( 59 ) 


IV. A fecond Letter from Mr. Colin M c Laurin, 
(profejfor of Mathematicks in the Univerfity of 
Edinburgh andF. 5 . to Martin Folkes, Efq$ 
concerning the Roots of Equations , with the De¬ 
mon fir at ion of other (Rules in Algebra , being the 
Continuation of the Letter publifhed in the 
Philofophical Tranfa&ions, N° 394. 

Edinburgh, April 19th, 1729,, 

SIR, 

I N the y ar 1715, I wrote to you that I had a Me¬ 
thod of demonftrating Sir Ifaac Newton's Rule con¬ 
cerning the impoflible Roots of Equations, deduced 
from this obvious Principle, that the Squares of the 
Differences of real Quantities muft always be pofitive 
and fome time after, I fent you the firm Principles of 
that Method, which were published in the Philofophi¬ 
cal Tranfatfions for the Month of May , 1716. The 
Defign I have for fome Time had of publifhing a Trea- 
tife of Algebra, where I propofed to treat this and fe- 
veral other Subje&s in a new Manner, made me think 
it unneceffary to fend you the remaining Part of that 
Paper. But fome Reafons have now determined me 
to fend you with the Continuation of my former Me¬ 
thod, a Ihort Account of two other Methods in which 
1 have treated the fame Subject, and fome Obfervations 
on Equations that I take to be new, and which will, 
perhaps, be more acceptable to you than what relates 
to the imaginary Roots themfelves. Befides Sir Ifaac 
Newton's Rule, there arifes from the following gene- 


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< tf) 

ral Propofition?, a great Variety of new Rules, different 
from his, and from any other hitherto publifhed, for dif- 
covering when an Equation has imaginary Roots. I fhall 
particularly explain One that is more ufeful for that 
Purpofe, than any that have been hitherto publifhed. 

Suppofe there is an Equation of (n) Dimenfions of 
this Form, 

x w — A 1 -F Bx * •— C 5 -j- Da** * 

— ILx” — ' 5 -f- Fa; — Gx Hx 

lx ”~ 9 + K* ” -10 &c. = o. 

And that the Roots of this Equation are, a y b,Cyd,ey 
ftg , h y iy ky ly &Cc. then fhall A — a-{-b-\-c-\-d-\-e 
-J-/ckc. and therefore I call a y b, c, d, e,f, &c. Parts 
ox Therms of the Coefficient A. For the fame ReafonI 
call a b,ac, adyae,bcybd,cd, < 5 cc. Parts or Terms of 
the Coefficient B j abc,abd y abe,acd,bcdy &c. Parts or 
Terms of C; abcdy abce, abcf. Parts or Terms of the 
Coefficient D, and fo on. By the Dimen/ions of any Co¬ 
efficient ; I mean the Number of Roots or Factors that 
are multiplied into each other in its Parts, which is al¬ 
ways equal to the Number of Terms in the Equation 
that preceed that Coefficient. Thus A is a Coefficient of 
one Dimenfion, B of two, C of three, and fo of the reft. 

I call a Part or Term of a Coefficient C fimilar to a 
Part or Term of any Coefficient G, when the Part of 
G involves all the Fa&ors of the Part of C: Thus 
abcy abcdefg are fimilar Parts of C and G j after the 
fame manner abcd,abcdef are fimilar Parts of D and 
F, the Part of Finvolvingall the Fa&ors of the Part of 
D. Thofe I call dijjimlar Parts that involve no com¬ 
mon Root or Fa&or: Thus abcy and defgb arediflimi- 
lar Parts of the Coefficients C and F. The Sum of all 

the 



( <5 i ) 

the Produds that can be made by multiplying the Parts 
of any CoefficientC by all the fimilar Parts of G, I ex- 
prefs by C G' placing a fmall Line over each Coeffici¬ 
ent.: After the lame manner D'F' expreffes the Sum 
of all the Produds that can be made by multiplying 
the fimilar Parts of D and F by each other*, and C'xC' 
expreffes the Sum of the Squares of the Parts of the 
Coefficient C, but C'xC, - expreffes the Sum of the 
Produds that can be made by multiplying any two 
Parts of C by one another. Thefe Expreflions being 
underftood, and the five Propofitions in .Phil. T'ranJ'. 
N° 3 94, being premifed, next follows 

PROP. VI. 

If the Difference of the Dimenftons of any two 
Coefficients C and G he called (m ) then pall the 
Product of thefe Coefficients multiplied by one ano- 

-■■■ | r 2 

ther he equal to C G 1 wq-ixB' H' -)--— X 

A 1' -— T > x Xl XK. 

X 1X3 

Where B and H are the Coefficients adjacent to the 
Coefficients C and G, A and I the Coefficients adjacent 
to B and H, I and K the Coefficients adjacent to B and H. 

It is known that C = ahe -j- ah d -j- qhe -\-ahfffi 
abg,l Sec. andG —ahcdefg-\-ahcdefJo-\-ahcdefi 
c d efg h y 6cc. and it is manifeft, 
i. That in the Produd CG each Term of CG # 
will arife once as a 2 h 2 c 2 defg. But 

x. Any Term of B' H'as a 2 h 2 c defgh may be the 
Produd of a be, and abdefghy or of a bd and 
ah c efg hy or of ahe and ahe dfg h Y or of a hf and 

abed eg h. 



( 6t ) 


abcdegb,oro{abg and abed efh , or laftiy of abb 
and abed efg ; fo that it may be the Produd of any 
Term of C that involves with a b one of the Roots, 
fjdfe/ig^ multiplied by that Term of G, which in¬ 
volves ab and the other five j that is, it may arife in the 
Produd C G as often as there are Roots in a 2 b*e defg h 
befides a and b , or in general, as often as there are Units 
in the Difference of the Dimenfions of B and H, that 
is, m "f" z times ; becaufe m exprelfes the Difference 
of the Dimenfions of G and G, and confequently in ex- 
prefiing the Value of C G the Coefficient of the fecond 
Term B' H' muff be m -f- z. 

3. Any Term of AI, as a 2 b edefghi, may be the 
Produd of any Part of C that involves the Root a with 
any two of the reft bg,d,e,fg,h,i (the Number of 
which is the Difference of the Dimenfions of A and I, 
which is in general equal to m -f- 4) multiplied by the 
Part of G that involves a and the other fix ; and there¬ 
fore a 2 b e d efg hi or any other Term of A' I' muft arife 
as often as different Produds of two Quantities can be 
taken from Quantities whofe Number is m -|- 4, that 


is m -p 4 x 


+ 4—1 


times or 


+ 3 


X 


-h 4 


times j and confequently in expreffing the Value of C G 

- | i 2 

the Coefficient of the third Term A'I' muft be- 

m 4-4 1 

X -— * 

3 


.4. Any Term of ixK as ab edefghik, may be the 
Produd of any Part of C that involves three of its Fadors, 
and of the Part of G that involves the reft, and there¬ 
fore may arife in the Produd CG as often as different 
1 Pro- 



( < 5 ? ) 

Produffs of three Quantities can be taken out of Quan- 
tities whofe Number is m -\- 6 that is, m -j- 6x — * - 

X 

^ - | T a 

X —times, and therefore the Coefficient of the 
3 

fourth Term, in the Value of C G muft be ^ X 

i 

m 4 - < m-\- 6 

-— X - 1 - 

a 3 

In general, in expreffing the Value of the Produff 
of any two Coefficients C and G, if x exprefs the Or¬ 
der of any Term of this Value as A'I', that is, the 
■Number of Terms that precede it, the Coefficient of 

, m „ , zx m z x - 4 - m — i 

that lerm mult be -- X - - - X 

i z 

ZX 4- M - z , . _ _ 

- : -- 6cc. taking as many t actors as there are 

3 

Units in x. 

Cor. I. If it is required to find by this Propofition 
the Square of any Coefficient E, then fuppofe «r= a, 
the Difference of the Dimenlions of the Coefficients in 
this Cafe vaniffiing, and we ffiall have E* = E' x E' -j-, 

iD'F'+ 3X — xC'G' + 4 x — x 4 x B'tt 

x x 3 

&C. =E'xE'-f iD'F' + 6C'G* + ioB'H' 
-f- 7 o A 1 I* -j- iji K. Therefore if E' x E / ex¬ 
press the Sum of the Produffs of any two parts of E 
multiplied by each other, we ffiall have E* = E/ x E f 
-)- i B X E ; , and therefore E*xE ; = D' F 7 -f* 
3 C'G' + ioB'H'+ 3 fAT'4. 126K. 

K 


Cor. 



( < 4 ) 

Cor. II. It follows from this Proportion that 
E*=: E'xE'+2 D' F '+6 C G'+z o B' H' +70 A' I' + zyz K. 


DF = - - D'F'+ 4 C , G'+i 5 'B'H , +j'< 5 A'I' + 2ioK. 
CG= .... C'G'-f- <JB'H'+2 8 A'I' 4 -izoK 

BH = - -.B'H'-f- 8A'I' + K 

A I ..A'I'-f 10K 

K ..K. 

C o r. III. It early appears by comparing the Theo¬ 
rems given in the laft Corollary, that 
E'E' = - E*— 2DF + 2CG—zBH-f 2AI— :K. 
D'F' = - - - DF— 4 CG-j-pBH—nSAI^-ifK 
C'G' = - - - - ~ - CG—<SBH+ 2 oAI—foK 

B'H' ..BH— 8 AI + 3f K 

A'I' .AI— 10K. 


PROP. VII. 

fl - — I ft — X 

Let / = « x -x-&c. taking as many 

% 3 

Faftors as the Coefficient E has Dimenjions and 

— . l - x E 2 flail always exceed DF—CG -f-BH 

— AI -f- K when the Roots of the Equation are all 
real Quantities . 

For it is manifeft that l exprefies the Number of 
Parts or Terms in the Coefficient E, and it is plain 
from Propofition V {See Phil, franf. N 8 394) that 

.-""I!— x £* muft always be greater than the Sum of 
x l 

the Products that can be made by multiplying any two 

of 









() 


of the Parts of Eby each other, that is, than E'xE,; 
but iE'xE, = E* — E' E' = (by the firft Theo¬ 
rem in the laft Corollary) iDF—iCG -f-^H 


— 2 A ; I + 1 and 


therefore fince 


/-—i 

u~ 


x E* 




2/ 


E* 


mull always exceed E 1 E,, it follows that 

muft always be greater than DF —• CG -f- B H — 
AI -f- K when the Roots of the Equation are real 
Quantities. 


S c h o l. In following my Method this was the firft 
general Propofition prefented itfelf. For having firft ob- 
ferved that if/expreffes the Number of any Quantities, 


the Square of their Sum multiplied by 



muft 


always exceed the Sum of the Produces made by multi¬ 
plying any two of them by each other •, and that the Ex- 
cefs was the Sum of the Squares of the Differences of 
the Quantities divided by 2/, it was eafy to fee in the 

Equation x * — Ax n ~~ l Ba? 2 —- Cx 9 ~~* -j- 
Dx"“* 6cc. = o. Since B is the Sum of the Pro- 
duds of any two of the Parts of A, that if / expreffes 

theNumber of the Roots of the Equation,-— x A* 

muft always exceed B; and this is one Part of 
the ^th Propofition. In the next Place, I compared 
the Sum of the Produds of any two Parts of B with 
AC, and found that it was not equal to AC but to 
AC —D from which I inferred, that if / expreffes 

K the 



( 66 ) 

/—. I 

the Number of the Parts of B then x B e muff 

zl 

always exceed AC — D j and thefe eafily fuggefted 
this general Propofition. 

P R O P. VIIL 

Let r exprefs the Dimenfions of the Coefficient C r , 
and s the Difference of the Dimenfons of the Co¬ 
efficients C and G, then B and H being Coefficients 
adjacent to C and G, n —. r — ~s x r C A G f /hallal¬ 
ways he greater than s -j- i x s -f- % x B > W when 
the Roots of the Equation are all real Quantities af¬ 
fected with the fame Sign. 

For taking the Differences of all thofe Parts of the 
Coefficient C that are fimilar in all their Fadors but 
one, as abc,abh> ab i , &c. and multiplying the Square 
of each Difference by fuch Parts of the Coefficient D 
(which is of s Dimenfions)- as are diffitnilar to both 
the Parts of C in that Difference, the Sum of all thofe 
Squares thus multiplied, will confift of Terms of 
taken_pofitively, and of Terms of BffF taken negative¬ 
ly. By multiplying in this manner abc — abh\ 2 4. 

abT^abi l 2 -f qbc — abk \*8zc. -f- abc —acIQ 4 . 
afc— — ack \* Sic, -j -Tbc—bcb\ % + 

abc —■ bce\ 2 -^-abc — bck j 2 5Cc. by defg the Term of 
D, that is diflimilar to all thofe Parts of C, you will 
find that a 'b'c'de fg willarife in the Sum of the Pro- 
duds rx n—r — s times: For thofe Produds may be 

alfo expreffed thus defgaffx c — b\*-\-c — 

SCcffidefga’c* x b — h\ -f- b — /{’ + b — k \ 2 Sec. -f~ 

defgb’c 1 



C h ) 

i'efghc 1 x a —/?[*+ /» — i|* + ^ &c. where 

the Number of the Differences c — h, c —. i y c — k 9 
dec. whofe Squares are multiplied by defga 7 b * is ma- 
nifeftly equal to the Number of the Roots of the E- 
quation that do not enter a*b t e 2 4 efg or a bedefg, that 
is,, to the Excefs of the Number of the Roots of- the 
Equation above the Dimenfions of a b c d efg,a. Term of 
G, that is to n — r — s. But in colleding all the 
faid Produds, n—~ r —s x a 2 b 2 c t defg muft arife as 
often as there are Units in r: Becaufe the Terms 
which are fubtraded from abc may differ from it in- 
the Root r,.as abb, abi, abk,dCc. or in the Root Z>, as 
a c h y a c i,ac k, dec. or in the Root a as be hjoC% be k ; 

that i?, n — r —a a t b , c i defg muff arife as often as 
there are Dimenfions in a be, a Term of C,. or as often 
in general as there are units in r, which expreffes the 
Dimenfions of C : Therefore the Term - a’h 3 e*defg 
will arife in the Sum of the above-mentioned Produds 

r x n — r — s times. 

The Negative Part muft confift of the Terms of 
B'H' doubled; each of which,as za 3 b 2 c defg may a- 
rife as often as there can be Differences c — d> e — e, 
e — f y c — g, d‘ — e, die. affumed amongft the Terms 
e 3 d y e % f y g whofe N umber is equal to s -J~ z that is ? j % 

J 1 I 

^-times j and therefore a 2 b r cdefg or any other 

Part of B' H 7 muft arife in the negative Part s -j- i 
X IT z times ; and fince die whole aggregate muft 
be pofitive it follows n — r ~—jx rC'G* muft ah 
ways exceed s -f- i x s -ft- z x B' IT, 


G O R«- 



( *8 ) 

Cok. I. SupptJfe we are to compare E'E' the Sum 
of the Squares of the Parts of E with D' F' the Sum 
of the Produ&s of the fimilar Parts of D and F; in this 

Cafe s vaniffies, and therefore n — r x r E' E' muft ex¬ 
ceed % D' F'. Let n — r xr=» and confequently 
n — r — i X r — i =m — n i i « — r — z x 
r —— m — z n -f- 4 j 0 — r — 3 x r — 3 — 
m— 3 0 9 j *—— 4Xf — 4 = *0—6. 

Since it is plain that n — r — q x r - — q = n r x r 

~-qn-\-q 2 , Then by this Propofition, fuppofing 

_wxE'E'-,. iD'F'= a' 

m — n- f-t x D'fii C'G' = l* 
m —{-4 x C'G'—3 oB'H' = c' 
m —3 0 -|- 9 xB'H' — 5-6 AT' = ^// 

*0—4«-f- 16x A'i' —90 K' = et 
The Quantities muft be always pofitive 

when the Roots of the Equation are real Quantities affed- 
ed with the fame Sign,. The Coefficients prefixed to the 
negative Parts are the Numbers 2., 11,30,5:6,90, whofe 
Differences equally increafe by the fame Number 8. 

Cor. II. Suppofing as before, that n — r x r = m\ 
and alfo that my m —n -f- 1 = /?/• niym — 4 

= m"-, m" ym — 3^+9 = m"' < 3 tc. it may be de- 
monftrated after the manner of this Propofition, that if 
«*E'E'— zD'F =za' 
m'ElE 1 — ixnCG' = a" 

m" E'E' — i x 11X 30 B' H' = a M 

m 1 " E'E' —- x x 12. X 30 X AT' = a"" dec. 

Then 



( < 5 ? ) 

Then {hall Sec. be always pofitive when the 

Roots are real Quantities, whether they be affefted 
with the fame, or with different Signs. The Negative 
Coefficients arife by multiplying thofe in the preceeding 
Corollary, 2,11,30,56,90, by one another. 

PROP. IX. 

Let a',L',c',d',e', and m exprefs the fame Quantities 
as in the Corollaries of the laf Propo/ition,and m E*— 

m n + "i X D F = d + V 4 " 2 c> 4 ” Sd’ -f* 14^'. 
For by Cor. ii Prop. vi. 

E 1 rsE'E'+aD'F' + tfC'G' + aoB'H'+yoA' I* + zfz K, 
and by the fame 

DF= - - - D'F+4CGjjhjt5-B'H'+jtf-A'r + MoK 5 
therefore m E 1 — m -}- n 4- 1 X D F = m El El 4 * 
w — n—i xD'F' -j-~ m 1 n — 2 X 2 O G' 
4 - m — 3 n — 3 X 5 B' H' + m ■— 4 n — 4 x 
14 A' 1 ' -f- m — 5 ;/ — 5 x 4 x K = (by fubftituting 
fucceflively for m E' E', m — n -f- 1 x D' F', 
m •— 2 n 4 - 4 x C' G', m — 3 » 4- 9 x B' HA 

m — 4 n 4-16 x A / i' their Values deduced from the 
firft Corollary of the laft Propofition) ■ == af *F b' 4 * 
2 o' 4- 5 d‘ -f- 14 e> > where the Coefficients prefixed 
to a'Q.c'fp, are the Differences of the Coefficients 
ofE'EVD'F', C' G',B' H', A' i 1 and K in the Values 
of E* andDF taken from Cor. ii. Prop. vi. being 1 — o, 

2 — 1, 6 — 4, 20—15, 70 — 56 and 252 — 21a. 


Cor. 



( 70 ) 

C o r. Since m = n —r x #* therefore i 




4* n 4. i 

r+ tXr and confequently ——x 


r 


r + i 

n _x E r rauft always be greater than D F the 

Produd of the Coefficients adjacent to E 4 and hence 
the Fractions are deduced, that in Sir Ifaac Newton’s 
Rule are placed over the Terms of the Equation, which 
multiplied by the Square of the Terms under them, 
rauft always exceed the Produds of the adjacent Terms 
of the Equation, when the Roots are real Quantities: 
For it is manifeft that the Fradion to be placed over 
the Term Eat "” r according to that Rule is the Quo- 
,n — r . n — r+i 


tient of 


f + 1 


r n ■ 

divided by— 


r 


P R O P. X, 

The fame Exprefions being allowed as in the pro¬ 
ceeding Proportions ft will be found in thefameman- 
nerthat as m E*—m+ nfT x D F=«/-f I4 e ' f 0 
m — n - {- ixDF— m+zn-\-Q. xCG= - b'-fWf -{-z%e' 
m — 2. n -)- 4 x CG—xBH— - - c’+f+zoe' 
m —3 «•'+ 9 x B B.—m-{-4 r n-\-x6 x AI = ... - d'-\- 7 *>- 
m — 4 m -f-i 6 x AI— mffn+z 5 x K — - - - - - . e . 

Thefe Theorems are eafily deduced from the Theo¬ 
rems given in the fecond Corollary of Prop. vi. and the 
firft Corollary of the viii th Propofitionj and the Co¬ 
efficients prefixed to a'f^c'f^e', are the Differences 
of the Coefficients of the corresponding Terms in the 
Values of E%D F, C G, B H, A I and K in Cor. ii. 
Prop. vi. 

Cor. 



( 71 ) 

Cor. Hence the Produds of any two Coefficients, 
asDF and AI may be compared together when the 
Sum of the Ditnenfions of D and F is equal to the 
Sum of the Dimenfions of A and I. Let the Dimen- 
(ions of A and F be equal to s and m refpectiveiy, and 

n — s n >— s — i n — s — x „ 

let p =-X- x -e>Cc. taK~ 

J-f-i J -{- x -f+3 

ing as many Fadors as there are Units in the Difference 

of the Dimenfions of D and A. Let q = -- - m . x 

n — m —■ i n — m — x . m H“ 1 

—-- X ---&c. taking as many Fad- 

m -j- x m - j- 3 - 

ors as you took in the Value of p. Then fhall x 

p 

D F always exceed AI when the Roots of the Equa¬ 
tion are real Quantities affeded with the fame Sign j and 
this Rule obtains, though the Roots are affected with 
different Signs when the Coefficients D and F are equal, 

PROP. XI. 

the fame ‘things being fuppofed as in the preceed- 
ingPropoft turns. 

1. ®E ! —»«-(- ixzDF-|-B2-f-4XiCG—-ra-|-9X 

2 £ H nt -j- 16 x 2 AI—• is-]- 25 x 2 K - - 

2. in — »-J-ixDF— m — n-j-4. X4CG-j-»z— ii-fy x 
9 BH —in —■ n -j-16 x 16 AI -j -m — n -J- aj x 25 K 

5. 2?H-4 xCG— sk— 2«-f9Xtf BH-j-%- znfis x 

20 AI -j-m —2 a -f- 25 x 50 K - - - - - - 

4 - m —3 ? H ~9 x B H —m — 3 a-j-i 6 x 8 A —3 x 3 5 K-=d'. 

<• - - m — 4a -j- \6 x AI — m — 4 a 4-25 x xo K = e\ 

L Thefe 






( 71 ) 

Thefe Theorems follow eafily from the third Corol¬ 
lary of the vi th Propofition. The firft eafily appears 
thus, a' — mXPE! — z D' F' = (by that Corollary) 

m E *—zm DF+ CG — x m BH+i m AI — zm K. 

— z D F -f- 8 CG — 18 BH-f 3 xAI— 50 K. 

= m E* — m 1 xiDF-j- m 4- 4 x z G G — 

m : -f- 9 x z B H -f- m -f- 16x x AI — m -f- z$ x iK. 
The other Theorems are deduced from the fame Co¬ 
rollary compared with Cor. i. Prop. viii. 


PROP. XII. 


The fame Things being fuppofed as in the fecond 
Corollary of the viii th Proportion. 

j.mE 1 —»-fiXiDF-f w + 4 X 2 CG — m -j- <, x ~t , 

2 B H m -f -16 x 2 A I—■ m -f- 25 x 2 K - - - 5 


i.m' E 1 •— 2 in U F -f- m —■ 12 x 2 C G — ni 
2 B H-j- nr — 240 x — — 600 x 2 K 


— 7 * 

- * -i 




3. m" E* — 2 ni' DF -f- 2 m" CG — m" 4- ;<5o x 
2 360 x8x 2AI — z»"' + 3<Jox 35X2KJ ~ a 


4.m"xE J —2DF-f 2 CG —2BH-F»' —750 x 28 x 
AI — z#"" — 7200 x 28 X2K ------- 

&V. 




Thefe Theorems follow from the third Corollarv of 
the vi t!l -Propofition compared with the fecond Corollary 
of the eighth Propofition. The firft is the fame with 
the firfF of the lafF Propofition. The fecond is de- 
motiftrated by fubftituting in m f E l W — 24 C / GT—- 
The Values of E'E' and OG' given in the third 

? Cor, 




( 7 ? ) 

Cor. of the vi th Propofition. The third is found by 
fubftituting in m ,f E'E' 710 B'H' = a ,n the Va¬ 
lues of E'E' and B'H^ and by a like Subftitution 
thefe Theorems may be continued. 

A General COROLLARY. 

From thefe Propofitions a great Variety of Rules 
may be deduced for difcovering when an Equation has 
imaginary Roots. The Foundation of Sir IJ'aac New¬ 
ton's Rule is demonftrated in the ninth Propofition, and 
its Corollary. The feventh Propofition (hews that if 

l -^~ xE* does not exceed DF— CG+BH — A I 

-f. K, fome of the Roots of the Equation muft be ima¬ 
ginary} and fometimes this Rule will difcover impof- 
fible Roots in an Equation, that do not appear by Sir 
Ifaac Newton's Rule. Thefe are the only two Rules 
that have been hitherto publilhed. But the Rules that 
arife from the Theorems in the eleventh and twelfth 
Propofitions,are preferable to both} becaufe any imagi¬ 
nary Roots that can bedifcoveredby the vii th or ix th al¬ 
ways appear from the xi th and xii th Propofitions j and 
impoflible Roots will often be difcovered by the xi th 
and xii th Propofitions in an Equation, that do not ap¬ 
pear in that Equation when examined by the vii th and 
ix th Propofitions. The Advantage which the Rules 
deduced from the xi th Propofition, have above thofe de¬ 
duced from the preceeding Propofitions, will be mani- 
feft by confidering that in the xi th Propofition we have 
the Values of the Quantities a\b^c\d\ e', feparately ; 
whereas in the preceeding Propofitions, we have only 
the Values of certain Aggregates of thefe Quantities 

L z joined 



( 74 ) 

joined with the fame Signs. Now it is obvious that 
if thefe Quantities be feparately found pofitive, any 
fuch Aggregates of them muft be pofitive; but thefe 
Aggregates may be pofitive, and yet fome of the Quan¬ 
tities a' } h\c',d', e 1 , themfelves may be found negative : 
From which it follows, that if the Roots of the 
Equation are all affe&ed with the fame Sign, and no 
impoflible Roots appear by Propofition xi th , none will 
appear by the preceeding Propofidons j but that fome 
imaginary Roots may be difcovered by Propofition 
xi th ,when none appear in the Equation examined by 
the Propofidons that preceed the xi th . If fome of 
the Roots of the Equation are pofitive, and fome ne¬ 
gative (which always eafily appears by confidering the 
Signs of the Terms of the Equation) then the xii th 
Propofition will be in many Cafes more apt to difco- 
ver imaginary Roots in an Equation than thofe that 
preceed it. 

The Rule that flows from the firft Theorem of the 
xi th Propofition, obtains when the Roots of the Equa¬ 
tion are affeded with different Signs, as well as when 
they all have the fame Sign, and it is this j Multiply 
the Number of the Terms in an Equation thatpreceeds 
any Term, asE# ”~ r by the Number of Terms that 
follow it in the fame Equation, and call the Produd/2?. 
Suppofe that -f- D,-— C, -j- B,— A, -f- I are the Co¬ 
efficients preceeding the Term Eat r >and that -f. F, 
__ G, + H, — I, K are the Coefficients that follow 

1 _ 

it; then if —m E* does not exceed m 4 - i x D F 

2 

— m + 4 x CG + »+9 xBH — m + i 6 x AI 
+ m -f- zy x K the Equation muff: have fome imagina¬ 
ry Roots; where the Coefficients m + i, w + + 9, 

£K.c. 



( 75 ) 

declare found by adding to m the Squares of the 
Numbers i,x, 3,4, Sec. which fhew the Pittances of 
the Coefficients to which they are prefixed, from the 
Coefficient E. The fecond Theorem of the xii th Pro- 

pofition fhew?, that if — m' E* does not exceed m 1 D F 


— m •— 11 x C G —1~ w — 7 x x BH — m l — Z40 
x AI +»' — 600 x K, the Equation mufthavefome 
Roots imaginary. 

For an Example, If the four Roots of the Biquadra- 
tick Equation x 4 — A#’ -f- — C# -}- D = o 

are real Quantities, it will follow equally from the 


v th } vii th , ix th > and xi th Propofition?,that —A* muft be 

3 8 

greater than B, and that —-G* mull; exceed B D. The 

O 


vii th further (hews that — B* muft exceed AC — D: 

IX 


4 

the ix th demonftrates that — B* muft exceed A C; but 

9 

our Rule deduced from Prop xi. thews that x B* muft 

exceed yAC— 8 D, the excefs being — a\ and the 

x 

Rule deduced from the fecond Theorem of the xii tf » 
Propofition Ihews that B* muft always exceed x AC 

-f 4D,the Excefs being ~a". It appears from fe- 

. 4 

veral preceeding Propofitions, that if the Roots of the 
Equation have all the fame Sign, then A C muft ex¬ 
ceed 16 D : Let the Exceffes $ B * — ix AC -f- ixD 
— p, 4 B 1 — 9 A C =f y, A C — 1 6 D = j j and 
it is plain that ^' ( = 4B* 10 AC-}- 16 D) = q 

— s 



( 7<S ) 

2 % 

— s = — x Tp’ZTs^ and that a 11 = q 4 . s = — * 
_ 5 $ 

ip + 4 s - Let us fuppofe, 

i. That s is pofitive, then it is manifeft that if ei¬ 
ther p or q be negative, a' muft alfo be found negative, 
and confequently that when the vii th or ix th Propofiti- 
onsihew any Roots to be imaginary, the xi th Propofiti- 
on muft diicover them at the fame time. But as ai 
z __ 

(= q — s — — x ip — s ) may be found negative 

when p and q are both pofitive, it follows that the 
Rule we have deduced from the xi th Propofition may 
difcover imaginary Roots in an Equation, that do not 
appear by the preceeding Propofitions : Thus if you 
examine the Equation x 4 — 6 x' - f- io x* — 7 x 
-f- 1 by Sir Ifaac Newton's Rule, or by our vii th 
Propofition, no imaginary Roots appear in it from 

either. But fince z B 1 — 5 AC -f- 8 D (= — a 1 ) = 

z 

zoo. — zio 8 = — z is in this Equation nega¬ 
tive, it is manifeft that two Roots of the Equation muft 
be imaginary. Let us fuppofe 

z That s is negative, and that from the Signs of the 
Terms of the Equation, it appears that fome Roots are 
pofitive and fome negative; then in Order to fee if the 
Equation has any imaginary Roots, the 1110ft ufeful 
Rule is that we deduced from the fecond Theorem of 
Prop. xii. viz. that if B 2 does not exceed zAC-f- 
4 D fome of the Roots of the Equation muft be ima¬ 
ginary : For the Excefs of B 2 above z A C -f~ 4D be- 

1 1 _ . 1 ___ 

ing — a" = — X q + s = — x ip + 4 a °d s 
4 4 10 

being 



( 77 ) 

being .negative, it is manifeft, that if q or p be nega¬ 
tive a" muft be negative \ and that — may bene- 
4 4 


gative when q and p are both pofitive; that is. This 
Rule muft always difcover fome Roots to be imaginary 
when the vii th or ix th Proportions difcover any impofli- 
ble Roots in an Equation ; and will very often difcover 
fuch Roots in an Equation when thefe Propofitions dif¬ 
cover none. For Example, if you examine the Equa¬ 
tion x* -f- 5x % - f- 6x* —— x — iz = o, you will 
difcover no imaginary Roots in it by the vii th or ix th 
Propofitions j and though AC— i6D (=.$•) be ne¬ 
gative, it does not follow, that the Equation has any 
irnpoffible Roots, becaufe it appears from the Signs of 
the Terms, that the Equation has Roots affeded with 
different Signs. But fince B 2 — x A C — 4D( = 
3 6 + io — 48 = — z ) is negative, it appears 
from our Rule, that the Equation muft have fome ima¬ 
ginary Roots. 

I might Ihew in the next Place, how the Rules de¬ 
duced from the xi th and *xii th Propofitions may be ex¬ 
tended fo as to difcover when more than two Roots of 


an Equation are imaginary, and in general to determine 
the Number of imaginary Roots in any Equation ; but 
as it would require a long Difcuffion, and fome Lemma¬ 
ta to demonftrate this ftridly, I (hall only obferve that 
thefe xi th and xii th Propofitions will be found to be 
ftillthe moft ufeful of all thofe we have given for that 
Purpofe. To give one Example of this; If we are to ex¬ 
amine the Equation x* ——4 ax* -)-6 a* x* — 4 ab’x 

*4* ’— ~f* — 

4- // = o by Sir JJ'aac Newton’s Rule, it is found 

4 - 

to have four irnpoffible Roots when a is greater than b ; 
for though the Square of the fecond Term- multipli¬ 
ed 



( 78 ) 


3 

ed by —be equal to the Produdof the firft and third 

O 


Terras, yet in that Cafe, in applying Sir Ifaac New¬ 
ton’s Rule, the Sign — ought to be placed under the 
fecond Terra, and the fame is to be faid of the Square 
of the fourth Term. The Rule deduced from the vii th 
Propofition {hews four Roots imaginary, when a is 
greater than b, and alfo when b 2 is greater than 1$ a 2 , 
but a Rule founded on the xi th Propofition, (hews the 
four Roots to be imaginary always when a exceeds b, 
or when b 2 exceeds 9 a 7 ; from which the Excellency 
of this Rule above thefe two is manifeft. I have faid 
fo much of Biquadratick Equations, that I muft leave 
it to thofe that are willing to take the Trouble, to 
make like Remarks on the higher Sorts of Equations. 

In inveftigating the preceeding Propofitions, when I 
found my felf obliged to go through fo intricate Cal¬ 
culations, I often attempted to find fome more eafy 
Way of treating this Subjed. The following was of 
confiderable Ufe to me,and may perhaps be entertaining 
to you. By it, I inveftigate fome maxima in a very ea¬ 
fy Manner, that could not be demonftrated in the com¬ 
mon Way with fo little Trouble. 

Lemma V. Let the given Line AB be divided 
any where in P and the Redangle of the Parts A P and 
P B will be a maximum 

when thefe Parts are e- A----p-B 

qual. 


This is manifeft from the Elements of Euclid. 


L e m m a VI. If the Line A B is divided into any 
Number of Parts AB, CD, DE, EB, the Produd of 
all thofe Parts multiplied into one another will be a 

max- 



( 79 ) 

maximum when the Parts are equal amongft themfelves. 
For let the Point D be where you will, it is manifeft 
that if D B be bilfeded in E, the Produd ACxCD 
xDE xEB will be 


greater than A C x 
CDxD*x.*B 



becaufe by the laft Lemma DE x E B is greater than 
D e X e B j and for the fame reafon A D and CE mud 
be bilfeded in C and D; and confequently all the 
Parts AC, CD, DE, EB mull be equal amongft 
themfelves, that their Produd may be a maximum. 

Lemma Vlf. The Sum of the Produds that can 


be made by multiplying any two Parts of A B by one 
another is a maximum when the Parts are equal. The 
Sum of thefe Produds is A C x C B-f-C Dx D B -f- D E 
x EB: Now that DE x EB may be a maximum ,D B 
mull be bilfeded in E by the v th Lemma, and for the 
fame reafon AD and CE mull be bilfeded in C and D, 
that is all the Parts, A C, CD, D E, E B mu ft be equal, 
that the Sum of all thefe Produds may be a maximum. 

Lemma VIII. The Sum of the Produds of any 
three Parts of the Line A B is a maximum , when all 
the Parts are equal. For that'Sum is ACxCD xDE 
4 -EBx ACxCD + ACxDE-fCDxDE;and 
fuppoling the Point E given, it is manifeft that AE muft 
be equally trifeded in C and D that ACxCDxDE 
may be a maximum by Lemma vi. and that ACxCD 
+ AC xDE -f-CD x DE may be a maximum by 
Lemma vii th . From which it is manifeft that all the 


Parts A C, C D, DE,E B muft be equal, that theSum of 
the Produds of any three of them may be a maximum. 

Lemma IX. It is manifeft that this way of rea- 
foning is general, and that the Sura of any Quantities 
being given, the Sum of all the Produds that can be 

M made 



.... ( *°.> 

made by multiplying any given Number of tliem by 
one another, rauft be a maximum when thefe Qpanti- 
ties are equal. But the Sum of the Squares, or of any 
pure Powers of thefe Quantities, is a minimum , when 
the Quantities are equal. 

THEOREM. 

Suppofe x n — A x n ~ l -f- B # *“* _ Cat”"" 3 -f- 

D < v’ , ~ 4 — E^”“ 5 SCc. = o> to be an Equation that 
has not all its Roots equal to one another : Let r ex¬ 
press the Dimen (ions of any Coefficient D, and let 

n — i n — z n —• 3 

/ = ny. -- X - X -otc. taking as ma- 

z 3 4 

ny Fa Hors as there are Units in r \ then pall ——x A r 

n r 

be always greater than D, if the Roots of the Equa¬ 
tion are real Quantities afteHed with the fame Sign. 

This may be demonftrated from the proceeding Pro- 
politions: But to demonftrate it from the laft Lemma¬ 
ta, let us afifume an Equation that has all its Roots 
equal to one another, and the Sum of all its Roots 
equal to A, the Sum of the Roots of the propofed Equa- 

tion. This Equation will be x -A I = o, or 

n 

n—i A* 

x* — A x n 1 4 - n x - X —r x ” 1 — n X 

z n 

n —1 n—z A* 3 _ , .. 

-X- X- x oCc. = o and if r ex- 

a 3 

prefs the Dimenfions of the Coefficient of any Term 
of this Equation (or the Number of Terms which 

pre- 



( 8 . ) 


preeeed it) it is manifeft that the Term it felf will be 

A r . ... 

lx -— x n r ’• But by the Suppofition D x n ~" r is the 
n r 


Correfponding Term in the propofed Equation, atid D 
muft.be the Sum of all the Produds that can be made' 


by multiplying as many Roots of that Equation by one 

. . IA 1 

another, as there are Units in r ; and-. rnuft be the 

n T 


Sum of the like Produds of the Roots of the other 


Equation ; which' muft be the greater Quantity by 
the preceeding Lemmata, becaufe its Roots are equal 
amongft themfelves, and their Sum is equal to the Sum 
of the Roots of the propofed Equation; and the Sum 
of fuch Produds is a maximum when the Roots are 


equal amongft themfelves. 
it may be demonftrated that 


By purfuing this Method, 
aii T 

— ~ . =t-= x / muft always 
nxn— 1 [ 


exceed the Coefficient prefixed to the. Term x*~~ r in an 
Equation whofe Roots are all real Quantities affeded 
with the fame Sign j providing that r be a Number 

--- ■— r 

1X3^ ~ 

greater than x ; and alfo that — = == — =—= x l 

8X» —IX»-1 

muft exceed the fame Coefficient, if r be any Num¬ 
ber greater than 3. 

It is eafy to continue thefe Theorems. 

The third Method which I mentioned in the Begin¬ 
ning of this Letter, is deduced from the Confideration 
of the Limits of the Roots of Equations-, and though 
it is explained by fome Authors already, yet as I de- 

M x monftrate 



(»*) 

monftrate and apply it to this Subjed in a different 
Manner, I fhall add a fhort Account of it. 

Lemma X. If you transform the Biquadratick 
x *— Ax* 4 -Ba ; 1 — Cat -f> D = o into one that 
fhall have each of its Roots lets than the refpedive Va¬ 
lues of X' by a given Difference e j fuppofe y = x — e 
or x = e -f- y and the transformed Equation, the Or¬ 
der of the Terms being inverted, will have this Form. 

e* -i- 4 e 1 y-U6e* y*4-4ey r -i- y 4 = q „. 

— Ae* — 3 Ae*y — 3 Aey* — Ay t 

-{- B e* -f- z B e y -f- B y x 

— C e — c y 

+ D 

Where it is manifeff, 

1. That the firfl: Term e 4 — Ae'-j-Be* — C^-f-D 
is the Quantity that arifes by fubftituting e in Place of 
x in the propofed Equation x* — A # 5 -J- B at* — 
Cx + D. 

z. That the Coefficient of the fecond Term 4^ 1 — 
$Ae* -f- zBe — C is the Quantity that arifes by 
multiplying each Part of the firfl: e* —. Ae J 
— C e -j- D by the Index of e in that Part, and divi¬ 
ding the Produd by e. 

3. That the Coefficient of the third Term 6e* — 

3 A ^ B is the Quantity that arifes from the preceed- 
ing Coefficient 4 e ’ — 3 A *?* -{- z Be-— C by multi¬ 
plying each Part by the Index of e in if, and dividing 
the Produd by z e. 

4. That the Coefficient of the fourth Term arifes in 
like Manner from the preceeding, only you now di¬ 
vide by 3 *?; and in general, the Coefficient of any 
Term may be deduced from the Coefficient of that 
Term which preceeds it, by multiplying each Part of 

3 the 



( 8 ? ) 

the preceeding Coefficient by the Index of e in that 
Part, and dividing the Produft by e and by the Index 
of jy, in the Term whofe Coefficient is required. 

Lemma XI. If any Equation x n — 4 

—• C x H ~* die. = o be transformed in the 
fame Manner, by fuppofing x = y — e or x = 

e -f- y, and confequently x n = e Ax "~‘ I = 

\Bx* =Bxe+y\” 2 dc c. The 
transformed Equation will have this Form, the Order 
of the Terms being inverted, 

ft —— I _ __ 

ne n ~'y^r «x-xe” 2 y 1 &c. =z o 

Z 

_ _ fl —•«. 2 

— Ae*~ 1 —n — i x A e n ~~ 2 y ~-n — i x- 

2 

_ ft *—• 2 

-j-Bc” - " 2 + » — 2 x Be” >y +« — 2X ^— -xBe"~‘ *jp&c. 

z 

— Ce"'"- J —»—3 xC e n -*y — » _ 3 x ^4<Ce”-* ^Sic. 

Sic. Sic. Sic. 

Where it is manifeft, 

i. That the firft T>rm e'-Ae""" 1 4 * Be — 

C^”“ J See. is the Quantity that arifes by fubftituting 
e in the Place of a; in the propofed Equation x”~ 
Ax n ~ l 4Ba?”~ 2 —Ca?“~ 3 die. 

x. That the Coefficient of the fecond Term 

ne n ~ ' — n — i X -j-®” 1 X Be n ~~ i — 

n — 3 X U<?”~' 4 5 ic. is deduced from the preceeding 
e»—Ae n - 1 4 Be n ~ 2 —Ce n '~ i die. by multi¬ 
plying each of its Parts by the Index of e in that Part, 
and dividing by e. 3 . That 



(« 4 ) 

3;. That tht Coefficient of the' third Term is dedu¬ 
ced from the Coefficient of the fgfcond Terft^by multi¬ 
plying. after the fame manner, each of its Parts by the 
Index of e and dividing by 2 e. In general, the Co¬ 
efficient of any Ter my r is deduced from the Coeffici¬ 
ent of the greceeding Term,tha.t is ofy r ~ 'by multiply¬ 
ing every Part of that Coefficient by the Index of e in 
it, and dividing the Produd by r e. 

Lemma XII. If you fubftituteany two Qiantities 
K and L in the Place of x in x* — A x J -f> B —• 
C x -f- D, and the Quantities that refuit from thefe 
Subftitutions- be affeded with contrary Sign.?, the 
Qiantities K and L mull be Limits of one or more 
real Roots of"the Equation x*- — Aa? 5 -{- Bat 1 — Cx 
-j- D = 6. That is, one of thefe Quantities mud be 
greater, and the other lefs than one or more Roots of 
that Equation. 

For if you fuppofe that a, b, c, d, are the Roots of 
that Equation, then it is plain from the Gene ft s of 
Equations, that a? 4 —. A x' -f* B x l — C x -f- D = 

x — a X x~ — b x x — c X x — d\ and therefore 
K and L being fubftituted for x in x — a x x —■ b x 

x—~c X x -— d , the Produd becomes in the one Cafe 
pofitive, and in the other negative; fo that one of the 
Fadors x — a, x —• b y x — c 9 x — d mud have a 
Sign when K is fubftituted for x in it, contrary to the 
Sign which it is affeded with whenL is fubftituted in 
in it for x, fuppofe that Fador to be a; — b j and 
litice K — b and L — b are Quantities whereof the 
one is pofitive, and the other negative,, it is manifeft 
that b one of the Roots of the Equation mu-ft be lefs 
than ones and greater than the other of the two-Qtian- 

3 tities 



( ) 

titles K and. L : So that K and L muft be the Limits 
of the Root b. 

I fay farther, that the Root whereof K and L are 
Limits, muft be a real Root of the Equation •, for the 
Produd of the Fadors that involve impoflible Roots in 
an Equation can never have its Signs changed by fub- 
ftituting any real Quantity whatfoever in place of Xj 
becaufe the Number of fuch Roots is always an even 
Number, and the Produd of any two of thefe Roots 
fuch as x —m — V —n , and a; —m -j- V — n is 
x — m r +.-*• which muft be always pofitive, what¬ 
ever Quantity be fubftituted for x while n remains 
pofitive, that is, while thefe two Roots are impoftible. 

Lemma XIII. If you fubftitute K and L for x in 
x n — A x” 1 -f- B ^”“ 2 &cc. and the Quantities 
that refult be affeded with contrary Signs, then fhall 
K and L be the Limits of one or more real Roots of 
the Equation x n — Ax n ~ l -f- B x ”~ 2 &c. = o. 
This may be demonftrated after the fame Manner as 
the laft Lemma. 

Theorem I. If a,b,c,d are the Roots of the 
Equation x* ■— A a; 3 -\-¥>x * — C a: D = o,they 
Ihall be the Limits of the Roots of the Equation 
4 a?, 3 —- 3 A a? 2 -j- xBa; — C = o. 

Suppofe a to be the leaft Root of the biquadratick 
.v* 3 — Aat’-FBat 2 — Ca; D = o, b the fecond 
Root, c the third, and d the fourth, and the Values of 
y in the Equation in the x t!l Lemma, will be a — e t 
b — e, c — e, d — e\ then by fubftituting fucceflive- 
ly a,b,c,d for e in that Equation ofjj/, one of the 
Values of y will vanith in every Subftitution, and the 
firft Term of the Equation of y, °oiz. e 4 Ae 1 -f- 
B e * — C e -f- D vanifhing, the Equation will be re- 
duced to a Cubick of this Form. 4 g 



( 8 6 ) 


4 e * + 6 e*y + 4 1 -f* * = 0 

— 3 A e 1 — 3 A — A/*' 

- 4 - iB £ 4 - By 

— C 

And confequently 4*?? — 3A^ i +iB^—-C inuft 
be the Produd: of the three remaining Values of y ha¬ 
ving its Sign changed ; that is, it muft be equal to 
— b — axe — a xd — a w hen e is fuppofed equal 
to a , it muft be — a — b xc — bx d — b when 
e — b ; it muft be — a — cxb — c 'x d — c when 
e — c • and it muft be — a — d x b — d x c — d 
when c = d. Now it is manifeft that thefe Produds 

b — ax c — a x d — a, a — b X c — b x d — b , 
~a — ~c x b •— c x d — c, a — d xb — d x c — d 
muft be affeded with the Signs — refpec- 

tivelyj the firft being the Produd of three pofitive 
Quantities, thefecond the Produd of one negative and 
two pofitive?, the third the Produd of two negatives 
and one pofitive, and the fourth the Produd of three 
negatives. Therefore fmce by fubftituting a, b 3 c 3 d 
for e in the Quantity 4*?* — 3 A e* -f %Be — C ? it 
becomes alternately a pofitive and a negative Quantity, 
it follows from the laft Lemma that a,b, c, d muft be 
the Limits of the Roots of the Equation 4*?* — 
3 Ae* + z Be — G =0, or of the Equation 4 x ’ — 
3 A x’-f- zBx — C = o. 

C o r . It follows from this Theorem,that if a 1 b' and o' 
are the three Roots of the Equation 4 x 3 •— 3 A x 2 + 
iB^—C =0, they muft be Limits betwixt a 3 b 3 c 3 d 
the Roots of the Biquadratick x* — AA”-f- Bx*—- 
Cx -f- D = o: For if a,b 3 c,d are Limits of the 

Roots 



( 87 ) 

Roots a 1 , l ', and c 1 j thefe Roots converfeiy rnuft be Li¬ 
mits betwixt : a r bi Q and- d. 

Theorem II. Multiply the Terms of any Bi¬ 
quadratick a? * — A a; J -j- B at * —- C # -f- D=oby 
any Arithmetical Series of Quantities / + 4 + 3 

l i m,l -\- m, /, and the Roots of the Biquadratick 
a,b,C)d will be the Limits of the Roots of the Equa¬ 
tion that refults from that Multiplication that is of the 
Equation. 

lx 4 — l A x * -f- / Ba; 1 '— lCx-\-lD = q 
-f-4 mx *— i>m Ax* -}-z mBx *— mCx 

Suppofe that fubftituting the Roots a t b,c,d of the 
biquadratick Equation at * —- A a; J -J- B at 2 — C a: -f. 
D = o fucceflively, for x in 4 x 5 — 3 A at * + 
z B x — C, the Quantities that refult are •— R, -j- S, 
— T, -f- Z; while at + — A a- J B .v 2 — Ctf-j-D 
is in every Subftitution equal to nothing; and it is ma- 
nifefl: that the Quantity 

“j— lx ^ ■ / A x * -j- IBx * ”/C a? -j— /D 

4-4^at+ — Ax* mftx *— m Cx 
will become (when a,b, c^d are fubftituted fucceflively 
in it for x) equal to — m R at, + m S a:, — m T at, -h 
m2iX ; where the Signs of thefe Quantities being al¬ 
ternately negative and pofitive, it follows that < 0 , b, c, d 
muft 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 Cubick Equation A at 5 —zBat 2 
+ 3C X - 4 D=o, and converfeiy, that the Roots of 
this Cubick are Limits of the Roots of the biquadratick 
Equation a 1 * — Aa^-J-Ba: 1 — C a? + D = o, for 
multiplying the Terms of this biquadratick Equation 
by the Arithmetical Progreflion o,*-— 1,— z,— 3, — 4, 
the Cubick Ax* — z B a? 2 -}- 3 Cx — AD = o 
arifes. N The- 



( 88 ) 

Theorem III. In general^ the Roots of the Equa¬ 
tion x n — A x n "~ 1 + B# "“** — Ca?* - ” 1 die. = o, 
are the Limits of the Roots of the Equation nx*~ l 
— n — i n —- x x Ba? die. = o, 

or of any Equation that is deduced from it by multi¬ 
plying its Terms by any Arithmetical Progrejjlon 
/ + d, l T ^ d, lE 3 d die. and converfely the Roots 
of this new Equation will be the Limits of the Roots 
of the propofed Equation x” —A a? *~ I -j- Ba? n '~ 1 
o. 

This Theorem is demonftrated from the xi th and 
xiii th Lemmata in the fame manner as the preceeding 
Theorems were demonftrated from the x th and xii th . 
From thefe Theorems it is eafy to infer all that is deli¬ 
vered by the Writers of Algebra on this Subjed. 

Theorem IV. Ehe Equation x n — . Aat"“ * -f. 
B x n ~~ 2 — C x n ~~* die. — o will have as many ima¬ 
ginary Roots as the Equation nx n ~ l — n — i x 
Ax n ~ 1 — n — xxBa?"“" 5 5ic. = o, or the Equa¬ 
tion Ax n ~~ l —> x B a? ” ~ * -j- $Cx n ~~ i die. — o. 

Suppofe that any Root of the Equation nx n ~~ 1 — 

n — i X A ac n ~ 2 -f-ar — xxBa , *~ } die. = o, as 
p becomes imaginary, and the two Roots of the Equa¬ 
tion a?" — Aa ?*"” 1 -f- Ba? a ~ 2 die = o, which by 
Theorem III. ought to be its Limits , cannot both be 
real Quantities j for it is manifeft from the Demonftra- 
tion of Theorem I. that if they are real Quantities, 

then being fubftituted for x in nx tt ~~ l — n — i x 
Ax n ~ 1 -j- n — i xBa:* - 3 &c. the Quantities that 
refult mult have contrary Signs, and confequently the 
Root whereof they are Limits , muft be a real Root j 

which 



( «9 ) 

which is againft the Suppofition. The fame is true of 
the Equation A x 1 —z B x 2 -f. 3 C .r"" * &c. 
= o, for the fame Reafon. 

Cor. The biquadratick x* —> A a? 1 -{- B a: * —. 
Cx -f- D = °, will have two imaginary Roots, if 
two Roots of the Equation 4#*— 3 A* 2 -|- iB« 
— C = o be imaginary j or if two Roots of the Equa¬ 
tion A at * —- x B a; 2 -j- 3 C a? — 4D = obe ima¬ 
ginary. But two Roots of the Equation 4 a? * —« 3 A x * 
-f. zBa? — C = 0 muft be imaginary, when two 
Roots of the Qjaadratick 6 x 2 —. 3 A a; B = o, or 
of the Quadratick 3 A a: 2 — 4B # -f- 3 C = o are 
imaginary, becaufe the Roots of thefe quadratick Equa¬ 
tions are the Limits of the Roots of that Cubicle, by 
the third Theorem j and for the fame reafon two Roots 
of the Cubick Equation Ax* — zBa? 2 -f. 3 C a: —. 
4 D = o muft be imaginary, when the Roots of the 
quadratick 3 Aar ! — 4 B a -f- 3 C = o,or of the qua¬ 
dratick B # 2 — 3 Ca? -}- 6 D = o are impoflible. 
Therefore two Roots of the Biquadratick x + — A X 5 
—j— B AT* — Cx + D = o muft be imaginary when 
the Roots of any one of thefe three quadratick Equa¬ 
tions 6 X 1 - 3 Ax -j- B = O, 3 A A* 2 -4. B At -{- 

3 C = o, Bat 2 — 3 -(- 6 D = o become imagina- 

ry; that is, when -r- A* is lefs than B, — B* lefs than 
J 8 9 

A C, or —— C * lefs than B D. 

8 

Cor. II. By proceeding in the fame manner, you 
may deduce from any Equation at* — Ax -j- 
Ba;””’ 4 —CA:’’"' , §Cc. = o, as many quadratick 
Equations as there are Terms excepting the firft and 
laft whofe Roots muft be all real Quantities, if the 

N z pro- 



( 9 ° ) 

propofed Equation has iao imaginary Roots. The 
Qtaadratick deduced from the three firft Terms x ” *— 
Ax n ~~ l -}-Ba?”~* will manifeftiy have this Form, 
n x n — i xn — x X « — 3 Si c. x at ' — n — i X 
n — x x » — 3 X «—4 &C. xA^-f^ — ix« — 3 
X # •— 4 X 8 — j <SCc. x B = o, continuing the Fact¬ 
ors in each till you have as many as there are Units in 
n — 2. Then dividing the Equation by all the Fact¬ 
ors n — 2, n — 3 Sic. which are found in each Coef¬ 
ficient, the Equation will become n x n — i x x* _ 

n — i X 2 A at 2 x i x B= o,wh ofe Roots will be 
imaginary by Prop. i. when n x n — l x 2 x 4 B exceeds 

- n — 1 

n — il’x 4 A*, or when B exceeds- A*, fo that the 

2 n 

propofed Equation mud have fome imaginary Roots 

;/ — I 

when B exceeds-A *; as we demonftrated after 

2 n 

another Manner in the v th Propofition. The Quadra- 
tick Equation deduced in the fame Manner from the 
three firft Terms of the Equation A a; "~ I — 

+ 3 C x n ~ } Sec. = o, will have this Form ~n — 1 x 
n — 2 x n — 3 Sic. x A x 1 — n — 2 xn — 3 x # — 4 
5 cc. x 2 Bat -f - n — 3 x n — 4x0 — y Sic. x 3 C = 
o } which by dividing by the Fadors common to all the 
Terms, is reduced to n —-1 x n — ^xAx 3 — « — 2 x 
4 Ba?-P 6 C = o,whofe Roots mud be imaginary when 
3 n — 2 

x-— x B is lels than AC: and therefore in 

3 n .— 1 

that cafe fome Roots of the propofed Equation muft be 
imaginary. 

Cor. III. In general, let Dx *— r * , —Ea? n ~ r -f- 

F x be any three Terms of the Equation,*-" 

A x 



(91 > 

A x n ~~ l -f- Bj? " 2 dec. = o, that immediately fol¬ 
low one another, multiply the Terms of this Equation 
firft by the Progreffion n, n — i, n — z, dec. then by 
the Progreflion n — i, n — z, n — 3, dec. then by 
n — x, n — 3, n — 4, dec. till you have multiplied 
by as many Progreflions as there are Units in n —~ r —1: 
Then multiply the Terms of the Equation that arifes, 
as often by the Progreflion o, 1, z, 3 Sec. as there are 
Units in r — 1, and you will at length arrive at a 
Quadratick of this Form, 

n —r-f- i x;/ — rxn —r— 1 xn — r — z dec. xr — 1 
xr—zxr —3 xr — 4 &c. Da: 2 

— n — rxn —• r — 1 xn — r — zxn — r — 3 dec. 
x r x r — 1 x r ~—x x r — 3 dec. x E x 

4 -n — r — 1 — r — 3 xn — r —4&C. 

xr-\-txrxr — ix r — z dec. x F = o. 


and dividing by the Fa&ors n — r — 1,» —- r — x,<$cc. 
and r -— 1, r — z dec. which are found in each Coef- 
fi cient, this Equation will be reduced to n — r 
x n — r x xx 1 xDx 2 — n — r x z x r x z E x 4. 
ixl xF+IxrF = o, whofe Roots mud be ima- 

# — y y 

ginary (by Prop i.) when-x- x E* is 

n — t 4” t r 4” t 

lefs than D F. From which it is manifeft that if you di- 

ft « . I 

vide each Term of this Series of Fra&ions-,-, 

x z 


n 


z n■ 




r 4- 1 « n — r 


,- by that which 

3 4 f r 4* 1 

preceeds it, and place the Quotients above the Terms 

of the Equation x»— Ax 4- Bx n ~* — Cat*“* 

dec. = 



( 9 1 ) 

Sec. = o, beginning. with the fecond: Then if the 
Square of any Term multiplied by the Fra&ion over 
it be found lefs than the Produft of the adjacent Terms, 
fome of the Roots of that Equation mull be imaginary 
Quantities. There remain many things that might be 
added on this Subject, but I am afraid you will think 
I have faid as much of it as it deferves $ and therefore 
I fiiall only add the Demonftration of fome Algebraick 
Rules and Theorems that are very eafily deduced from 
the xi th Lemma. 

I. The Rule for difeovering when two or more Roots 
of an Equation are equal, immediately follows from 
that Lemma, Suppofe that two Roots of the Equation 
x ™ — A* n ~ l -f B#"- a — Ca?’** 6cc. = o are 
equal, and two Values of y (which is equal always to 
x — e) will be equal. Suppofe that e is equal to one of thofe 
two equal Values of x; and two Values of y will va- 
nilh,and confequently j/ 2 muft enter each of the Terms 
of the Equation ofy-, and therefore in this Cafe the 
firft and fecond Term of the Equation of y in Lemma 
xi th muft vaniflb, that is, e n Ae n '~ I — 

Ce n ~ 3 Sec. = o and ne n “* 1 — n — i X Ae"~ 3 -f* 
n •— z X — »•— 3 x Sec. = o atthe 

fame time; and confequentlythefetwo Equations mud 
have one Root common, which muft be one of thofe 
Values of x that were fuppofed equal to each other. It is 
manifeft therefore that when two^Values of x are equal 
in the Equation x n —A x”^ 1 -f. B;v’ , '- 3 < 3 tc. = o, 
one of them muft be a Root of the Equation nx n ~~ l _ 

*— * xAx"~~* -f> n~% x Bx n ~ 3 Sec. = o. 

If three Values of x be fuppofed equal amongft them- 
felves and to e , then three Values of y (= x — e) will 
vanilh, and the firft three Terms of‘the Equation ofy 

in 



( 95 ) 

in Lemma xfi will vanifh , and therefore n x n — i 
xe n ~ 2 — n — i xn — ixAe"** 4. »—ix»—j 
xB^"~ 4 < 5 cc = o; and one of the equal Values of x 
will be a Root of this laft Equation, and two of them 
will be R oots of the Equation n x n ~ mt — n — 1 x 
Ax n ~' i + n — > 2 xBa?""” 5 &c. = o. In general, it 
appears that if the Equation x n — 

&C. = o have as many Roots equal amongft themfelves 
as there are Units in S, then (hall as many of thofe be 
Roots of the Equation nx n — 1 x Ax "“** 

4 -»— xxBa?*~ 3 <5cc. = o as there are Units inS —1; 
as many of them (ha ll be R o ots of the Equation 
n x n — x x a? ”—* — n —• 1 x 0 — x x Ax n ~' i -\- 
n — x x 3 x Bat”~ 4 <3cc. = o, as there are Units 
in S — x j and fo on. 

II. The general Rule which Sir Ifaac Newton has 
given in th z Article de limit thus Equationum for find¬ 
ing a Limit greater than any of the Values of x im¬ 
mediately follows from the xi th Lemma •, for it is mani- 
feft that if e be fucli a Quantity as fubdituted in all the 
Coefficients of the Equation ofy^ix. in e n -~ Ae n ~~ x 
6cc. ne ” —t — n — 1 x Ae n ~~ 2 -f- n — % 

n — 1 _ _ n *— % 

xB*?“~* Sec. n x - x<?” 1 —n — x x -x 

x i 

_ n — 3 

Ae ”~~ 5 + n — xx ——— x Be n ~"* 3 c c. gives th*. 

Quantities that refult all pofitive j then there being no 
Changes of the Signs of the Equation of y in this cafe, 
all its Values mult be negative j and fince y is always 
equal to x — e it follows that e mud be a greater Quan¬ 
tity than any of the Values of x ^ that is, it mud be a 
3 Limit 



( 94 ) 

Limit greater than any of the Roots of the Equation 
x n — Aa?"“‘ -f" Bat”—* Sec. = o. 

III. From this xi th Lemma fome important Theorems 
in the Method of Series , and of Fluxions , and the Re- 
folution of Equations are dernonftrated with great Faci¬ 
lity ; it is obvious that the Coefficient of the fecond 
Term of the Equation ofy in that Lemma is the Fluxi¬ 
on of the firft Term divided by the Fluxion of e; the 
Coefficient of the third Term is the fecond Fluxion 

of that firft Term divided by ze 2 -, fuppofing e to flow 
uniformly. The third Term is the third Fluxion of 

the firft Term divided by z x 3 e *; and fo on. There¬ 
fore fuppofing e” — Be”~’ 8 cc. = c, the 

• • • 

C (/ 

Equation for determining y will be c-\- j/J_ y* 

e J 1 xz e 

• • 

c 

_j- - y t 5 cc. = oj and hence, when ^is near 

I X^X3 ^ 

the true Value of w. Theorems may be deduced for ap¬ 
proximating to y, and confequently to x t which is fup- 
pofed equal toy -f- e. 

IV. Let AP (= x ) be the Abfcifs and PM (= z) 

the Ordinate of any Curve BLM; and fuppofe any 

other Abfcifs AK = « and Ordinate KL = c, then 

• • • • 

ffiall * ( = ?M)=c + ~y y 2 + —— vi 

:: e ze 1X3^ 

J-£ . - y 4 SZC. 

1 % x 3 X 4 ^ 4 

For let .s be fuppofed equal to any Series confifting 
of given Quantities, and the Powers of x, as to A x» -f 
Bx r C x t dec. and fubftituting e+y for x, we ffiall 
find after the manner of the xi‘ h Lemma, 

3 z 



( 95 > 

zsz Ae** nAe*- 1 ,? + »** -xA/-’> &c. 

z 

r — i 

4 . Be f +rB* r ~'y 4 rx-xB* f ~*xy’&c 0 

2 # 

4-C^ : F > yC^-’> + ^x —xC^-’r fcc. 
cfr. £fr. $V. 



But when x~ £then x =c = A^ n -fB^''-|-C^ 
6 Cc. c = n Ae n ~~ x e -\-rBe r '~ 1 e 4 J'C« f "’ £ SCc. 
c=nx n — 1 xA^-'^+rxr— ixB^'^* 

4 j x ^ — 1 xC^-’i 1 * &c. and therefore ^ = cq: 
• ♦« •’« 

JL 1/ -1 - v * X —— 1/ * &c. After the fame 

r T 2f !/ t 2 X 3 ?' 

• •• 

-S & 

manner you will find that c = ^ + —r- jj/ —J-— 2 y* 

OC X OQ 

• • 

± ——■■ . ■ ■ ■ i/»&Cc«for c=i Ae”4- Be* 4 SCc. = 

1 X 3 ^ 

f f ^ 

AX^±jf4Bx«'±jrl H-Cxx+ji 1, 6 Cc. = ^ 4 ; ~ 7 ~y 

iA* 

O 4« 



( 9 * ) 


4- ~^y 7 &c. The Area KLM P is equal to the FIu- 


2 x 


ent of zy or of c'y, but 

• « 

cy=zxy± ^-yy 4.— yy± -~~j 3 yy &c. 


IX 
• • 

c 


x 

XX 3 AT' 

.V 

£ 


and *j, =c>T + 

And confequently by finding the Fluents 

. •• •*» 

KLMP =cy + —y 2 4-——rV’T --—r-/ 4 &c. 

a<? 1 ZX 3 e* xX3X4e 5 


or KLMP = zyy- —. y 2 -\- - ~y 3 i -- 3v 4 &:c. 

~zx zX3 x* ZX3X4X 

This lafi: is the Theorem publiflied by the learn¬ 
ed Mr. Bernouilli in the Afda Lipfi# 1694. It j g 
now high Time to conclude this long Letter; I beg 
you may accept of it as a Proof of that Refped and 
Effeern with which 


I am, 

SIR , 

Tour moft Obedient, 

Mofi Humble Servant, 


Colin Mac Laurin. 



