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# Title: Leibniz-Handschriften zur Mathematik LH 35, 14, 1
# Folio 235r, page id 00068201:0469
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Aliquot jam i plagulas insunsi examinandis formulis, uli aequationes datis similes ex assumtitiis, in quibus
sive par
mn, vel m n, alioque potestatum ab men summae reperiuntur, excitantur, e observari, in facta
ejusundi aequatione quinti gadus, praeter ipsas m. et n. haberi indeteminatas 1, al vero et totidem opus esse aequationibus
subsidiariis, quibus superfltua paria tollantur, quare sic nulla superforet, ad resolvendam aequationem datam
preter uicam mete mM.
Cujus defectus causa est opmod, quod muleae fractiones accessere quibus tamen omnino
est opus, in aequationibus e D. et O . Nimirum quililet terminus aequationis sive D. pine C potest adhuc acuper
a numerum eundem, demta unitate indieterminatarum quem paeter (non connumeratis tamen m. n. Jam habet Unitale inquam
unum lem
sse superfluam, nam multipticata fractione moltiplicata per B, fiet:
en
hec 0 + h
tam nominatore quam numeritore diviso per (, fier:  in +, unde patet tres tantum esse arbitrarias Q
idem in exteris exemplis patet. Itaque
. L
in  habebinus indeteminatas mvas 3. 2. 1. . 2. 1 p 9. sed
ex his novem rursus complures insumentur in novas ut ita dicam impensas. Ninirum a sit aequatio data X+ pA A s p
cumqua comparari debet facta ex ipsis D et, C per fractionum, seu potius nominatorum adjectiones reformatis; quas duas ita
exhihebimus. D Xt (A + b m +C X +(9 d rem +S Ho. et O X Aegn th X
CB) m +B
 unde fiec multipluanto
X +
(1)  vm +s
CX + EX +  x2
1+ 1c.
+ AG
puc
Er + EC.
53
X + pX~ + qX3 +Y X + sX + +,
subsidiaris
0
Sed in qualibet aequatione corlatitia, aliquot opus est aequationibu s, qualum
si innirum ineamus, apparebit, an aliquae supersint indeterminatae
5
(X) im 10
fuln +pn +9 X + 7Y m rsm A tm +V
scilicet,s prima collatitia opus habet duabus subsidiariis
secinia macio V tertia aunero B, quarta nunero g. Quinta nundero 2. Quarum summa est 29. opus ergo
foret aequationibus subsidiars numero 29, cum habeamus indeterminatas, (preter m. et n) tantum 23. Sed hinc malo
 remedium per olegans praesto esse videtur. quo effici potest, ut suppositis fractionibus non tamen novis opus sit
subsidianis preter illas 14, quibus etiam sine fractionibus, opus est, quod fiet si in qualibet aequatione collatitia
nominatores sint inter se comncensurabiles, sive potius, hoc loco congrueules sit ergo: FHB. Et H D DF _ B F. ex coll. 4
collaetitia
et M BBH  (id est  B3) n D F (quod iam patet quia p B2) ex collaitia 3. Et BM  YH2 (id est  B+)
Unde omni illo volimine lucraremur non nisi unicam: eamque nihil auglius puo stantem, qua si aequationem explicuissen
in determin
Dossumus ergo fractiones  saltem lationales omittere.
Si aequationes duas alteram quadrato- quadraticam alteram simplicem in se invicem ducamus habebimus arbitratias
in quadratica, 5. 4. 3. 2. in simplice 2. facit 16. Eruis duae Sipermmerariae, sed ac exigni usus, quia tantum ex multiplication
vel anumento raducis proficiscuntur.
eaque vera
Sa Aequalio X HX + qX + rx + sX B F. Ponatur nota ejus radit una, v. erit
VF+ qu + rV + sv B + B X= + 9X + rX + SX. sive +N + qX + r N + sX A quae aequatio
divisaper X - v, dabit:
X
DX3 + vX + v X + v  
ed 1110. eut X - s- 910
+Y-q
qV.
r
+ qu.
+rV +
12
XH Xt B2 +7 as+ 4 9 + (0.
+
X F + qX + Y5
+ b +c Y-t
s-
obX +1b0
x + 2b,
1
X. Ad +
1-1
Ax (1)  r Am 1
(M) n 5 Fom +o
92
unde fient aequationes. Clatis fractionibus
0
C EB + AF H FB p.
(3) GBFD + ACHD + CHBF+ HBFB
(3) LBHDF+ AGMVF CEmBH MBHDFY
(4 ALDH + EGBM  BMDHs
(Cu DM+
X X GX EB V X + qV
-r
X B0. sive X + YX + v B C
vel pro vera subotituendo falsam, fiet:
Xt LVX
+ v 0, adeoque X - Y
c Xn X + F. 2 v - 9
+X3.
X7 +2bX 4400
+2e 1. +q-1