of Algebra. 49 
works I have had access, make no essential difference between 
some parts of Addition and some parts of Subtraction of Algebra. 
From an attentive consideration of the subject, I feel persuaded 
that the operations of Addition should be restricted to quantities, 
whether like or unlike, which have like signs. That part of 
Addition which is employed in collecting quantities, whether like 
or unlike, which have unlike signs, should be classed under the 
rule for the Subtraction of simple quantities. Our authors de- 
fine clearly enough that the sign + denotes Addition, and that 
the sign — denotes Subtraction; they then blend these signs, 
or blend the quantities to which these signs are prefixed, and 
sometimes call the mixture Addition, and sometimes call it Sub- 
traction. Dissatisfied with this procedure, Mr. Bonnycastle re- 
commends new names for these two primary rules; as if there 
were some secret charm in a name absurdly epplied. If any ob- 
Jection lie against the term Subtraction, as Mr. Bounycastle sup- 
poses and affirms, that objection may be obviated by remoying 
the cause, whether real or imaginary.‘ The incongruous mix- 
ture,” as Mr. Bonnycasle styles it, may be removed or avoided, 
if offensive, by transposing the negative term or terms from the 
minuend to the subtrahend, and by transposing, also, the nega- 
tive term or terms from the subtrahend to the minuend ; by 
which means, we shall have nothing but positive terms in the 
minuend, and nothing but negative terms in the subtrahend. 
Thus, retaining the old furm of writing the quantities, if from 
4g 4+ a— WU 
wetake 42+ b — 8a, 
we shall obtain, by trausposing the negative terms, this arrange- 
ment, viz. 4xn+a4+3a 
—(4xr+6 +b) 
or this, viz. 42 +4a—4x2 —2$b=4a—2)— diff. required. 
For, the difference between any two quantities will remain the 
same, whether we equally augment or equally diminish them. 
Thus, let a excéed ; and, to avoid ambiguity, let a, as well as 
b, exceed c; then will a—b=(a+c)— (b+c) =(a—c)—(b—c). 
Therefore, if from 42 +a—J 
we take 4x + b ~ 3a, 
first, augment both quantities by J, and we obtain this arrange- 
ment, viz. From 4x+a 
Take 42 + 2b —3a., 
Now augmeut both quantities by 3a, and we shall have this 
arrangement, viz. From 4x2 + 4a 
Take 4x2 + 20, 
or this, viz, 42 + 4a — (4a + 21) = 4a—2) = diff, as be- 
fore, - 
Both quantities, in a manner analogous to the method em- 
Vol. 59, No. 285. Jan. 1822. G ployed 
