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ALGEBRA. 
P A R T IV. 
ON THE APPLICATION OF ALGEBRA TO GEOMETRY. 
THE figns made ufe of in algebraical calculations being 
general, the conclulions obtained by their a(Tidance are uni- 
verfal, and may with great eafe and convenience be tranf- 
ferred from ab ft raft magnitudes to every clafs of particular 
quantities; thus the relation of lines, furfaces, or folids, 
may generally be deduced from the principles of Algebra, 
and many properties of thefe quantities difeovered which 
could not be derived from principles purely geometrical. 
Simple algebraical quantities, their products, powers, 
roots, &rc. may be reprefented by lines. Any line AB, 
may be taken at pleafure to reprefent one quantity a , but 
if we have a fecond quantity b to reprefent, we mud take 
a line which has to the former the fame ratio that b has 
to a. Inftead of faying ; 1 B reprefents a, we may fay 
AB—a, fuppoling AB to contain as many linear units as a 
contains numeral ones. 
When a feries of algebraical quantities is to be repre¬ 
fented on one line, and each of them meafured from the 
fame point, the pofitive quantities being reprefented by 
lines taken in one direction, the negative quantities mu ft 
be reprefented by lines taken in the oppofite direction. 
Let a be the greateft of thefe quantities, then a—x may, 
by the variation of x, become equal to each of them in 
fuccellion. Let AB be the given line, and A the point 
from which the quantities are to be meafured ; take 
P 
A 
d e 
AB—a ; and, fince a — x muft be meafured from A, BD mu ft 
be taken in the contrary direction —x, then AD—a — a- ; 
and that a—-x may fucceflively coincide with each quan¬ 
tity in the feries, beginning with the greateft pofitive 
quantity, a- muft increafe ; therefore BD, which is equal 
to x, muft increafe; and when a is greater than a, BD 
is greater than AB, and AD which reprefents the negative 
quantity a—x lies in the oppolite direction from A. 
Con. i. If the algebraical value of a line be found to 
be negative, the line muft be meafured in a direction op- 
pofite to that which in the inveftigation was fuppofed to 
be pofitive. 
Cor. 2. If quantities be meafured upon a line from its 
interfe&ion with another, the pofitive quantities being 
taken in one direction, the negative quantities muft be 
taken in the other. 
If a fourth proportional to lines reprefenting/i, q, r, be 
taken, it will reprefent —; and if p— i, it will reprefent 
P 
qr ; if alfo q and r be equal, it will reprefent q 2 . 
If a mean proportional between lines reprefenting a and 
b be taken, it will reprefent y ah , which when a— i be¬ 
comes y b. Hence it appears that any pofiible algebraical 
quantities may be reprefented bylines; and converfely, 
lines may be exprefted algebraically ; and if the relations 
of the algebraical quantities be known the relations of the 
lines are known. 
The relations of furfaces to each other may be exprefted 
algebraically. -Let the lides AB, AC, of the rectangle AD, 
. _ contain the linear units a, b, 
" lb refpeiStively ; then ab will 
be the number of fuperficial 
units contained in the area. 
For every unit in AB, or 
a, has units in the area 
correfponding to it; con- 
fequently there are upon 
the whole, ab units in the 
area. Thus ab is a pro¬ 
per reprefentation of the 
rectangle AD ; and by reducing other furfaces to rectan¬ 
gles, their algebraical values may be found. 
Cor. Hence the product of the two quantities a and b 
is often called their rcElangle ; and, when b is equal to a, 
this produft is called th o fquare of a. 
In the fame manner, if a, b, c, reprefent the linear units 
in the three tides of a rectangular parallelepiped, abc will 
be the number of folid units contained in the figure; and 
confequently folids may be compared by comparing then- 
algebraical values. 
If the line PM move parallel to itfelf upon the inde¬ 
finite line AP, and at the fame time increafe or decreafe, 
the point M will trace out a ftraight line or a curve. AP 
is called the abjcijj'a, and MP the ordinate ; and the ftraight 
line or curve is tkid to be the locus of the point M. The 
nature of the curve depends upon the relation of AP to 
PM; and this relation, when exprefted algebraically, is 
called the equation to the curve. 
Having given the nature or conftruftion of the curve, 
its equation may be found. Let BM be a ftraight line 
cutting AP in a given angle 
at B, the relation of AP to 
PM is exprefted by a fimple 
equation. Suppofe AP—x, 
PM—y, AB—a; and fince 
the angles at B, P,and M, are 
invariable, BP bears an in- AT 
variable ratio to PM, let this 
be the ratio of b : c. Then 
fince BP—AP — AB—x — a, we have x— a :y :: b : c, and 
by—cx — ca, or by —rx-pra=o. 
Cor. A fimple equation belongs to a ftraight line ; be- 
caufe by altering the values of b, c, and a, and taking 
x and a, pofitive or negative, as the cafe requires, the 
equation by — cxJ^-ca— o may be.made to coincide with any 
propofed fimple equation. 
To find the equation to the Parabola. Let a point 5 be 
taken without the right 
line CB, and let the in¬ 
definite line SM revolve 
about the point S in the' 
plane SBC; alfo let CM, 
w hich is perpendicular 
to CB, cut SM in M ; 
then, if SM be always 
equal to CM, the locus 
of the point M is a para-B - 
bola. Through S draw 
BSP at right angles to 
CB, and, if Sfibebifec- 
ted in A, the curve will 
pafs through A, as ap¬ 
pears by the conftruc- 
tion; draw MP perpendicular to BP, and let AP = x, 
P M—y, A S—a; t hen S P 2 +PM 2 = (SM 2 =CM 2 — ) —BP 2 , 
or x-a] 2 -\-y 2 —x+a] 2 ; that is, x 2 — 2ax-\-a 2 -j-jy 2 — x 2 -f 
2ax-\-a 2 , or y 2 —\ax. 
To find the equation to the Ellipfe. Let two indefinite 
lines SM, HM, revolve, 
in a given plane, about 
the points S, H, and cut 
each other in M in fuch 
a manner that SM^MH 
may be an invariable 
quantity; the locus of 
the point Mis an ellipfe. 
Bifecl SH in C, and from 
Mdraw MPperpendicular 
to SH, or SH produced; 
let CP~x, PM—y, CS—c, 
SM-\-MH—2a. Then ySP'-f PM 2 —SM, and yT/A+PM 2 
=//M; therefore y SP 2 -fPM 2 -f. y HP 2 +PM 2 — SM -f- 
VT- “ 
MH, or 
-x') 2 -J -y 2 -)- V Af-x } 2 -\-y 2 — 2 a ; 
V c+x|2 -J -y 2 — 2a — V c-j-x \ 2 -\-y 2 , and fquaring 
hence 
both 
Tides, 
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