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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 439r, page id 00068196:0873
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=873
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# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 101 lines, status machine
# Lines: 101 recognised, mean confidence 0.82
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XLB L M LN  BL
uXEC
CGBC CG 2 BCdB ex data cycloidis dimensione
(BAD F BF NB probandum  FIGF
NBeD EA CRyp Cin. CAFG, circ - F iG
F1H CG- AF Ergo FIBACG. A0 EFBB DL DFHE
(E) DFiAO
er rectangulum FIGYB CGBAC - AD, AC
Eogo FVGFB F1GY - F1GF B CG, AC - AD, AC, = BL DF NB
Ego et CAiG ( CG, AC) = F1GF (N BL D F 1) + BLDF NB
+ BAPB
ALN arc BL. DF p an BLd. C PS  arc BLP
LN+ psam BL parcBLP. et arc BLD) arc L PC.
ergo LN + BS  arc BL B + arc ZYC F,  semicircumf.
seu BLc CG. Er idemque fit ubique ergo
Omnest N. exc. seu  Trilin. BDCGFB(d A= ACG
seu circulo. Et haec omnium simplicissima est cycloeidem
metiendi via
porro LMFi KL. et Pq P. etc.
NX  CG = KL - LN
p- (13) CC
exiaetius Ergod PSf NX + X,  Bklix urtu 7gr 13
Ergo NX fiS.
et circum ferention
XL f Ps- pQF  Q5 entius
Eodem moto MN Zs7.
Et NXGN, (20) B BQS B.
AB7 r.
CG - AD DF ergo
CP5-
(22
sit C peripheria sit 1, aBD CG
c= r a . Ergo E1 - r. Sx FI GF E per 1809.
GIF p  BMEFNB per 20.
ACGI ( . per 21, 22.
ACgI = FiGSF + B ME F NB f NC ACGI per 18. Ergo per 30 e 2g. enit ACGHS ACKFEL
Ergo pr 30. 29. erit ACGSF + BMEFNB F r. seinci BE DBFYC per 21. 22.
apegente circhimedis theoremate. jam BCD
2BCDB Ergo per 32. cur
(3
BDCEB. per 19. Ergo esut
BCEBF rCo Ergo per 31. erit BCEB, ACGSF+B EF
utrique communt AEZC si utrinque auferatur, fier B AEMBF ECGSF + BMEFKB. seu
BEG) B. sed hoc jam notum
si a circuer duplicato BCGH auberas circulum ACGi, quadrantem BADLB; et B H IFB
restabit BD F NB.. jam BFY BH) IFB F GHABFG = GIF
e Rursi si a circdupl. seu a BCGH. cufers
e
BMEFNS, BDEMB sen) BA DLB + BLd FN
BD si acirneo supticato BH auferas BAIH, auferas
Ergo Circ - quadrans smilte +G1FSG,
simicire.
IBM KAB N BIDFAB (B LBEMOsu, quod
Jungantur du recta BVF et BD. Triangulim BDF erit quadrans ergop DABL d. anferatr
communis atrique, BVLD. Ergo seguentum Trianus. BAD ips  DFVLD = seum BVB
superficies hemisphaerii est aequalis superficiei cylindri ejusdem altitudinis et basis adeoque semicirculo
Junus aus
e Si a Cycloidi BFG, cujus vertem B. basis B
ur
si semi cycleidi occurrar recta per centrum sirculi generatoris transiens basi
paralleta. j  segmentum recta punctum occursus et verticem
in generulons
sit semicyclodides BFG. cujus vertex B. altitudo BC. basis CG. centrum carculiq odi
meratoris Aper quod transcens recta AF basi cGparalleta, occurrat curvae in
segmentum  figmae cycloctdatis BF N B. aequari triangulo BAD
TrianguloB AD semel seu seriiquadrato radi quod ita demonstro. Jungatur. DF, ae quatur quadranti a incumferentia
erg triangulum ABF cujus basis arc quadrans circum ferentiae, altitudo AB, radius
ipsi AD= B DLB quadranti circuli. porro Retorta BDF B aequaturdu
qaoitur
qulediaso
siu radii, ut a scriptoribus de Cicloet de patm epreso nnutur u ostenditur ut  Wallisio et
ponorato Fabzi, ex aliis ut de Honvillaeo facile duci potest His postis facile reliqua absolvema
Triangit BDF escosiB ADL BB+ BF HB  _G segment cycl. BF NB  aequa B segniento
quadam circut BD. LB a (Retort. B DFB seu) quadr. Trianqu BDF + seymento cyc lo 18 B F NB) aequatur
B segmento Circuli BDLD + relortae, BDFNB. Nam ut alterus it BDFNB. pro Triangulo
13DF substituatur ejus valor, quad cans BAD LB, vel rursus pros quadrante, ejus vator, Triams. AHD + Segm. circ. BD
denique pro retor quoque substituatur ejus valor quadratum radii seu bis  bis Triangulum BAD. et loco prioris aequacitas
semento crculirob a bis tua o is du
segmento cruli BdLB + Cretortie BL BF AB cu) vis Triangulo BAD, ablato unobiae egnidtanto ieu
ut 
Bri
padatisteu elides tium
jam 
F1
(23)t
ergo FE B a
ACGHF
Y
sit jam Racdius seu
i
BC
EB)
BAEMB
Trapez CAFG) ra-  Trilin. IGFS
genhatori
restabit segaculum
cyllordalesue FAB
aequale
AE.
gorat
407