346 
poet, not to begin his epic at the origin of 
things, but to hasten on to the event 
proper ; consequently, I shall not go back 
to the Egyptians, Greeks, Hindoos, or 
Arabs, but at once proceed to the advances 
made in the present century. 
One of the first results of the differential 
notation of Leibnitz was the recognition of 
d 
dx 
ferentiation and the ordinary symbol of al- 
gebra; later the same analogy was per- 
ceived to hold for 4, the symbol of the 
calculus of finite differences. Guided by 
this analogy, Lagrange and other mathe- 
maticians of the French school, which flour- 
ished at the beginning of the century, in- 
ferred that theorems proved to be true for 
combinations of ordinary symbols of quan- 
tity might be applied to the differential 
ealculus and the calculus of finite differ- 
ences. In this way many theorems were 
enunciated, which appeared to be true, but 
of which it was thought to be almost im- 
possible to obtain direct demonstration. 
Gradually, however, the view was reached 
that the logical connection amounted to 
more than analogy, and that the common 
theorems were true because the symbols in 
the three cases were subject to the same 
fundamental laws of combination. This 
advance was principally made by Servais, 
who enunciated the laws of commutation 
and distribution. 
About the year 1812 a school of mathe- 
maticians arose at Cambridge which aimed 
at introducing the d-ism of the Continent 
in place, of the dot-age of the University ; 
in other words they believed in the practi- 
eal superiority of the differential notation 
of Leibnitz over the fluxional notation of 
Newton. Their attention was naturally 
drawn to the questions which had sprung 
from the differential notation ; and of the 
three founders of the school—Babbage, 
Herschel, Peacock—the last named took 
the analogy between the symbol of dif- 
SCIENCE. 
[N.S. Von. X. No. 246. 
up the problem of placing the teaching of 
algebra more in consonance with the views 
which had been reached of the nature of 
symbols. Peacock considered algebra, as 
then taught, to be more of an art than a sci- 
ence; a collection of rules rather than a 
system of logically connected principles ; 
and with the object of placing it on a more 
scientific basis, he made a distinction be- 
tween arithmetical algebra and symbolical 
algebra. He treated these names as denot- 
ing distinct sciences, and he wrote an alge- 
bra in two volumes, of which one treats of 
arithmetical algebra and the other of sym- 
bolical algebra. He thus describes what 
he means by the former term: ‘In arith- 
metical algebra we consider symbols as rep- 
resenting numbers and the operations to 
which they are submitted as included in 
the same definitions as in common arith- 
metic; the signs + and — denote the op- 
erations of addition and subtraction in 
their ordinary meaning only, and those op- 
erations are considered as impossible in all 
cases where the symbols subjected to them 
possess values which would render them so 
in case they were replaced by digital num- 
bers ; thus in expressions such as a+ b we 
must suppose a and 6 to be quantities of the 
same kind; in others like a —b, we must 
suppose a greater than 6 and therefore 
homogeneous with it; in products and quo- 
a 
= 
multiplier and divisor to be abstract num- 
bers; all results whatsoever, including 
negative quantities, which are not strictly 
deducible as legitimate conclusions from 
the definitions of the several operations 
must be rejected as impossible, or as for- 
eign to the science.”’ 
Here it may be observed that Peacock is 
a 
b 
impossible when 6 is not a divisor of a, asis 
a — b, when 0 is not less than a; in neither 
tients, like ab and —-we must suppose the 
not true to his own principle ; for — is as 
