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# Title: Leibniz-Handschriften zur Mathematik LH 35, 8, 11
# Folio 5r, page id 00068106:0009
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G
erae possunt solvi pe plures Tmuscendenses tollibiles ppossunt solvis per unam. Torcibiles sunt Exponentiales, Leg; et tegagon
Nequabionem propositam differentialem rationatem, lentemus resolvere in peuet ia
compositam ex pluribus firmulis quee tetragomisticis, ut x du v + deo  + dg etc
posito V, Cd, _, dari per X et y. ubi quidem potest cujus libet harum literarum valor ex x et
fingi rationalis, et communis divisor horum valorum. Porro quia DV rursus datur pec V, et
quidem, si placetrationaliter videamus rursus quorsumnam reduci possit formula
rationalis. Et quidem si in tegra est, verb. gr dv ve + +y om erc. non portet aos uetra esse
selicibs; sin sit fracta vel ehi cta fractio habebit simplicem  numeratorem, vel com–
positum. Si sit compositus ut a + bv + cvV,: + motnor. potest resolvi in simplices, nempe
in a: lmoturo, et bv: limutnrr, et cun: lime+nor Consideremus ergo antea
superest ergo ut consideremus fractiones simplicis numeruloris. , ut cVV: (+ m+nvo
ed Reducatur. Nominator in suos divisores. Et reduci poterit (Methodo alibi a me ostensa
fractio talis, in hujus modi
fermulas, pv: h+v plures inter se compositas, vel
e una rum rV.: v. id est Y. 1b vel Y: pcb
n
talis  ut v 10 vel dv: v reduci p ad C
prineudo est summabilis
n
nn +-
fiat V = C, fiet  V
in
= Cd
sed formula quaevis
sco idque rationaliter
et dv =. R. ∞-1.
mn
V du = n C0
et
do. si jam m ponatur data, et mn + n - 1 = O, fiet
ion +n = 1 et n = 1: mt ergo si sit V = co fiet V do = deo vel co =  B
mtis
postremo du. p. V: h+v potest reduci ad inferiorem nam pro h+v scribe E fiet dv = d3.
mat
et si e= 2, per exempl. 60t V: h+V = 8 - 2hE + hh,: E = E - 2 h + hh: E
itaEt si altius sit, verb. gr. si e = 3, conlinuatur resolutio res eodem modo se habebit,
et ita tandem vducitur res ad solum terminum, ubi constans dividitur
e variantem.
Et iba formula omnis ut   dv rationalis, sive
integra sive sracta aducitur ad compositas.  c. d8: g + fdx: Y etc
modo R et W, etc non sint = 
+ m. A   m N da + n M den, ete + h AIF: N. + R dov + s, etC. ita
ut ipsas 
E et. A, n RC; H, V etc. tantum opus sit interpretari per vim
5
qV +i, vel similiter id est  lationaliter, integre simpliciter
si quis tame
velit omnes ex his summabiles, reapse summare, seu facere dio
quia datur d5i Va rationaliter ex A (n, H, a
posito e
hoc quoque pt
numerps integros affiritatis
Ergo et datur lationaliter
ex v, ergo formula  dv. generalis rationalis reduci  reduci pt
ad d+ c d8: O + fdc: E erc. posito cd,  dari rationaliter
integre ex D.
S
d
et
jam si plures siut formulae
reduci pt in tales ut d5i + c d8:  + fdE. E
 dY 13+
var, cdde, quaelibet harum
 fiet ex do, et similiter
redneratur
a5 dβ: β + 6 dγ: V et. fiet ex Cddeo, jam d51 + dG etc. ad
quodemnque  posito 5, 8
β V, etc. dari ratronaliter integre per D et co, ea adeoque per x et y, qua
suppono V et co, eiei per X et y dari
Ergo hoc denique colleriums anes puldherriu
verte)A