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# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 26
# Folio 10r, page id 00068071:0019
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Hoc est fusius quam quod ad du. jac
Quadraturae irrationalium
Simplicium
um quaeritur
SeDdx, posito
Desse 10+11X + 12XX + etc et  esse 20+21X
+22XX + etc. res quidem semper obtineri potest
ponendo hanc summationem, esse aequalem
huic quantitati OD + (2 DD ax, ut 2 sit
formula simplicior quam 4, quando id licet
opusque est. Nec alia potest haberiquadrature
inderinitae hic formula, quia oportets vantitatem
et ejus summatricem ejusdem esse Ambignitatis
seu aequationem in qua una earum sit incognita
habere sor radices quot aequatio in qua altera.
earum incognita est, et proinde ambo quantitates
irrationalitate hoc praestante afficiuntu
Bernoullium misi
quantitatis uno radicali vinculo comprehensae differentiale
per illam ipsam irrationatem multiplicatur.
nam deDaest dD YB: et etiamsi D rationalis vel
et possat inseri
non rationalis tantum ut hic, sed utcunque irrationalis foret
Itaque ut tollamus nomina denominatorem,
faciemus (3 = ed8 et erit d(ed 8YD=
(+ 1) 43 + eD d8 + (e+ 1) PdD, ∞. Hoc jam
differentiale oportet cum dato Elemento summa
tionis O dx comparare, vel ut ego loqui malo
coincidentiare accedente alio si opus simpliciore
summationis Elemento consimili  sed simpliciore
2 VD ax et et Aequatio effectu identica
tommulis 8 et 2 ita assumtis, ut coefficientes potestatum
ipsius X quantitates constantes suit arbifrariae auniliales determinandae
ope aequationum auxildarium coincidentiantium ejusdem
terminos ipsius x in aequatone du in oppositis
lateribus aequationis e D d 8 + (et 1) 8L) +2 =C dx
occurrentes, vel destruentium omnes coefficiente M
cujusque potentiae in aequatione ipsius X in aequatione
(eta  Y18(C) d8 + (et 1) G d D, B: ax) + X = G
Usi Tponendo 8 = 30+ 3V 32XX + et C et exponentes
AS
graduum summorum ipsarum formularum D
t
vocando respective, a, (B, Y, ideo cum utile sit sumere
e quam plurimorum licet graduum, terminorum quiqu uin
vinculo summatorio reperitur, tantoque plures arbitrarias
suppeditat, fiet ) = (3+1 -a, eruntque ipsius 8 termini,
adeique et arbitrariae (2C (+2- a, sed in universum
arbitrariis indigemu, β+1, tot enim prodeunt termini
coincident iandi, ergo desiderantur adhuc arbitrariae
a - 1 quas suppeditabi B4, adeoque erit terminorum a=1
itaque si sit D - 20 + 11X + 12XX + 13X+14 X, erit a - 4
et num erus terminorum . ipsius 2 erit 3, nempe
titu
mu
vel 40 + 41X + 42XX, vel 41X + 42 XX + 43X3 vel
io tet 2a0 pe te pes interve tres ans contiresiten eu etram
si e sit constans
in effectu identicae)