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# Title: Leibniz-Handschriften zur Mathematik LH 35, 3 A 18
# Folio 1v, page id 00068231:0002
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summ. rodiC.
summ. quadkadic
(W aequ. 0- 29
va
sumum sidesol a radre
sum. Cub a radic.
SW aequ. 0 - 309+ 3Y
et (W aeg. 0 - 5, 59 + 50r + 5. 09 - 5. ot - 5gr, 5V
Hinc X quidem e seu 31. Sm + In + Sp  continebit rationale SW: 5 seu 0:5
Waeg. 0
uud rari
seu nihil. at X coseu quadrutum ab
Et culg
m + Sp+
continebit pationale es
 omnes rarives
continebit partem rationalem: 3r. 5 quodum si quadratum cum sit
ee
5()2 2f M et
uter in Ae
S12 nihil contineri rationale, er (nam si l non est extrahibilis per radicem quadado
surdesolidam, nec l erit extrahibilis per eandem, nisi ponatur l’esse quanti radix quadratica
dea tanquam hic specialis non videtur ratio habenda in generali consideratione) conseq uens
erit omnem rationalitatem inesse ipsi SIm, nempe 4. 5. non tamen dicere audeo
continebit,numt
rationale 3r: 5 et quia cubus ab l est a SS est sen ab X est SI  
43 (esim
3s Imin
hinc jam porne in quirendum esset an tat seu 53I +3 S’ em + 6 si tran investigandim
esset porro in quarram haum summarum possit sprationalitas, illa contineri.
Et
4 seu SSI + 4siem X - 4 (istun X = 2 quadr. sssim + 4 (Slmax continebit qua
quantitatem rationalem - 299+ 4+: 5.  secgntqus entum terminus in orpattione pro g.t.  -19q
X seu (S Y P feu hocest) Sl
enim
vun
aepigendum
is imux. p
quae continebit quantitatem rationalem =59Y45V:5 seu + gr -B: B
fieret: SIlr aequ. 9. 5
X  - 3s IIm X + 55 Stun X + 5qb. d.s.ism X  Ssistm s Ima
X X
+9X
a
+ 5Ss map. - 1
4+tx ej
et Ssmn aequ. r: 5 et s stmup aequ. A.5 + . 25 et
sli aequ. gr: 5 - v
auderemuse
Quod videamus an consentiat cum altera hypothesi audaci, si ponere illa ipsa rationalia 
quae in in istis sim Situn VImnp contineri deberent, ipsis istis summis aequari
seu summus istas fieri rationales nempe in X seu in fAL ss( +1 (estuo co
quoniam circa radices veras et falsas aliquid difficultatis resumam
sit sequatio X X qX+ YX + +X + v aequ. 0.
erit SW aequ. 0. et +sWVy aequ. 4. et _ svyzaequ Y et + swyzco aequ. t et
SW aeg. 0.
si ae s f siv aequ.  Siv - 2 sy o stoa
(unde ut obiter dicam ppatet si omnes radices possibilas q fore negativam) siv aequ. 
Hinc cum pro X (sit aequatio
= si X erc. 1equo et pro X aequatio inveniendis sit aequatio
X -sWV X erc. aeq.0. et pro X inveniendis, X - (IV X erc. aq. 0. et pro
N’inveniendis sit X. X exc: ag. 0. et secundi termini coefficiens mutato signo
divisa per numerum dimensionum contineat sit pars rationalis indicis. Ergo radicis
seu valoris in aequ. X et seu valoris ipsius X  parsrationalis erit +sW: 5 seu -29:5
Et radicis in aequ. XSetc seu valoris ipsius X pars rationalis erit + (W: 5 seu  (3Y: 5
Et radias inaequ. X2o erc seu valoris ipsius X pars rationalis. erit + (V. 5 seu 299-4+:5
Et radicis in aq. X2s ete, seu valoris ipsius  pars alioralis erit
4505: 5 seu + qu B - v
Pativantem cur terminos secundos ii squationibus X etc. et X et et X et es X et scripserim
fW. + (0 =(102. + su. haec et, quod
ponendo jam x aequ II + Im+ In +sp
fitHH +f ubi pars radionalis est 0. seu nihil, et faien fiet XX gq
aego () I +2 m. et ponendo in I nihil reperiti extrchibile per surdesolidum, tunc quicquid rationale contine
seu x aequ (YC
tur in ex continebitur in 2 Pes (M. et si audeamus facere ssm quantitatem rationatem, fiet 2s  l
aequale illi cationali parti ipsius X2, seu ipsi - 29: 5. ergo erit sSm aequ. - 9: 5
At X siet aeq;
9 sestm X - 3 sSImn ubi potendo in  cujus pars rationalis est = 3Y: 5. ubi ponendo nullum
 aribilo per ss et YEm esse rationalem, sequitur omnem partem rationalem contineri in = 3ss Imn. et
ay. Smedlit
Imn aequ. r. 5.
At X + fit aequ. s3s (4
si audeamus facere tmn quantitotem rationalem, fiet
eq sim e et sistmp puinde  abit exthralitieat. e pereualite gunrers im
up al t audtauus  faeraati
et sint radices ejus. N. y. Z. Ca (5
seu 0229 seu
Wy2 seu sW’aego - 3Y et sW aequ 299 - 4t et  ag. +5gr - sV
lndis
XY aequ +lo
+5S SIm X, - 5(iImn X -5 quadr sim X + 5 sstm s simn
Ubi si auderemus hanc aequationem
comparure cum data
denique
t
(W aequ 0- 304
3 1