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# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 1
# Folio 95v, page id 00068194:0190
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Extractum opta ecriptae a wallitio ad
Collinium oxonio Martii 29. C673
Sthack jou sorse papers f Extraits ont os Brassers aud terguions
Alyebra. His very true, est those Cubie aequations of cor qua giue m
stance, may vers velle sall onder cardans rule, notiotestanduu ye impossibi
lity os = (yesquare root of a negative quantitus) in duch cases as ye hinos
nual cube voill adnit of a Cubic roots extraction (Sc: a + V -- 1:) because thon
ge mipossibla part of ge 20ot (est is V-:) cours in ye addition tole extinguistit.
of coch) oas besore very vell avare. Put r maiu difficulty is extractin
yt Cabic 20oz oil ale Rinouiels do nos admit. yet thy difficultes is not sd
great as may at finst viero be thousat; non is it greater in this case deau un
other cases e gesaure rule, Wdhere such V. dote not ocuur. Hsind by yr
papez. est they have a method, oeu colat it is slenoco not; eut quesse it tobe
zes aue cote est os des-cartes in hiy Epistles. part. 2. est. 43. Out loug besore
Jhad seen unythniys os des cartes, or lndu auy thuiys et carvans rule (at
loast by est naue) J had long suice (cohile J Wvas yet but a gouny Hllyebrin)
in ye year 164S sound ont gesame by my ocon inqueries (os cous grue ac,
count in my presace torst agsa ncibomius.) Aad meetmiy thon cotl ye difficitts
os extractnius aje Cube-200t es a Binomial, Jtook this eans aud obvivus way
to epedit mysoes (∞ Woch 1 did in gesame year 1640 gine au accountr at hus re
quest, to nr Bmith thon Mathematie professor at Cambridye, in ansverto a
Lettez oflis.) oce is ouly, 6y reducinu ye surd into atqe rational aud ivratio
nal part; o2 (och is zesaure) by clariny so muit os it as e rational Boue
je note of radicalites; as sor instance, for 247 J vouto quit yry; aud. by ye
same reuson, 5or V= 243, J Woould but 9r= 3. For, by thys means instend os
VC: 10 = V- 243 Croch ys ist first in stance) J Thould put VC: 10 + 9V27. Aad
au precentey sure, est either V=3, or some multiple s it Cy a rational num
bez (integer or Gracted) is seirrational noure ofye binouial root; Cas in
se present case, 22 + =r= 3.) Red, cohat est rational multiplier is, vill not be
hard to discover: Seniqs est ve are sure enougle prouc ye construction of a bino
mial cubes supposnius ye root tobe a +Vb, ye cube must be a+3a= Vb+3ab
+6Yb; os coch ye fast aud y part vre contained in one, aud ye 2d aud yte
part in ye other meuber of ze binomeul; or, is je rest should chance tobe