# Leibniz Legible — https://leibnizlegible.com
# Piece: https://leibnizlegible.com/api/records/35242/text
# Catalogue record: 35242 — Note
# Incipit: Les inventions belles et ingenieuses
# Date: 1675-10-19
# Akademie-Ausgabe: AA VII (assigned, not yet published)
# Other printings: Hofmann, Über frühe mathematische Studien, 1970, S. 104 (Studia Leibnitiana Bd. 2.1970, 2, S. 81-114)
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=35242&katnr=Cc2+1077
# Work: https://leibnizlegible.com/work/00068124
# Title: Leibniz-Handschriften zur Mathematik LH 35, 8, 30
# Folios: Bl. 123 (from the shelfmark LH 35, 8, 30 Bl. 123 (=Cc2_1076)): 2 pages, canvases 247–248, page ids 00068124:0247 to 00068124:0248
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068124&page=247
# Lines: 182 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.
# 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 123r — 00068124:0247
# Source image: https://digitale-sammlungen.gwlb.de/content/00068124/jpgs/default/00000247.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 70 lines, status machine
quoq quoyqe
imitiles ne douvent pas estre moins
estimées par cequ’elles sont des aaccest des exeples de l’ait d’inventerj donnent quelque
les inventions
belle et ingeniense,
heplematu
Ouadraturae peooreminai tetragonstita
nonnelle veule pour pousser plus’avant cet art. a
gocto
er
teneratia ex tatigentibus
A centrum Hyperbolae AC. abscissae ex Asymploto CBordinata
BD langens occunens alteri Asyriptolo in D. AD sit BCE recta
ita ut CE HAD. idque ubixis factum intelligatur. Curva FGH
transire intelligetur, per illa puncta quae etro,  cui etio
DA et AC protuctor Asimmplosiciunt) Sputium e respondens
Eit exempli gratia spatium mixhilineum ECLGE, aequale
trilined AABH BAHB. ut exatibi Creoremate meo generali
gstat. Jam quaeramus calculo AD vel CE. ducta DBM, erio AC M
aa
AD,2CB. sit ACGX. erit CB+ = et EC
X
Hinc demonstratuur illud sant pulcherrimum quod sector Hyperbolicus ABH, equetur duplicatus
aequetur ipsi ECLGE, id est duplo BAHB, id est quod sector Hyperbolicus aequetur portioni respon
denti adi ymptoloque. Ducatur Axis APg. vertex g. axis transversus ARS. sit DR.
adem nonnalis, ob angulum RAD semirectom erit AB
AR HRB
A.
1 S7
qurbolae
2a
X
PL
4a
2
 AM 2X
AR + ROV
 et RD AR, et AR +412 Eode modo ST, plrgendicularis
AD + DT. ergo SY (2a + DT. jam 5+
∞ ad a. ergo 5+
et
quadrilinem
dupla AC F sive AC CM. ex natura Hyperbolia aquilaterae
ergo
2
et Asp n
a
ad A
)
se 24X+2 quae etiam pendet.
AI.
24)
X2 - qu
X -AX
ideo ex quadratum
quod tamen et atunde demonstrai potest, quia si addatur, Hyperbola commensurabilis,
st 2atira, fier 2ax 12 + 2a Si
no video erorem merm, non X seu
Y
ADH 2a H = seu in duplicata ratione
a
DTH 2a + DT. Ergo _ X3DT - a XDT (, 2at. et DTp
2atX.
+ais
2
quae
et dicidendo per Xta, fier; 2ac, ad Hyperbolam
sede AC. sed Ay. sumenda est
X  a
o a. pu pd11)

## Folio 123v — 00068124:0248
# Source image: https://digitale-sammlungen.gwlb.de/content/00068124/jpgs/default/00000248.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 112 lines, status machine
s
(L Hincatiud notabile. si et) sit curva, intervalla
4
tangentium ab ordinatis in axe BF
applicentur axi in BL, (B) (L). et in punct
L, (L) ipsae BC, ipsis BL norinaliter
applicentur, in LM, (2 (M). erit curva
spatium LM(M) (L) L spatii C7
iplum
ABX BC+ y H a EA AE, YAB. EF BBC
erit C
ergu XF Z. et EF
aesi
g. 
nempe
ABCD.
+Y
quadrilineo CE Mc duplicato unde VCBE+
per puncta B. C. F disranseunt rectae Lndefinitae, BL, CM. FC
l
e
s
 DGK, transferatur
FA ad X9. erit spatium DGKgDO duplum spatii
BL ad DO, CM ad GD.
casus, quo BCF, est lince recta est illo qui tat therrema quod antea consideraveram
BDGK 1 FC B.. Cujus Vheorematis
Alter casus fingipotes, simplicessimus quo fingitur DGK linea recta qualis RS.
D, ubi cuurva BCF cum langentes BD, CG
DGK repraesertanti in punctis DGX. et transtatis
gratium ODKGD
ralii BLK
et
X
sit in eon rem
X occurunt rectae RS curvam
BL, in DO, CM in GP, parallelismo servato
CB duplum. nec refert quem angulam adeas faciatrecta
Rs. em Nec refert sitne curon con BCF concava an convexa, vide figs.
ide plura
2au octobi
1
Hsperbola memorabile quadrilineuum CBN MC. aequam hic
sit alia curva DGK, cu in punctis DG,K cin in punctis D. GX langenses prioris
tam
ut DL, CpM, KN sint lungentes curvae D
alia figura
ert
Ep 2X
2a unde illud ruifin
revrema illud intum memorabile, jam video non nisi Casum esse uncum aa simplicissimum generalioris, insprcefiguram 
 sit curva _ BCFI, cujus laterm infivite parda BC, CF, Li S
ECFG, HFIK occurrant
. ubi et
illudamolatur, si ordigatae acuva BCF, ad recto RS ut FY quae sunt
scitficet aud angulos, applicebitur ei dem ad angulos
langentes currae B
in
W. erit spatim 
t
ABB KFCB duplui
a.
et
ubi
sH jt
rectos, sed ibi ubt curma
RSVW
ipsius 
.Quae mita alis thcorempla f sunt et
primm in figs. 4. Cujon BCd. longentes BDCH, B
APE K9
spatium
qavallelae et aequales DoB
ODKQ PO duplum spatii K.
C1
et Do
LB, MC, FN. er cura
FK I
BCF
sit vigs. ABCF quadrens circuli cujus centrum R. N
erit DLD ad LB, ut LB ad AX. sive LD+ a prti NL erit NDd
ubu
et La
D y+xr
que
a
q2
x
= X  v.
NB tonstutae ad LH sunt ad Hypertotum
simplicissim) et ex stocharnctercistito longentes insistentes arcubus, aequantur portioni iuperbolicie seu sectralibus
675 C ad axci. sed hoc obler. jam et perum
arl
XB.
et. DOH Sae-ad
Commensurabailis sive. a+
patia angulisivumo genea sunt
s
si =b, transferitur ad D0, eum sit  a  et ZHNd a ent
s
a TI a5Z- a, et currucum aequatio: 72C
L r
alaco H 2 sive h
Sa co
e
st
2y2
4.
taZ -a, euit harculo
7. perdet ex inculi quadrotura et et illa ipsa cujus
2 videl elissil 24. octobe
.