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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2a
# Folio 109r, page id 00068197:0217
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es artificio problematum transcendentum
Mumericom
et de usu eorum in quadraturis Et de problemate inquam in
dalo ad rect raois
invenire curram quae sit in ratione dala ad
suum axem
Vdx + dy. = bx sit dx edu = v x
unde dy = dx Vv0- Tantum ergo res eo reducitur ut ita
explicemus V,  per X ut fiat tam v, quam
raetionam
VVo- summabilis. Et quidem ut V existente integro fiat
Vvp- rationalis integem generali constructione, fieri non potest.
fieret enim vv -1 = 44 quia deberet _ vo, et 1. aequari
vel ipsi  u-min vel ipsi 2Cm alte utrum
Nealio modo potest esse rationalis integer
illud dispiciendum est, an v existente ssumto X
possit fieri X - 1 quadratus rationatis integer. Et
lt solvendum erit generaliter problema tale: dato
dati numeri potentiam invenire, quae unitate auctu
faciat quadratum, fiet ergo solulio generalis ista
X
ergoe Hinc Hoc problema numericum hic per differentialis
aggrediendi retio mira detegitur, quae fuit
forfe arcanum Termati nam alicubi innuit sibi
Melhodum Tangentium pro numeris servire. Nempe
X Z em X = mm +, ( posito in dari per X legre Y
fit
= 2dm + 2mdam
posito in dari per X integre  rationaliter)
ea 2
= =man a et rursus c. C-1 X
et ita porro confinuando differentias, differentiarum
e X
habsnu aequationes quol lubet, et si non serres non c
ingruila problema solabile in numeris. Co asumendo
jam forgulam, locosm,  et componendo invicem aequationes
muttiplicatas prius arbsrarie. Componenda est sandem
en
equatio idculica, si res est solubilis
Ita anclussicam tande raoem invenimus problemrata
in numens hauscendentia tractandi, ope jangentium
Et quotres hoc artificium
LCadvalu curtarum:
ad quadcolurus applicatur, sufscit desideratum
e
C
sc
V
in in numbis sauscendentibus inencri ut
irrationalis nihil sit nociturum