we 
328 : On the Calculus of Operations. | 
successive multiplications by unity, or the involution of a geo- 
metrical unit, will be geometrically interpreted by so many succes- 
sive productions of the linear unit to the right of the origin. This 
informs us that the vth power of a linear unit is a line of n times 
the length of the line involved, or that 1° =4.1 by this method. 
But if, instead of a linear unit, we take fora geometrical multiplier 
the nth part of the circumference of a circle to radius unity, the 
performance of n successive multiplications will transfer the multi- . 
“plied body (multiplicand ) around the circumference, by ” succes 
sive steps, to its point of departure; at which point it had the meas- 
ure unity (the radius of the circle), aud therefore we now have 
1"=1, or the powers of geometrical and arithmetical unity may 
e made to coincide. Inversely, if the entire circumference be 
the properties of the four algebraical signs ; for which, see the 
second and last chapters of the Calculus of Operations. In all 
cases the distances traversed have a dynamical value, as meas- 
ures of the respective operations of describing them. 
Division is an operation compounded of addition and subtrac- 
tion; wherein the quotient, which is a pure ratio, is retained as 
the measure of the result of the operation, which consists in ef- 
fecting an equilibrium between the forces of which the dividend 
and divisor are the measures. 
In the fluxionary calculus, the various functions or complex ra- 
tios are treated by the analyst as though they were primitive eX-— 
isteuces, or productions furnished by the hand of nature. Just 
s the chemist collects his specimens of mineral substances from 
the bowels of the earth, and proceeds to decompose and analyze 
them by the application of fire; so the analyst takes his given 
functions, and proceeds to analyze them by differentiation after - 
the method of Leibnitz, or by derivation according to grange. 
The calculus of operations, on the contrary, proceeds synthetl- 
cally, and actually shows how to construct the ratio, or how the 
function may be generated, and thereby furnishes an irrefragable 
h 
gebra, and has an ample bearing upon our views of the philoso- 
phy of nature.’ By this theory alone, unaided by and uninterfe- 
ring with the notion either of infinitesimals or of limits except 
so far as merely to show their necessity for the production of cer- 
tain particular classes of phenomena), the theorem of Taylor, 
which is the acknowledged basis of all mathematical develop- 
ment, is directly established in its most general form, a0 the 
analogy between symbols of operation and of quantity recelves 
its fullest generalization. Lagrange’s theory of derivation shows’ 
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