182 On Fluaions. (Serr, 
variations as a calculus succeed fluxions in the order both of nature 
and of invention, the proper appellation, perhaps, would have been 
subfluxions, with a suitable notation, It would be improper, how- 
ever, to propose any change. 
es 
With regard to the fluxional notation, = seems as convenient as 
Ey, while the latter d is preserved for algebraic operations; and f 
seems as convenient as s for marking the fluent. In a philosophical 
point of view, there is no comparison. 
J sometimes hear mathematicians say, We ought to adopt the 
foreign notation. Would not such adoption be to attempt, as far 
as it is in our power, to efface the knowledge of one of Newton’s 
greatest discoveries? Would it not be also unpatriotic? Inde- 
pendently of a natural patriotism, and of the respect due to Newton, 
would a change rather unphilosophical be a change for the better ? 
stx A 
To some it may seem a digression, that the formula : Ser Ty ti 
Sd x A 
Beco tes is derivable by a boy from the simplest operation in the 
Rule of Three; that in the eighth ofa line it contains Euctlid’s 
fifth definition of eight lines in his fifth book; that it comprehends 
all proportional quantities, whether commensurable or incommen- 
surable; and that Euclid, it is probable, thus deduced the defi- 
nition. 
The mistake of a very able mathematician, Carnot, in his Méta- 
physique du Calcul Infinitésimal, where he endeavours to show that 
the differential equations are imperfect, seems to arise from his not 
distinguishing sufficiently the differences or increments from the 
fluxions or differentials. 
From all that has been said we may conclude, that no demon- 
stration ought to depend on motion, if motion can be avoided, but 
that motion is either mathematical or mechanical: that no demon- 
stration of the fluxional problem can be rigorous and satisfactory 
that depends on infinitesimals and on vanishing quantities: that 
though, in compliance with custom, I said in p. 331, line 24, 
* vanishing quantity,” yet it is not strictly a vanishing quantity, but 
a guantity which, by the continued bisection of the increment of 
the abscissa, may become less than any assigned quantity how small 
soever: that in my former paper I might without fig. 1 or 2 have 
stated and demonstrated by fig. 3 the doctrine of fluxion in the 
form of a theorem; or in the form of a problem thus, prop. pro- - 
blem, to find the fluxion of any function of a variable quantity : 
or thus, prop. problem, to find the rate, &e. To find the rate of 
change in a quantity and its functiog. This procedure would have 
been more scientific and elegant, not more intelligible, than that 
which I employed: that Newton’s Jemma consists of two parts ; 
first, of the conception of the generation of quantity by motion; 
and, secondly, of the demonstration which relates neither to mo- 
