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# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 1
# Folio 283v, page id 00068194:0566
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quia
y sep tembr. 1680. Mirum inventum pro perfictione methodi
tangentium invexsae
SX da aqu. X. Ergo X aequ Z SRX d
7+ 1
Hinc omnis potentia tolli potest adhibitis solis summis et differentiis,
ac summis summarum et differentiis differentiarum quantum satis est
replicatis: sit vig. X. erit aequ 3 sX dx. rursus N aequ 2 )X dx
ergo fiet: X aequ. 3 sisrax dx aqu 6ff 6 Xdx
et X aequ 2, 21, Z re x Unde nova miraque cujusque potentiae
ete
expressio per summas summarum ec continnatas per numeros
seliam continuos in infinitum continuorum pro ductos multiplicatas. quod
ibit in infinitum, si Z sit numerus non-integer.
Contra d X  egu Z Z. X d. sed priorem expressionem praefero
qua potesta, reducitur ad summas, quia ita semper diminuitur, quanquam haec quoque
quia jam omnis aequatio differentialis liberari potest a potestatibus, et ad solas summas
Seravire possit pro propessu in infinitum
et differentias horumque producta reduci. Considerandumj am porro est, semper
eliminari posse sumuas rei omni reducta ad meras differentias. Ita delbandem
habebimus calculum constantem ex meris differentiis, ex differentus differen
fiamu tam ipsius x quam ipsius y varie inter se compositis, v.g
a daus x dadus aequ ccax dus + dax sed ponendo tunc:
X esse ae arichmedice crescentem fiet: ddx aequ. 0. et addo aequ. 0.
et fiet: dx  dadyquorum summa dabit x. Quod si jam ponas
contra y esse arichmetiae crescentem, tunc posito possibilem esse hypothesin
d
debet prodire atrobius idem. An ergo semper talis hypothesis possibilis
licebit aliquando et ponere x esse progressionis geometricae. Hoc utile
ab initio ut theoremata constituamus pro summis simplicium, ut pro (dddus
ddius
Nota sufficit ipsius X aut differentiarum ejus aut differentiam
differentiis potentias hoc modo tolli, potentiis alterius relictis, nempe
 uta  tlian aripsis  discnid oeasius et surelinu ue sss posunt
a sNam et ipsius a diferenti dem potentio relinqui possunt
tparte