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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 3 (shelfmark LH 13, 3)
# Folio 109r, page id 00068200:0219
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X+dx- = aequ.
23 Mortii via revocandi ailias differentrales
1682
adquadraturas. Et evitatio exaltavis
ad comparendas aquaves requisitae
in curva datur relatio inter AP et A∞. quaeritur
relatio inter AB et AD. sit AP, Z. et AB, v.
et AB, X. et AD, y. t BP. HC GW, dx. WL, dy.
ob Triangula similia  GWL, CBP, co DC, erit:
BP: BC ::  WL: GW, sive BP: y: : dy: dx, et BPerit y oly
eodem modo DD: DC:: GW: WL sive DV: X:: dx: du ax
et DU erit XdX. Est autem AP aequ. AB + BP, erit
dy
(1)
ergo Z aequ X + 3 az similiters  aequ. AD + DC erit
ergo v aequ. y + X dx. denique ob triangula
y
i
 AP et GWL similia est APO: AP:: GW: WL. sive
 v: Z 3: dx. dy. vel dx aequ. v. quem
y
v+do v
valorem inserendo in aequationibus 1 et ~ fiet:
Zaequ X + 32 ex aequ. 1. seu Zaequ. vX et
+)
v-y
vaequ. y + XB ex aequ. 2. seu v aqu. ZV vel
=X
exaequ. 6. fiet: ex quae aequationes rcoincidunt
nempe 4 et 5, , cum 6. et 7. fit enim utroque modo:
vZ aequ vx +Zy. Unde differentialis aequatio
9
erit: Zdv + v dque aequ. vdx + xdo + Z dy +ydz
sit jam data relatio inter Z et v ex problemate
L(101
v verbi gratia sit: aZ aequ no quem valorem
psius Z inserendo in aequ. 8. fiet: taqu ax +vy
y+vax (XD. xdo) v aequ VX F vvy. seu v0 aequax ty. cuius
a (2)
Ergo L x aqdixdo differentialis: 2V dv aequ. adx + vdus + y dx. Et quia
vv.
+X + X + d
conpalier
+xvIex
+(HIXdo
sit +s
epsit
et () sit t
ex qualibet ordinaria derivari possunt duae differentiales
fiet scribamus loco aequ. 1,
(14)
13)
vaequ ax +y. fiet:
do aequ a x axtaedx + dy
jam ex aequ. 12 
1
Vvnaxtvy
do aequ. adx + vdus ex aequ. 14 est dv aequ. vvdus +avax
20- y
(15)
quos duos valores aequando, erit:
(17)
v0+ ax
tat;  is avvax (+ VLy) - aa xdx-axvdu aequ. Q dus (2avod
constanter
durnando
ut simplex
Lwvydy tavydx
ho un ponamus valorem ex aequi 11. fiet;
majore (2(3aex dx tavy dx (aa x t a xvdu aeqiavrau ravelus
X a
seu X+dx
se
M (17)
taax dy tavyd
Faxyayue. Vyy chyavydi
detraratur avi das vuo ayt aaxt axvduque Ebvodus a vvix .y vedu avydx
vy y ovdy
+
op qua aequatione 17. facta explicatione et destructis destruendis redit
aequatio II. itaque inutiles sunt duae differentiales. Omnia ergo redennt
ad aquationes 2et 1.  n
nempe
1oq vertatur