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ALetter from Mr. John Collins to the ]{everendand Learn- 
ed Dr. John Wallis S^vilid.n Prof ejfor of Geometry 
f/^^ Univerlity cf Ox'ord, giving his thoughts about 
fome Defedls m Algebra. 
TO defcribe the Locm of a cubick Equation. 
A Cardanick jEquation convenient for the purpofe, 
(liz., fuch as (hall have the dioriltick limits rational) mull 
have the Coefficieat of the roots to be the triple of a fquare 
3 
rmmber fuch is ^ ^—^v- 
Alliime a rank of roots in Arithmetical progreffion , and 
raife refolvends thereto ^3-48^— or refolveads. 
R N 
T 48=-47 
8 96=:88 
27—144=: 17 , 
64 -192=1:128 
I2 5'-24C = II5' 
xi6--x88==7i 
^ flx-g84="^l2 8 
. - 72-9-4^2=t297 
Draw a Bafe line and a perpendicular thereto.and from O im 
the Bafe line prick the negative rejohends downwards, and the 
affirmative ones upwards, and raife their roots upon them as 
ordinates, a Curve palling through the fame is one Moity of 
the Cur^e or Locm on the right hand for affirmative rootS; and 
the other moity on the left hand is defcribed in the fame man- 
ner by afluming a rank of negative roots, and raifing refol- 
vends thereunto. The Cur've Fig, 4. may give a refem- 
blance of the thing. 
And 1 6 the third part of the Coefficient of the roots cubed 
is equal to' the fquare of -i half the rejohem!^ or dtonpck \ivmt. 
Which in compoling of Cardans canon is always fubftradted 
from the fquare of half the ahfol^te^as in the example following. 
It I were to find the root belonging to the refohend 197 
The fquare of half thereof is 2205-1^ 
The fquare of <i4 half thedioriftick Limit — 4' 
The difference isi79j(5i 
And the rule is 148 > vi 79/5^. 
. . I48it Vi79fdi. 
That is in a quadratick Equation, if 29 7^ were the ium of 
the two roots and 6\ the root of the Rebta^hzle then if from 
the fquare of half the fum , the re£tani,ie be llibduded, 
there remains the fquare of half the ciifterence of the 
Z roots* 
