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# Catalogue record: 59825 — Hic analyseos aditus pro serie Assignabilium in polygonis accedente, sumtis chordis, vel tangentibus, vel triangulis aut rectangulis aut trapeziis aequalibus curvae ascriptis
# Incipit: Si polygonorum inscriptorum vel circumscriptorum partes vel latera
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=59825&katnr=57535
# Work: https://leibnizlegible.com/work/00068196
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folios: Bl. 348–349 (from the shelfmark LH 35, 13, 1 Bl. 348-349): 4 pages, canvases 691–694, page ids 00068196:0691 to 00068196:0694
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=691
# Lines: 380 recognised on 4 of 4 pages, mean confidence 0.82
# 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.
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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 348r — 00068196:0691
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 349v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000691.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 78 lines, status machine
Hic analyseos aditus pro serebas a serie assignabilium in polygonis accedente,
sumtis chordis, vel sciti, vel triaugulis aut rectangulis aut tr. metiis aequalibus purvae ascriptis
investius anda secii summarum prohis aequalibus, rrumerum semper mithmetice augendo
Si Polygonorum inscriptorum vel circum scriptorum partes vel
atera semper possent commude sic assumi, ut fierent aequalitas haberetur, inde facile
hraberetur summa. id fit in triangulis et trapetiis sectionum Conicarum. Sed videndum an res
erius promoveri possit. Exempli causa datis duobus curvae punctis A, B, quaerature
tertium C, si ut trapetia DAEZ et DC DBFC eut aequatia
hoc utique fieri potest Itaque similiter datis punctis A et C, inveniri potest
tertium G ut trapetia DAHG, et DC DGic sunt aequalia. item datis
punctis t B et C tertiam K ut trapetia BCLs, et DK MB ex
3 DBMK, et DDLC sint aequalia. Quod si ergo contingeret ex
natura curvae, ut semper etiam ap) aequalia inter se essent aequalia
duobus alteris, velut oGic ipsi DK LC, res esset in plano
sed hoc
esitr
AC, BC sint a.
il L pK, sed non est im pot
Cum ergo ins per intra circumscriptas, vel intra-
inscriptas obtineri non possit, ubi nempa latera polusgoni
prioris posterioris cadunt intra priora, nec ipsa secant;
cogitavi an non res obtineri queat per transinsrit vel
per novum inscribendi aut circum scribendi genus, nempeper
ransinpripta, aut trans circumscripta, ubi latera polysgoni sequentis
priori polysioni hatera secant. sit chorva AB, et in arca cui subtenditur punctum
.Dum sumatur punctum D tale, ut aequales suit
cto AB, DE, EB. e punctum F tale, ut aequalius fiant AF, F S. Ses
vatuor rectae AF, FC, C H, HB. Et ila porro
quorenda de hoc modo eae AF seu
arum AB, AC, AF, et c.
 nonnihil
In eo videor mihi connihit
invenire difficultatis quod inscriptiotrium
aequalium, non admodum juvat ad inscriptio
niem quituor aequalium et ita porro.
Aisi forte hoc modo, inventis quao
N lateribus ADEB facile invenietur htuo
AdDB, EB facile invenitur quartum 
Ergo et Al. Ergo ex data AD.
invenietur AD ergo et vicissim
ex data AB habetur AD, seu mod
inferibendi quatuor aequales.
sit curva HHHD, in qua
arcus ACD, cuju ita divisus
in C., ut AC et BC suit aequales
jam centio B radio BC
describatur arcus circuli
curvam HAd secans in E
jungatur AE
AB sit L et
et ABbiseget
bijeceru
jungatur FC
curime, LM
E punctis A et B in axemp N B0
demittantur hormale
quibus ut et ex
rectis LN, 10 determi
ata erunt puncta A et B ineunde
nem e pu
F et C demitantur
P, CK et ex F,
nornalis FGaequali
ducatur AI aequalis et
Np oecurrens ipsit Pa
NO
eesimilia erit CGiG
 AI = 1O= LN,:2 et Fi-
AACqu= AIqu + F. 1
et F odcatur a sB nobma
ficue qu Eit
semper nolus promiffere non possumu
EEK
C tale, ut fiat AC = BC
et curidie
Le
FI. C et sit Lia
 e a

## Folio 348v — 00068196:0692
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 349r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000692.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 99 lines, status machine
8
Inoestigatio chordarum aequalium e curvam datam dividenti
tit curva HAd HAK, in qua sit arcus ABD arcus cujus chorda AB. jam
centro B radio BA, describatur areus circuli, secans curvam in C. fungatur AB AC
jam pro calculo referentur puncta A, BC, ad Rectam indefinitam 1 Ar demuis
rectis cormalibus Axi. Ob data puncta A ob dator puncta A, B dantur LG, LN
abhictae, et ob naturam curae resario inter LG, GA, et inter L1, 1B. ob naturam
sirculo excentro B radio BAde scripti erit  B0 qu +Co qu= ABqu. et quia B0 = LN - LI
2LN. 21 + Negut 1Bqu - Z. NC. 1P4 ABqu.
ek NC = 1B, fiet. L LNqu.  Lqu
Alimnendo datu qua exhibet Ex naturi curvae relationem
inter LNoet NC, habebitur utraque halum rectarum. Rursus AC, GNqu + (NC- BA) qu, seu
comper pro
AC qu = (LN - LG) qu + (NC=GA qu).
exprimenda curvae natura scribamus GA = LG et IB - L1
et NC = LN fiet LNqu. ( LI qu) - 2 LN. 11 + L 4 4 ( 21. 
et (0- 
pea- du. VG a thldect
N- 1G) qu + (LN = GA) qu
Habemus valorem AC, et duos ralores AB, C
et ACeju. =
valores ABqu. cum qua Regu
tres literas LG, C LI, IN et harum trium
aequationum ope tollendo LN, LI, LN
prodibit aequatio inter LG, AB, AC, vel
Itaque hinc vicissim datis punctis A et C datur B
qualem ipsi BC vel AB) habebitur relatio
datur ex LG et AB, et BC est aequ. ipsi AB
 et posset tollendo LG relatu
C1. FC
(si quis id maltet inter LA, AB,  AC
n
illodem
BV
Et quia L1
dabitur relatio inter LG, A
Sed sufficiet paucis assum deseripto r
inter AB, AC, Bd. postremo quarendo Ad, poterit relatio haberi inter LG, AB,
vel et prsse
inter LB, B
vel inter AB
Dqu=
DE qu =
et
ep+ Nequ+
P1
1. c
50
2G
Hsc
determinari D, et hinc duci AD, atque unde sublatis reliquis quaeri
+
 circulo centro eira
1y
3
23
en
lationem inter A in Ellipsi or 
dio
qudud
AB = LGqu + L1 qu - 2(G. L9 + GA qu + 1B4
BCm = ABju = LIqu + CNqu - 2 L1. LN = 1Bqu + XC qu = 2i B. NC
LPqu -2
tL
u
He quesqu LAqu
Dqu=
ed
Z
in = 2 dB
ZN+ GA u  Nc
2
l. L
In Hyperbola H Aequilatera sit ubi y = aa : X
1, 2X LN, EX. LP, AX. LQ sX
Aqu
GA Gu 4)
sit RGiX.
quabesa
qu BC qu
123X
 2X - 1X) + qu aa - a)
2) ju (Ex - 2X) + qus au- aa
qu. CD= qu AB - qu aX. 3X) + qu a - au
M
In AE = qu ( 5X 1
ess (5X aX) + qu (aa- aa
 a
-0a)
porre
17
a
NC, a
qui d E= qu
IX)+ qua
(AX- IX)
AB qu = L
 s
Ju Ad

## Folio 349r — 00068196:0693
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 348v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000693.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 90 lines, status machine
jam ope aequ 12, 5, tollantur 2X et 2X
in Ellipsi  vel Hyperbola sit L centrum fietus LG AGAG ad G
ge 2rx + pXX = yy sit XA  Z  X +4 seu X- (. 9 fet 21 Z 19+ 7  (2 4r 
Hinc
seu yy = xZZ- Y9 seu AG. AG = r (LG. 1G) - r
Ce= sint LG, LI, LN, LP, LZ, Et GA, 1B, NC, pd, 
W X 3 Z
(V), (1) (X), (3)  (0(v) 00- r
ee = qut B = vv + W01 - 2∞WV + Y vD + ruI 29 - 2 ((rVV - (, 19, Y 0u -r9)
qu. BC = qu AB= WW + XX = 2WWX + Z WWW+YXX -2r9 - 2 +(rWI -193 Z XX- 19)
LGe. D tG + LIqu en  G. La) C . 2. 15) - (15 - 1A) (2E1 4 7 + C)
ff = qu AC = vo + XX - 2 Vx gF 00 + Z XP - 2rq = 2 +(Z vV - r9, Y Xx r4)
ope harum trium aequationum tob debent W. et x
Ex aequ fiet Cel  c v irq  at vrrrvo- 2
ee = (1) v0 + Wo - 2VV + (V(V) + (1) (10) - 2 f 11)
ee = (2) WWW +XX = 2WX + (o po) + () () - 2 () (X)
ff = ( VD+XK = 2VX + (v (V) + (X) (X) - 2 (V) (X)
Oste rarum trium aequationuu posunt tolli W0 et x; sed
remanente v. sed video jam praestare omitti tertiam aequationem
solis adhibitis prima et 2da ita ut maneant v et X, sublata
H
vn F W0 -2u +(V(V) ((10) 20) = 2(∞(10) = (Wou XX BrWOX ((o (Co  ( (a) - 2 ) (X)
1) + (V) = B
 et 2 C  ) -1
vocemus am
sum solum media W. Erita
int n 2p () e et q posito
 
seu un-rxho+ (V ()- 2(∞120
compendii causa vo B() () + XX + (X) (X
Wo esse radicem, fint aamm- Jamno + annioro = ap uto). F
V) - 2(X)
in Hyperbola est (1) (10) = Wco = rq et fiet aamm -4amnuo + annioco = apporo
3C
o) (u)
e5lX=
=4 pprq compendi causa aa seu Ce W0W + tamn W = aamm + IPX YI
op. 4n pV
pp - nn =(X) + (vYV) - 2(X (V)  - XX -vn + 2XV
seu (XX - r9 (t  vD -rg = 2(4) (V) (XX vV) + 2xV
App ann arb
in =IL 2rg - 2XV)
et am = - vv C∞ - v) v) + Xx + () () = vv + 2XX (= r9 Frq
am = X+V, X- v et =amn =  ~ + v)nu X+Asi
sit fiet: itaque positis X+V, l; Et X V, n; set (X) -(V), p. et = am = 1a
seu am = 2 (v0 -XX)
Inn
2rg - 2Xv + 2() (1)
fiet oi 
W
+) Una + rg que ) rare
 814 _ 8, r9 - X~ + () (1)
lavenietur e per alterntiam
1 vel 2. ita datis punctis extremis et et C, habetur medrum B
inte invento ergo valore ipsius. W, per a g. 4
sit jam tertius valor ipsiis ee aequatione sta
adeoque et AB et CB aequales.
nempe
ope harum trium aequationum 1, 2, 5 tolli debent literae W et X. jungantur aequationis
ee -5) Xx + gy
=Eyx +  () + (5) e3) - 26 (s)
2 et 5, fiet 18 1d + XX ad instar aequrtionus 3, Wov - 21oX H(o ) (1) - 2 (0 YX) C y - 2 Xy ~ 1) 17)- 
d = jungatur aequ. l et 5 fiat eeraroro - Wto 4 (r) C~) H ) 10) 2(1) (o) = XX 49y - 27X + 1Y) (Y) es) () 2B
unde Wno + (1) (1) = = VV + 2vW- 1) L 2(V) (1) + XX +yy. 2yX 2(1) CX) + (y) Cy) txH)
 + 210X 4 2(10) (X) + gy - 27. 9 + (1) () (Y) (7) q 3
Ex 6 est (20) = (X)+ (9y- 2Xy +(XX + (7) (7) = 2 (4) (7) = 1010 + 210X)
qui valor conjungatur cum sore aeq0 (((3)
Aequandoque duos valores fit:
hinc quaratuo
valor ipsius ro
V 2v0 + (n) (n) - Xx = 2op C (X) (∞B - 2(X 1V) +(3( 6) L (yy - 2p7. etc)
Pro V, 10, X, 9, Z scribamus 12, 12, 13, 14, 15 etc L. v + X = 113 X B = 13) -
2, 23, 24, 25 et 5- (∞) (X) - 213 (∞ (∞) = 231 -
Ex apo Ex arq. (3) fit nB-2MId,: Ep = (ut seu substitulis numenis
21
2(∞) - 2 (X)
(1), (20), (X), (1), 2)
Ex 1et 2 e L    113. 131 Y 213. 231
Et ex 2 et 5
similiter
mus
12t. 192 +224. 242 - 2. 132 1
135, 1534235, 253 - 2 153,14
 = 2. 131. 12 - 2. 231. 22 -0
2.292. 27
2. 262
146. 164+ 246. 264- 2. 164. 15 - 2. 264. 25
Co
qury qy 1
349

## Folio 349v — 00068196:0694
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 348r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000694.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 113 lines, status machine
loco. v, W, X, y, Z vex, () (20), (X, (7) (2
Scribatur 117 12, 1, 14, 15, etc 21, 227 23, 24, 25 et
1t B = 11(1B -g - 13 21+ 1 = 213 23- 21- 29
et similiter
113. 131 +213. 231 = 2. 131. 12 = 2. 231. 22 = 0 seu 13. 13 -11. 11 +2323 -21. 21 = 2. (13 -11)12 -2. [3-21)22 = 02
124. 172+324 242 = 2. 142. 13 - 2. 292. 23 =0 1439- 12.1242924. 32. 22 - 2 (14- 17) 13 - 2. (4 - 32) 23 0
135.153425. 253 = 2. 153,14 = 2253. 24-0 Ah. 15- B. 13+ 25. 25-9. 23- 2(15. 12)54 - 2 (25- 23) 2930
Et eodem modo 22. 2+=12. 12 - bb
1616- 14. 14 26. 216- 29. 24 4 1 6- 19)15 - 2 (26. 24) 25 1l0
et 23. 23 (4) (13. 13 - bb et 24. 24 - (9), 14 = bb
Iis valores substituendo in aeq9. 1, 2, 3, et fiet  EK aequ. 1. 13. 13 - 12. 11 t 13. 13 - rg - X 11. 1 + r9-
=2. (13- 11) 12 = 2 (23 - 21) 22. seu sit 1+8 = lf eritque 5 (13. 13 - 11. 11) - (13 - 11)- (23 21) 2
Harum agg. 11, 12, 13, (4 etc.
Eodemue modo ex Lerit f. (14. 14 - 12. 12) -  (14 12)13- 824- 21) 23 =
mequ eramni quantulore, 13 per 1 et 12 Dx ~g. V) ex3 f (15. 15 - 13. 13) -  (15-13) 19 -7(25. 25) 24 =
14 per 12. et 13 X aequ. 12 et 15 per 13et 14 ex a, 3. Ep4, f(16 1b- 1414) (16. 14) 15-8(6 295 
Et da porro. quia autem 23 - ((13. 13, g) ideo utto
id tel daZ ad habendum valorem ipsius 13
et ita porio
assequemur, si faciamus +(413. 13-bH) ex trahibilem. sit nempet 13. 13 - bb - 913. 13 - 2913. 3345
opus foret ex altatione. Sed in fatsor brevins quaesitum
seu 2.913. 33 = bb + 33d seu 15 1 9. 13 = bb + 33. 33, : 2. 33 et 23 = 913 - 33,25 23
23= bb 4 33.33 = 2. 33.33,: 233 seu 23 = bb = 33. 33,:2.33. itaque ex aq. 11. fiet
C(b + 2. 33 bb +351 = 499. 35.11) - 5 29lb33. 12_ +ag293312 449935. 11. 2
= 49933. bb22 42993322 44993321. 22 = 0 Est autem 5 = 99+1.
Et ordinando aequationem F33Y +299. 22. 337 + 25bb33 = 29bb12 33 +fb =0
4915.. = 299bb 22
+ 4g911
ag12.
ultra pa sed 11 etiam ascendisset ad 1142, et 21, 12, 22 ascendisset ad quadratum, quod hoc modo evitatu
Equidem attntis 13 et 23 non
Rlura emersissent membra, nec 13 assutrepisset
+499. 21300
Hinc similiter ex aeq. 12 faciendo 914 = bb +34. 34, : 2. 39 fint 24 = g. bb-34.34, : 2. 34
et f34 + 299. 23. 3_ + Ef.bb 34 = 29bb13. 34 + fb =0
+ quooutue
+499 22. 3) fict
= 299bb23
perians ds
per inansos valores
534+ +gbl. 35) 54 +2 flb3+ 34 - b1. 534 + fb
24tg9  bb33
Ubicum 33 assurgat ad quadratum, ideo x
= 2bb. 3. tgm
ope aeq. praecedentis pro 33, et hujus tollendo  33
ascenderet 34 ad octavum gradum fiet et haberetur
relatio qua cujus ope haberetur punctum  B ope
a anre
+291g bb. 37)..
+2ubt: 33)
-ggbl. 33)
gybbogue.
-21199225.
ad gradum B. sd si omissa explicatione ipsius 13 ex altassemus aequationem 11, ut habete
sin AB ex datis A et D, sed quae assurgeret
quadrat 23, Ldinabimus sic 23. 22 = f5 (13 - 2: 13. 1 11 +. 1) - 251512+25 1315+25 11. 13N- SII. 12
+ 2517 2 32 A = 2515. 21. 22 4 13. 17 -7. 13. 11. 12 + 11 12 + -2. 13,. 12. 21. 21 F. 11. 1. H. R
213. 22 - X bb 22 et ordinando seundum
15 - 25l21)  - 255 15 13 +4f 1112 13 + 5511
videndum an haec communislex determi-
2 ur
possit ex seriibus aequation qua lubitia 2
fandumentatiam
an  equo
2. 11. 12 21
a2. 12. 21. 2..
Quodsi ergo depressio melior statiar in aegq. ar. 1ra
H depressiones inseremus fiet
ef 23- 21 = 1(X 13. 13 - bb) - Fr1. 1 bb)
= 2f11312
=2f1. 11. 22
22, 11. 12. 11. 22
+ bb 22
Calculus institutus est, tanquam manentebqu semper punctis praecedentibus
nA, B, etc var epuncta C, D, etc. semper alia ali alia essent
sed scopus est rem ita risdlocia, ut manente semper netimoprimo et ultimo nunc
nunc tres nunc quatuor ete. hodae equales inseribantur. Sit ergo
punctum determinatur per H1 et 21  altimum determinetur per 19 et 20
Eodem habeditur
ut hoc aequatio es
inter 13, 22, 82, 21, 11
habelituenter 1a 29, c3, 22, 12
sed ili tae quae etia
est aei gnd prn 14 sed
ut tollatur 23 per 13 assur
getur ad gr. 8, et 13
ad gr. 4. unde cm
13 in duabus aeq
ad cendat ad e.+
Follendo 13 altissime
assur pet 14. ideo
metibr videtum de
pressio
eteaque pro duabus chordi inseridendis habemus aeprationem has ex cet  sutus parpinae 419. 19 429.29 - 29- 11) M =  dantur invicem 11e1 21
ergo ut et quaeritur semper punctum B, s seu recta AB, bis, velter) etc. repetenda
 -2s29. ∞1
 pordis aequalibus achitende aq. Ba 2 hocmodo 113. 13 823 23 = 2 ()- 11) tt9 19 + 2929 - 2(19. 79730
13 qu
1. 1 =21. 21 = 2(2] -_ X
2 422. 3 - 2(292 21) 23
ae quatibus adhivead 5 1. 2. 3 = 11. H = 2 et = 23 2g L 12 ,  - 2(243 d
e15.13 deste - 2(2923)G
ubi solum ponuntur aeqq. 1, 2, B, 4 etc nisi quod semper in ultima pro summique abscissa ponitur B9 et pro
Tamma ordinata 29. Itaque produabus a tribus, quamor, etc. chaedis aequalibus pro 13, 23; 14, 24; 15, 25 et espective poni
 debent 19, 29. Quibns a44. accedet demum proprietas invae exhibens relationem inter 11 21, velM, 2, vel
=.13, 23, etc vel 19, 29 ordinata. Semperant -e = 11. 11 + 12. 12 - 2. 11.12 + 21. 21 + 22. 22 - 2. 21 22
 I1 autem et 2_ item r9 et 29 semper sunt datae, itaque tantum opus inveniri 12 vel 22, quod non potest si
sed t cum chordae u aequales arcui inscrbendae sunt plures duabus nob plotest fier nisi intervenientibus pluribus, ut 1 item
penste pro tribus chordis, ipsis 223 pro quatuor etiam 14, 20 quio et etium 15, 25 et ila por
quacam seestio ducbus e
versaeassi
22. 213,225 3 14, 324, 19, 29 et it. portos ila quaeronda progressio.
ea relatio communis aiquot ternor oinuore