330 On the Calculus of Operations. 
order uniformly, etc 
B : 
wer of the fourth order, which generates power of the third 
., ete. ; 
eduction it is found, in the case of the power of the third 
or unit measure of operation or distance. And from these con- 
siderations we find that the generated power of the second order 
increases as the duplicate of the time, the velocity increases as 
the triple square of the time, and the distance as the cube of 
the time. 
terms of the fourth power of a binomial. We find that the gen- 
erated power of the third order increases as the duplicate of the 
time, the power of the second order increases as the triple square” 
of the time, the velocity increases as the quadruple cube of the 
time, and the distance increases as the biquadrate of the time. 
These examples are sufficient to show that the process may be 
extended to any required positive number x; and thus when the 
operations performed are genetical, our calculus shows the direct 
genesis of any positive power of any given number 7; ‘while we 
have seen that when the operations were performed by a power 
of the first order, they corresponded to successive multiplications, 
and served merely to determine the powers of a geometrical unit. 
Now when the successive multiplications are effected by semi-rev- 
-olutions around a centre or origin of measurement, we exhibit 
the successive powers of negative unity ; and similarly by com- 
bining the principle of revolution with our genetical operations; 
we are enabled to show the genesis of negative and fraction 
powers of zx. Other conditions are readily assigned, which de- 
termine the several forms of exponential series, together with 
their trigonometrical and logarithmical relations. «ge 
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