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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 3 (shelfmark LH 13, 3)
# Folio 114r, page id 00068200:0229
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A Martii 1682,. Tangertium inversa per calculum dffferen-
tralim gene ralem.
per calculum
irca Methodum Tangentium inversam multa praecedenti
cheda notavi singularia. Quae nunc prosequi placet. Et quidem quando
datur dae aequ _ axx + b xy +c3y
dy
(4)
seu dy aequ.
dy
ex + fXy + ggy
4 = Eequ +X dys = y du
Ergo d =  +x +3
ds aequ
dx.  jam ex equ. 1. hiet da (5) a XX + b = + c
quo ratore substituto in aequ. 4.
dy aequ 
linca datur
u
l
ays aequ
compendii causa faciendo;
e Xx + fx + g
ex
et faciendo X aequ. 7.  habebimus.
Z+
yy
te inia duci potermnt. Nam in quavis culva data firgi
erit aequatio, ut gralis 1. et proinos et inveniro
a22 + 1~+ C
ezZ+ fZ + 9
quaa
tiam aequatio
ut x aqu’y at recta
auque solu inconnitae servent legerus homogeneorum, sed talis equatio revera non est nisi)
orum,, nisi
cicdat, ubu
Cujus curvae inventio havetur per quadraturas rationate
gaegua
et datatali ae
tivire differentiaei potent inu
itaque habemus Theorema praeclarum quandocunque duae in determinatae ine
se m
inve
pro literis, Teliquis vero pro numeris, lex homogeneorum observetur ni
loga
haberi potest solutio per quadraturas, et siquidem differentialis rihim
aequatio simplex, est rationales
imo hic subit
e
fine
aequationes quae tales non sunt
in aequationes tales transformare, ut lex homogeneorum servetur
sit
unde
ponatur
dx
dy
aequ. y atbxrcy +Xyt siat X aq  Ay aqu vtZ +
fiet: 4 a+ b x + C v+Z
v-
R+ X +Hy + X7+ PXerq vaequ. XZ aequ. Z 29
ege
fiant: X aequP + 2 Z + 3 et y aequ4v +5Z
6v + 7.
experiri licebit quaerendo v et Z ea ipsis X et y.
rago i vaqo Z
quod an succedere queat
fet, WV + 2ZZ + 3ZV aequ. X e
Et  sed praevideo id fieri non posse, revera enim nom tam duae sunt
literae v et Z, quam una  cujus valore bis in vento haberetur relatio
inter X et y. quae tamen non hinc haberi sed ex problematis natura
inquiri debet
x aequ. E
lx
(10
seu du aequ
quam aequalio
Ae d=
X
a
X
e2
dx ex oqu.
+ X dy per aequ. 4
et ix aequ + l y 3
61)
posito X. qu Z fict: dus aequ f X d= + Z ex 1pop. 8. et ex aqu 4 a) d2 +. dy hop. 9
video ex his duci.  sit . qu. exx + sxy tgny x + hX dy, quae est aequatio itque et ex aequ. 1. erit
go
dy aequ. dx f y d,  et aequando
valoros
Item ex aequ 10. erit dx agu dy L xdi et aequanto ex
cor1, 2. fiet
(15)
x bos 4y dq + a7d Z
aequ.
2 vulores Mleter fia: dy aequ X = xd; 4+ yZ dz
Previus ZX aquy et Zy apx. Ergu ty aequAX Z x
et dx aequae  et y 4 + Z dus sit 3 aequ v. fist ves aqu Y,
sist Zaego - et vequ e
E.g. 1Z
(5) X. fint Zy aequ X et ∞x aequ. y
N _.Z ly +2Y dZ aequ. dx et 3V Ax +4 X du aequ. dy
ave
Habemus ergonalorem ipsi
Z aequ X2
 qu. dx -. Zbys aegu 4u - 3 v ex 3rgot, dqu 4 du. x + (2. 3. VAet.
Ergo n agt
r
 eet
da. Soraa 2buat ,et 2 ZY 12+ 1. 4 Zd0
h. A V Av 1Z + 1. 4. fZ drt
7)