1815.] On Flucions. 181 
easy, but circuitous and difficult. Thus the learner may think with 
regard to Lagrange’s process ; but the learned will admire its gene- 
rality, vigour, consistency, and important applications. Why is 
not the Calculus of Variations, the noble discovery of Lagrange, 
admitted into our initiatory books ? Much of it is quite elementary, 
and its nature is easily apprehended. 
It appears to me also that much of the Méchanique Analytique 
is elementary, and may be taught early. Can any thing be easier 
and simpler than the two formulas, the one for statics, the other for 
dynamics? How delightful will the study of that comprehensive 
treatise, and of Laplace’s masterly work the Méchanique Céleste, 
be, if the learner previously understand, as he easily may, the 
parallelopiped of forces, the three perpendicular axes of rotation, 
the three perpendicular co-ordinates, the three co-ordinated planes, 
the principle of virtual velocities, and be accustomed to introduce 
by substitution the sines and co-sines, &c.? Nothing will allure a 
learner more than to study the way in which Euler, vol. ii. of In- 
troduction to the Analysis of Infinites, employs the sines and co- 
sines in changing the position of the co-ordinates. May not the 
student also learn early, in that fine performance, the generation of 
curves from their equations, and the progressive induction of those 
equations without end ? 
I wish Lagrange had been more precise in the titles of his two 
books, Theory of Analytical Functions, Calculus of Functions ; 
for, as his Theory does not include geometrical analysis, it relates 
to algebraical functions only, and not to them all; for it does not 
relate to common functions of known and unknown, of constant 
and variable, quantities ; it therefore relates to derived functions 
enly ; and not even to them all; for let any one consider Arbogast’s 
Derivations, and he will see that it does not relate to derived func- 
tions where the operations, not the quantities, are derived from 
each other ; it is, consequeatly, the theory of fluxional or differen- 
tial functions direct and inverse. 
Here let me remark, that the views of perhaps all the writers on 
the important subject of fluxions relate more or less directly to the 
; Mh arent ah. aN , 
doctrine of ratios, ye ee fi 2 = Ts? according to Lagrange’s 
own statement; for, in every fraction, is not the numerator the 
antecedent, and the denominator the consequent, of a ratio? 
The observations of Lacroix and other eminent mathematicians 
may remove the difficulties which learners always find, in conse- 
quence of the differential and the integral notation, as the differ- 
ences of the absciss and of the ordinate are not employed, nor the 
integer of a fraction, nor the sum of quantities; the notation, how- 
ever, is extremely convenient, and will not puzzle a learner, if its 
defect be supplied by a very careful explanation. 
_ Even variation is not a very happy word, for variation may be 
either starting or continuous. Fluxion is the happiest word that L 
know, as it marks a continuous, not a starting, change; and since 
