ALGEBRA. 
3 2 * 
y are impoffible. But if 1 2 be —4 ae be alfo =x>, then y— 
therefore the curve becomes a 
-b+ cx ±\/ b 2 —4 ad 
right line. 3. If c 2 — 4a/ be negative, the curve has no 
infinite arc, for, when is infinite, the values of y are 
impoffible; hence the curve is an ellipfe. 4. If c~ —4 af 
be negative, 2 be — yae=zo, b 2 —4 ad be alfo o, or negative, 
all the values of y are impoffible ; in this cafe the ellipfe 
wholly vanilhes. 
On the CovJlruElion of Equations- 
The relation between the abfcilfa and ordinate of a co¬ 
nic fedtion is expreffed by a quadratic, in which for every 
different value of the abfcilfa, there are two correfponding 
values of the ordinate; and if the abfeiffa be fo drawn and 
the conic fedtion fo conftrutfed, that its equation may co¬ 
incide with a propofed quadratic, the two ordinates will be 
the roots of that quadratic, which may be determined to a 
tolerable degree of accuracy by adtual meafurement. 
/ 
Let MCM be a circle, (a figure more eafily deferibed 
than any other co¬ 
nic fedtion,) whofe 
centre is A, and ra¬ 
dius AM ; take AP 
an abfcilfa, PM an 
ordinate, meeting 
the circle in M and 
M ; join AM, and 
draw MB at right 
angles to AP ; let 
AP—x, PM—y, AM—r, and the cofine of the angle APM 
(to the radius t)=rr; then i : c :: P M : P B—cytPM—cy, 
and AM 2 —AP 2 -\-PM 2 —2 APy^BP, or r 2 —x z -\-y 2 — zexy, 
i.e.y 2 — 2cxy-l~x 2 —? 2 —o; which equation may be made 
to coincide with any propofed quadratic. 
Ex. Let the roots of the equation y 2 — py-fq—o be re¬ 
quired. Here 2 cx—p, and a 2 — r 2 —q , and any one of the 
quantities c, x, or r, may be taken at pleafure. Suppofe 
. P , it 2 
a—1, then xz=.~, t 2 =a 2 — q—~ — 
2 4 
and, fince the cofine of the Z APM— radius, PM coincides 
with PAD ; let, therefore, a circle be deferibed with the 
and 
Ip 2 
T — -? > 
radius 
NM 2 , or n- — x- 
2 by-\-b 2 
z.n z 
arranging the terms according to the dimenfions of y, we 
obtain y* —2 pci — p 2 .y 2 — ibp 2 y-{-p 2 Xa'f-b' — n 2 — o, abi- 
- tt II II III III 
quadraticrequation whofe roots are PM, PM, PM, and PM, 
and which may be made to coincide with any propofed bi¬ 
quadratic wanting the fecond term. 
Ex. Let the roots of the equation y*—if-\~ry —t=:obe 
1 
required. Alfume/i=ri, 2a — i=zq,ova— -; —2 b—r, 
or b- 
■; a 2 f-b 2 — -n 2 —— s, or n 2 ~a 2 -j- b 2 -f-s, and con- 
fequently n—f a 2 -\-b 2 -\-s ■, deferibe a parabola whofe pa¬ 
rameter is 1, and in the axis take AD— -~ - ; draw DC at 
2 
. —r 
right angles to it, and —-; from the centre C, with the 
2 
q, cutting the line DAP in D and C; take 
AP— -, and the roots of the equation are PC and PD. 
2 
The interfeftions of two conic fections may be determi¬ 
ned by a biquadratic equation, and, if the figures be fo 
drawn that this biquadratic coincides with a propofed 
equation, the roots of the latter equation may be found by 
meafuring the ordinates which determine the points of 
interfection. 
I> III 
Let MAM be a parabola whofe axis is AP, MMM a cir¬ 
cle whofe centre is C, and radius 
CM, cutting the parabola in the 
/ // III 
points M, M, M, M ; from thefe 
points draw the ordinates to the 
axis, 'MB, MP, MP, A IP, and 
from C draw CD perpendicular to 
the axis, and CN parallel to it, 
meeting PM in A r . Let /ID—a, 
DC—b, CM—v, the parameter of 
the parabola—/?, AP=zx, PM—y, 
then px—y 2 ; ■ alfo CM 2 —CN 2 -I r 
-j-y— b\", i. e. x 2 — 2ax-\-a 2 -\-y 2 — 
y2 
and, by fubflituting x for its value —, and 
radius \fa 2 -\-b 2 -\-s, deferibe the circle MMM, cutting the 
./ II III 
parabola in the points M, M ; M, M, then the ordinates : 
tiie axis PM, PM, PM, and P M, are the roots fought. 
When DC reprefents a negative quantity, the ordinates 
on the lame fide of the axis with C reprelent the negative 
roots of the equation ; and the contrary. 
Cor. 1. If the circle touch the parabola, two roots of 
the equation are equal; if it cut it only in two points,, or 
touch it in one, two roots are impoffible ; and, if the circle 
fall wholly within or without the parabola, all the roots 
are impollible, 
Cor. 2. If a 2 -\-b 2 —n 2 , orthecircle pafs through the 
point -A, the laft term of the equation p 2 n 2 — o; 
therefore y—2 pa — p 2 .y 2 — ibp 2 y—o, or y— : ipa — p 2 .y 
— zbp 2 — o, a cubic equation, which may be made to coin¬ 
cide with any propofed cubic wanting the fecond term, and 
nn in m 
the ordinates PM, PM, P M, are its roots. 
Cor. 3. If a 2 f-b 2 — n 2 — o, and alfo b—o, the equa¬ 
tion becomes y s — 2pa—p 2 —o by means of which any 
quadratic wanting the fecond term may be folved. In this 
cafe the circle pailes through the vertex of the parabola, 
and its centre falls in the axis. Thefe folutions may he ob¬ 
tained, and nearly in the fame manner, by means of any 
t\yp of the conic feftions. 
If the roots of a cubic equation x 3 — qx'-\-r—o be poffi- 
ble, they may be found by means of a table of cofines. 
Let DAC be an angle whofe coline, to the radius nr, is .v; 
in AD take AB—m, from -p 
B as a centre, with the ra¬ 
dius BA, deferibe a circle 
cutting AM\\\ C, and'from Tp, 
C, with the fame radius, 
deferibe a circle cutting _ 
AD in D ; join BC, CD, and A. K. C 
draw BK, DM, at right anglM to AM, and CL at right an¬ 
gles to AD. Then, the triangles ETC and BCD being 
ifofceles,. the angles ETC and ECT are equal, as alfo CBD 
and COS; and the perpendiculars EA", CL, bif.ecf the bafes 
-AC, BD. Alfo /_DBC—.Z_BACA-/_BCA—2Z.BAC, and 
ZiDCM— z CAD + z CDA—Z CAD y. z CBD — / CAD y 
zzCAD—iZCAD. Let CM, the cofine of ,/OCM to-the 
radius m, be called c ; then, from the fimilar triangles 
ABK, AC I., AB : AK :: AC : AL, or m : x :: zx : -—— AL , 
and AL — AB-- 
-m=:BL ; hence TO, or ALf-BL, 
4 V 
m 
-m\ again, AB: AK :: TO : AM, or n? : x : —- 
-■ m • 
4 -N 4 **f 
• - ; HF 
V6l. I, No. 21, 
