# Leibniz Legible — https://leibnizlegible.com
# Piece: https://leibnizlegible.com/api/records/1121/text
# Catalogue record: 1121 — quomodo concursu curvarum formentur curvae, tangentium proprietate quaesita. concurrentibus rectis, circulis, conicis, ordinatim positione datis vel compositis
# Incipit: Si concursu rectarum TY, (T)Y formetur curva Y(Y), erit calculus talis
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=1121&katnr=58671
# Work: https://leibnizlegible.com/work/00068072
# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 27
# Folios: Bl. 2–3 (from the shelfmark LH 35, 6, 27 Bl. 2-3): 4 pages, canvases 3–6, page ids 00068072:0003 to 00068072:0006
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068072&page=3
# Lines: 190 recognised on 3 of 4 pages, mean confidence 0.84
# 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.
# 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, 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 2r — 00068072:0003
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 3v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068072/jpgs/default/00000003.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 75 lines, status machine
quoniodo cncnisu curvaum formentur curae, tum
tangentium propuetate quosita concurrentibus roctis, circulis, conicis, ordinatim positione datis vel cisitis
Y8)
FY, (T)Y
si concursu rectarum formetur curva, erit calculus talis AX, X. AT, e
CX  x- e et quia TX: TY = dx: dx, fiet (+mo) x.D. y= tra
Iuo  xalg: y = dx: dg = (230) per compend (taa) Bto) X -e,: y
Et aequ F differentiando fat pro concursu, erunt +, et e variabiles, sed
X et y eadem, et fiet at: a =- de y seu y = (210) - ade: d+
ponamus ergo rectas, positione datas, ergo datur ratio TX ad XY ex AC
cordinatio
sit XX TX: 7Y = t: a. Ergo (1md) X- C,: y = +: a
quae est aqualio adcurm TY, quam differentiando secundum e et F, non
rectam
secundum x exy x et y hababitur concursus curvarum, e
fiet
(2d0) - de: y = dt: a jam +: a = dy: dx. quia rictae YY sunt tangente,
0
to
curvae Y(Y) Hist Quod si ergo detur relatio inter t et e
etiam ex aequ. 2. tolli poterit de: dt Et habebitur aeq in qua e et
quarum duarum ope cum qua Et ope agg. 1, 4, 5 sublatis C et F
et facien
halebiur relatio inter y et X, vec opus us aeq. 3.
si sit aequatio data in qua dy,  dy: dxe x, y a
ope . dy: dx, = t: a, et x- e l: y = t: a, tottendo dy: dx 2t
seu e = X- y(aX: dy) ergo si detur relatio inter + et e, seu inter dy: dx et
x - 1(dx: dy) seu si ope aeqgg. dy: dx = f: a et x- e;y = +: a tollendo dyidx
pt X, ut maneati solim abitrariae suno variabilato a sit a e
habebatur solutro sit ex. gradx dx le: dydus = FaC X - y (dx. dys)
e + 4ax et d = H(yy + 4aX) - 7. Itaq si
quaecunque sumatur quod ut alllro quoque calculo comprobemus tollamus
a
 V Lgy + 4a e+ 4oyta
t
dysus
ex hac aequ. jam dy: dx quam  X, futi
et fil 4aa et tgy fiet 2t +y = Syy + tac + 4Fy,
seu H= ae ut an te
AH + 4 Fy Lyy = (7y+ qa e (+ aty)
pro Circulis suo contactu curvam formantibus, sit AX, x
XY, y
AP, e ponamus duri pY ex AP, seu circulum tangente
esse positione datum, dabitur curva. Nempe est dy dx
dx; dus = Y:yNPA C -X yi, C-X, rursus yy + ee + XX - 2eX = pp
HHini si ope harum duarum gg. ex aequdata differentralis secundum x
et y simul possint tolli sublatis dx dus et X simul tollatur y.
caeterum dat, et
caeterum ex 6,qu. 2 differentiando proconcus su ductum circulorum fit
habebitur quaesitum; seu relatio inter C, et 
ede = xde = pdp Hinc si (5to) data esset aequ. ses. e, P de, dp. Et
ope 5 et 4 tolleretur de: dx, et ope aeq,. 1 et 3, sollerentur e,
L
ponamus conutu descriti e curvan aesdescubi evolutione curvae ve
VY, p.
habebitur 6g. ad X, y , dy: dx
primum  AX, XE XY, y. 1Z, Z
Ex natura circuli cenlro Y, racho VY descupli est V F2+ FY = p
+ y- v  pp et quit concursus ductum circulouue
habeatuo fiet Z-X d2 + y vdo L pap ex, natura circuli est
seu Z
dy:dx: dy = y - v: Z-X seu Ax : dus Zx. Z, :, y- v. quod et
patet differentiando sec. X et  quod si jam curvi sprunt est tanus
dz: dx  dy: dp dx
re
2 + y- V
V (Y) simuil est langens curva NV) fiet
(didza dedo
Hinc ose 699. 2, 3, 4, 5, 6 llendae X, y, dyi dx. sed una est
et fit dp de= x  + d dB et
sit dy dx = XxEgy aa fit = dy. du
t dz dv = X -Z3:, y -v se
XZ = (y-v) d  d) seu X = Z + (is v) (. de) -  (pp - y. 
est superitua, nam quod aeap= diaze+ avd, jam est consequentia caeteror
seu 099. 4

## Folio 2v — 00068072:0004
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 3r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068072/jpgs/default/00000004.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 70 lines, status machine
 sint circuli ordinatim datri, quorum
Centra in curva V(V), et, dalo puncto curvae i,
 inte Horum circulorum intersectine
formetur curva  Y (3
Exnatura cioculi AX, X
CyV v. VVOr. ZZ E=X + y -o L) rr. et pro
concurtu circularum (2 - X) dX + (y- 0) du  rar Et ex
Circuli langentit, (Z- X) dx + (y- V) dy =0 seu Ddx: dy=
Z-Y
soeiitis
3)
v.y fat sit jam aequatio data: Ax: dus = t: a
Z- X9:, v- y  (6to) F: a. jam ex aqu. 1,F, 1 et 2, inveniantur
X ot Y, et substituantur in aequib Rursus jam ex aequ. 1. et 2. Invenians
E
X et y, fi et substituantur in aequ. 6, habebitur aequ. inqua
solum Z, v, Y, dz dv, dr., Nempe ex 2 est
v
y - v = Y ar - (Z- X) ax, :dv  = 1(rr -(Z -X) 2) et seu
YX(drdr: d vdv) + (Z- x) 2 (dX dx: dvav) C 2Y (Z X) (drdx: de dx)  rr-β X) =
seu (Z-X) = - 2rd rdx (y- X) + erdrar dede guextrrade + rrdrardudx
qdx+ axdip
12) (ax X+ dvar) ( (dx dx + dvdv)
BY,  drax + savire  L + dB Fararae 3
et fit R Z-x =  s 
10
seu X = Z = Y
Cardx + sevdv (de d3 Favdn fF ardrdZaZ) x de + dudx
az dz + dx do
et parijure V -g = Y ardio + fl 2de C   +dvdu) + arar av ilo)
az z+ av du
seu y 0 - Y. drdv + (ddqze (d2A Z favdr) + drar dudv)
dz d2 F dudo
g, et 1I, fiet
et fit et ex aequ. per
+ (3) araque + Y (do dv (d Zdz +do dv) + drdr dzd2)
a
in aeg. 13 drdv F Y (dZdZ (dZ +Z + dvdv) + drdr dvdv)
Et quia explicando  per X. a, y: a et ex datis secundum aequ. 5.
et in aeq. ut fiat aequ. 14 et in ea pro X, y, substi tiendo valores
ex aequeo 10 et 12. habebitur aequ. inter Z, v, du, dv, r, dr,d
si si b + a expli compendio sid r d Z + (+a (d 2 1Z+ dvdr) + ardr dezu)  dsu
et idem permulatis solum dZe et de sit, = (16) dgdus et dZ dv + dv dv sitsos) - dede
fiet ex 13, t: a = (18) afdf: dgclus et  = 10, X = (19) Z - Y (fdf: dede)
et ex 12, y = (20) v- r (dydg; dece) aequatio data 5, t: a Xxtgy,iaa
Au (22
seu xx. per 18, XX+yy, : aa=dfds: eg dodus, et fice et ope
e +rdsErZede d, dodo =aa dsaf de 4+
Edg+ Zero Eyd9s c
aegg. 19 et 20, ex aeq. 22. fiet 
videamus an licuat facere (24) df =0 seu de
seu ex
15, Card raz d2) = dZ dz (d ZdZ+ Lodo)  dr+rdzdZ et fiet
vel dZ du Ya  quod est impossibile, vel aq
seu ex eg. 15. Cdrdr dz d2 = dodo (d2d, + dodo) (+ drardydu 
adeoque de = dz f- 1. sed ia fieret et de=. unde praestaret potius
onservare df. sed video posito de=o tamen huc tandem veniti d
aliter = faadtas XXdgs idg , seu Ly = Ma d fa sde - 27d0.  2Er dedeatat 
22
:dg = v - r(dgda: dede)
si loco aqu. 22. fuisse
fieret ex ig et 20, Z- 2+ ZEde +rrds - 2Zr dededsls  rta: +X 
24)
vode-rrdgt +vY dydegus d agag=na dfils do +
ubi faciendo Z v = 0 fiet YYdf4 - rraq, dodus = aa d fd+ dC
Ex 24 sit dZ = dv
et dfdf. per 15 = C 17) draz + de N(2 dxd +drar) = (28) iudusddb. Et ex aequ.
25 fit redes rrdSE er de sed ob aeg. 29. in ipsa 25 dvavescent Omnia, et
fiet af sel de =0,  de non dede = 0 non licet Ergo fiat df =0 sit ita fieret y=0
et X -Z et y- X. quo non licet facere

## Folio 3r — 00068072:0005
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 2v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068072/jpgs/default/00000005.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 45 lines, status machine
progrediamur ad sedionem Conicam generalem, ubi adhibetium
illipsem sub qua alligi pt mutatis signis  hyperbota, 
Axis AXZF. curva duae VV et GG, quarum ordinatae
ad axem Iq et GFqu sitque curva puor, locus unius foci
et altera GG, locus alterius foci, summa foum longitudo
distantia foectum VG, sit m et sili longitudo nempe
VY + GY sit 2Y. seu semiaxis transvertis Y
G. AX, X. AXY, y AZZ ZV, V AF, + FG, g.
qu v4 F7
L+rum   vi Y V + GY facit ZY. et
jam VY _ = 4 (9-V + Z.X) et GY = (y.9 + f. X) ergo fiet
(y- v = +(Z. X) a) + + Y((y- 9) = + (f x) )  2Y Ex his aejus prima
non ingreditur calcurum. Sicunda du ilici modo differentiari p vel ut genim
unicitur X, y; gneminentur caeterae Eterae, vel contra. pripre, modo est
quod si r sit constans quod in pte est, facilior ert tatenlus, et fiet
(y-v du + (Z. X) dZ
Ny v + Z=X2
(9L v) dy  (Z- X) dx
y-9du + f. X +5 =0
Y(y=9 + f =X 2)
+ (y- 9) dy - (f -x) de
si 
si r constans
 seu fiet
N(y. + ZX
(y-a + f)
 aequ 2. et 3 invenit (G - et + (- X) 2s - v)  sg + Z. X) 1y-9) dy 
161 
. ly -g +  X) (7. x) + Vsu- p + Z-X) (f - x) (= dx: dy = t: a)
posto 1. a) = (7) X. y. a dato Er ope aequ 2 et 3 inveniatur valor ipsarum y et X
Z, dv, df, dg.
e habebitur aequalio dans stelationem pro ipsis 2, 0 f.9
ius : lx = XX +yy,: aa sit ila mstors fit seu sit lx constans fit q dilus. dx = (2X Ix + Eydy). 1X + (∞X rgy) ddx. re posito delus.
cadhus Xxdax -X dxax (
seudus aadyqixddx ved dx q Axardus + 2X
seunadiseinus = x  XXCxx ax a aadus ex +. XXdx dx Z: 2dx Vaadus xxax
2dX iiciau. Excx
imo et quaevis alia
priore modo
posteriore nod
et tollatur inventus ex aeq. 7
fit =Ry = XX dix F yydux F2XdXdX,. Eydx, se
Ailiter aa dus XKdx,: dx = yy. seu y = adus  xXdx): dx et dy-
= Haa dy - XX dxx)  Acx
cd

## Folio 3v — 00068072:0006
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 2r); this page is its left half.
# No recognised text on this page (recognized).