On Addition and Subtraction of Algebra. 117 
_ lead, precisely in proportion to that high degree of estimation in 
which his writings are held, All Mr. Bonnycastle’s three cases, 
in Addition, exhibit a mixture of positive with negative quantities. 
Now, this mixture is contrary to the nature of Addition, for its 
operations should be limited to quantities (whether like or unlike) 
which are either all positive, or all negative, By avoiding this 
mixture, Addition will be greatly simplified, and rendered con- 
sistent. When positive and negative quantities are opposed to 
each other, Subtraction must inevitably constitute a part of the 
operation. Nothing can be more certain than that the “ incon- 
gruous mixture,” in question, should be transferred to the rule 
for subtracting s¢mple quantities, in which the operations will 
require no change of signs, because only those quantities require 
to be subtracted, to which the negative sign is prefixed. . 
A change of signs is applicable to the Subtraction of compound 
quantities only, and to such only as contain a mixture of posi- 
tive with negative terms in the subtrahend; for, when the sub- 
trahend consists entirely of negative terms, Subtraction may then 
be performed by the rule for simple quantities, since the nega- 
tive terms in the minuend (if any there are) will continue to be 
negative terms, when transferred to the snbtrahend. 
To conform to the old rules, out of mere politeness, is to vio- 
late reason, in an instance, in which, to exercise reason is our 
professed purpose. By absurdly admitting a part of Subtraction 
in Addition, and by confining the nominal rule of Subtraction to 
a mere change of signs in the subtrahend, our indulgent authors 
apparently justified cach other in the impropriety of prefixing 
the affirmative sign (+) toa compound quantity, as l3— /b+ 
(5—4—/2). . 7 
But, should future authors perceive the propriety of making 
those now arrangements which I have suggested; then, either 
some new sign must be substituted for (+), when it is used as 
a prefix to certain compound quantities; or, a mose extensive 
definition must be given to this affirmative sign, than that which 
is appropriated to it when applied to simple quantities. 
It is often fruitless to search for the origin of vulgar errors ; 
but we may with probability suppose that our authors derived 
their erroneous ideas of Addition, from the operations necessary 
‘to be performed in finding the final product of some factors in 
Multiplication, Thus, suppose it were required to multiply 
Le +xy—y", by x—y. 
Here (a* + xy —y?*) x (x—y)= 
(a+ ay) 2— (2+ 2y)y—(2—y) y= 
(a3 + xy) — (ay + xy”) — (ay? — y°) = im 21y? +’, 
which 
