— On the Calculus of Operations. 327 
Arr. XXXV.—On the Calculus of Operations; by Joun 
M.* 
Paterson, A. 
In the algebraical and other forms of calculus in vogue, the 
Various quantities, numbers, symbols, functions, ete., constituting 
the matter of the science, are understood as ratios of magnitudes 
merely as such. The expressions z, x”, ©*, for example, are ra- 
tos of the magnitudes zl, x2. 12, x3, 1*, to the unit magnitudes 
1, 1°, 1%, respectively; the units 1, 12, 12, representing a line, 
a square surface, and a cubical volume respectively. But as all 
geometrical magnitude is confined to one, two, or three dimen- 
Sions, it follows, that while such expressions as 4‘, x*, etc. are 
Yatlos of 7‘ 1+, x5 15, etc. to 14, 15, etc., these units of compari- 
son can no longer represent geometrical magnitudes. ‘The ques- 
tion arises, what do these units signify? ‘They are complex units 
of number, But, we ask, numbers, or number'of what? Wherein 
does 14, 1°, etc., differ from 1? Number is a general idea, and 
therefore must be applicable. to several particular instances, If 
then a particular instance of geometrical unity can be found, to 
Which all other powers and roots of unity can be referred as a 
base, we shall be prepared to answer the preceding question. 
The Calculus of Operations professes to have discovered such 
3 base, and to have broached the method and traced the route 
which leads to the most general interpretation of the functions 
Of unity, By this calculus, all functions are regarded as ratios of 
the measures of operations performed in space and time; for ex- 
ample, the expression x‘ is the ratio obtained by comparing the 
Operation whose measure is x‘ 1‘, with that whose measure is 
. 1‘; the latter being a compound unit. of operation, ultimately 
teferable to a simple unit of space, a linear unit. 
_ Addition is an operation, in which we take for unit measure of 
Operation the distance through which the added body is trans- 
fered in the unit of time; each unit of the recorded result ex- 
Pressing the performance of one operation of addition, and indi- 
cated geometrically by the unit of length drawn to the right of 
an origin. 
Subtraction is an operation diametrically opposed to that of 
addition, and will therefore be indicated geometrically by the unit 
of length (measure of operation) drawn to the deft of an origin. 
Multiplication is an abridged method of performing a series of 
additions ; so that when the multiplier is unity, the operation 
coincides with the performance of one addition. Consequently 
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“an This, article was received by the ¢ditors along with the recent work of Mr. 
Paterson, arated “The Onleala of Operations.” 184 pp., with 4 plates, Albany, 
eateety and Sprague. The merit and heii’ views of the work wilt be uth 
‘red from-this exposition of the subject by the author, : 
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