# Leibniz Legible — https://leibnizlegible.com
# Piece: https://leibnizlegible.com/api/records/5497/text
# Catalogue record: 5497 — [Aufz. zur Analysis transcendens]
# Incipit: Inveni alibi Theorema seqvens magni momenti ad Analysin Transcendentem
# Date: 1691
# Akademie-Ausgabe: AA VII (assigned, not yet published)
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=5497&katnr=39705
# Work: https://leibnizlegible.com/work/00068195
# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 2
# Folios: Bl. 202–203 (from the shelfmark LH 35, 12, 2 Bl. 202-203): 4 pages, canvases 403–406, page ids 00068195:0403 to 00068195:0406
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068195&page=403
# Lines: 385 recognised on 4 of 4 pages, mean confidence 0.76
# 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 202r — 00068195:0403
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 203v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000403.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 112 lines, status machine
ec.
le quei 
∞( )cc + () EC -1 + (c0) () n = Y9 2 -1
inter
Dic + Y1 1 +  (1)1 = f 4)
u complementi
AHhuius cb 0 ad 1
aniu patet enim
Ambitus polus
aterum quia mensum dimensio aircus
Anum complementi
seuplicuri ambitus go feri ex chordis partia ectu
On multeplicati densivee sectionum Crco multiplicatis.
per numeri
junicu qvene
sut ergo polyquai in arcu AD uscripti ambitus ex arcd ) selis uusque ad AF secto erit:
Co. 1C. quem ambitum rocabimus Ip. et cn+ C erit B. pro
et denique co 30 erit 2p.
1 p aequ COn. 1c
2C aequ 2C + 25
2e + 2f aequ 1c
(2
Aequationes exurgent talis
2+c
P aequ C0. 2c
3C aequ l 3C + 35
re2f aequ. 1.
d q 3p ego Cd 92C + 2 fli c
(C0t n+ 2 1:co
3p aequ C0 t 3C
(8
equ ex aequ 3. 5. 8. 9.
jam ex aqu. 6 et 7. invendemus 2e et 2F 2p aequ Co
et ip aequ con 2c + 5 a
1:C0
3e equ 2e
(9)
E qur
3faequ2f
ret 25 ex 2544
ex aequ7 est 2f aequ 1: 10. quod substituendo in 6u rationibus 10. 11. 12 fis
3P aequ Co 2 Cl:c
(1
ne
nico
2C
110)
cot rC + 1.28
et 1 p 2e9. c, 2e + 1: 2e
seu   ed pro resribamus v. siet:
2:0
11
aque   N equ. s seu re - p e l + 1 eq. 0.
C
12
ce
(17
28 - 28 28 + 1 eq0
(19)
vels a aequ. 3
(20)
1: c0
e re = 1P2C + 1 aegu. 0
ait
20o
Co
.2e: re 20 +1
coa
a
 (18)
peris C0 aq 2F 2,: re + 1
eit n ad 
a
verum si ita jungamus aquationet 19. 20. 2 . svenitur ad multipticiter
affectas cum per . 17. 18. facildus veniri indeatur ad paras.
c
Y- 
ue
Ergo ex oqu 1b. est de aego )
Ld ex 20 et 21 tiet
ex 20 et 18 fiet:
2oco
20 4
2C
(2n44 + 1
((1)) 2
 aequ
2C + 1
1 laqu 2P
P
1p. C0 2Co1
10 -
quarum duarum aqua
debet 2C
3p. C0c0. 2
quod si ob implicationem potentiarum sublatio nostra ista non procedit. poterim
2e L,
aberc ad huc duas altas aequationes pro ls. 2p 3p ponendo Ep. 3p. +p.
H
202.to pro cd co prtendo cd. co. et pri 2C.
+1 
2e27 71
3P
ricoed +
d 22  fiet
4p. C02C
et talia quationum paria adhuc multo plum o p
   itu ti eit 
1. c010, -
se)ec 
Metisi 3
tionum ope tolli

## Folio 202v — 00068195:0404
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 203r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000404.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 92 lines, status machine
pro re scubamus v fiet:
id vV + 1. ((2
vo + 1
aequ
2pv
vr +1 aequ.
pve + 1
EPV
ep co. V
 3pCoco V. seu ia
 v’ +V ocetuo M. fiet
Jam y + y aequ dat radicem y aequ:
+ Blico
seu V
+Y
jam y + gy aeq. Y dat y aequ  Y+ S: er + 1, 9 + V1r-
 jam se fit ha si sit ysit v et a sit 1, et Y sit
fiet v aequ co
1 mip + si mnis
11c0
2 ipa
cos
V’ipipase
+ V2
rursus si sit y aequ V, seu YV. et q sit 1 et r sit
fiet
V aequ 
s
(D an lD co. Erus
3s+ mot + S. mait
2
Vhat
+ V
2peoco V + apopco 37 +
3ico  plico aq
3pc quacitur V
Assumantur rp. 3p. 4p.
3c aequ 3e + 3f
4C aq. 3C
AC aequ 4C + 4f.
4f aequ 3C lic0
p
3p.
P+ V
ipipcot
paequ co 2c. 3p aq. Cd se opan  c
3e 13F aequ. 2C 303 fag. 1
Ergo cum omnia data
sC ∞ sint ut ante, nisi quod pro ip. 2p. 3p. C. Ct Co+2 1c. 2c. 3C
ponuntur. 2p. 3p. 4p C cont cont 3 2C 3C 4c
fer pro Cd (at c0, et pro C fiet C0 scilicet extra ex ponentem
at pro V seu 2e scribatur 3C, seu 2co
ergo in aequo C substituendo fiet:
 aequY
 L1:c0, 3c0
2p cocos
1:W0W
+ V
ee
seu pico co
2pcoo
seu  v
+
Ep u V + V aquu 3ic plico 
 lico
ubi praecise redit aeg. D. nisi quod erior alicubi in caleulo deberet
serlicet loco apcoco esse Ep
imma
3pcoo
habernimus vilorem ipsius
ipsius 12 3ico
93c0
eo
na
aequ
ipsius V, sine Ed in exponente. Et atbaego n 
g b.
2 nAY possum in venire ve aequ a-
ergo co aeqo la aequ  uys lo co
1c. Em lx aego la lb.
et lx 10 aq. 1 seu. 1∞ au 16 et 20 aequ clb cgo lalb
glicoco ago. c. per: colv aeq. la lV aequ. 1b. et lV aeque lc eo mrors
coco
V cqu  ipsius v3:coco Y1: coc0 aqui et
1:c0
ita porro, videamus an licebit ne hinc invenire valorem
a
2C
t
3p. cotd
xipcoc2
esp

## Folio 203r — 00068195:0405
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 202v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000405.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 102 lines, status machine
+ v ipcossaegu V +D spcoco
BI.co ssum
ipasqu C 1c
paego co
2C
Cd aego ip: 1C aqu, 2p: corC aequ 3p: cocor 3C
Ap
conc
2F aequ  co2c Ego et ix aequ
3p
coco. 3C
seu 1p: 2p: 3p :: 1C = co 2C1 Coco. 3C
1p aequ cd 2 2 -c
2p aequ. Cd. 2V2 - 2c. Sp aequ et 2 123
ip: 2p: 3p:: Y2- 1c: co. S2- 2c: coco. Y2-3c
ip: Ep: 3p:: 2- 1c: d 2 2c: cot. 2=3C in qua analogia continentur
est seu : F: 3 B: 4p e :: 2 - 1c : 2- EC : 2 -3C, 2-9c
eus aequationes.
2 -1C
ipco
c04
lC aequ _ +rf rer aequ.
2C aequ 2C +2f
Ee + 2f aequ 1c.
2C2f aequ. 1
Se aequre
p. Co
3C aequ 3C+3f
BC aequ
4C aequ
β3C + 3F aeq. 2C
3f aeqo 2fl:c0
2C aequ
1g. y + 1 8 quolis
facis
3C3f aequ 1.
2.
. +
..... + 2)...
3caeq. 2 = 3P3F 2- 1c
ipip cost
ipip Cd
F2-1C
epip coco
vib
vyu 2 - 2p3p
ceq 2L∞ 2-15
ipipco
s in ultiuro scilicet casu cum potusquam est
infinitaagulum
nu
Nota in eodem casu ultimm fit 2C aequ 1c
que adeoque fit 1C ∞g. 2 = 2pp
 1.1 1. 1 +b 1: 1+2b 1: 1+. 3b 1: 146
van
ap
2P 2 - 1c
1+b
ipiped
b A+ Eb 13b. 1+ 4b Ego tC aequ Ec b
sunt ergo 1C. 2C. 3C. progressionis termeture decrescentis.
+ V aequX
X aego  y aeqo Z. Ergo  aqu Yy = aeq.  Z  aequ 
 pet X aequ 1y. YZ aequ 32
nitus le
2-1c
tndiu nota cum C aequalis ponitur 2. fit aequa tro identica
a
in 20 posito Z esse infinitam
ergo aeq. 
)
5. Co
VEc + c +- S+ - aequ 2 - 2p2 2 -1c
coco
ad hoc
si nissot eries 2C aego 1eli cor2 , fli cerB et 3C aequ. 1  l. c r +1, 1 a + 1+
1:c0
d
X. y :q:: 1:
N 2 aequ 1C
Ergo  tre aequ lie: co
de
aequ 0. fict. n ar. 0
seu ln - lo aequ 0 lb
lifaego lif.co com lit lic
lif ) ce
ptico scribam  russu IiC + lit aq. 0 l2e + 12509. 0
 10 + ct +l hico +fm Kicaeq re  +oro i
BM. C
 1c aeque
ad matur quod n est inis a et b c et bErgota Et la lC ccaegienit dctevrt.  bolet   12  15 +l: 1C + it   nifrni. H: A: K en 1t
sC87
1e +  x lay. 12 + g
+ ixtlaqC aqu
quaeritur y
EC aequus 2C +E)
eit 
+1+ ly
3paego Con. 3c
2C
2 2Y 2-39
Ep ipco

## Folio 203v — 00068195:0406
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 202r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000406.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 79 lines, status machine
ita vienendum opinor ad tangente
Inveni alibi, _ Theorema sequens magni monenti ad
Analysin transcenidentem si sit arcus circularis AF ad
ad arcum circularem AD, ut 1 ad . (modo ambo sint
quadrante minores)  posito siu complenentica
C
et radio Y. fore sinum complenenti L ac arcus AF,
quem vocabus(quem
sium CGq vocabinus Caqualem  a + S -  + Via  an. 
Quod si continuetur ratio arcuum, et fialt
cum in besectione quim fruectione
ee polygoliorum retadem quaerans. Arcus AD. AF. AH. AM continue proportionales
erit et q
erit series hujus modi: CE aequ a
si = a + Y aa -  vocetur C, et La- S aa vocetur D.
fiet: CE aequ ( + g0
CG aequ ( 4 gcot
CL aequ (  9c=
CN aequ. 1c = + 94
CG aqu. Vatiq aa   Da - yna
 t
0
Ct aquo. Viat BEaa.  +  La - Y aa vr
ex  +........
. taque ratle nsqua arcus AD secatur et suffecatur existente co, numero jectionum factarum
existente C, ex. 9. pro arcu AM, 3, erit sivus complementi huicaraui Am, reones, 
arcui a, respondens C N aequ. ( ed 7 cd s et t ioH
eit
le
10
)Con cont seu vet
1ed
ed i
2G e
.1
ped er
tam
. d DEoge  OD ap. 1
 q2  
sunct e sublata n, item
et de
ce. 0 + 15 91 aqu 1e d  1+ O 
 Id e vyste b
scu 
vatio 1C aequ 1e+1
er iet is tollenda reclus 1fago 1C - 1C 15 aequ 2C - hre: H. a
 =R et 15 aegit   1( K 1cd = 2CH KI C): h = K seu 1S equ ECmhoC t si
2C = R1C i h = K et eidem mogo 15 aequ 2C F - t1c.:
k-
C = 16. D,: 1.0 - 1: D et yfaqu. 2C - 1C: 0, . 1: 9 - 1.
qu. 2C = 1c. ) _O = 1: co pco
vel etiam 1e aego , 3C - 15. D 
+rc  1. D :. 5 - 1. 0
qu  1C:0- 2
 3Caeg. 2C 3c = 1C D + 1cO 2C( v e 9-00
cedit autem divisio per 5= O: et. fit thaotema Mremotibile 3e. eqqu 2C. O D - 10
seu 1C aq. rrsc  9 + 9 20 + X. 3C quae est relatio generalis
inu complementi, quodu carcus 1C 1C, 2C, 3.C quorum
respondentes majon meco medius, minor sunt in ratione ac
continua ed ad 1
fbus quibus hibet
betur  quantihesi ex omnibus simt complementi eodem
tempe
+Dco
ub
G) exstente 
formata
oposita
1. 1
et Ot Dry
lobet si cos2 s tentem ac
BHI) pErr-inC
1st
an sum
peum
nogit cons
uthnetica
endrfferentias insente partas
iem oideamnan univerale aliqud abrchi