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# Catalogue record: 59322 — NB. NB. Hic prosequenda quaestione pro tangentibus inversis ad quadraturas reducendis. Hoc melius alia scheda in fol. Jul. 1690
# Incipit: Quaeritur dx : dy = yy - xx :, 2yx - 2/3 bx + 1/3 cc
# Date: 1691-07-22
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=59322&katnr=57518
# Work: https://leibnizlegible.com/work/00068196
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folios: Bl. 317 (from the shelfmark LH 35, 13, 1 Bl. 317): 2 pages, canvases 629–630, page ids 00068196:0629 to 00068196:0630
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=629
# Lines: 117 recognised on 2 of 2 pages, mean confidence 0.81
# Catalogue: Leibniz-Katalog / Arbeitskatalog der Leibniz-Edition (https://leibniz-katalog.bbaw.de/), ed. Arbeitsstelle Potsdam I der Berlin-Brandenburgischen Akademie der Wissenschaften, CC BY 4.0.
# Machine transcription. Not an edition. Errors are expected: about 8% character error rate on Latin and French lines, higher on German.
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# After the first blank line, page by page: a "## Folio <label> — <page id>" line ("## Canvas <n>" where no folio label is recorded), "# " lines naming its source image and model, then its recognised lines in reading order, one per line.

## Folio 317r — 00068196:0629
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000629.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 52 lines, status machine
N A. Hicprserenon quae ssue iose meligsatia invertis  A int regrecendis
Quaenitur dx: dys =yy- XX;, 2yX = = bx + EC fingamus:
dZ Z + deo co + du 6. 20 et faciama esse
5D
Z  G. X et co = y. X et v = y. X
quibus valoribus
Substitutis in aeg. 2. ibt ex aeg. 3. 4. 5. peibt  debet prodine
aeg. 1. ita Tangentes inversae erunt reductae ad quadraturas.
sit egoRem melius exprimamus:
dx: dy =  X. 9: X. ) 5q. data
d2ZF
C, (2
quaeritur dx b: v + dZ Z: Z + deo Cd: co oita ut ponendo
= (3
(4)
Xy
  et  v= Xy: F y et Z = X J: X
et c0 = X. y: F. y et has valores substa atque ita collendo ∞, g, co
ex ag. 2. prodeat aeq. 1. porro eb +.3 est du x y dx, 1, 4  du
ab ag. 4 est dz  XB dx +, F XG dus ∞ ob aeg. 5 est deo  d X. y dxtax. y dy
et ex rofiet A pri 5.g. X 3: kg: X.
F. FG. Xy
do = X. 1 ∞. y ax + X. y = X y dy - X y d F.  ax = X. y = X. y dy: 1 xy
1Z2 1X.  + X dx + X , _ Xty dy =xiy d X.  dx = x  = F. 8 dy. LC X y
similiter aco. Sed fortasse re im licebrt traclare bieoius, hoc modo
dx: dy = a: b. de v: P + du Z: 3d + dco Co: =0
sit v = m: n ∞ = p: 4 co =r: t du- a centm dx tn d ay
du nam dx + n d Mdu -m dn dx - m An dy,:  un 2 exaegue
dy 2) 4 d  dx + 4 ap dy - p d q dx - p dg dus, : 99 3. 4.5
U
de + dr dx + + drdy = Y d + dx -y at dus,:
Ex aequo 3, 4, 5, 6,7, 8, substituantur valores in aeg. 2. fiet
m: m an: nn, m:n: m: n, .. C- dm. n tumdn:un, min: m. n
+ d p. g- p dg: 44 p. q: p. qu (per = ap: q + p d9: 99, p. 4: p. q
+ dr: t - rdt: H, rot: r: F ,)
l - dr. t + Ydr: H, F: F: Eat
) ay: dx)  b: a ubi explicaudo m n pg
ubi explicando m n p, qrt perfstmutas rationalese sounatas X, y
ignificare etiam formatione cocle intequ
et ponendo c et e esse signi et s et g, et  et
colegiam, potent fieri as explicatio generalis. Sed brevitalis causu eligamus casum specialem
simplictorem, cum n= a = + = 1 unitati, tunque enim ex aequlo fiet:
am m: m + ap Pd: pd + dr F: F: Bm m: m +XpE: p + = rY:
dy: dx) =b: a et redudis omnibus ad coem denominatorem, fiet.
e1
 compendii distinctionis causa, pro C, C, f, 9, n, scrib. 3, 4, 5, 6,78
am  m dp + d   ita ut soperioribus respondeant un pares, inferioribus pares
fiet: am pe apmr am npr +ap mr+ r
id est tomtis assumtis formulis in p, r, (ete)  ex Xety
-dg: dx) = - b: a; ratimd ser interre sopiis, t rursus formuli, x
item ex p, itam ex Y formatis, significalis per n
quicqud in formulae in difereutia multiplicatur per dx, et in quicquod inea mullipercatur pera

## Folio 317v — 00068196:0630
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000630.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 65 lines, status machine
 habebitur aequata 4 quae de2 formula aequationis 14 quae
debet coincidere datae forta a ib, ubi a, et b sunt fa formula
datae ex y et x ravaliter integreque formatae
(17
Remlicebat statum ab initro sic facere: dx: dus = a: b
do v: v + dz. X:  + dc co: cd  Ct O posto v, 3, cd, dari per X et y
jam de= =V dx + dv du et du= Z dx + dZ du et deo dco ax + X dus
fiet de v + dq ZV a + dcocov Z:, dv. VZ c+ dVdo
Eet -dy: dx) = -b: a. latus autem suistrum aequavis. E vel  exhibeamur
aectus ipso ponendo formationes per Xiis esse rationalis e integras, et numeros
4, 5, 6, 78 significare contidem foatee formulas rationales integras. Monamus
p 10X + 26y + 1 et Z= 30X +40y+ 91 et cd = 5oX +60y + 51 erit d v =  10
et Xv = 2 et dZ = 30 et dZ = 31 et a c0 =50 et d0 = 6 et V =  70+ 71
et  = 50  81 et R- 40Z + 91 et 23 = 1002+ xF 10) et co =10 11
et 108 = 12O4 + 121, et fit ex 84. 6: 10. 70. I0XF goy1BI +71. 100. 3OXH0y1 31 4 01
120. 50X + 60y +5) + 121 + 30. (80. 10X+ 20yKII +81 90. 20X+40
16. 50X +60y151 F
450. 80. TOx +20y+1I. +8]. 100. 30X + 40y + 31. = 10) : 20, 70.10X et 8100. et 20
+10. 80. 10X.: 9030X.. 170,50X = 60. 8010X, 100. 30X. 11050X.. = - b: a
unde perinde est ac si parbamus 10 E 150X16B 170~9 11OXG +151X (1 161 y
+17+ yX + 152X + 162y + 153 et,, 430, in 190X + Ly CM, +50 in 230124
indivis. per 20, in 130X)  eu  + 40 in 90X + eu, +60, in 230X +  et,
vel Tubi -b: a alcendat
dB. . Ryeux. py ea
E
 in immentore et numeratore eunt prii numentor stae ipsi ae quandae aequetur
edus non sulori, Tabemusue literas conpartioes intrliendas decem
iseum fiat s in numeratorem compatatio, non
pec
 que fia
biunl litere
e
a
, b bia
sit inferior, poterit multiplicando Pattollilad p. dele in habeautersit
liferae conpantitiae. Nota: NZodet e   190 et 100, iem 10 et red finsti
noniunsuntur sufficit hotum nagna sed et 1 st abjici possint, et pro tis poni mutas
terns iier padfecturae sindt 15
xperopretium etit prosequi quont  s hac
 smplici positione, multiplicare fecta adendorm mogrdilliaat
proxinni est ut V, Zvela ex  et y ormentur Hormula integra rationat adrasica
sed ipsae 3 56,78 significent autum somalicas formaoes ubi musidebine
e quousus racinave proficiptua nostieno videbim quid fiat si utem modis 
stimand si quadratic soel estiam ost sint altiores aequales aut
in aequailes Et haec vietur compendiosissima ratio tractandi hanet
minam inde  siiliter plegeuad differentiales formutas
irratomales.
uquollies s cb 
dod  imaloeissimis manesa quosties quol tatemu literas ut v, Z co, toties 
EKhabemut E larbitrarias proficiente
ec literarum. jan in quovis quadu o sunt
. Et ad Tradum ascendimeus aequale numtro
ant termin
n+I nt 2,: 2 posito n numero
gradus seu ex pomento, et arbihariae sun  sunt 5n.   quibus D
tns
 alius gradus ipferior pe
epantio itaque numeu terurigore 
SM +
verum bis instituenda est
debet este n+ l. net de un
+1. pF2: 2 dee essae  nf1. diFE  posito n. p = gradii
a daule exaltioe sed pet sme exsitratione dato gradu a, et gradu b, 
n,vel salle
 et ex hiis ndiginio (4, siet ntqt 1. nq non major n, et dato 9. qutur na