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# Title: Leibniz-Handschriften zur Mathematik LH 35, 2, 1
# Folio 65v, page id 00067993:0130
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Valor aequationum affectarum, nam sumatur aliqui
series pro X incognita, hudius epotentiae varidi per literas
quasdam arbitiarias multiplentae invicem addiantur, ut
inde denius destrucendo fiat finita quantitas. Sed quia sere
series in se invicem ducta facit etiam ex arbitrarias
exultari hinc itur non est facilis exitus
calculi hac via, alia ergo via generales serres
dare possumqus pro qualibet aequatione, et quia exitus
illa via semper habetur rationaliter, priore vero
degenitur ad aequationes valde exaltatas: hinc com
parando divesas illas vias habentur innumea theoremata
disfantea seu modi quibus aequationes exaltatae
rationaliter indeniuntur. Siidus locum habet etiam
cam glares incognitae sunt in aequationibus. Unde
innumera theoremata isoschetica pulcherrima imo
Pheorematum Tabulae prodibuat.
Iuerea nihiloni hinc aperitur ex hac contemplatione
(etsi ad problimata seu priore methodo, involutionum
(tametsi ad inveniendam sariem apta non summam
seriei indentendam apta non sit) via deminstrandi
impossibilitates.
Excmpli causa demonstrari poterit, an possibile sit
 +  = L + L - L etc. esse numerum
11
rationalem. aut an ejus quadratum sit numerus rationalis
aut cubus, t etc.
Nota autem ad
hoc faciendum non sufficere,  exaltatione facta
destinctio non appareat, sed productum comparandum
esse cum d seriei differentiali absolute
sumta; exempli causa, an  - = + L - L et
sit terminus rationalis judicabitur si reducatur expri
motur series per formutam, omnia ad un omnes terminos
communi, relatione ad literam X. ex primentem (posito
eam esse numera integri termini sertei naturalis signi-
ficatlrcem seu esse 1 vel 2 vel 3 vel4 vel 5. etc.
et inde facile judicari poterit an haec formula sit
differentia alterius formulae. Imo sumenda hant
generaliorem F-  + - t crc. etiam poterit
ad formulam finitam revocari locet transcendentem
qua exprimatur quivis ejus terminus, exempli gratia
haec: XX
-
X+ 2
+ X eta redit ad hanc +- F + f t et
e
qui an alterius formulae, ut