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# Piece: https://leibnizlegible.com/api/records/62488/text
# Catalogue record: 62488 — [Auf. über die Ellipse]
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=62488&katnr=60035
# Work: https://leibnizlegible.com/work/00068160
# Title: Leibniz-Handschriften zur Mathematik LH 35, 10, 7
# Folios: Bl. 4–5 (from the shelfmark LH 35, 10, 7 Bl. 4-5): 4 pages, canvases 7–10, page ids 00068160:0007 to 00068160:0010
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068160&page=7
# Lines: 287 recognised on 4 of 4 pages, mean confidence 0.80
# 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 4r — 00068160:0007
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 5v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068160/jpgs/default/00000007.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 76 lines, status machine
AP, h. et AB X
44
du+
50 ag
2
BG 3 182B dx. D2G, dus
PG, v. GL dv Leg r
 ibas   A ei  ph A
2
36.
2er
Ex E Vdr Jydu ah dx + xx per differcutaalem ipsius aeg. 2
(3
Et in aeq. 3, pro e substicuendo valisem ex Lq. 2: fiet
seu vdL Xdx = hax + du NN-hi fhxx. seu
du L sit 25 ∞ = 4y + 36h - bis 6h2X + a x
cujus differenttialis
SV Zdo  2X Ci X six it 2 XIdY +
5V4dv = 3y8dy  - 6h, dx + 2X1dx
porro Cd 64 dus + 1dX ( 1b dv + 49 22
Ea e Ex c9. 2 tollatur 3y. fet per aequ. 1. fiet
4 5V4 dV +6h1 dx  = 2X 1dx  8 diu 125V - 36h + bis 6h2X qu
rursus ex 5g. 4 tollatur 8 dy compe aeg. 3. fie
5V4dV +6h1 d∞ - 2. X 1dx =
V25V - 36h + b 156h2∞ +X. 16 dV + 49Y 1cx
a sit jam sit  in Ellipsi g bis  xX
sit a = 3 et seu  97 9 = bis ag2X a +X fit 974:  q2X- 4X
sit A divisite per C sit g - 18 et C, _ seu c 06. fit 8+ et qu C g et
 pet _ g18: 1f 18 . M sit divisibile per X et per qu2
ergo 4- 4 et fiet gy 49: 492X-4X =ga seu habermus
.39-2a et.
1= 2.39:-239-9
seu 9= 2. 13:69
ergo a = -3912
et 9 = 2. 13: 69 f39
34
78
101a
6NV
1679t
44
cujus differentialis est: 9
et hujus differentralis (39 dy 37gdy = a idx -  2X 1 dx
29
in Ellipsi sit C y vel alia
jam sit Ellipsis vel alia sectio Conica; cujus aequatio  est
24X +   a a: b XX = yy. et _aax f a: bxAx = y dy.
u
Est antem A, distantia a foci a vertice h =  + B + sB + ab. Ergo
qoaque aequanda quadrato b9V; V 2am + aem et f  posito
l’et fiet 9 = 2 m9 - am; seu 9 = a m: 2m - 1 sit autem a = 2 m - 1 fict -
et firt ma - 2m L1 = GH - ab et SG- ab = m - 1, et h = m  - 1.
ergo leis 2m - 1X 4 am -1: M XX = yy sit autem y = 2M 1. 3, fet
Itaque 9- - 2.3942969: 13
3
o
a
ga 1 dx - 32  2X 1dX =y sRy 5 gy = bis  a 2X - = 4X
Eugo 248. Batqeg
et a = 12 Ergo
20
qy bis 9a 2X - 34 4X
18.+ - 9 = 4446
6 - 3 = A16
9 = 4a-44
24 = a - 2a
Et 2qa,
39. 2 = 78
61ng
621
y = 2m - 1: 3. fiet bis  X + 1: m XX = 2m =1 : 9
Ed +4 = dy
tit dus 8
55- 16299
sit av AC
sit

## Folio 4v — 00068160:0008
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 5r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068160/jpgs/default/00000008.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 51 lines, status machine
inveniendu est calculus
sol solas eductas
differentialium
es umbilicis
Ellipseos per solum
AV sit 9 EF sit e
EG Y FG q.Y
Datis lateribus tuanguli DEF, datm
lateribus singulis tamen prius di-
summadimid _ +r + g- r = _+9
3tia
fitanea eva
sit area, q  in C - 2r-9: 4
3 Summa 2r-9 et C, F. diffia eora rectanguli
ex mediis proportionalib, inter duas primas et duas ultimas
pars qudta est area trianguli, et dupla dimidia
parsest area paralle logiammi s quae divisa per
Est autem tr ducatur GN bisecans Angulum ∞GM, patet fore
ror pollus ia MGad MG+ _ GF
MN ad FM, ut EG seu Y ad EG+FG seu 4. Ergo
MN = ...... .. : 29. Est autem GM=
=Y = FM = L 4Y. q. Y + g C. C - 2Y -9)
M: FM: Gd seu AB. Y: DL seu dr
Y: FM ( 1) = ab = drr: drY
in C - 2r - q  =  4 a 0 - d Y : dr y
sigq. rq si t - pr 4  r un - hhec +hhpp =
gc+0
12
= 4rrqq- 8r4 + 4r=  hhee + app. +
trer 9r + 9t- aer++eq equ
c
 +r - 4 tr = Erte
 -r +g- r Etg - 2Y
=4 + r + 9 - r = q -
qe 4: 4
seu si ex ponattur: summa e et q2 differentio eorundem
quar 
seu 84
area, quae sit ex di ducta sumnci
et omnium differentiarm ex tribus
midiatis nempe-su
diff. pha
2da
seu EG dabit FM seu F M = .. . . . .. . .. 2Y
il
F-4.
seu GM
rE
et p = r- na
fiet p=2
et hh = 49- et

## Folio 5r — 00068160:0009
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 4v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068160/jpgs/default/00000009.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 53 lines, status machine
vens ha e= p: 4 = aY. et 27 - = s Se- p. 2
344+5 =125e. 2=6)
3+ 4 -5 = 2; : 2 = 1 (136
3-4+5= 43
= 2 =
2
3+4+5= 6; : 2 = 3
et FM = hSe -p: 2Y.  C et  sit FGseu q. Y = n. fiet: GM = Inn -F p
seu = S  99 447+- h C21 kp Sarrnn- hhee +hhpp : 2Y
jam GD: DL:: MG+GF. H MG et GD = ad : r. fiet DL = MG. aB: Y AnGF GF
Triang.
seu DL = MG. ab: Y MG+ n multiplicetur per GLe tam num erator quam
nominator per MG+n. fet DL = MG. ad. Y MG   ad-MG 4 n MG: r. MG = tin
jam MG + nn = FM et fiet DI = 4Y aD FM= nN +n MG: hi  et p
jam seu D1 = ad ln ee pp erena tarrumque. rte cp vel si MG
vocemus m. fiet: DL FM vocemus m, fiet et MG, g. fiet: DL = ab po-g
d
jam alibi invenimus esse DL = ad Seepp - ggrr: hrp ergo scribendo eepp. qur-H
aao
fiet DL = ab T: hrp. et duos valores aequeendo fiet ng gg: mn .: L. hp 
vel nghp-gyhp= umtt. debent autem aHolli ad quadratum et L
ad quadrato quadratum, ut totlatur irrationalitas, et fiet:
nnpp= m (E+ mmtt ggtip + g thpp. Et rursus compendii causa
runsus scribatur nt compendii causa mee pp- ggrr: hp vocetur f
et fiet ng-g9 = t debet  ita attolli ut 4 assurgat  ad quadratum, et f
ad quadrato quadratum, tg fiet ng = t + gr. seu ng = V + 169+
6.
4
eda
est1aa
c
rursus nngg -  - st = 2 tg. et fiet (4 g = n g - 2nnggt 2 nng+ l( 2tg14
al
2V4
seu 2Fg+  2n g t +2 = ntq + ++ g
C2
est n seu 2 i cerp qqrr. lip. arrun-hhee +. hhpp: 167)
a
a
r t ceti 
2n. 4Yn - E C +h p: 4Y. m ep - qr. ti p 
n arru ee +hhpp: 167+
P = r- n su N = r- p
X +ist = V. X = v - Y, v+ y = 2m V- y = 2ln fiet X = dnn
nam v- tm + lm,: 2 et y - lm - tm :2. sit l= 2 et m= 1
fiet  2 et v = 4 et y et v = lm + In et y -lm - ln:ll
sit  l= 1 m = 2. n = 1 fiet v = 45 et y = 3 et X = 4
sit l= 2 m = 2 n = 1 fiet : v = 10 y = 6, et x = 8. itaque si caeleris
manentibus variatur l, manent triangulum rectangulum idem specie lant
Ergo semper ponamus l = 1. et nunc sit m = 3 et n = 1 fiet,
a V = 10 y = 8  X = 6 nam 15 +36 quod rugus simile priori
sed si sit l =01 m = 3 n = 2. fiet y =5 x = 12 v = 13
23

## Folio 5v — 00068160:0010
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 4r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068160/jpgs/default/00000010.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 107 lines, status machine
Y+2Y - 423r-9
r- 2r +g = q- V - p
X44 5, c. 3
eg. 7. 
fi et 6= 10. 15. 7 - 27.
 1+ 10. H1. e ex
aeq. 10 fit .. 6=
12. 131.7. 7, : 12. 13-1
ee pp) - 4qrr = (eCpP) - 2 CpV + v. fiet ep- vvr. ggrr: 2V
seu ep = v: 2 + ggrr: 2D sit Yv par, seu = 2d siet
sit Gr av seu v- gr: 2. seu co qr s+ pe pares
et 2 seu 4co =  4. fiet co=4: 4 sumamus ergo a = 4 fiet
pan
(0 =1. et v = 2 t et fiet ggrr: 2V = 44 , 6rr. 2. 2 = 4r2
et fiet ep sit e =1 et fet p = 1 + Fr ar? Ergo cepp- ggrr
1+8r + 16r - _16Y = - 8r + 16Y cujus radix erta - 1
Opp- rr et respp - ggrr
et see- pp exprimenda in numeris.
et 9q ee
quod fiet si
horum factores sint quadrati  se  aut quadrati ad idem
latus applicati p+r =  42
e+ p = 6. 7. C- p = 6. 7.
ex + qu = 4. 5 ep - qr = 4. G
etq q+e = 89. et q - e = 8f. 9. Ergo 9 = 8.9 +5. 9: 2
si contenti simus aequatioibus
usque ad 10
et ex
f
et C = 8.9 - 829,: () et e = 6.7 +63752) et p = 6.7 = 6. 7,: (39)
et p = 2. 3 + 2239: () et
(1)
n= 2. 3 - 22. 3: (2). Ergo fiet
 ob e fiet 8.9. 9- 8(.2 =
6. 7. 7+6(.2), et ob p fit 6. 5. p b:
2. 3. 7+2(2) e et ep fiet 627 1- 62 (4) 
2
fil 6. 7, 7- 6(. P2
quos duos valores aequando et 9r = 8. 9, 2.3. 9+B. 7- 2, dui paulo ante eunt ex = 45. 57 4. )
connescnt 7, et fit
10, 11  1. 12. 13.-1=
et gr = 4. 5. 5-4: (2) ergo rursus habemus: 627. 7 - 6 (2)4. 5. 514
5
et 8. 9. 2.3. 9+8. 3-2 = (2) 4. 5. 5.4 tantum ergo efficiendo
(6
si jam aequationem sunt aequationes 1, 2, 3,4. proaeq. 1. fit 8 = 10.6, et 9= H,7, fiet
12. 13, - 1. 1. 410.11
divellamus in duas,
10. 72 =1 L0, et
12. 1  = 0. fa
10  V: 11
et 12= ri
pondtur cpu e
ponatur 10 = o et
12-e2. fll
20
Are
20. 11 =  12 13-1 =
= 22. 1-. 1+20. 11
sit 26. 11 +1 = 
19. CB 52.k-
et
ex aeq. 1. 10. 11. 1B11.7 - 10.6 =7+6 rursus ex (2) (it 2 = 12. 6. et 5- 13.)
()
1928. 11 - 1, 127. 13 - 1 =
se
(2s
12
fiet. 12. 13. 13.7 - 12.6 = 7 -6 ob aeq. 3 fit 4 = 14, 62, et 5= 15.2
12
fiet. Ob aeq. 3 sit (es 4( 14 6os. 7 6 et 5 sit 15, 6. 7. 46,
e
et fec en aqa fiet: sit 4 14.6 et 5 = 15.  p. siet
-62 (2). 14. 15. 146+ 15. 72
i
postremo ob 0g. E. fili
14)
10. 6. 11. 7. 12. b. 13.7. 10.6+ 11. 7. 13.7-12.6=(2) 14, 62,. 15. . 15,-146
Superest ergo tantum ut aequationes
13 14 in numeris grantur
sed maxime aeq. 1 (3. ubi fiet _6
1 4157: 1H)14
debet esse v 14. 15= v. 1+ (2)14. 15 sit 14 = 164 2b et 15- 22
siet , 16. 17 16(2) 17. 22 - 16. 22 - 21, 17 = 21 (2)17. 22 - 21 28119
ete
= V + ()  57. 16+(2) 16. (21 1 et + 22. 162+22(2) 16. 21427. 21
2 v+ 2) st 4. 21 posito jam B=
fit e 22  () 24 sed non debet esse v= i difficultas autem
et f fiat, 21. 22
est priori similis, sem aeq. 15. ipsi aequationi 17. sit re eamus ergo
ad aeq pp vel 16  In aeg. 16 faciamus V = 1+ 16. fiet
(1-14. 15 1 1 (1) 16 4 (4) 1B (1 + (2) N. 15 + (2) 16+ 167, 1 + (2) 14. 15
sit si sit 14. ise 17 14(g. 19.1 sit 1+ (4 15 17. 14. K
fieteec cujus aequationis ope hibetur 15. si jam hunc valore
p(2)42 15 = h. 15 Sit  L Sit f1 1bq
sit (2)16 + 16= 17. 15 tollitur tera 16 ex ag. 18. et simul via
assumtum sit
in ea dividi possunt per 15 et fit 14. 15= (2) 14. + + (2) 1 1. 17.
e
quae aequatio dat f15 sed solverta adhaque aeg.  19. id est devet
17. 15 + t aquadi quadralo = (4∞ 14. (2) IY. 18 + 1718,
hec in duas, seu ponatur
et fit 15 = (2). _ 18 + 17. 18 (2) 16 + 16 =d seu 16 = -2. sed hot non ot H 
v
Dividatur aequati
in Numeris
~22. 13 + 1, _ +20. 11