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# Title: Leibniz-Handschriften zur Mathematik LH 35, 2, 1
# Folio 46r, page id 00067993:0091
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impossibilitas quadraturae circ. Universalis
Quadratura duplex est, universalis et particularis: Universalis, quae
regulam exhibet cujus ope quae libet circuli portio possit mensurari, seu cujus
ope ex data tangente (vel simu) possit inveniriarus sive angulus. Particularis
ortionem
quae certam circumferentiae portionem, (et eas, quarum ad hanci nota est ratio
exhibeet. Uende et si quis totum circulum totamve circumferentiam
exhiberet, non vero nisi eas partes, quarum ad circumferentiam nota jam
tum est ratio, is quadraturam, qualis desideratur, universalem non dedisset.
Itaque Quadratura perfecta et generalis includit sectionem anguli
universalem, sive Trigonometriam accurate Geometricam; quae longe
inajoris sunt momenti, quam quadratura circuli per se spectata.
His intelligentiae causa positis, ajo quadraturam Uenversalem, seu
relationem generatem inter Tancentem (vel Sinum) et arcum dari non
esse, nisi qualis a me exhibita est, per aequationem compositam
ex terminis effabilibus infinitis. uln inveni enim radio AB posito Y
seu unitare, tangente AF, F. A, 2
Arcu AC appellato C, tunc fore
9- +1
etc.
Hoc ergo demonstrabo Theorema: Impossibile est
Relationem generalem inter tangentem (vel sinum) et
arcum exhibere, nisi per analysin transcendentem.
Unde et sectionem anguli universalem seu in data ratione
et generalem cujuslibet circuli portionis quadraturam alia
quam qualem dixi ratione analytice exhibere impossibile esse
sequetur:
Analysin autem Transcendente.
voco, cum valor incognitae non per aequationqs ordinarias, certarum
dimensionum, exprimitur, sed per infinitas, aut finitas ineffabiles, de
quibus alias.
Jacobus Gregorius neque particularis neque
universalisDvadraturae impossibilitatem evicit
Porro erheoremate isto semel a me demonstrato infinitis
Taboribus inutilibus imposterum supersedebunt Geometrae. Nam
pleraeque omnes methodi plansibiles, quibus hactenus in Quadraturam