(J 7 ) 
Roots a', V, and ^ tliefe Roots converfelv muft be Z/- 
betwixt ^and' Z ' ’ ^ 
^TheoPvEm II, Multiply the Terms of any Bi- 
quadranck A x ^ + h x ^ ^ C x + D 'Jo by 
any Arithmetical Series of auantities I 3 
+ X + ^) h and the Roots of the Biquadratick 
a,t>,c,a will be the Lmits oi the Roots of theEqua- 
non thatrefults from that Multiplication that is of the 
Equation. 
7 ^ 4 — 7 Axi+ 7 l^x^ — lCx+ ID = 0 
^^mx^ — SmAx^ + zmBx^— mCx 
Suppofe that fubftituting the Roots a,b,cj oi the 
Djquadratick Equation x^ — C^v-l- 
D = o fucceflively, for in 4 at * 3 A x^ + 
2 B a; — C, the Quantities that refult are R, + S 
— T, -f Z ; while x ^ — Ax^ -f B.v> — Qx 4. D 
IS in every Subftitution equal to nothing; and it is raa- 
nifeft that the Quantity 
- +/^4 _ /A*’> + IQxi —ICx ID 
-f-4i*?x4 — im Ax' ^%m^x‘—mCx 
will become (when a,h, c,d are fubftituted fuc'ceffivelv 
m It for X) equal to —mKx, mS x, ~ mT x, I 
mLx-, where the Signs of thefe Quantities being al- 
ternately negative and pofitive, it follows that a,h,c d 
mult be Limits of that Equation by Lemma xii. 
Cor Hence it follows, that a, b, c and d are Limits 
of the Roots of the Cubicle Equation A a; > iBat* 
-T 4^=0, and converfely, that the Roots of 
this Lubick are Lmttsoi the Roots of the biquadratick 
Equation ac 4 _A a-* + — Cat + D = o for 
multiplying the Terms of this biquadratick Equation 
by the Arithmetical Progreflion o, i r j ^ 
the Cubick Axi - i B + 3 C; _’aD = o 
antes. jq The- 
