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# Title: Leibniz-Handschriften zur Mathematik LH 35, 3 A 18
# Folio 8r, page id 00068231:0015
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contineatur
in potentiius radicum aequationis datae, ita investigabimus
quadrato
sinde solidae surdesolido
Quid rationale anveriatur in Badinis datae aequationis, nihil rationale
quadaticae, qu ita investigabimus vel cubo datae aequatimis cubicae, dc ita investigabimus
continantis quadrata aubo, suodesolido, itca ita inveniri potest. Sit.
XX aequi pX - 9 sitque pars radicis rationalis 1. irrationalis H hoc valore substituti in
aequatione sirte XX aequi p 1 e p. 9 Ergo pars ipsius X rationalis erit p1. 4.
datis partibus rationatibus
ag inveniri potest ex eo quod est  potentiaum inferiorum
ejusdem radicis ut si scsit aeauntro X pXL GX’ErX + x + D aep o.
posito rationatem partem radicis esse1 (nempe P. S) ipsius X esse 2 (nempe pere p  96)
in radice aequationis simplicis quadraticae, cubicae, continur etc.continetur pars
seu summam omnium radicum per coderum earum divisam) portem
rationalis est terminus secundus aequationis mutato signo per numerum gradus divisus.
seu Aro aequationis N X+p aequ. 0 X+ pX etc aeq. 0. X + pX et C aq0. X  pX CH 6g. 0.
erit pars rationalis - p. 1 = p: 2 - p: 3 - p. 4 etc.
Sit jam aequatio X + pX4q eq. 0. ergo id quod rationale in pd X continelitur erit 
 id ob X aego pxq erit aequ. p in p. 2 -q seu pp: 2 -q
273 Videamus quid tunc X contineatur sit aequatio X + bX + c aequ 0.  pars rationalis contenta
ui x est pp: 2 = q mullipliceter per 2+ dsiit X3 + bX + ex
(9)33494. - (9)32.34. a) 32. 37.84
quae
X +G)32. 33X 4(9) 312X. GBC. 33 si sit aequalio X + pX   + r aeq. 0. quae scimus in X contineri rationale p: 3
1 +db + de
quaeritur quid tunc
x +9 rix
Ca3
es
in se continentur, divnentur eu per X fic
se jam sit aequatio quaemumue X X p X aX qNr
rationale
erit aequatio pro N, 
X
2C-2
p X.
38-3
X E p R
ambinatoria
Quos terminos sumsi ex tabula alias
+29
+5p9
SpY
4.. .. X  -4 ... XY -6 etc.
i
509
+ 5pt
+5qr
concinnata unde etiam termini
-5’esse rationalem scilicet
Unde patet summam omnium quadratorum a radicibus, esse rationalem, scilicet p - 24
relicti vacui suppeeri possent
cuborum
quadrato quadratorum
9. r. F. v. sunt combinationes radicum, nempe  unnum summa uhiduum, qenuma 
surse solidorum
Y summra ternionum, +quat summa quaternionum, V, ipsa quinio unica
Hinc po Hinc jam sequitur partem rationalem unius radicis esse p: e.
et po 
binionum,
pp- 29. NC
posito scilicet aequationes X etc. vel x etc
vel X etc. non esse tractabiliores quam aequationem,
Si valores isti bene inventi sunt, succedunt inveniendo in aequatione data quadratica
ipsam X. Quod de generali intelligendum.
cubica, aytica, surdesolida  id quod rationale in quadrato, cubo, seto, sundes olido
continetur, perinde ac si id non jam nossemus. sit Xx aequ pX - 9 cum rationale contentum
(nam e eo caso valet 2) erit rationale in X contentum p in p. 2 - g.  seu pp. 2 -g
qui valor consentit cum priore pp- 24:  seu p pp+ 29: 2 seu pp C2 - q
sit aequalio, X3 aequ p X = gX +Y cum rationale in Xt contentum sit p: 3, in X sit pp: 3 - 29:3 ideo
rationale in X3 contentum erit P :3 - 2p9.3 = pg: 3 +Y seu p: 3 - p4 + Y seu p - 3p4 + 3Y, : 3. ut aute
in X sit p: 22
to aorius que que am strite pui t neu is iu incares pote
pro d
C
36=9
etC
2c. 4
ps
+4p9
apr
-299
+4+
4.... 7
A A.... X
4C = 12
ete
se- 15
ete
4C-8
Se-5
30-10
p- 3p9 + 3Y
P - 4p9 + apr 429q. 4L
P - 5p9 + 5pY +5p9. spt. 5qu
quadrati
cubi
jgti
dicet p- 344+3,. 3bC
Cp- rpotaprtq. at: se