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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2a
# Folio 67v, page id 00068197:0134
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proponitur ydo = do SCafvr+ ag,: Wtahi) siat  v - bbc + a 2,:R et yX ca + a 5,: 7. 
a, F, B, h sunt datae constantes, b et C sunt quaesitae constantes. Q, R, S, T sunt variabeles simplices ex certa aliqua
assumta vartante, varinti si etset R= 10Z + na, aut simile quiddam.
Affumatur sit Cd = ILM + AP + K. sed
compendiosioris calculi cnusa put posse K= 0. Ita dabitur sc ubi L, M, N, P, K, erunt etiam variabiles simplice,
ex certa aliqua Z. Itaque seo az haberi poterit per quadraturas rationales. Quia autem quantur ydu
seo CC+ a S,: T, in, bdR dcoib: R) + bc d (1: R) + ad (Q: R) palet quari non tantum eodzue multipl. (per rational
axque debere summari; sed et  codco, et cocodZ. Unde patet debere summari  Y LMNB, sed
hic nimis assurgeremus, nisi fiat M = , et  summati tantum debebit dz. M +LP.
itaque omiso tentabimus an omisso K, liceat facere co = YLM +M. Utque hinc tollatur irrationalis,
fict coco =  LM + MN + 2 M LN et ct + 2 Lmecoco + HN + L n = (+ LmmX)
DELMAN
Apurjus gy = af votag, :, vorah, st et proy et v substiturno valores, fiet ct coco + 2ac Sc raa ss,: TY =
af bbcoco + 2af ab ZW + afaa QZ + agRR, :, bb coco + 2ab Zc0 t aa QQ + ah RR. Unde fiet
bbcc W0 + 2abec QCo3 + ag 29CC coud
taasshatiea
+ 2abbc Su + quabe sq coco + aa Z_traC Sco errEF 
ah RR
+ agbb 53. dt
ah RR
~ bbcc cet + 2 LmibbcCoco + LLmm IH.C
pata
MKS
+ MMNN
= 2LMMN)
 seu S =  cib Et ob co fiet
ac 245 + hc AR5 + ab 55Q_ = A6f TY Z = 0 ubi pro S = ponendo valorem, fiet
Et has aequationes idextificando,
fiet olo c9+C=0
-ac 22E hc RR P + ab El 2q = _bs TY =0 sentabcc Dcck RR 1 2cf-
a0 2Q + CkRR + ab S5
Aequatio secundum cdco, seponi potest, quia ejus ope habetur LM + M N
superest aequatio secundum cd., in qua si posset obtineri extractio, haberetur LM - MN, it uterque valor
non continet nisi formulas rationales integras secundi gradus, Aequatio uttina explicatis set T
a  E + ah ZCRRQQ + acc htRREQ + acc R gR = bbcc, qu (LM - MN)
bbF si veliuus habere exfrahibilitatem
seu aget + 2ah RRZQ + aYhg: 5) R = (4 qu (LM - MN) et fa h = vatg: f)
ita stabit:
seu
aatiti = a hih = atig: f vel h = ag: f, seu g = hf. a. sed hoc non licet, quia
ita haberetur afvo+ aeg: en +ah, fieret constansu
schibatur at L + 2ah QQRR F aahh R+ = b qu (LM - MN)
(+ a hg: f R =
-Z. aath.
a qu (a2Q L hRR) = bb (LM - MN) + RR 1 (4 19 - aahh.), Nbb(LM- MN) = RR + (a b9 - aahiti)
t
 sad et possemus agg +kRR, aequare uni  invicem ductor (mutiplicato per K: a, et alteri multiplicato per a: K
sed ita fiet q et R inter se habent rationem constantem. Sed  Itaque sae ambage, idemforet si siumpliciter assume remus
R in ratione  constante ad Q. Et ita LM = MN, haberet rationem cistantem ad 2, Et LM + MN, haberent etia
rationem constantem ad G. ergo LM hatebit rationem constantem ad s et MN, etiam, quod non licebit, nam fierent
sLm et Yme extrahibiles.
VBl a
U+ bl g +bbC
17 hb = et
l=
R, a +2Bh 22 + B
dape
ab + (abboac
sit =m +f
+ mmt
mits
itn q
9a
a RR
h9 - (n (a l mo ( du .
 - lina), 22 + h - ((in 1b8q), a  - b MM, qu (L - X)
tota formulo Z +... etc. in quadmor fuas
redoes resolvatur necesse est  ut ut mobis sit rndix rar aequalis, finge enim omnia explicari per Z
aa aass)  RR
et B esse  1oGtua et simuliter explicare M et L et sper Q,, et quandam radicen multiplicari per
recesse est aequari bis radices, quod sne mulatione determinatatam conlicet
Aliter  tentemus: Retineamus K et facamus
5= 0 fiet 2 = 2Kbia ex co
a Atetbbi au. - af s1. f T =  bbCCKX - 2bb ce E
acchRR - af bb TT = 2bbCCKK = 2 b. bCC (LM + MN). sed video nos hrac ratione
et ix e seponamque aequex, et ex co fit acc ZFtR
estiam eo tandem redituros, ut formula ext binis inter literis inter se legem homogeneorum
servantibus sit aequruda quadrato, quod tali via non est impte. Itaque apparet S, Q, et L serrandetere
cu
video revera nihil protasse K, nam si pro co-k ponas constantem novam, ut W,  tunc lanelimodo
alitisoganss
at BBa1+ Daf abZco + affaaqueque
quid si faciumus x Bco +aQ,: R. fint Wesvotau = agna coco tragaa pco + a graXR = 9).  cotrTM
et  C1otaZ, :, c0+ T
vvtah
cocoL2RC F_ RR
BB c0ro +2 aB gco + aa 
st et B Bco+ aZ fiet fuvnaqu =BBC2aDque at Coco + rah RCo + ah RK
FBBgCo + FBgco+ aat eZ Sa PC + aR
gy = cro +2 ARco + ungra
exeocotre Bocoha e 2
Ccotas Aps cocogrtissio raati RR
00 +a
Ad
sed videotioe in variabilibus seu unversaliter praestari non pots
RR
namen set sit Q in valoibus yav, imnalit poserunt
sit y= exD + aa
a
aa
F
a
et-CC oco + 2a esco + nuss
Ddcoto + EADFOF ATY
eovtaat eccoco +2aCRWgnAAE a
Ad fgu CB ataqu in qu ( eto taT) = equ (BC + aq) in qu (C a + a 5)
fit gque pco + ap)
ubi p et D suntoconstantes
aspondant Bet O, s et R
C +t D, 5et T
haipc+aR)
ut desmantur Coh et cd, sint 5BBDGCCDBBDL T FGADAT = 2eBBCaS + 2 lpp CaS
seu F qu (Bcotaque ( qu Dco ta r) = e ((cot a5)) = qu P a taB), (AquC Cco + a 5) = gqu (DCO + a T)
quia jam hoc coincidentiatadum cum _ Co + 2LM dco + qu (LA- MN) = b necesse est ut
hoc quoque divrti possit in duas partes, quaum quaevis habrat qualtor rabiees prioribus coincidentes
eersa ereta
 BC +2CLa coc +eucom mo naC   ss
3t
 . A dh aquacedt
t
+g
+bon que