# Leibniz Legible — https://leibnizlegible.com
# Piece: https://leibnizlegible.com/api/records/42840/text
# Catalogue record: 42840 — [Math. Aufz.]
# Incipit: AE aeqv a AB,x. BC,y
# Date: 1682
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=42840&katnr=39662
# Work: https://leibnizlegible.com/work/00068195
# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 2
# Folios: Bl. 133r–134 (from the shelfmark LH 35, 12, 2 Bl. 133r°-134 (39663)): 4 pages, canvases 265–268, page ids 00068195:0265 to 00068195:0268
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068195&page=265
# Lines: 217 recognised on 3 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 133r — 00068195:0265
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 134v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000265.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 65 lines, status machine
aa + Xx aequ. yy.
XX XXF 1X (. B aep V2 2ay B
y+ 9
360600
173
77
Ergo + X ax aequ y dy (et rursus dy aequ. + fxpx
et iterimus aa + ra qu. g. seu ag aequ. aadly exader 2fclys
37
Var +x
TaX AE aequ a. AB, X. BC,y
yy aego 2ax + x + da e
I dy a
y sit aequi e+ fX + g XX + h. X + K X . du aequ. f + 29X + 3hXX + 4 KX eto
ydy aequ. fe +HX + fgXX + fh X + fK X ec
+ge + 2X +2ggX+ rgh X +  K X
aequ. +3AEX +fGC +  + 3th X + 3h K X
+aFex +AKfX +GKGX +AKhX + AKK X
fe aequ. a. seu faqu. a. et aa + 290 aequ. 4 1. seu 9 aequ 4 e 28
A. a7.
a
et
212
rce
2C
13 aec +2a K aequ. 4. 3. a e + a + 2 e 4ceaaFa
4
205
16
seu firt. K aequ. 2. A +  a + 3. 4 a C f 3 aaee + 3. 4. a0+ 
206
AC. CB. Bd aequales si AF FEd ciculum langant
erit AF aequ. AE
Traxrqua
sit AG aequ. X.  AC aequ. a. sit: AG AG. ravegua
x aqu
fier AE aequ S2ax, et GEaequ Trax - XX
jam AF:  AD:: EG: GD. seu AF aequ. AD. EG
G7
Ergo AF aequ. 3a Vrax-. Accedente jam
3a = X
tangentum proprietate: est DG: GE:: GE: + X + a seu 
seu DG aequ. _ 2 aX - XX. Et AF aequ 3a, +X - a
X = a
srapux
equando duos ipsius AF valores, fiet: rax (=XX ee qu 3 a X - J aa fXY
et 2a Xaqu3aa. Et X aeqo 3 a quod verum est
+a x
Tentemus in idem accommodari possit ad Hyperbolam
F AFaqu AD. EG aequ 3a Y 2ax + XX
Gd
.3aH +X
ya xta
aequ
sract
Ergo (aXtXXeqo (3 a X + 3 a a (F XD + ax. seu X aequ 2 a
se
L
EGaequara + 9 reqo E L2 ga AErgo Zaa + 2L aa
se AE aequ : a (30; et ACaqu. 43a t21 , 3a + 2a
seu AF aequ. a sed inde oritur impossibile, fit
Cum EGaeqe a Y23a +gaa sunde
patet nullam ex d posse duci tangenter ad Hypertolam, quod
est et verum est
X

## Folio 133v — 00068195:0266
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 134r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000266.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 86 lines, status machine
Logarithmus rationis b +n ad b est sit 
 ete
2b 3b
n aequ l + V + 13 + 14 etc
12 b 1. 23b 1. 21. 4b3
Logarithmus rationis B m ad b sit aeq
Logarithmm b = m ad b sit aequ. A
erit
 aeq. m + m + in + m etc
m aequ. A - N + N - 1 erc
2 1.1.3 123. 4
Quaeratur curva cujus area sit l+ B + V ete
14
1. 1. 2. 3 1. 2. 3. 4.
A. 2 123. 4 F2. 3. 4. 5. 6 etc.
area sit 1 + 1+
At ququ a + bl +Cb + d l + a l etc.
dq +q. b +rcl + 3dl + 2013 etc
ACaequ Y, ABaequ A.
AE aeg. n
BD aeq. 1r. a VA BdVay. S d
ron
EF aeq Vr et VAEFrS da
rn
Si Y +n sint progressionis Geometricae, Sre du erunt propossiGeom
rtn
ut logatittui
Nam si ra progass. Geom. erit de siedu purge Arithm. Eit  in aeque ac
rin
quae co ponitur contans ergo dn aequ r+n co. seu eist da ipsi r+n propor–
tionalis, ergo progress. Geom.
sit co aeu. d. quaeritur relatio inter
et sit naeqo a + bl + cl + 1 + f12 etc. eut du
erit du oy
rrsus du y.
rursus duaeque bAn aeque
ece
YL M
Si AB sint pro pessionis iui geomelticae seu numeri,
ut Nuin eri
posito raq. 1.
x. Ergo eut
b +2c1+ 3el +4f13 etc. dl) quos duos ipsius du vulores compa
+1. + bl + cP + e13 etc . dis
+a
fiel ergo naeq. e l + 1, + 13
rando, fiet: et ponendo
a a9.0.
frt bag. P. et. Caequai et eaequt et sayt, 19q
17
123125 et. posito laeg. Ser aa
Ainc AB aq. et, si AB sit n, R YABdV, E, eut naeq. l + 1 ∞ B etc
1.23
AE sit n et VAEFIAL, sit n aeg. l - V + 13 e
1.2
+n
1. 25
Sit AGaeq. BD. seu GHaeqe CB erit VABdY pu VAGHV, quod sit l, et AB, n, et AG 9.
e
3
2 1.23
14
 et 9 aeg. l - 12 + 13 - 14
 seu n + 9 aeq. L + 13 + 15 et.
1.2.3 1. 2.3. 4.5
7.2 12.3 1. 2. 3. +
et n- 9aeq. l+ 14 + 16 de
Sed quidi si n.g sit quantitas egativa orietur absurdum.
1.2 I.2.3. V 1. 2.3. Qs 
Reassumto ergo calculo faciamus: frran aeq. 4 l ita scilicet, ut si + significet-
rin
unc et logarit huu ponatur quantilas negaliva fiet F aeqn I
t fict. dn vu b + 2 + 3et + 4fPer dl
annu In  aeg. 1 F bl + cl + e B et al, 7 et fier brg. 1 cag ea
+ a1
Ergo n aequ. I + V + 1 etc sive sit  sine
1. 2.3
soe sit l’aego san sive ago san quod est absurdem neque enim haec
+1
1- n
duo possunt coincidere. Itaque puor methodus bona mnebit nisi
quod non habet locum quando, q majod quam m, nisi forte velum
scribere esse:
ry l+1 seu
1. 2 1. 23.4
et alia cujus

## Folio 134r — 00068195:0267
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 133v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000267.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 66 lines, status machine
N + G vocrmus b, et n - q vocemus C
arit b+ cl ae lb + hc aeq hb + 5 aeg. hb + et
erit hb + el aeque hl + h13+ 4 hI
1 +
+ h1
1. 2.3 1. 2.3.4. 5 1. 2 C)
12
+
1. 4
+
1.. 6
et facre ado h aeq. - (3 fiet  - 3b + c l aeq. - 2l  + et
et omissis reliquis faciamus: _taeq 3b + Cl∞9. El frit laeq 3 b
2 +C
x
est autem Bqu  bg bag. BGBA, et 9 aequ.  posito esse
AK aeq. AtG et CG09. Bd. Ergo fiet tru
Est autem Ray AG AB, et 4aeq. AG posito CGaego Bd..
baeque HX B A + AG1 seu BG seu BK: et si sumatur A (in AB) AM vy. AG
fiit fier BM aeq. n. 4 cut dimidiur n - 4 erit AN, est autem CA, 1. erit rg
erit ergo n-4+1, (s CN) aq. e seu CF c eut CN. et fiet
laeque 3BK.nin produ
2+eNV
Constructio in producta Ac sumatur CZ dupla ipsius CA et
erit QA, 3 et QNverit 2 + C. sumatur NZ paralleta AV aegits si
pragatur Zp secans A H ver
cujus reci
aequalis ipsi BK
alita
) Buviter: CB est Namerus equaeritur AL Log aribhmus sumatur G aequ Bd, et
3t et AN dimidium differentiae inter AB et AG et Np. parilleta Bd aequalis dimidiae BG
3) et  CZ dipla CA juncta QP, secubit AV  (parilletam Bd) in LC
quodsi velimus explicare difertius laeq.
qaqui + l = 1 + N
+n
b, 36
2tC
1et K
bagu ntqe Gta
n + q aeu 2n + nn 
+ n
et baeque 2n + nn  n = 4 aeg. n - n aequ n en -n aeq un, et n -9
in
e
I+n
2+ 2H 1
1+n
1+n
et Caequo un r + 2 + 2n, et 12 + errgo nn + 3n +3 et fiet:
laequ
 1A t 20
2 + 2n
6n + 3 un, son  aeu = 2+ 32
nn +3n+ 3
n
un +1 + 2+3n
Alier l aego n = n + nd et El 4.  1 + n ay 1 + 2A
A sy. n + n + nt
(+  aeq. n + n +ns
l.Aag n4 nt +n5
L. e 2+ l aeq. log 2+ 2n aequ ly 1 +en, ut D est  loy ian
ergo 2 +l aequ L Ergo 24l fit ex 1+2n ut l’ex n
seu 2+ laeq. 1+ 2n - 1gnim 4 1+6ntA+SN ea
seu 2 + laqe 12i1i eu A 22- 2 + sn  = 8 1470
na
2 722

## Folio 134v — 00068195:0268
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 133r); this page is its left half.
# No recognised text on this page (recognized).