3^2 
4 -* 3 
ALGEBRA, 
-x—AM, and AM- 
-ac-cm = X t — 3X = zC , 
w? J 
there¬ 
fore 4* 3 -—3 nPx—nPc, or 4.x 3 —3 m 2 x — rrP-c—o. 
Let the equation 4.x 3 — imAx — m 2 c~o, or x 3 — 
3 m mfc 
, , . 4 4 
=0, be made to coincide with the equation x 3 — qx-\-r—o ; 
. , 3W 2 nPc 
that is, let-— q, and —»—— r, or m- 
4 4 
A 
4 q 
3 ’ 
-- ] then, from a table of cofines, find the angle whofe 
9 
3 r 
49 
49 ■ , 3 r 
— is greater than - , 
4<7 • , 9/ 
or — is greater than — 
3 <r 
that 
Let;/”— ax -f b.y n l -\-cx 2 -\-dx-\-e. 
y 1} “—&c. =0, be the equation to 
the curve, reckoning the abfcifTa 
from A\ alfo, let AP—q, AQ—r ; 
then the eauation in the two cafes be¬ 
comes y n — aq-\-b.y n ~ 
Sec. —o', and lince, 
and c— 
cofine is-. to the radius \ —, and the cofine of one 
9 3 
third of this angle, to the fame radius, is one value of x. 
Cor. i. If A be the arc whofe cofine is c, and P the 
whole circumference, c is alfo the cofine of A-\-P, or 
A-\-2Pj therefore, the cofines of ———, and-———, are 
, 3 3 
alfo values of 
Cor. 2. Since the radius is greater than the cofine, 
fegments of the abfcifTa, PBytPCx PE>X PE, will be to 
the redfangle under the ordinates, PMxPNX POX PQ, 
in an invariable ratio. Let y n — ax-\-b.y n — ‘-j- &c. . . . 4 - 
gx n -\ r hx r ‘ '-Vlx n 2 -j- &c. —o be the equation to the 
curve; then gx n -\-/ix 11 '-\-lx n ~' '-j- &c. — PMx PNx PO 
X PQ-, alfo the values of x, when y~ o, are AB, AC, AD, 
AE, that is, the roots of the equation gx n -\-hx ri r~ l ^-lx n * 
1 lx ^ 2 
4- Sec. = 0 , or a:”-]-, -h ' ■ -f &c. “o, are AB , 
AC, 
hx n ~ 
AD, AE, 
- lx h ~ 2 
and 
8 8 
confequently 
the quantity x n -J- 
&c. —AP—AB X AP — ACxAP—AD 
q 
is, — is greater than —; therefore this folution can only 
27 4 1 
be applied when the roots of the cubic are poilible. 
General Properties of Curve Lines. 
A curve is faid to be of n dimenfions when the equation 
belonging to it rifes to n dimenfions. 
Let y n — ax-\-b ,y n 1 f. cx 2 -\-dx-\-e. y n 2 — Sec. 
gx n -\-kx n 'fix' 1 2 -\- See. —o, a general equation of n di¬ 
menfions, exprefs the relation between the abfcifTa and 
ordinate of a curve, then for every different value of x, 
there are n values of y, therefore the ordinate will cut the 
curve in n, or inn—2, n —4, &c. points, according as the 
equation has n, or n —2, n —4, &c. poilible roots. 
Cor.i. Hence, if the equation be of an odd number o f 
dimenfions, the curve will have, at leaf!, one infinite arc ; 
for whatever be the value of x, there is at leaf! one pofiible 
value of y correfponding to it. 
Cor. 2. axfrb is the Turn of the ordinates, cx 2 fdxf-e 
the Turn of the products of any two, Sec. and gx n -\-hx n —‘ 
<-\-lx r ‘ “4. &c. is the product of all the ordinates. 
/ //> / HI 
If two parallel lines, MM, NN, be draw n in a curve, 
and cut by AQ, in fuch a manner 
that in each cafe, the Turn of the 
ordinates on one fide of AQ is equal 
to the fum of the ordinates on the 
other; all lines drawn in the curve, 
parallel to thele, will be cut by AQ 
in the fame manner. 
~‘-f Sec. =20, and;/”— ar-\-b.y n 
in each cafe, the fum of the pofitive 
ordinates is equal to the fum of the negative, aq-\-b— o, 
and ar-\-b—o, and by fubtraftion axr — q~ o, or a— o, 
hence 5 =o; therefore whatever be the value of x, ax-\-b 
■—o, or the fum ot the ordinates on one fide of AQ is equal 
to the fum of the ordinates on the other. The line AQ is 
called a diameter of the curve. 
If the abfcifTa APE, and ordinate NPQ cut a curve in,as 
many points as it has dimenfions, the rectangle under the 
3 
XAP — AE—PBx PCxPDxPB ; and gx n 4 bx 11 1 -f 
lx n ~-‘+ Sec. —g X PB X PCxPDxPE—PMx P A'X PO 
X PQ ; therefore, PB X PCX FL> X PP■ PM X P^X PO X 
PQ:\ 1 :g. • 
Cor. If n—2 the curve is a conic feftion ; and, if the 
abfeiffa be a diameter, or the ordinates on each fide of it, 
PM, PO, equal to each other, the rectangle under the feg¬ 
ments of the abfeiffa is to the fquare of the ordinate in an 
invariable ratio. 
If there be n right lines, BM, CM, DM, Sec. and PM t 
PM, PM, be or¬ 
dinates to the ab¬ 
feiffa AP, the re¬ 
lation between the 
abfeiffa and ordi¬ 
nates will be ex- 
prelfed by an equa¬ 
tion of the form y v — Ax-\-B .y r - 1 -\-Cx 2 -\-Dx-] r E.y n 2 —- 
&c. =0. For if AP=x, then PM—axfb, PM=zcx-\-d^ 
PM-=zex-\-f, Sec. where a, b, c,d,e,f, are invariable; 
that is, the values of y are ax-\-b, exf-d, exff, See. there¬ 
fore/'—^^..; X* + bfd+f... x/ l ~~* 4- &c. = 0, and 
if A-a+e+e..., B-bfdff..., See. y*—Ax+B . y«— 
4 &c. —o. 
If a curve have as many afymptotes as it has dimenfions, 
and a line be drawn which cuts them all, the parts of the 
line meafured from the afymptotes to the curve will toge¬ 
ther be equal to the parts meafured, in the fame direction, 
from the curve to the afymptotes. 
Let y n — ax-\-b.y n ’+ Sec. =0 be the equation to the 
curve, and y n -Ax-\-B.y n ‘ 4 * Sec. =0 the equation to the 
afymptotes; when x is infinite, the former equation be¬ 
comes;/”— axy n ~'+ Sec. —o, and the latter y n — Axy n * 
4- See. —o, and thefe equations coincide; therefore A=za ; 
alfo axf-b is the fum of the ordinates’ to the curve, and 
Axf-B, or ax-if-B, is the- fum of the ordinates to the 
afymptotes, in all cafes; hence the difference of thefe, 
b _ B\ is an invariable quantity, whatever be the value of 
x ; and at an infinite diftance this difference is nothing ; 
therefore it is always nothing, or bz^xB', confequently, 
ax-\-b=Ax-\-B; that is, the fum of the ordinates to the 
curve is equal to the fum of the ordinates to the afymp¬ 
totes. 
Let 
