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# Piece: https://leibnizlegible.com/api/records/34605/text
# Catalogue record: 34605 — Linea Berthetiana
# Incipit: De curvis protensis. Linea quam R.P. Berthet imaginatus est. Centro A
# Date: 1675-11-03
# Akademie-Ausgabe: AA VII,5 N. 045
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=34605&katnr=42000
# AA VII,5: the volume in the Repositorium des Leibniz-Archivs (CC BY-NC 4.0): https://www.gwlb.de/leibniz/digitale-ressourcen/repositorium-des-leibniz-archivs/laa-bd-vii-5
# In the Akademie-Ausgabe: AA VII,5, p. 9 (PDF, Repositorium des Leibniz-Archivs, CC BY-NC 4.0): https://www.gwlb.de/fileadmin/Leibniz/repositorium-des-leibniz-archivs/VII5B.pdf#page=9
# Work: https://leibnizlegible.com/work/00068124
# Title: Leibniz-Handschriften zur Mathematik LH 35, 8, 30
# Folios: Bl. 12r (from the shelfmark LH 35, 8, 30 Bl. 12r°): 1 page, canvas 23, page ids 00068124:0023 to 00068124:0023
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068124&page=23
# Lines: 116 recognised on 1 of 1 pages, mean confidence 0.88
# 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.
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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 12r — 00068124:0023
# Source image: https://digitale-sammlungen.gwlb.de/content/00068124/jpgs/default/00000023.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 116 lines, status machine
nec
3. Nodemt. C75
Berthetiana
Quadratura spatii.
gale Tria
mpnit
AD crescere
a+
proptit
intula
ADF spatim compleres
secta ergo curva arus circuli
aequales patet
Rectas crescere aridhm
na
inthmelice
ny
sunt a+ ZB
est: a+L
sive a  2a2 + ZB. summa omnium 
et VCem AD
est
q. Summ. omnium 2
sive simima omniu
Denique summa omnium
Had. diane ham
Duplum ergo suatii AB(D
est rectangulum (D) (C) A
A(D) (C) ductum alio
+ a2
a
rectanguto quad
ustiur
id est versus B
sit ad ipsum
ut quadratum a
ab ipsa e
est ad triplu
rectangulam ) ACC)
ve _ aZ + Z + Z posita Zn (et V
er an RC)
BIBLIOEH
HIAAEOVKBME
De Curvis protensis
Linea quam R. P. Berther imaginatus est; ceta
centro A. radio AB. i circulus BCB. producto radio AC. producto
adjiciatur recta CD  auni BC et ducatur curva quae transent
per omnia puncta D. designetur. Hujus cum primum
visa mihies haec curra satis elegans ac digna in qua methodo simeas
experirer.
Ac primum quaeramus ejus langentem in A(CYD sit A(C) F ACD. jungantum
DF, C(C) D(D). (Et (D) D proeducta occuriar diumetro AB in G
Ertur
 quaeritur recta AG. Habemus trim gula sinulin (D) GA, (D) FD.
Ergo ilem DFA, et C(() A. Ergo DF   a F Z Bat  BC. Ergo
(C)AB
DFH a + Z β. ex (D) FNB Ergo () D SB 
et A(D) Z + B
(0) G
Ergo DG
(0) D
(D)FAB et BGpB ZB. B Zar
(D) D 1)
Sa+B aque
Itaque per punctum itaque li punctum curvae eD
Z+aFBAII ducatur tecta anus camti genentioris tangeri
Ct parallela in qua summur HF aequalius radico circuli, jungatur
AH fintque DG ad DA, ut HF ad HA, erit DG tangeri
Antro D radio DG descriptus circulus secabit dametrum productam AB in
G. erit G juncta DG tangens quaesita. Sed bretiior longe atque
pulitrior constructio haec est. I producta Ercus geneutoris ton-
gente  in (C.) sumatur (C) L quae sit ad (GD) arcum ut DF est
ad F(D) id est ut D F est ad C(C), id est ut ut A sumua
arcu
nen gens adii est ad ratium, constructio ergo hic redit, in tangente
d) Errculi producta in eam partem a qua incisit  curiae generatio
sumatur (C) L quae sit ad crcum exlensum cF, ut summm exem
A Cmmma cucus extensi et radii est ad radiam CA. juncta (EL
est langens quaesita. ob Tringula similia (D) F D (D) CCZL
Si curva fit alia ab BC. non sit arcucirculi, eadem
et crcus sumatur in curvae, perpendiculari prdncta Eadem
locum habebit constrncti no 
sed punctum A. non erit constans
sin punctum A sit constans, et arcus sumatur simplicita
in recta AC. productu, qualiscunque illa sit, tunc loco langentis
a punc
t
der debent Narisi
cularem ad ipsam AC. in A. nn sun 
erta
A f
fixum
Gdescriptio hujus modi currau uno tractu
sit flum curvae circumplicatam, et in regula mobili
circa centrum A, ad verticillum sine orbiculum raeniens
quem ambit, et inde retlexum in ipsam AC, rongitudinem exten
ditur et in quoddam mobile desmit, quod semper in repta
Ac progredi conatur, Mota ergo regula cuca centrum A
ebet verti si veridicullus esset immobilis et AC non semper
eadem (quod in solo circulo contingius) ali describeretur curva, et
distantia versicilli a curva pratensioni arcus decederet, quas curvus elia
calculone io considerare licet. Sid vero vertiallus ita sit mobilis
susquedeque in recta AC. ut semper sequatur curvam materialem
ia tenteu in esta et detibetu ciriva desubrum
non sunt similia
quia angulus (BFB
non est rectus. at
inguies. parallelus
est cangenti arque
nento
et, qui ademdum
non ea nosmolis
ad hurc radium, ACC).
nendo Z. aricmatice AC. F
ipsae autem DF
E