus, both of which have exercised a beneficial influence on the 
progress of science in England. He also wrote several original memoirs on the subjects referred to in the text. 
a man of congenial ability, has written a short but pleasing biography of his friend, 
Mr Leslie Ellis, 
(33.) 
