292 
ALGEBRA, 
In two ranks of proportidnals, if the correfponding 
terms be multiplied together, the produdls will be pro¬ 
portionals. 
Let a : 6 :: c : d 
And e :f:: g \ k 
Then will ae : bf -.: eg : dll. 
(1 C CP CIS CP 
Becaufe and -=y, therefore or ae : bf :: 
,, b d j h bf dk J 
eg : d!i. J 
This is called compounding the proportions. The pro- 
pofition is true if applied to any number of proportions.- 
On Variable Quantities. 
In the inveftigation of the relation which varying and 
dependent quantities bear to each other, the conclufions 
are more eafily obtained, by exprefting only two terms in 
each proportion, than by retaining the four. But though, 
in confidering the variations of fuch quantities, two terms 
only are exprelTed, it will be neceflary for the learner to 
keep conftantly in mind that four are fuppofed, and that 
the operations which this method admits are in reality the 
operations of proportionals. 
Def. i. One quantity is faid to vary direP.ly as another, 
when their magnitudes depend w'holly upon each other, 
and in fuch a manner, that, if one be changed, the other is 
changed in the fame proportion. Let A and B be mutual¬ 
ly dependent upon each other, in fuch a way, that if A be 
changed to any other value a, B mufl be changed to ano¬ 
ther value b, fuch, that A: a:: B : b, then A is faid to vary 
direftly as B. ThisfignCX, placed between two quanti¬ 
ties, (hews that they vary as each other. 
Def. 2. One quantity is faid to vary inverfely as another, 
when one cannot be changed in any manner, but the reci¬ 
procal of the other is changed in the fame proportion. A 
varies inverfely as B, (Act.-—,) if, when A is changed to a, 
B 
B be changed to b, in fuch a manner that A : a;: 
or A : a :: b : B. 
Def. 3. One quantity is faid to vary as two others jointly, 
if, when the former is changed in any manner, the product 
of the other two be changed in the fame proportion. 
Thus, A varies as B and C jointly, ( Act.BC ,) when A 
cannot be changed to a , but the produdl BC mu(i be chang¬ 
ed to be, fuch, that A: ai: BCt be. 
Def. 4. One quantity is faid to vary dirclilly as a fecond, 
and inverfely as a third, when the firft cannot be changed 
in any manner, but the fecond multiplied by the recipro¬ 
cal of the third is changed in the fame proportion. A 
varies diredlly as B, and inverfely asC, ( Act. “,) when 
B b 
A : a :: ■ - ; A, B, C, and a, b, c, being correfponding 
values of the three quantities. 
In the following articles, A,B,C, Sec. reprefent corre¬ 
fponding values of ally quantities, and a, b, c, See. any 
other correfponding values of the fame quantities. 
If one quantity vary as a fecond, and that fecond as a 
third, the firft varies as the third. Let A : a:: B: b, and 
B:b::C:c, th'en A : a \: C : c ; thatis/fccC. In the fame 
manner, if A(XB and Sa~, then 
If two quantities vary refpedlively as a third, their fum, 
or difference, or the fquare root of their product, will vary 
as the third. Let AqlC and B&.C, then A±Bcx.C, or 
fABciC. By the fuppofition A: a:: C: c :: Bfb, there¬ 
fore A : a:i B:b, alternately A: B:: a: b, and by compo- 
fition or divifion A±B-. B a±.b ; b , alt. A±.B ; a±:b :: B 
t 5:; C : c; that is, CotA±B^ 
• Again, A: a:: C: c 
And B : b :: C : c 
Therefore, AB : ab :: C 2 : c 2 
And f AB : fab :: C : r; that is, CCf f AB. 
If one quantity vary as two others jointly, either of the 
latter varies as the firft diredfly and the other inverfely. ' 
Let Fcx FT, then 
V V 
Fa^rorTa—. 
7 F 
Cor. If the product of the two quantities be invaria¬ 
ble, they vary inverfely as each other. 
Let BxP be conftant, qr BxT’OCi; by divifion, 
If four quantities be always proportionals, and one or 
two of them be invariable, we may find how the others vary. 
Ex. Let p,q,r,s, be always proportionals, and p inva¬ 
riable, then sCXqr. Becaufe ps—qr, psQf.qr, and, fince p 
is conftant, sQuqr. 
When the increafe or decreafe of one quantity depends 
upon the increafe or decreafe of two others, and it ap¬ 
pears, that, if either of thefe latter be invariable, the firft 
varies as the other; when they both vary, the firft is as 
their produft. 
Let See Fwhen T is given. 
And Sex 7" when Lis given. 
When neither T nor Lis given, See TV. The variation of 
5 depends upon the variations of the two quantities T and 
L; let the changes take place feparately, and whilft T is 
changed to t, let S be changed to S 1 , then, by the fuppo¬ 
fition, S: S' ::T : /; but this value, 5‘, will again be chang¬ 
ed to s, by the variation of L, and in the fame proportion 
that Lis changed, that is, S': s:: F:v; and, by compound¬ 
ing this with the laft proportion, : S's :: TV : tv, or 
S : s :: TV : tv. 
In the fame manner, if there be any number of magni¬ 
tudes, P, Q, R, S, each of which varies as another, L, 
when the reft are conftant; when they are all changed, V 
varies as their product. 
On Arithmetical Progrejfion. 
Quantities are faid to be in arithmetical progrejfion, when 
they increafe or decreafe by a common difference. Thus, 
1, 3, 5, 7, g, See. a, a-\-b , a-\-2b, a-\- 3L See. a, a — b, a — ib, 
a —3 b, Sec. are in arithmetical progreftion. Hence it is 
manifeft that if a be the firft term, and a-\-b the fecond, 
a-\-2b is the third, a-\-^b the fourth, Sec. and a-\-n — i.b 
the nth term. 
The fum of a feries of quantities in arithmetical pro- 
greftion is found by multiplying the fum of the firft and 
laft terms by half the number of terms. Let a be the firft; 
term, b the common difference, n the number of terms, 
and s the fum of the feries. Then 
a -\-a-\-b -\-a-\-ib .... a-\-n — l.b—s. 
Or a-\-n — \.b-\-a-\-n — i.b-\-af-n —3 .b . -\-q~s. 
Sum 2 a-\-n — x.b-\-ra\n — \.b-f2a-\-n —&c. to n 
terms, =2!. 
Or 2a-j-n — i.bX.n—2s 
And $—2a-\-n —i.^x-* 
2 
Cor. Any three of the quantities, s, a, n, b, being 
given, the fourth may be found, from the equation s= 
n 
2 a-\-n —1 .by,—. 
2 
Ex. 1. To find the fum of 14 terms of the fe ries, 1 , 3, 
5, 7, Sec. Here a—i, b—2, 72=14; therefore, 5=2+26X7 
= 196. 
Ex. 2. Required the fum of 9 terms of the feries, n, 
9, 7, 5, Sec. In this cafe, a—11, £=—2, n=9; therefore, 
5=22—16 X -=6x-=^7. 
Ex 
