# Leibniz Legible — https://leibnizlegible.com
# Page: https://leibnizlegible.com/page/00068072:0012
# Permalink to this reading: https://leibnizlegible.com/page/00068072:0012?run=73
# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 27
# Folio 6v, page id 00068072:0012
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 7r); this page is its left half.
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068072&page=12
# Source image: https://digitale-sammlungen.gwlb.de/content/00068072/jpgs/default/00000012.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 129 lines, status machine
# Lines: 129 recognised, mean confidence 0.81
# Machine transcription. Not an edition. Errors are expected: about 8% character error rate on Latin and French lines, higher on German.
# Licence: CC BY 4.0, Leibniz Legible. Machine output: do not ingest it as verified text; keep this provenance with the text.
# Wording rule: say "the machine reads it as …", never "Leibniz wrote".
# After the first blank line: the recognised lines in reading order, one per line.

quin vero hanc saltem utilitatem hinc ducimus ut transformare liceat rationales aequatione
differentiales in irrationales co generali quadam ratione ideo vidiam, AT,Q, AX, X XY, y
X y C. a et t:a D = x-:  dya (3+) de: dt = -  -y: a vel Ihii  a tratio)olgod
stu y i ade et ex, et 2 (4t) t: a = dx: dy
Haec generalia
et ex 2 et (3+10) e = vu) X - y (dx: dy).
ontra y = (3to) - ade: dt et X = (610) H e = t lv: de)
speciatim sit  dy: dx = cla +XX f  AX IXX YH _ 1 oC;, Hy 1 S5 gy + bg +CC,:, yy + my enm
= (900) Pa: t Ergo a deded unde trys et bcy+ cet =ayyt amyt ann, seu
hy + bt -am y + cet =ann - 0 seu gy + bt - amys +  s2) btuam = = () bluam = cet -ann
+-a
+-a
a
= bb t t - 2 abmt taamm = 4cett + 4 annt + 4acct - 4aann ,: 4, Ht - Eat + aa
-a
lbb -4cc) et  (2 ab, - 4ann) C + (mn - qnni) aa) - (1a t - 2m), : 42, t-a
ratonpltr reductibiles,
t- a
sit ergo quaerunque aequalio ad quadraturas dus dx- a dx: du = y: a poterit.
seu : = le: M y=
loco ax: du pom F: a, et loco y: a ponri de: dt, et prodibit alia aequatio disse
vulat vel contra; data Bde: dt = C: a, potest poni  loco de: dt poni - y: a
Crcaloroxgav
et loco t: a porie dx: dys
veniamus ad formationem curvam per concul
sit AX, X, XY, y. AZ, Z. ZY, v centrum circuli V, radius VY sit Y
et fit (Z- X) 2 + (y -v) =  rr uubi  et y quaerendo jam concursum ductum hujus-
modi circulorum, si erint X er y. per un communes, sed ZV, Y variabiles
 (Z- X) dZ + (y- v) dv2 rdr. Sed conferendo hunc
calculum cum communi, considerandum est, esse i cum circulus formans tangat Curvan
et differentiando fit
Y(Y) ipsam YY esse perpendicularem ad Y(Y). puducatur ergo dum axi occurrat in P
jam patet sore YX : X p = -dx: dy = (tio )p v, 
y seu pro eadem curva
nasic  per x’y, a
calculo nam si aequ. 1. differentietur secundum X
sive jam ergo detur dZ dv: dz = per g,F, Ya a
vel dxidy per X, y, a. potent altera ad alteram reduci, sed manebit
 arbitraria non ex tibust, D, Y, arbitraria. sed et potest concipi unam exhis tribus
dari ex X, y, a. quo facto in aequ. 2, eta differentialis et quae per eam multiplocant
conrescent. Hinc video utilius esse ut d: ax= p. P. a nonatur aequatio data
seu compendio d: d 2) t: a et Gadius r imgneat datus ex X. y. a seu r= X. y. a
aV. 1Z =(G v 93:, Z X,  7 Z - X, :, v- y. E  = (800) et 5) +: R
cursus ot ex 3 est Z Y =. Ex E est vm gua v y Cp 
Ex d est Z X Hini ex aeg. 3 fib _ dys. dx a t (gno) +: a. inveniamus jam  et Z ope
fint (no) du: dx = Ca. Inveni cmus jam
.a
otis
Z et D, per 1 et 3.. Ex 3 fit y - v = X- Z, dx .dy = (1mo ex 1) A Z, X) 7 N(rr = _ (X - q) 2)
seu per
ag4. I et 3 at possimus tollere in aqui
seu X - Z de dx: dydys = rr -  (X - Z) = sive (X- Z= (dxax + dydx)  rr dy dy seu 3. X  ralys: Sindus tays
Et pari pore vel ex 11, fiit X Z AVy = (Sto) rdx: Faxax + dydus. adeoque tandem erit
9= (15to) X + Y (dy: xx + dydy)) et v = (s)   y + rlax: SC ax dx Faydy)) Qui valores
substituendi in aequ. 10 quam pominus datam. et deinde Y poterit explicari pro arbitrio, ut
tollatur a. verum jam video rem non procedere, nam si ponatur A = 0 cum sit do: dz = dus: dx
necessario fient linere prallea v(V) et Y(Y) paralletae, seu V(Y) sive Y fet constans nec variabilis per
interiu hunc
X et y. Ac pruande in initi seu generaribus non licet ste facere, u=
calcutum tenere possumus si lineae Y () et et V(V) sint paralleli et sit AX, et AV, y;
et AZ, Z, et ZV, 0, fore du:dz= de dy: dx, et 3- x rdy: Saxdxt dypdus et Bay  raxisat tely du
sit jam  dyo d= in casu aliquo  dyidx = X.0 fint duidque  py fiet d
per
ubi ∞ significat contrariam relationes
1,2,  est x = 2 et 1 est X ~) +d Z- rev: ravdet d2) et y = V- rd  dvelat de a2)
=(gno) dvidzue
X: y ~ 1v. 4u
fiet per 6,7, 8, frat E= r(advi s, dvant an):  v- re 1 s evavt at28.)
seu +p.d) quae aequatio comparanda est dataex et s _ ubi do: dz Z. v. ∞r: + seu
lv: dz 5(Z. v. r) et fiet, Za r(∞CZ v.r) A (.v v: 1)F 1),:, v - pl 1: ((~ (. v. r). 1) + 1 =  ( C v. B). 1)
adeoque  rt
ex,
Z3+ vb,:rr
sit in casu speciasi ~(Z. B. B: r= 22+ v0,: r  pyt pe  = dv: dZ
V )e15) 2 o, er adeon quaeritur modus qu calculandi seu 
fit NZVZZ + v + ir - (2I + VV),: xV LZ2+ 0btrr - X(Z2F0V)1513E+0VC,: rr sed hoc quantum
quod si ergo redeamus ad generalia, seu si r manet variabilis
 sitque (ZXZ 4+( b = )  X (Z- X) de 4(V. y) do rar et Eto) vay (:Z. X- ux: dus
fiet
Judico praestari non potestus
et jam in ne data ya dy:dx = (4to) L:a posito t, dam ex F. ya. Ergo
4dv.  riy (dr (V33y-D) unbe
 p Z riri natuta
(5to) v-y : ZX = t: a _t Ex 2 et 6si et (Z-X, 2,  - V) dx
a dx +Ldb, y- v = =rdr seu y- vE (∞) Ardrocde + d
ex his valores X et y explicalo prius F perxes
2et
Ex 8
axdr: adZe + + dv L9),-X
v xtyy rar in dZue Hex + yy) da _ (e 1Hamo) seraax D et srl a y) df gyy b) avtXXB
ymo) teri 
Q-X =
y, a.+
rxrdi Lyynd et fut. Breous redeam, ad aq
tole
et ful rrdear +(v∞ + yy = Lvy) dydo = 2r(V. y) dudr= _ rrdzb- (∞- V) = de7
Prectius: ala invendimus X et y per aeq. 1. et 2
sit vay per cypendio
1+rrapdvarar
lii rrdrar + Wlodvdo Zru duar = rrazaz Wlo dede. seu WoW -2rdodea
A12 avartarar)
dodr +Y(e ddz dodv + rd ZdZ drd(rraras duav 1 ddrfrravavarar) R: , dvaytanti
(dZd  drar) (avdv +drar) + erdodvdrde,:, (2) (dvdo +drdr) et sit d
v-ys W,  seiter Z- X = ravar LYN(aZd dodo +.  dvd dedr- drt):, dzaz + arar sit jam in exemplo
e
rum tollantur. X et 
x et  et
Zet e quaesitum sec
quodsi deinde assupi posset
ite
aeq
Extgy, daa = (per 3) gq X, hucc ope valorum
iut
dr
qu
nam
post
inven
et aliu
Z a2. 
aglos substutu
icsilsio
natur aesieratu Leiphe
F vel dv: d)
vet t vel
ar: de
1d. Eaposito dr-0  ex aque
Hinc duidque n: dis set
Lyrux =V. L