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# Title: Leibniz-Handschriften zur Mathematik LH 35, 8, 11
# Folio 5v, page id 00068106:0010
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a calculi compendia et universalitatem. Quod o omnis
dumma quotcunque formutarum  telragoni siearum rationatium,
e
ut
do + co deo, et, posito ∞ et co, dari rationaliter
 utcunque per x et y, possit reduci ad formulam talem
ed
+
ce
dN + c dp: P + fde:
e
1.-
etc. posito X, 8, E dam
rationaliter per X et y. Ubi proinde licebit omnibus illis valoribus dare
jam si sit P = O: D fiet d8 = D dO -  dD,: DD. Ergo dp: P - Dd(7 - Cd
divisorem communem.
seu dD:  = D d0 - OdD,: OD: itaque si co sit = 
DDO:D
itaque si sit N = O 7: D et 8 = 0: 5, et E = 2:d  Rete cntq
3 sit dN + cd8: 8 + fde: E ete =0. fiet Dd. 7 = 2dD + cDd0- Od3 + f342 -7 dD etc =C
C
F.
s
e C D AB  seu d3 dD e+ 5 occ  4+B + c d + f DdH
 2
 posito e
2 et D,  et e et O et 7, etc.  davi rutionaliter per x ac y. 
cari ratimaliter seu d3 -  C+ 5+9, etC dD - 3d2 1 c 3d0+ fDd + g Dd et C.
d
seu CAd2 - 348  - e+ frg ete dD + cd + f d4 + g dh  etc = 0
D3
21
id. est d3 - 7d8 - ctfrg ere log. D + ctog. O + flog 4 + 9 log. tr etc -0. ja Quemadmodum
d s supra erat N + clog g + f log 2 etc = 0. jam c log =16 log. 9
eo fit d s  log Ergo log. clog8 + f logE = log g. + log
jo fiet N + log. G)
C
id erc = 0.
f
3 = log. 8.
Erpont dem postremo sit Dc es etc=
exponentes C F, et
D non sunt semper rationales integri
E
ubi tamen
Ergo si res per neris quadraturas rationales, et explicationes rationales prostari pt
 N + log G
ret
= 0.
omna reduximus ad hoc dN + log B = 0 posito X et  dari rationaliter
Itue haec omnia praevideri poterantus nam (V dv, in rationalibus reduci pb ad
sc quoad in determinatas per X et y.
Eg cognita ut logarithmos, invicem additos. similiter caeter seo deo s et V qualiscunus
it semper tamen exprimi, pt rationaliter per x ety simul. Quod si P daretur
irrationaliter ex _ v, nifallor laneci isidem res reduci ad rogarithmos et veros scilicet
si sit S 1 o log  log. D =0
si sit Xi + log. O quaesita oportet assumtam dy: dx 2
vel curginarios more meo.
u si 
c
X,y si ad hanec
reduci pt, cum ea esse  coincidentem, non tantum consentientem, qua differentiando aeq. 1. satim tollibum
exlog: et seu transcendentalitas, itaque si lolendo dis. dx otrobiqu prodiret aequadqu si non coinciderent
( + SV + log + log9. Coy D, 20 differentrando, et aeq. (3) in qua restabl log. ), et dy: dx ope 2 et3, tollatur
seu essent cdentica una posset tolli qy:
aa eiya
ay: ex halebitur aeuas in qua restabit log D. solum Hanc jam rursus differentiando collendoque transcendente
debet prodire eg. identica cum aeq2. si quaesita est transcendens. Eadem arte ose datae et assumlae ordine
sit aq. data dy:dx = X, Y. sit assumta
N + YlogO +c log e log. = o ergo ft diff. f et dsi +bd0: 0 + log8. dD: D + C logD, dB: = , quod dovidendo
5. Omnes transcenda itales quantitates tollemus
D. unde tillantur diferentiales per aeq. +. fiet aeq. 4) ubi efficiendo ut unus logant hmus set per
se, et differentiandus talletur illq in ari (5) restabit uuicus. Ex aey. 5. Vollendo diffrateea
pe
ficiatur ut stet per se atque differentietur
 socum erpit loy uneatem loypD, qu
itra derit asid ublema sest trauscendense debet cm
 hadsceridens.
et
+5
ta
E