# Leibniz Legible — https://leibnizlegible.com
# Piece: https://leibnizlegible.com/api/records/10302/text
# Catalogue record: 10302 — [Math. Aufz.]
# Incipit: [1mò] x+...
# Date: 1696
# Akademie-Ausgabe: AA VII (assigned, not yet published)
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=10302&katnr=39187
# Work: https://leibnizlegible.com/work/00068255
# Title: Leibniz-Handschriften zur Mathematik LH 35, 3 B 5
# Folios: Bl. 64–73 (from the shelfmark LH 35, 3 B 5 Bl. 64-73 (dazu Bl. 79-80)): 20 pages, canvases 127–146, page ids 00068255:0127 to 00068255:0146
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068255&page=127
# Lines: 1,406 recognised on 18 of 20 pages, mean confidence 0.80
# 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 64r — 00068255:0127
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 73v); this page is its right half.
# No recognised text on this page (recognized).

## Folio 64v — 00068255:0128
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 73r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068255/jpgs/default/00000128.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 81 lines, status machine
IImo) e = O1
xtum = Oe xinn -∞D
fit xelle xrtam, xrim =  g2
ui aequando rrimum valorem cummedio sis F
2do) Hinc X = O =l = 0 tum = D- an q Hinc  C, D. Gset m et similiter
aquando ultimum valorem cum medio fit Z, D- O) pinmm. Unde sequitur tertia aequando primum
et  extremum valores inter se, sed sufficit nempe CD. Il=2D-n.  sufficit ergo duas
priores aequationes praestapi. ita ut C et  et D, prpractentur eadem modo et l atque n
itidem eodem modo; sed D et m, mediae quantitates a reliquis distiagventur
O = 12 Lvy + 113 IrZ + 123 fyZ + 100 p
P= 212 1yy +213Y07 +223Yy2 + 2009
F = 3I Woy + 313 YvZ + 323 VyZ + 300
9= 413. 523Z Nvy + Am2. sez y l02
unm cujusque numeri pone
veluti
sessit F1Z.  id desopeetoggu2
123. S12. 323. 7e2 caeferis f125, 25 sus
2+ 33n = 1 
= l12 135
+ 12. S = 1ig1) +  = Co2sB et ita porro ctatt
1312. q = dutos = 3uem - 31 2s s +s 1r. 11 - 3 u1
L aequales fit . 100. 512p + 12 500t + lr13 Z= 0 100. 513p + 113 500F. + 11 23 y = 0 100. 523p. +. 123, 500F + l 213 V = 0
compendio scribamus uA P F 12L +137 =0 2ip + snt + 23V =0 et sip + 3it + 33y =0
Significant Sir I12 12313 S3
Ex duabus prioribus aequationibus inveniamus valores ipsarum p et t
FI1. 21 p) + 12. 21F + 1. 2Z =0 fit .t=
Es pari jure in notis dextris numerorum permutando L et 2 fif p = + 12. 23 Y,:
omissis intermi 100 et F
soo quia semperre
ouo sutlt intelligi
12. 23V: +12. 21
quibus valoribus substitutis in aequatione tertia 31p + 32t + 33y 20 fis -12. 23. 31V 4. 11. 23 32V + 12. 21 g9 20
413. 22.31Z = 13. 21. 322
em ipsorum 11, 12, etc. substituendo valores in aequationibus 1, 2, 3 hujus articuli
velu B: Suc3. Unde permutandu 1 et 5 sit 5e0F =  S12. 11h12 
13. 123137 nam ta t tr13, permn
nam l2g13 vel lau permutatis, 1 et S manent, sed S1223 vel similis) permutatis 1 et 5 mutatur in = 51223
+ si2. linvi:, Si quibus vatoribus substtutis
A3 523 11313 Z
=112. 513..... - 123- 512
+ 523121. y0 vel y = Sir D +515131Z, : Si2 231
(2) +712. 3129: .712 etce
et 72133V +713233Z 472313y= quam equationem conjungendo cum aeq. 6 articuli
et pcum Y, et t cum W, fit Bo Y= +343. 319
praecedentis  ut tollatur Y, fiet + Sirrl. 723123 D + 513221. 723123Z (+ Saur I. 70u3V
=GagEl. 71273 V - 52312(. 7132337 - 523121. 7131 231)
Unde quidem haberi posset v: Z, sed quia hoc non licet, neque enim eorum ratio debet esse data, alio qui
1VZ non foret irrationalis in indeterminatis, contra intentionem. ideo hanc aequationem oportet
(6to) 523,3t. 712293 - 51222l. 72413
1009- 100 P seu
adtem haber o deprassio. Ham mulliplicetur a q. 7 per aiti c. praced. per aiqu 4 aniculi Eum
ppermutando 12 et 23, fiet 122+ 522 um si i. ca. sed 52
32313, 51223. seu 123. 32313. 71223 + 523,12313.71223 -323.32313. 5123- 723. 32313. 51223
autem habena  prisstio. tam mullislictur aq.  pr ic. pacud. per aequaata
fid Erres He(7mo) Sc2I(si: 72233Ct1)
quam aequationem rursus divellendo in duas, ut destruantur V et Z ne determinetur v: Z, fiet ideo ut tollatur Y
 112. 1121Z. 71223 +512- 11213. 7.223 = 312. 31213. 51223 -712. 31213. 51223 60 seu 112 +512 ) 3icg. S1213
12
et) ut patet ex artic 3. seu 2009= 100 p + 700 W, ut patet ex artic. 6. itaque fit 500t + 300r= 100 p + 700W.
jam considerandum 500C esse 2009-300Y. seu
et et similiter ut tollatur
ope harum aequationum
pari jure destruendo irrationales in D, -O,  ex permutando initenles latd; cum 3167.
seu Si Sin3l. D + 51323I. Z
fietg (610)51213l, Jng.
in aq3 hujs articuli fis denique ex aeq. 3. explicatis 11, 12, ete fit +r23. 513. 12313Z + 11Z S12 ln13V 4. 112. 52(y = 9
Itaque tandem, = 4 123. 12313 2
F 11. 813 = S12131
6T jam in aequ’artii  destruamus irrationales ponendo coefficientes eorum nihilo
aant  in halor ipsis d, P.
= 11. 21 p) = 11. 22. + - 11. 23V
Bo  = a + . 123 Z
divellere in duas
ergo fit 123 +523
(5) in hoc aequatione primum destruamus omnes crrationales, faciendo coefficientes learum niilo aequeles e. peruntando 100p cum 300r
et compendii causa faciamqus
+112. 523,312.723 = 171223
+112. 523. 312. 723 = 711223
saudu s
51323. l = - 52313)
B - 12t  + 3

## Folio 65r — 00068255:0129
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 72v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068255/jpgs/default/00000129.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 2 lines, status machine
ua.
eeeeeeea

## Folio 65v — 00068255:0130
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 72r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068255/jpgs/default/00000130.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 103 lines, status machine
Itaque ex HvD H. 31 VY + 123. 5232y = 1G. S1Z vZ +100p 500t U-mm
HB. 51, 13) i. 523L H3. 512
2t513  3 5 1M223S
Hmm) 11223. 5123.
10rb destructis jam irrationalibus in aequatione articuli 4 restat in  eadem aequatione pars rationalis
112. 512 vy + 13. 522Zy + e) + 100 p. 500 F + 113. 513VZ =nill oum, ubi substituenus
valores y et  et 500C inventos articulo 6 et 7. jam  100 p = M. H2V + 12313Z: 5123 et
seol=  aq si ul jam 100β = - 12. 112. 513 V + 123. 123, 513 Z,
12. 527 et 5ot permutando 1 et 5. fi
 seu  500 = + 512. 11213V2313Z.: 51223. Ergo 10dp 5oot =. L812 u2IZ vB + 112. siZ 11213. 12B3VZ,: 5122S
Seot - - 312. 113 A~, .
Siz. 123  - 523. 11
Rursus ex artic. 7. est y = 1121y. 5I213V ~ 12313, 52313Z.3:, 11223,51223: Elyo
+II2. 5I2. 51223, 11213.51213VV - 112. 512. 51223, 12313. 523131)
= Bur2z. Snz si (ill- im m)
123.523. 52317. 11213. 51213 --. 4 123.523. 52g1. 12313. 5231377
ubi 123  permutantur 12 et Ncum 23 et Z
12. 512, 11223. 11213. 112B3VV f 1122211225. 112B. 123B ... = 123, 523. 131 2 1291. 123132
l.Fid est 2. 12. Su 12313. 1113)
2312 2se - 51223
jam coefficiens ipsius vo est +112. 512. 11213 (151223. 512I3 = 11223. 11213). A 11. 512. 11213 fa - b) (1. r)- + b) m) seu
fr2. 123)SI2 C N13- sIL. 113) -
as. i 3  4 113, 513 11223. 51223, 52312...
1 su ns, , tr 2.112. 515. 1∞21Z, 12713
2112. 512. U2B3Am + bC) seu- 2. 112. 312. 113a3(112 SI. 13. 3241. 512: 12I
us in quibus siru, hac
et prius quaerat nempe meuorum primum secundum et quartum. Et quibem proprim
 siS qu - 112. S1 (12313. 32343 dat 112. 512. 51) seu H 2. 512: (123. 513 115. 5252),
 123,523. 51223, 11213. 5213 dat 123.523. 512131 seu + 13. 523 (112, 55- 115, 512).
uttermia terminus ( 13. 513. 11223. Axs. 1M23 dat
pro sccuntus terminis qua rumus
+ 1125. 11223. 293 1_B da
112. 523 +SI1) 113. 313
112 13. 12313 =, 112. sey + u3. fIr, tBt H s = siL S23. 113. pE. 14. 113 +
certius denique
+12, 512. 5257112, 523.
=112. 512. 525 123. 512X
= 123, 523. 512, 112. 523
f123. 523. 512 123. 512
+512. 523 112,52)
45(2. 527 125. 51
n2 SI2. 523. 51125 113
3. iEius sasis
(42. 512. 513. 112. 123. 512.523)
seu 2. 112, 512. 525 115. s s M h ar. 12.
. _. 123. is
112. SIL. 125. 5I22
. 12. jur
se 2.512. 523. 112.523
-113.513 si. S25 = 123. 512), 51223
 (12   en. se 1) 11227
SB B- 12223. 5p23. SI22Z 113, 515
11. 525 = 1232 51 + 2. 112. 123. 9n. 523 2 113. 51
Machigue. bes+ i ac FEcitus oege
12) Ergo tandem aequatio explicata sic. stabit
= 2. 112.515. 512. 52. 1B32 ∞V = 2. 125. 525, 512. 51. 113
 12. r3.. 513
11223. 11223. 11213....
4 4. 112. 123. 512.523. 113. 513
Z2. + 2. s12 113 1pi vis. 115
+ 2. 112.123.2
I23 H3SB +4 12. 123. 512. sig. 1B S
supere (t coefficiens ipsius vZ.  quem ut explicemus, ita procedetur
B. HB S
+ 11515]3, 1123. 51223. 12312
seu 4123. 513 (112. 513 - 113. 515), Su2er u3)..
+ p227. 1229. 11213. 12313 seu + 112. 123.513 + 51. 513 11 Q 11223 B
5(2254 51225 + 4. 112. 13 512. 523
et coefficiens ipsius vZ est 25. 5 l1223 S. 113 + 2. 12. 123, sf2 + 4 (1r2. 123. M2. 523. 113. 513.
A st23 + 11225 est 2. 112. 513 + 2. 123512 )eu r. Js E
112). itaque tandem aequatio ex rationalibus explicata sic ctabit
25512. 523. 132 (+IIE. SIE VV
 I223.15. 5B, +125.523 2
 +E(tirS(Sn 6C3. 113
D. Hrar 5 (312. 52 3 113 + 112. 123. 5152) = L. H1223, 112. 123. 512. 51313513
quam aequationem compendio scribemus 120VVI Z IV voaa - 0. Et cum ex
seu dividendo ft 112. 512 VV + 123. 523
 13) Alia similis aequatu fiet ex ex rationalibus subtatis riationatibus valoris ipsius E, d- O, tantum
1 et 5 troturum initialium permatando cum 3 et 7.  quam compendio similiter
seribemus, 220vV 202ZZ+ etvZ+ 200 aa 0; Ias duns aequationes
latere radix esse ( 112. 512V + 123. 523Z et fiet - 2. 112. 512. 123. 523(2 11213. 12313 =  l2X 235(,1 12G. 
se fiet L1c. 523. SI2. S23. 113 + 2= 112. 525 1 123, 515 + 1t 525. A. 1223, 112 1)SIL. 313. 113. 513 =
+2. H12. S1]13. 513... . ..
+2. 112. 5121359. 504.:
B1 en sis quae aquatia q
quae aequabib dividi potest per u2 23 seu per 112. 513 + (3. 512 et fi
.512. 523113 + in 2312. 123.515 + 4. 112. 133.512. 523. 113. 513=0
Et pari re modo
tractando G, D. O fiet 31223. 712723, 315 4 31213. 312. 325. 7132+ 4. 312. 323. 711. 715. 313. 713=0
IVZ = 11223. st. 13. 5231 2(l. m C
l 2. 1123. 112. 123, 512. 529. 113.513
quasi ab uno
=12. 5I2. SI213. 1313 32313 seu - 1. sir (125. 51 - 15. 113), Su  Mn (SB + sM2, Z13
= 123.523, 51212. 11213. 51213
seu -113. 513 (1123. 523 - 113. 512), 51229 
11223. 13. 51348
Et coefficiens ipsius ZZ est + 123, 523. 12313 (4 52312. 52313 - 12312. 12313) sin 1stus uesn (ib. a f t. )  (B Y a) f19) - - ( c m nus vidi sotas
 sed  b9 + af = 123.523. 113. 512 + 123. 523. 112. 513. Ergo fil coefficient ipsius Z2 -2. 123. 52512313. 11213
je surorest coefficiens ipsius ez. ubi primum excersamus et explicenus ter
ipso omisso, entice
et ordinai
p. S pr13. 123B3 BV +
vq  i edre auimm

## Folio 66r — 00068255:0131
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 71v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068255/jpgs/default/00000131.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 172 lines, status machine
Ad Ex articuli 1g aequo 27 fit per aequ. 25
+(12518 123 126i 5
7123
sis errin
sas
23712
+M (237120
(12543 (23612, 1723. 12529. 23512 4 1712. 23512011723
R712
cplalis n lems tots
+23on brr) 1971 F
nnesuoq - 7u2 (2 5 23) 23113)
+, (272 (137125) = 1723 23512. 12723 - 1712. 12 523. 23712
(12523. 23712)
His ita correctis Numerator utinque solo differunt signo. Ergo cum natsferendo mulando
(12523. 23714)
seu fit namfil = 7123. 12523. 14723 + 7112. 23512. 27712 (28) = 7123. 12513. 12723  - 7112. 23512. 23712
saequ.
in A (D 17Z. 1 523. 23712 4 1712. 23512. 1272)
+1723. 23512. 12723 - 7112. 72513, 23712
itaque mutatis signis alterutrius aequantur denominatores, fietque
112. 12513. 23712 4. 1723. 42512. 12723 = + 1712,23512. 12723 F1723. 12523, 23712
sel Atq adio tandem  seu 1H2B,  125 23. 2372 + 23512. 12723 aequ. p 23~12523. 23712 + 23512. 12723
aut denique  e 1712 = 1723, 12523. 23712 + 23512. 12723=0. unde ( vel erit 1712 = 1723
ispi
Eodem modo ut ex aeqque. y et 13 articuli B duximus aequationes
(progressus
ipterin sic ht donce quid praestet ostendat
et 4’articuli 19. ad eoque et equ. 23.24 ejusdem
14 et aequ  art
articuli 19, ita possimus et  consequentias instituere ex aeq. 12 et 14 apticuli 18. conferendo cum adiil  lsian. , iorii
ait Nempe  aequ. 1. articuli 18 conjuncta cum aequ. 12 artici dabil fretque 1723. A. 7_ - 1712523. X2,:
15 nempe fit 512 = - (15: 1) 23512: 123 et
17. 4) 28712: 327 et723 f(97: a) 12 3: 3M quibus substitutiis in aequ. 30 fiet
:, 12. 133, 12523 + 23S12. Tollamus jam 12.3723, 312, g3. per 10g9. 8. 10 articali s 4 et aequs 1 l. 2. au
et 523 =  (15: a) 12523: 112
1722 (15: a). 238(2. 131. e) 13
 1712. (5: a) 12523 (57, a) 23712. . 312 (15) 12523, 23512, 12523 - ∞512
15d =
(112. 1239, 12523 + 23512
- 1723. 23(12. 17 252. 12. 113 + 1712. 125251 23712. 1 312:, 7HB. 27115. 19. 312 4 712. 12725. 12. 323 = 12523. 23312, 12 523, 23 SI, :, 125137 2512
e
(31)
+7 23. 128. 63 
alterntra per + muliplitato
permulat
+7u2. 128)X, 
s + erch. 23515. 12725. 1 2. 323,:, -7123 1  qu as  ish. 1 . s. nd s
ationem
c. 15 data
A(413 ∞12. 12
ol. 5 cum d et 7, inde 71 mutant signa
ue rerliced aqu fiet
3, Hsu3. 1911
sSS. An rS Aum nchtsIG= ( 23.33 512) 25ig - 13513: nsuB+.
atae dat - 1713. 22512. 12723 F 1 7C 12 523. 23712,:, 4 713. 4 23512. 23712 + H12. 12513 p273 = (3tio ) 12523 - 13512,:, 12523+ 13 5 12. Et parisare permotundo
 pari jure
fietue - 173. 12845 29)
interse latera
18 conju
lis 33,24, articuli 19 Et pr
3 123. 312 p
umus exa
13. X35 12. 2572 - 713. 12513. pr2g -B 4to 1927 23, 20712. 1213 t enia  ivilio 
iistrosum denominato res non differunt in
signis. Erso n
eligimssiriorem aequationem
12238.
2tsi2. 27E3 = 171L. 13 23. 2372 = 41727 12523. 12712 A 1712. 3 
Unde erit vel 1723 = B, 112 vel 12523, 23712)=-
et fil 172
SI2. 1272
d aequiq cum dictis huc
quepur
endo si cons 3 et 32, quod frisset optimur
uantur
us suneratores
ce grmus daq.29
+ 66t0)   te a  3 ad 33
3o
oex 6. f. 24
fiet: - 1723. 12527 23 112 + l12 233
+g. 123252(3512. 12723 + 712. 2331E(1523 2372 , l, 1723. 12523. 23712 = 1712. 23512. 1272L (12513) 63 ∞12 125
Si ae49.
fit L7123. 12923. 12723-712
oportecisset incipere et.
arius
ase 194. 31 ,82, 36, 37 fui
ut aequatae Aequabimus jati
l. Jam vel 34 et 3
omnibs per by12.5323a fiet
112. 5y2 4
tu
1323Faa  (5328- aa
v 1712. 1723 = aa seu 1712, 172) = 1. fietti. 12+
progressus doc
123+523, 1g 323+723
r. 115
512 et 523
235123:3 12523423512. lir
, 112. 123a + 15, 12523, 31. 323a-37, 237
112. 12
te  et 17, quae to dant idem. qui oilere aequalio
23a +37. 12
10) 112+512,:, 312 +712 = (112. 123a - 1
os duos natore
nendo sit +eras    E
EL
(2)X112. 1923 4. 711
jam  123 - 713 fieret
intati
et multiplicatis
di oii ate i iies t tres dide
5325 t suhaata
sitaa
ligonum
eodem icto
 ee pr
aten multiplicato per + fiat  L. 1725. 235 P. 12r25 + n. neriar
 (2) 1712. 1723 = 4 7123 duHes
7H12. 712320
512313at
ne tractare aequationes,
. 12527. 235I2. 17223IE L 12. 112323,3 2772 + 1122,7112. 12523. 12725 - 7112. 12523. 3512. 12723. 237 12.
dividi possant omnia per torum quudratum, et fieret = 1723 (1712
Erro fiet 17i2i
= 42 st 13
a
ny seh
S2-fy23. si
int itrois qrtira
ai ttores per atdiu
 siia
13712).
i12. 123,:, 312. 323. quod f et secundum, in stothes (t) a sed
in video alter
eore pednolonam er pncei 
aequari alter
vel 3t tuileq. 3.
piset e
49. 2, 24, 36,37
el 34 et 36. sed prius pla  5 +
+723 vel ouendio 522 +
potam
93+523= 112. 17) a + nes 15. 13
. 323 a + 37. 12725)112: 312
7. 12723,: 312 Hinc C
p2. 123a + 1512513,:)
ii (4) sit - fiet 5712. ∞2
Sir, ayt si ) t eyemo
iutem praestet, rectaagulumque an fa ara - 5. 2351
sit 42+ st2 (2d0)) 112 + 512
et 323 +723a-
i
25sZ + siaa el sye cr a-  sintrs s
d Possent ad hus saquari  
nem quae determinavit esse
et fit 5312. 5723
312. 323at
at
Exaq
71
1)723. 1712. 23512. 1523. 12725, 2371
2351211
12523. 23712
Seu Malialiando per + et trauspe
sid. ag. 7. per ag. 7
nnoliiltto
ex artiaiti ib iqu ue 13 dupinus aqu. a 23
piticuli a5
alteutae aotem putter volclisemus tractare
deinde ixi

## Folio 66v — 00068255:0132
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 71r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068255/jpgs/default/00000132.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 74 lines, status machine
(14) Ex articuli 13, a99. 3 et 4, quaeramas 513: 113, et 713: 313. Et quideme
aequ. 1 3 fit 513 + 4.112. 123. S12. 523. 513
11223. 112, 123 113
4. 112. 1232. 512 +23 = 112233112. 123,512. 523
seu 5B + 4112. 123. 512. s2Z 51
115 11223. 112. 123 113
seu D12. 123. 512. 523Dhh-S) ()Uh = (3+ 0) 11223 sh et ex 1 et 2 fiR extractioni fecta fi
defunctis = 112, 123, 512. 523 f  112.523hh + 123. 51 hh = 2, 112. 123. 512. 523hh seu 51223h= f~- 112 123. 51. 5C3
sed ut in eg. s succedat extractio sit Hinc sequitur 112. 123. 512. 523 (7 mo) esse quantitatem negativam
utjam amm a qu S succedat extractio, fiat
ex aeg. 5 firts b(f. ath)  11. si h.  Ex 9 est 123. 512 = bb (2h- H): M. 523 et 
ex 10 est 123. 512 =  (f - 4hh) (b: h) = 112. 523 quos deos valores aequando fit 12. 523 - (, 4hh) (b: h) 112. 52
(11m0) = + bb (H- 4h4) seu 112. 52A - (f- +h hi bh) 1 2. 523 +  A stlbe bbf - 8bb ffth + 6bbh (: ahhil
=at bbf obbsthh (+1bbh) - 4bbhh ( 16bbh s: atk  bbff (5 - 12 hsC): ath
seu (12m0) 112. 523= 65 Fs5- irtih qbs- att):  = bb bs hc 166h aF f - 166d: +tih
textratendo f 112. 523 = (12mo) bf ss-ath + _b (f5 - 4hsi): 2h. Tantum ergo superest, ut fatiamus
H- 4his  aequalem quadralo f- 2f8 + 85 seu f = (13tio) 4 sh + 88,: 28 et fiat sf- +hh=
V 16ht + 8hh 48 + 84-88hh,: 484)- (.26.) 4hh - 94,: 28. Adeoq ex 12 et 14, fiet
112. 523 -fista) by 16h (  1bt2 - 8titis P : 8995 seu 12. 515 =  ah-  h8,: q
Ergo per g fit substituendo valonem in 10 fit 123. 512=)  (( og, f1)  16 + 4tt9 f. 4h 8
seu 132 123. 513  (b ) Clot) ) (b:h) , 88 - 4 hp. unde 112. 123. 512.5, 16= bb  4titi: 8
unde vera reperitur ag. 9 per 14 set ex 15 16, fit 11223 7 (1800) (b; ahg): 16h otha, quod itidem
consentit cum aeq. 10 seu fit 1122 (b: h) (T (+th - _8 : 28). Et ex 6 per 17 fit 51223  b. + 1bt : 484
am M 23= S1223 =, Z. M. suq = Lt (b: h), as 2- g + 16 H + (9  l 1 5h2,: 44q
+ ∞ + prodit valor ipsius i 112. 523; sin sit + ∞ X, prodit valor ipsius 23. 512. Unde patet n
in 3r esse 51223= fb, h), 16h- 8: 4g non mutati signo. posito ss aha esse
4kh - 8: 2; spe inen emne non contraria signa ut in aeq. 14. non vero contrario signo.
lldeta Ex his jam s. )B = ubi possumus sponete in aegq. 15 et 16 repertas esse 512
et 525 5EG. per 12, et (12. Et ex aq seu ita in valore ipsius 51Z: (iC3 ex aeq. 4
seu 513: 113 = 112. 173. 512. 523(f - 2h). 1122. 112. 123h ficc
erguees 2t. 8
etr 1 = 8 bb,   glih - 4
seu dema SB3 (21)  2ta - b, 2h-9
jam in aequ. 3 articuli 13 pro 51 subssi hanus
bb ima. ge, (49.  s
hunc numeratorem, pro 13 hunc denominatorem, et fiet
 + +th- 4g v 6 b Et. G
quibs valoribus +58 h 21, 19, 17, substitut
nes qu. 3. articul
res 112. 113, it
aeg9. 15, 16, 17 19, 20, 21,  pro quibus continentur vasores rationales ipsorum
b,- ric2. 10 b4 (i 1123
1 1 )m n m t  ug sinae (1
mucon
pari jure ob 9, D- 0 t, ut praestetur rationaliter aequatio 4 articuli d, fie
L =uo retinendo h, et pro  f, b, 8, ponendo 9, C, V fiet 713 (1m0) = L. 31. 323)
(c. t- hY, et 123. Sir AB to) (b. at) gg 32 1C do) (c. YY) asch- 1)
et 2 et (200) 312. 723 = (Ch: YI), 4hh-  et (3tio) 323. 112 = (C: 4h),  +hh - Y)
et Ato) 312. 323. 712. 723 = - (C: 477), 4hs -  et 31223 = (5t0) (C: 4hs), Anh - V et denique
(610) 12 23 = (6. 4671), 1 166 7 4
Extractione igitur facta ex aequatione articuli 12, fes 
S1223( mm ). Etpum jare 32. 712 B F 33. 721 2 = dp2s f rrera un Amm)
12. 512 V + 123. 523Z = 5l223 + 2. p23. 12413
seu compendio 112 12V + 123. 523Z 4 600100 a=0 et
312. 712V + 323. 7133+860. 800 a3 (4to) =0
et sibtulis a tutis pro tollendis absolutis fit  p. 71. G0 aV- 31. 714. 600 a2 2 600. 800 a
quam reddendo identicam
fiet 805 2600 = 3121712: , eL 800: 6052 33. 723 :, 127. 523 unde fit (800) 312. 712:, 12. 312 = 323. 123:, 123, 523
seu (gno) 112. 512 313. 723 = 123. 523, 312. 712. sed aq. 9. sequitur ex ipsis 6 et 7. ideo, duae ex his
112. 12351 + sir. 525 H3 F 1BE3 NB. SiZ quod multipl. spr 112. 13: , 113. fit 115. 12y. s1 P us. s2 siB  vES. N2 rB  ortcissed Iter
bb 4- 8 16f +9
859mon acu p 17
 in aequ. C arsic. 16. potent explicari (. 1212. 1231) Aer 13, e et per VE. 123. Nam vir C 12 A 103 
 = bb A  . b atih. ps (112. 1 -b  
ahhrgaligh aeguets
in Bs S gl g Ahsi +b. G(, 4 Ah 29 a haer  tq  b si (+ Pha q d citrh. 212
unde U D seu (ir. 125 bb1  
adeoque 11213 a 23B =. = 113
Ubi ut irrationalis reddatur extralibilis fiat 4.115. 123. S12. 523 - 11225. 112. 123. SA2. 523 = (2do) i. 123., 12. 513 f: hh,
513: 1132 (112. 123 512. 523 (f - 2h):, 11223(. 12. 129h Sed quia in 11225 latesit explicando 11223  ut
in aeg. 3 fiet 112. 123. 512. 523 (4hh, 5f) = 11= 5() 112. 5237+ 115. 5125, + 2. 112. 123. 512. 523 hh seu
ut imaginariae evitentur, et ptroinde in aequ. 3 debet esse smajs  quam 4hs
SM Si 11Z. 13512. 523 = bb (Ahh - ss et
ahh - 98, pi2. 113. 

## Folio 67r — 00068255:0133
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 70v); this page is its right half.
# No recognised text on this page (recognized).

## Folio 67v — 00068255:0134
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 70r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068255/jpgs/default/00000134.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 92 lines, status machine
19) Antequam evolvamus aeqq. 11. 12. B. 14 articuli 18,  explicemus aequ. 8 articuli 16. tollendo 512, 323,
712, 124. Nempe per aegy. 9 et 10 articuli 14, et per aeque 1 et 2 articuli 15 fiet (1mo), 512 = -(15: a) 23512: 123 et 523 15. a) 1eseg: 12
et 712 = BF  - B71a) 23712: 323 et ry23 = (7. a) 12G25: 3it. quibus insertis in aequ valoribus in aeq. 8 artic. 16 subistitutis
quae erat 112. 512. 323723 = 137. 523. 312. 712, fief -112. 15. 27612, 323. 37. 127152 - 12g. 1, 12535. 312. 37. 23712. seu fiet
N2. 3237, 235122. 12725= 1232 3122. 12525. 137 12 sen 12. 323;, 123. 312 = + 12523. 23712, :, 2352. 127 25 = 33. Xn:, 512. 729
et 112. 723: 123. 712 = 112. (12723: 312) = - 123. (23712 -323) se uE 129 13 12 = 112. 323. 12723,:, 122. 312. 23712. Unde per 2
5x3. 312,27512 323. Et Porro u12. 512, -, 123. 523-
S2Z12 23512, =, 1232, 12523 =(mo) 31. 712, :, 323. 723
eris 112. 723, :, 123. 712 = Ato) + 12523. 12723, :, 2351r 63712.
et 17. 512, : 312. 712= -112 (15: a) (23515: 123), : , 312 (37. 4) 23112. 323 = 112. 323. 15. 23512, 5, 123. 512. 3]. 23112. se per 2
(15. 12523. 23512, i,37. 12725,. 23712 = (9n0) 123. 523, :, 323, 723. Et 112. 313;, 623. 712 = 123. 312,: , 512, 72
(20) Ex aequ. 13 articuli 18. At per aequ. 17. articuli 14, fit - 7123.  21 1 4 712 St2. 3C5,:, 1723. 123. 312 - 121 . 143. 5  
23 12523. 23512, 12523 -235127:,  12523 + 23512
itidem quaeri posset, et 112. 723,; 523. 712 =. 123. 312, :, 5.2. 323
-(5, a). _. 12523, 23312, 12523 -2351, :9 112.123, 12523+ 23512 su
L7123, 125232, 123. 312 + 7112. 23518, 112.323,:, 1723. 123312 - 1712. 112323=
127 2),: 1723-1712 (1i2. 322 seu (+ 
. 12717do) 1Stosa
vel denique + 7123, 12523, 12723 f 7112. 23512. 23712, 1:, 1723, 23512. 12723 f 1712. 12523. 23712
vel + 1n3. 25 E x a u.  ( 
13. 712-1712. 512.723,:, - 7123, 123.712 +7112. 112, 723
Unde post destructiones 12523. 2ysI2,:, ~B125 23H23BI2 2, 1723. 123. 312 e 15 27. 23112 - 1712. 17. 13. 2952. 1275, :, 4 7p p. 123. 82. 237122 + 712. 112. 315. 12725
5: a)12523, 23512, 125C3. 23512, i, d2 423, 11523+ 23SI 15. 12925 72. 22712 15  sp. 22. 127 25
UIB EY aequ. 14 articuli 18 fit per ac
bel Denic 4t) 125a3 - 275125. 1 12523+235I = 1723, 12523. 23712 4 1712. 23512. 12723, :, 7123. 23512. 23712 +7y2. 12523, 12723
iculo coprobavimus
1723. 12523. 2371241712823512 12123
. . . . . . . . . = 17 2 3. 125 23. 2371 ( - 1712. 429. 
(6t) 12723- 23712:, 133t 25t12 = 1723. 23512. 13723 41712. 12523. 23712, :, - 7123. 23512. 234712 7112. 12523112723
erdantando 15, 3,7
5t) 1272323712; , 12123+23712 ( = - 7123. 12525,12723(L 7112. 23512. 23717
e 1723. 23512. 12723 + 1712. 12523. 23712
Et ex aequ. 12 artic. 18 fit per ∞eg. 6 artic. 15. ad instar aequ.q
tem illius coincdere numertori hyius, Itm   et nomina oqu   alis
123, 12523, 12723 47112, 23512. 23712)
Has Utamur jam egg. 2, 4, 5, 6 articuli praecedentis et dividendomultiplicemus aqu02 et 6. nvi-
in labenta deptris n aequ. 4 et 5 numeratorem illius coincidere nominatori hujiu
(200) 12523- 23512,:, 12523 t 235125
Ubi notandum pui aeqq. 2 et 6 no
1schCZ - 12525. 23712 15 - 23512. 12525 12725 4 235I2 12523. 23712 +2723
Sed nondum hinc aliquid magnopere apparet
12323. . ZZIIL. HB 4. 125C3. 27SI2. 23712.  +23 51. 25712. 12723 +23515. 21712
12515. 1272524 12523. 23712 127234 23512. 13523. 12723
+12523. 12723(12523, 12723 27712 4 23512. 12523.127154
si aequ. 4 et s eodem modo conjungas ut aeqg.
et  prodit idem nempe aequ. 1
(22) Quo brevior autem imposterum sit scriptura pro
et ex aequ. 1. articuli praecedentis fit: 200)
12523, 23512; 12723, 23712
Conjungamus jam et aega. 2 et 5 articuli praecedentis, considerando
 tm i ifferre signas,
523. 275. 127 + 1712. 125. 237 ideu podis si dividas aeg. 
712. 235. 127 erq cpant.
Ex his jam fubus equatur nibus articuli
deu piait quod aequando et 
(lujus nempe 2, 3, 4. solvi possunt omnia
eadem praesturi possunt quae ex aequs. 4, 5, 2, 4, 5, E. arliculi 20. quoniam ex eo quod composita diverso modo
et fis oteo dividendo s per aegr fit 
susterestat aequemus aeqg. 2 et 4, quo case
7123: 7112, et 1723. 1712 quorum valores substituti in aeq. 4 dabant aequationem ali illis liberatam
7127. 115. 127 47112. 235. 237
Haec ergus ut praestemus pro, aequ. 2 pa artic praecedentis per constendrum scrbatur (e
1723, 231.12771712, 125. 237
ndicum  est leas non nisi tribus aequisollere
100, 210X + 100. 20, y. seu X: y = op. 20. it o 2
+(25 + 235, 127 + 237)125.127 + (125- 23531271 57) 195. 137
Et pari modo ex aeq. harticuli praud. 1 B 4 (ir5 +2)5, 127. 23 ) .3 5. 127 4 (125 335, 12 H . 11  25 57 4 ande X - + 31 4d
noK LIBV,: 210X+ 201y  100, 200
indi Adeoque restituento priores quantitates 2
1(125. 235, 127 +231) 235.127,. Cu p2oG 17 - 1 ) 125. 13), uti tip 2 et R - 7123 (t  - 7112
articuli, 20 icet enum illa non sit comved homoschetia, tamen vel ide
quae obssant, aut prodibit quod sicebit duplicare percuutando et
7123 et 171E tolli in aeque. 4 artic. p 22
et ex aequ. 2 et 3 713 = 9 7112 (17A= Z 1723
D-struS, etcL 6, D, 8. Gsehabent
ata meVI mostimoci
25 1u, (125 - 236, 17- 257) + 23 5. 23, (125 F235, 713, 6- + (125 + 235, etc
Cinsirk niress, urtn) F 135. 2Bus, antIE((125H235,127-237) 235. 127 4 C125- 225, 127 1297) 125. 257
1((125 + 235 177+ 297) 15. 127 4 (125. 135, 127 - 237)235. 237
ys127 (V2st2y5, 17 - 237) 235. 127 425 - 135, 127+ 237) 15. 2y, )
5. 27.G +u5. 235. 23F + 295. 237. 127 f. 235. 251
4 12s. 23 (125 - 237, 127 + 23)) 235. 127 + (123 +235, 127- 237)125. 257) cue
odeamue an non nihil contyahere liceat C = +, 1251. 127 = 125. H7. 237 - 235. 125. 127 225.12i2t 
D = 1252 127 +125 127. 237 + 235. 125. 1272+275. 125. 127. 137 +125. 235. 237. 127- 125.95. 23,-235. 257. 11 1275.9)
8 = 7, 25. 275. 127 = 125. 235. 127. 237 +2352727- x352. 127. 237 + 125. 237. 127412522372- 235. 125. 237. 127 -235. 125. 237
 = 125. 235. 1272 4 115. 235127. 237 -235t. 127 - 235. 127. 237 4 125. 237. 127 f 125. 237 +235. 125. 237, 127 = 235. 125. 737
rnnsae fr u2 23512. 25712 um n am mst
(12523. 423512, 12723+23712)12523. 127234 (2523 - 23512, 12723 - 25712) 23 512. 25712
cem prodit mo) oratroon 
y123 4(12523 -23512, 12, 23- 23712) 12523. 121
Qua bavi, autem imposterum
137I

## Folio 68r — 00068255:0135
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 69v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068255/jpgs/default/00000135.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 73 lines, status machine
tos
supere st ut in aequ. 10 excolvamus 112451C, 3231723.. jam _5i =,    _ - (5. a) 23512: 123. ergo fiet
2to) 1t SI2 = 112. 123 = 23512.157: (23 et 32 +712 = 312. 323a =23712373:323 et 123+523 = 112. 123a +12525. 15,: 112
et 323+723 = 312. 323 a + 12723 37:312. Unde jame fiet
unde jam piet
(4to) 12 +512, 323+723 = v12. 1234 23515. 15, 312. 323 a + 12725. 37, (,   123. 512 112. 318 C. . E2 22. 22712
teur
351323A2 nSL3. 12723
23512. 127233
12533. 23712
C2ye. 2371
+ 12523. 12723
articuti praecedentis. Ex bus auterioris nempe  decimi y
512. 712 A seu
+0 1292 3. 23712
(2523. 12723
12. 13a + 12529. 15, 3 . 329 a - 23 712. 37  F 1 233n. 107E3,  3or. 12712 
( 112, 123a + 12525. 157 312. 323a - 23714. 37
112 + +12
 et 7123
M2+ 5I2= 312+1 3F 12 +512
323+7 (Z,9 383+ 23, 23+ 513
13e  13.
+23512.
12523. 1272G
sed videtur aequitio c0 vel sabtris 3 nunc aeddi posse tractabilior, ex quo repertum es sse
3. 212. 723 + 13. (512. 323. 712 = lis 112. 723 + 123. 712 (Exl ergo (29no) al sernetrum geluditur s15pE. 22712 + 21512. L2p.
112 +51Z = 5+2+ 712
eL:, 12s u Zmmnunrhtn us. 2n
t
1712 f ry23. unde per aequ. 2 vel 8 ejus demarsiculi fit  3112= 7123
m  2
Padentis 112. 123 a 4 12513C. 15, 312. 32ha = 2371~. 27, ., 11 2, 3235123. 312, f, 29512. 12723
itaque ex aequ. 10 per initium articuli hujus et per aequ. 2 et 4,
qq. 23 et 24, et ope aeq 2b
2
112. 123 a = 23512.15, 312.323a +127 23.37
qu. 25. fdebuis et jam institur articulo ficerente post 6 post eg. 8.
134523, 312+712
 112. 123a = 23512,15, 312. 323 a + 12725.3
exglutione indiget. Nempe. per
Ergo ex ag. 28 auiculi 19 fitt ee 
Restat
2idendum an haec aquatio contrahi possit, conjungendo cum ea aequationes 26 et 28 catculi
1234523
13 (213, : 9513. 313 = (20 mo) = 17: 53 vel distincstius 113 713, ., 319. 15 nes 5reyc . mis alimus ipsum Meaerato er
adhuc alisiu
Ergo per aequ.
esse 172
28 articuli 19 st
12 + L(23 (2 + 712
21) Sed video non fuusse lus opus post quam invenmus grtic r fine
C3 4523431 1
sub finem articuli ag
= 4 12523, 12723
Vid artic. 10 aequ. 1
quia aeq. 1 non daret idem
Haec vide sripta disictius in
jogna avera articuli 14 initt
seu Ar12 1Sig. 123+ 71I. 235IL. 25712
123. 14 523. 13712 41712. 23512. 12713
nra - 1712,  4 7123 1. 51Z. r2Z - 712. 2] 5I2. 17712
41723. (3312. 12723 - 1712. 1252
osse sol differdat dignooiet
aqu. + 1712- 12513. D
uovo teisigu frist
251 2. 1gn =3 F 13512. I2713
13. Su3 C
35f2. 2377Cd 123. 3go
= +3 12928. 1274
33512. 13712
2ys1r. 28 3iZ 1Y2 3. 1273
anteor

## Folio 68v — 00068255:0136
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 69r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068255/jpgs/default/00000136.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 98 lines, status machine
27) Hine patet etiam ib
S1. 113 aqu. p7 aricali 14 et valore 713. 313
0 et 7112 =0cessare valores ipsarum 73: 313, et binos valores
ipsarum 51: 313, quos quos, alias dedissent aequationes
11, 12, 13, 14 articuli 18, sed
loco eorum valorem prodere inde ras aequationes quatuore (723 12. 34 = (1mo) rG. 3pe 1712, 123. 312 e
ex 13tia, lire ex IIma 3 1 1723. 112. 323-1712. 123. 312 =0 vel e 28 ve
(1mo) 1723: 1712 = 1230312,:, 112. 323. et ex 13tia, (2d0) 1723: 1712 = 112. 323, :, 123. 312, et ex
12ma, B3tro) 1723 : 1772 = 523. 712, :, 5121723, et ex 14fa, (4 t0) 1723: 1712 = 812 13 :, . 523.712
Utroque modo colligitur 1723 = (+ C +)) 1712, seu vel summam eorum vel differentiam, nihilo aequari
ut jam habuimus cgrticuli 26 aequatione 9. Cter etsi valor hujus signi ex aliis determinam possit
per inventos jam valores ingredientium ipsius 1723 et 171E. Ex aequ. 1 et 2 huijus, fil 123, 312 = 112. 323.
seu ax aequ. (2 articuli 19 (6t0) 1252. 2372 = 235t.  1272 quae est aequ. 7 articuli, 26.. Et ex aegq. 3 et 4
ot. 122. 729. et substituendo valores ex aegus. initialibus articuli 19, fit utique
hujus fit 523. 7182
5). 12  (37) 28
125 seu 1254. 237, :, 235+ 127 = (8v0) 112. 323, : 123. 312 sed per aequ. 2
aes 1 (a) 31E articuli 19 est 115  112. 323, :, 125. 312 = 125. 237:, 235. 127 Ego
ex. aequ. 8 hujus fit 15 295t 1 1252. 237:, 2352+ 127 = 125. 237:, 235, 12. seu 125. 237 = 235, 127, ut
jam habuimus articula oto. Itaque aequationes 3 et 4 nihil dant, quod non jam dederitt aequ. 1 et L
Assumeramus autem ali  (aeterum omnis calculus ab aequatio articuli
18 inclusive huc usque erpori innixus fuit, sed felici, nempe in . 18. qu F aequ. 4
assumti A, 11223. 71223=51273. 31323, quod putarem permutando 1,5  9,  1 et 5 cum 3 et 7
unum ex altero prodire. In quo errami, sed feliciter. Nam hac una hypothesi assumta prodeunt omi
quae malto calculo huc asque indagavi Ae Nam per revera ex aequ. 13 artiuliia et 4 articuli 15, bit
13. 7 3 et 13 articuli i9 et ar 5 articuli 15 fit.  11223, 71225 = ) 2 15. 37, 125t 25, 17 + 2, 2
3E, 125 + 235, 117. 2372. Ergo posita, aequ. 1. fit 125 = 235, 127 + 237 = (Gto) 15 4 235, 123 - 299 2
quam aequationem effecibs expliate ponendo fies 15  (125. 12,+ 125. 2372 - 235, 127 (- 235. 237) = (125.128- 25. 233
 +95. 117= 235, 237) unde 2 125. 237 = (2), 235. 127 quae est aequ. 7 articuli 26, vel
297 Rocolligamus jam inventa (1mo) 125. 237 = 235. 127 per aeg. 7 articuli 26, vel  jam
aequatio 6 hujus.
+ en aequ. 4 artiC. 25 est 112 +512 = 312 + 712 Aa a - (2ho)
seu 323, 112. 123a - 15. 239 2 adot 125, 312. 313 a - 3). 2372 et ex aequ. 6. artic. 25, 113+5 23 = 3134 723
per aequ. 4 articuli 18.
Ecqts 312, _ 112. 122a + 15. 125 . 12 5e. 313a + 37. 12722 jam vero antea ex aequ. 8 articuli 16 vel
aequ. 2 articuli 19 l 1232. 312, :, 112. 323 = (4t0) 235. 127, :, 125: 237.
Accedunt jaem qu ex
 ex aeqq. initialibus articuli 29, vel ex aequ (H B. 13. 14 aegg. 9 et 10 articuli 14 itemque aegy. 1. et 2 articuli 15
valores sequentes 5i = (56o) - 15. 235 : 123a et 523 = (6to) 15, 1252: 112 a et 712 = (,mo)  ~ 37. 237: 323a
37. 127: 312a  Deinde ex aeq. 17 articuli 19, 513: 113 = (9n0) - 15. 1125. 235, 125 -235,:
et iD 713 313 ex aequ. 6 articuli 15 fit 713: 313 = (10 M0) - 37. 127. 237, 127 - 2373 :1, 312.313, 127 +237. sed
et in q. ex aequ. 10. artic. 16. fit 152. 125t. 2352 :, 372. 127, 2992 Hm0 s A : H H = (+223, Imon;, ester. 123 139 C
et 723 = Bv0)
= (51225. 1L3  100 H 200MM, :,  2 H213. 13 13 id est per a94. 5 et 6 articuli 1(7, 
articuli 17, (11E 232 (100l-200 mm, . 113. 513): 31223 (300 nn -200mm, : 313.713). sed per aequ. 14 articuli 14
2y 112. 123, 125 + 235
or quem valorem inserendo
in valore ipscus IN: AI fet et duos valores aequando, destinitur utrobiue 15,37 et fi R 125. 235, :, 127, 237
et per aequ. 5 articnli 15 est 11223: 312232 = 152, 135 - 235, :, 373 127 -23
= 313.713, 125-2350 (105 - 200mm) :, 113. 513 127 2- 23, (300nn -20d mm) seu fit tandem 313. 713;, 113. 513 = (11 mo
12) ((125. 235, 125-2371): (127. 237, 125- 2351)) (300 nn -200 nm, :, 100 11- 208 mm) fuide quae scripta pagina aversa articuli 261)
sed  ex aequ. 1. hic, sequitur esse 125. 235, 127  - 237 = 127 3377 125- 235 vide calculum peractum
pagina aversa articuli 26. seu est  12720237,:, 1252 235 = 127-237: 1252 - 2392 seu denique
et Z13 per aeq. 9 et 10
et similiter
113. 513(5 - 15. 115, 125. 235, 125 - 235, 29, 112. 1239125+235 = - 15. 1152, 125, 235, 125- 225 3: 4 112. 123, 125 - 255
 et post per aeq. 13 fiet
odnna adonn
3172. 71s (6 135. 112. 15. 127- 13 
152 313, 3232 125-2354 = 1124
325 ponendo per aeg. 4, 123. 31 : 112 ert 115 1233:, 315. 325 = 112 5:, 312.seu fiet
,2s. 277 sobma. vpu d 31 2. . 1. g
ex qui artic. 16 fiet
iults insiquis qu s canticuli 17. fiet 12. S12V + 123. 52)E = 11229 Spoll 2orum, c2. 113. 513), ubi
1300 nn -200 mm, 15=it. uu
1t, 12y 522 vatorem per ea 14 articuli 14, et
ratorem ex aqu.15 Exius et ponendo
(275 ergo fit S13: 113 = 15. 125- 238: - 12. 13Irr
et pari modo 713: 1132- 37, 12 - 237:, 312. 323ser et per ( 11, aequ 15 sit as 8 113 51 = 1, 13G 125 155, 12 13 
per pro 133 substituem
125 235 12 +295
q et a est 513: 1Z2 -15. 125. 235. 125-232: 1112. 1231 1
(12, 31Z, 523, 71, 712, 723, consideremus, ex aequ. artic. 3 esse _(m) 213 = 519 + 313 7 13 13
512 + 312 = (280) 312+112 et 223 = 523 +323 = (3tio) 723+123. Hunc explicando inventos.
et C2 r3: 513, (21) 15 27. 323 i 155
Ias duas aequationes conjungemus cum
93 235 4 31 213. 323a  37 125, 15 1 1 13 valores iteram ipsarum si, 523, 812, 723
(30) Habilis jam valoribus
prodeunt it pli summa aequs. 1 pro literis 8, quantiratibus 8, seu potius pro rationibus, 7. Quia in ae99. 2 et 3.
et similitur l 212-
uticuli 29, quibus si acceda
erticuli 29 signa sunt apubigua, adeo ex aequ. 2. fvi ibi, facianus  112f + sif = (4to) 312 9 + 7129. posito 9
valores in artic ag fiet
et 3. conjungemus cum tribus aequ.
itatem signo affectam, adeoque gy =1 g, idenque ff =. et ex aequ. 3 ibi faciamque  12yh +513h (sto B2k 1r23A
 itidem sunt unitates signis affectae
512 = 2(2- 312 et 712 = 212 112 substituerdo in aeg. 4, fite
3EB +9) 2(f-9). Hinc si f et 9 sint sig nondiversorum
odem modo (7mo) 213 = β27 -123) (h+K): (h- K), unde
112 F, +212F -3l2f = 3129+ 219 - e129 seu 212 ver
 conscatit aequationi l’artiali 2g. posito inroe intria ex aequiat u et 
125+ = 235 : 1252 235 2 17 = (12mo) 127- 237: 127. 237 Unde tandem fit (13tro) 313. 13, :, 113. 513= zoonn- 2odinm, :, vooll. 20 en
nisi faciumus 125 23t =0 et n 23X a0 quod minus se jam in hac eu Ex hac aeg. tollamus
et 315. 713 = 37. 3134, 127, 237, 127 - 2574,:, 3122. 3257. 127 - 251. Unde per aequ. 12 o
itemque 115. 5132 = 15. 113+, 125.3359 (25- 2357, :, 112- 123, 125 -235
 Z 12N. 127 esC
100 l - 200 mm, 37
fieret

## Folio 69r — 00068255:0137
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 68v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068255/jpgs/default/00000137.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 95 lines, status machine
Ex aequ. pro aeq. 5. articuli 19 inveniendas
oay aqu rur c pl siis mno Crnginid  uis onsce
-23712, : 9, 127234 237126312 923)
esCgala  A 712 2. 152 2723 f 2 p2. 22SI2. 2342, :  2513. 2372 F 1712. 23 512. 12723 - 13723- 14712, ., 1 723 + 23712
(2. i se quigrlut
eo bd q ur es-l uinderiu
pro aequ. 6. articuli 19 invemiendam 
it isi
  r
+ 1729. 23512. 12723 4 1712. 12525. 23712,:,7 7125. 23512. 23712 4 7112. 12513, 12723 = 12723 - 23712,:, 12723+23712
Ergo tandem mundando
al + bm me cemeat Ct a b. ad
Ad artic. 21. aequ. 1.
F seu in t a = mt 1 
aat b + bbr Bla + Hmb = fnu  F Ib fit a = B5l-Bgm, et
ast= atm = 6fl - bgm
tam aglebum = afm+ us
fmx + fly- - glX. + gmy
siu colas fma = f1b gmb seu 4 E
Aid artic. (23. pro aeq. 2 vel 3 arlic. 22. scibatur
=Cx + my erpo seu X, =
et 292 = les 3,2 4+ 5e2 = 3in - (= 15, 125: 122a) seu
212 (6) 3 12123, 312 a = 15. 235, : 123 a et 22318 112. 323a - 15. 125,: ma. seu se quia
1232 (323, et 112=8,12, et 15. 235 = 37. 235 et 15. 125 = 37, 127 hinc ut omnia enuntientur
et 123 + 313 = 3131723 ex artic. 3 articuli 29
isp secundum justitiam fieb: 212= (123. 312 + 112. 323) a - (15. 23524 37. 2372), : (123+ 323) a, id st _ 212 - 12 123 a - 15. 255,: 123 a
et 3232(112. 323 + 1230312) a = ((5. 125 4 37. 127),:, (112 + 317) a dest 223-712 323a - 37. 175:, 312a
ope ergo aequationum 6. nempe duplici aeg. 2 hic, et duplici aeq. 3 ha, et ope ag9. 2, et harticuli praecedentis,
id est 14 et 15 huius habnimus, 2, et quae sunt 6, habuopus ae99. 8, et 18 et 16, et r7 859, E0, 18
8, 18, 16, 17 et 11, et 13, quibus invenimus nondum habitos valores, ipsarum 312, 323, 212, 223
37.15, et 237: 127 vide post 1 aequ post aeg. 13. ubi feliciter evenid, ut valor ipsius 2377. 123, sit idem
superest um ut praestemus
qui jamante fuit repertus, evitaturque contradistio, in quam alias incidi potuisset. 2p
513 f313 = - 13 + 113. suastituamus valores
s -2t 12. 123. 313Ar d = 37. A7 11579 431 323 19 7
quo aqu qui valor aequandus
equ. l. articuli hujus pro 213, 513, 313, 713, 13 In aequ. + segtago
ipsarum 513, 713, fet   1seteS as per aequ. 20 ut usqe articuli 29)
valori aequationis 18 articuli praecedentis 29, in qua evanescit 312 et 12 quia aequales  et sir explicando
seu 313 :13 = b9no) 15, 125 - 235 + 312. 323 Trr, :, 37. (27-237 4 112. 1231YY,
secundum aequ. 12 articuli 28, prudit nova aequatio ex his duobus ruboribus, per quam non video quid impediat
determinari aliquam ex is Inveniri aliquam ex lis quae supersunt quantitatibus 15, 37, 112, 123, 125, a
235, 127,  237 de quibus nihil ad huc habetur quam 37: 15, et 237: 127 si in aequ. 19 7)
Hic jam tentemus periclitemur novalo H pothesie, et faciamus (20mo) 6 125. 235 =
39) Hic jam periclitemur nova Hypothesi, et faciarmus  15. 125 - 295 = 37. 127-23,, ita enim
ob. 312, 323, =  112. 123 freiet per c9. 19 artic. 30. fieret 313 = 113. unde (3ho) ex aequ. 18 artc. 20
faeret (125 - 235) 115 1300 nn -200 um = (45o) 12 (127 - 237) 537 11ool-2oomm. sed hoc  non
licet nam ex 4 et ) hujus, fieret 14390 nn -200 min = S100 l - 200 mm quod est contro hypothesdi
prestat ergo redire ad aequ. 19. et explicando, et faciagnas
n :a et ex ev. 18 artic. 29, fiet
ampendio 125 - 235 vocemus   25 et 127-237= 27 erit
 a+ vere  vt + uvoreu
17
l’ex aequ.
Yise Y300 nn -206 mm, 8 15 :, 100 ll- 200 rum, 37)
quod est contra intentum
per et 125+ 235 21 et 127 -237= 23 erit per 12 ee arstic. 29
125- 235, 4,:, 127-2770
19 artic. 30 fiel 15. 25 +  112. 123, 21. 25  ) B, :, 37 C  37. 127. 237. 27 4 312. 323. 23. 27 a  313: 113, C
introbique dividi potest per 25 et 27, fiebus
27 + . 923. 23. as
 37. 127. 13
15, 125. 235, 25 + 112. 113 2. a
i2 (idest ur. 123 vel 3123d9 
unde tandem habebitur n2. 123
Ere. 125. 235 - 37. 27. 17. 237 ita taudem
. 23 ab - _2 2) a 
es in quibus nondum implicantur et reliquae i
seu 312. 313. eu fiet 112. 123 4 312. 3
25, 225, 17, 237
(ex quibus pendent 25, 27, 23, 21, B, V) sumi possunt pro arbitrio, quantum patiuntur jam habitae deter
112, 123 vel 3(2323, habemus per Pitera
nn = 2od um et similiter
minationes, et scopus. licebit etiam ponere 15Brest DDag:, 3de et
atu
inventas quantum
quat conditiones postu laut, determinatas  112, 2n, 31E, 18, 213, 313, 123, 213, 323; jam absolvamus
substituendique erunt hi valores ubique pro 37 et 15. opieri pretins upere
u
utile erit i exprimi tam vec v quam et
171et 3 artic et init. artic. 8((pro 4e, esp
(19to),   , 3
explimere ipsas v, Z, y inde terminatas ubi fors
et dendo aeq. 1. et 2 artic. 16 pro ut tandem obtineamus irrationales Suy, Y0Z tyZ cum ipsis pqdere
Mer y, adeoque et reliquas vide antur aequ4. artic.
 en
artic. 30. et nitio
deinde an sabis fiat initio articuli 10, b. tum pro valore
possumus intir dissimulare quae instituimus
riationabs in aquiutio sata Hi pu n
idalonis pratio dantur irrationab, in aequintio dati His podib, um q cemratius
parte ulie  eiupari
aibenta salisfacunt eq altuonali
12723- 23712.23
12723 + 237125312. 313)

## Folio 69v — 00068255:0138
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 68r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068255/jpgs/default/00000138.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 118 lines, status machine
Quemlibet jam valrem   ipsius 513: 113 repertum in 8gy. 13 et 14, articuli 18
aequabimus valori invento aequatione 16 articuli 14. Et similiter quem libet vactorem ipsius
fient aegq. (1) et (2) articuli hujus. Unde so ex quibus consequitur  aequatio qu (3) hujus quae oritur ipsos
duos valoris in aequ. 13 et c4 articuli (8 aequando inter se. Et a similiter quemlibet valorem apsius
713: 113 repertum in ag9. 11 et 12 articuli 18 aequabimus valori invento aequatione 6 articuli 15 fient
094. (4) et (5) articnti hujus. Ex quibus consequitur, aequ. (6) hujus articuli, quae oitur ipsos duosvalores
in aeqq. 11 et 12 articuli 18 aequando interse.. Et has 6 aequatiomes distincte scuibtre supersedeo prate
3 et b. ubi videndum an non compendia prodeant Nempe ae
0 aeq. (3) R. 1123. M (23. 513. 312.712  7 17 2. 112. 512.323. 723 + 7112. 7123 (112. 523. 312. 723 + 123. 512 . 523. 712)
C15) 1723, 118. 512 323. 723B1 712. 123.522. 312. 712  1712. 1723 (112. 523, 323. 712, f 123. 312. 312. 723)
et -71 23 + 1723 2) = 1212 - 71   et
sed ex aeq. 8 articuli 16 est 112 Si2. 323. 723 = 133.523. 312.712
et -1712. 1723 f 7112. 7123(2L - bis 12+512, 323+713, 123+523, 31 2 F 7 (2
lurgo ex aequ. 3 sic explicata
1234523, 3121712 1124512, 323+723
Lit ) 12. 513. 312. 712 + ()12, 512. 317. 723 = (2) (1124 51, 323 +123 4. 123+ 51Z, 312+712) (112.523. 312. 723 = 123. 512. 323. 712) ̀ qu. 0
Et pari jure ex aequatione 6 explicat a debet oriri aeq. (11.) nitilo differens ab aequ. 10. sola permutatione
+523, 312 +712
112+512, 923+723
inter 1 et 5 cum 3 et 7 in notis initialibus, sed ita pulchre evenit ut aequ. 11 et aeg. 10 coincidant
quoniam aequ. 10. eodem se modo habet ad 1 et 5 Ut ad 3 et 7. Unde cum aeq. 3 resultans ex 1 et 2 idem
Det quod aeq. 6 resultans ex 3 et 4. consequitur aega 1 et 2 itemque 3 et 4, revera non tam pro quatuor
quam pro tribus aequationibus esse habendas. Unde ex his 4 aequationibus possent excerpi decussatim duae
ex 1 et 2 aiticuli hujus
et eis conjungi aequalio 10. commode, etiam sic procedere licebit, ut loco aio  psiqus 713: 313 per valorem
excerpendae, consideremus dividamus ex vt a si C D.  S. sIE S valorem primum
aequationes 13 Ipsius 513: 113 ut fiat ceq (12) hic, et nursus ax artic. 18 dividamus valorem aequationis 12 ipsius 713: 313 per
valorem aequationis 14 ipsius 513: 13 ut fiat aequ. (1) hic. nempe rursqualor novo  ipsiq. 13. 7B3; E3 Hos duos valores
sed aequando cum velore  ipsius
aequando inter se fieret  (15) quae tamen videndum an non redeat ad aeq. 10. hicjuse
113. 713, :, 513. 313, ducto ext  17 aiticuli 14 et ex ag. 6 articuli 15. habebimus aegq. (17) et (18) hujus ita
aeq. 15 non coincidit cum ae. 10, poterimus conjungere 494. 10, 17, 18 onquam sufficientes ad omna determinanta
videndum tamen an non ex i’et 2, itemque 3 et 4 huju non possit fieri aliquid simplicius quam aequ. 19, 17, 18. imo
 re considerata apparet nit posse fieri melius; ob aequ. 8t articuli 16 nobis profuturam. Nempe dividento aequa per aeq
112 (2)  2123. 112a723 = 2112. 123.712. sed per aequ. 8 articuli 16
fit 112.723, 1,523. 312 = (1440) 123. 712 :, 512. 323 = (1gno) 17. 55. 18t   ex aequ. 12  Denominator voloris
ati aequ 1 articuli 18 per q aequ. 13 ejusdem fit
2. 323,512
scribi poterit 7123. 112. 129 (53 : 17) - 7112. 123. 712 (53.17). Ergo jam praeclaro tam exitu evanescit numerator aequationis 2
de primitu denominatorem, et it
jam ex articulis
= (17mo) 171= (pero, 31   1d. 727. a qua
 Et ex equ Himc itaque aequ. 15 est denitica  nec quicquam faci
(12. 123. (12523 +23512)537(12723 - 23712) 12723. 23(12
(312), 323. (12723 +23712), 15 (12523 - 23512)12523. 23518
523. (312
=15. 4) 12523: 112 (21mo) et 223 = (3724) 127235, 3 12 et ex eq. 17
articuli 14 e st 523f
qu. 9 112. 127232373 : 312. 12515. 15
unde ex aequ. 17  tandem fiet
123. 312 (12323+29512) (12723 - 2512) 12523. 23712 = (2284) 112. 323 (12723+ 22712, 1252. 22512) 12723. 23512
non differt aqu. (18.
723: 523=
je, aequatis cum aeq. 17 articuli +4, et aequ. 6. articuli 16 oriendis, et fiet - 7123. 342. 523 +7112. 323. 512,:, 12. 323-
+2251Z ubc Ro esta
1712. 123. 312 = (1mo) (51. 13 L) - 15. a () 12523 - 23512 (12519 - 13512) 2, 12. 123, 12 315G
ubi pro 523 et 512 substituendo valores (15: a) 12523: 112, et - (151a) 23512: 1237
 ex aequ 1 fie 2 00 equ,
+ 7123. 123. 312.  12523 4712. 112. 323, 23548 (2 3t0) 12523. 23512 (12532 23512)
+ et stiuendo 3 et 7 fet 12523 + 23512,, quas Aegq. 23 et 4 seponamus et
deu 12. 1C13  (21 2523 = 123 23712, :, 323. 385 1 at a qull ent Eili s egrienti dug +123    s
(2sto
ex t. eZ isto  B523. 21712,: 52723 23512. 12724
(113523 + 235I2 C, 127 (3- 23712
under rereausi 
seu 12523+23512, :, 12623- 235(22
223+ 23712, :, 12725= 23712
12628 a Eeden
2tme ) 7123. 112. 323. 12) 23 - 7112- 132. 312. 23713
fiet: 7129. 123. 312- 1252547412. 112. 34. 23512
23. 112. 323 - 1712. 123. 312
et27 437(2, 1725. 13. 3M - 1112. 1123 13
Ar9. H nug un r f1235 17 27r2 eq
12727 Toestuendo ubique 1igi B3, fiet
(1251 1  (23312), inum n +
manenim
12523. 23712 171t
a
=(272 (23712
seu 2Z. hur. 25 S12. 1371 = seu 13 1s  23 f 1 
vel Yi
1  (sis)
L 112. 92. 232 17
 11 123- 23512 X a) : 123 3. 112. 123 + 1252 65. a),: 112
H SI ZBa 127L3 323. 37
 312+712, 1323+723 = 312. 32 a -23712 15 (2. 375:, 31 4 5 a t. 725. 313.37
f312 a, 112. 285(2. 12723, 123 (312) 12523. 23712 HE 2572 (3123h.1. X35IX. 12723 :, (31da. 15. 312. 12525 2p112 + 1272 3. 52 57. 113(19 1251G X(112. 112. 1 2512. 12713
Sen f 12. ct. 128. 12. 14523 ~ 3B12. tereBESI 2 M. 31. 2s En A has EsuIE.n reS. HE. Msi. AEYEa H 12. 128. 37 ag 3 3sr2. 1425. 23 2
ed per aeq. 25 323 = + 123. 3(2. 12523, 23712, 7 12. 23512, 12713 quo subst luto sit  312. 123 a. 12523. 23712= 112. 312. 25715. 23M. 12723. 37: eue qo
2a = 2351R: 112. 15,:, 112. 12Bat 125 . 123. 15B L 11I 45I2,: 11345,8
12322(127~). 712
pr anenter
ulbamus et p2At 13 + CB in eadem eu. 10 funt jam 112 = 31Y. 120 1E. a et 523 (1 ee. 12) is. a aambus
19.7 123. (31: 4) 237122. 313
es + in  as   seufps)Binm cm
Ear. 12725 = (ob 25 hic) 4 12. 922. 12725d. 12523. (237 12 = 4 12523. 
12 magu
Nunc superest ut coy. 10 evolvatur, et ut alterutra 23 vit 24, sola. Et pringum pro aqu. 10 tollamus 512, 523,
53. 2  et ibi habebimus inter alia +13 Z
et 712 = 65: a) 2351 enfunde 123. (12712
2.73 1(2 (37. a) 127237312
utAI. . 2IEy B 22.  1 
351226s tuendo valaem ex aequ. 25
unde ubi pro 3F 123. 312
173812. 23512
723+ 23712
123, 711
27iZ. 112. 93 1212 _ -2112. 15. 3I2. 23718  1ngrg. 23712 (12733 -25712) Se Redeamus jam ex aeq. 22. proqua consideremusquod
1723. 112. 323 = 1712, 123. 312, hujus
fed ex aeg. 14  hic pe
112. 323
placet tamen ad huc a actu ipso exhibere aequ. 1. hujus articuli quoniam hactemis  quator aequ. ex . 11. 12, 13 articali
fiet: n seu
ipusa
13. 712  7112. 42. 723, ubi  in dem de nominatore rursus ob a9. 14. 15 pro 512. 323 ponendo
ex aequ. 12 fit
123. 712 (53: 17), et pro 523. 312 poneldo
et

## Folio 70r — 00068255:0139
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 67v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068255/jpgs/default/00000139.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 98 lines, status machine
124) Sed quia nondum satis hinc compendii, conplegams, consiteremus quod sit: 1723 - 7123  57 np
et per praecedenlim articulis, aegg. 2. 3, est 1723 - 171. 8: 8. et 15, 1725 = 1712.62A 17123 = 31120.  et 75 = 7128.5
Ergo fit  1723 - 7123 = 17p 6 V = 711284: 1 4aa, ubi  ascribae aa, ad implendam legem homogenemta
S - 171259 4 4a20 4 =  4aa 3 9r. seu
 19 a) ita possumus eligere
sed per 1  1. hic est 7112 = 1712- 4aa et (is
 rm vero  et proine e  lic sit 713 = 0  et quoniam
sed idem efieret calculus pro 1712, et 7112, si modo permatentur qui pro 1323, 7123, sa modo inter se
O; D, itaque fit etiam BE 173 4aa; - 58 = 6to) 0
et si iterum eligimus 125 qaa mop fiet 7133 = 0 seu 7123 -0 si jam 115 
(25) Tentemns au liceck facere 125 = 0 (mo) 7123 =0 et (2do) ex 71a ex privie siet
1723; 7123; 9; 8; dC
Gtiv) (123 + SiB  323 +713) i (323+ 723, :, 123+523) = 0. jam si (a 1b)- b: a = e fit aa = bb
unde (450) 7E fit 323 + 29 2 (4), 123 +523. Et ex aeg. 2 pari june 312 +712- (+)), 123+523
aa S,  5= O:, 5~-0∞2
fit 7112 = 4 aa c 7 4 aa D∞) 4 aa 54  40a08,: 87 58 seu fe fit fit
(5to) 7112 = 4aa DD, 8 - 6, : 165 - 07. it a enim 1712 - 7uE - 4aa, (E D 55(5+ 56:, 6109 -naa t detebat
itaque sit etiam (5to) seu (4to)  fit 1712
Nam tantum opus est ut
cidn svi poteste catelis ee  3, sed idem caludus fieri potest pro 1723, 7123.
4a6 E3: 45- d6 seu +aa 3, 8-8: 6Y -0 g ita salis facit aequationibus positis h
2a4 5, 0 5,:, 80495 seu 4aa 53 5-8, 85s
inter se permutentur respective
25 Q ex inspectione ae hypothesin quandam assumendo, quae si satisfucit, nec ideo plures riuntur quam
Quoniam autem aequationibus salis facimus, non tantum quaerendo valores
hic positis videamus an non
alterntio enim assumta modo ne fiat Cd vel 7=0, eriit
 non corit nos plures assumere aequationes, quam initio habuimus
vtam radicem quod genus hypothes eos necessariae satis, nihil
b altera 20 nam fiet 5 f 2)0.
liceat facere 7112 vel 7123)  esse
62 qt0, adeoque et posito. fie
=sibelin vicure ina
et (2d0) 7112 =0 et (3tio) 2123 =0
a = + b et fit 312 + 71E = 4) 12 + 512
et per 2, fiet 12 s1t. 31ri2 E
=0 seu aa + bb =0 seu a = b f-1
5 et 6 pro 512, 712, 523, 723, substi
(7.a) 23: v L (4) u2 -e5. a). 255.  123, se
623.15. 235. jam a ut ut ex gu
in aequ. 7 permutabis 12 et 23, fiet
poro ob ceq. 3. fit similiter, 323 + 723 = (66o) (+γ, 17 +523
52D  esset (4) ex (9 fieret ex C99.
esset 4(+)  (+D addendo
patus qui habentur in Lpad) X) iio 3 ac
ex 7123=0 f9quitur 325 (8v0) 323+723= (+BaXC
Bu2. 12i - 123. 237) 37_ (12.S-313. 235) 15 sed si esset d (+) ∞C
(+)) I2. 127 1. 137)37 ((f) 312. 135-323. 255)15 In quorum nentio
asenvicen subtnctisu i ) o7-(3 15 313. 255) 15
(Gno) n2. 323a 113, 37. 127 (+ 1) 13. 3 a
iustitiae
simplicius et servatur ponendo (+) ∞(+D servatur lex
et ex qu quarta, velus (2d0) vel raisus 15 2350 vel
ex aejus. 7 et 9 subtiahendo, ( 123. 31a H
incommodi indeo prius autem paulo
 vel in ros ro vel minoni   vel ut riptdhet asque ru no vel momo rrti a s u on +. mnerr - 0
vel rarsus 1723. 125. 237 4 171. 23
liti isa M iat o. sb
23 235. 327 4 1712. 125. 237 =0. Hinc jam patet utilius esse n
ve pussus 127 237 =0 vel Et ex quartani
123. 175. 237 4- 1712, 235. 127 - 0
rita qua ur rvera
satis facit per facilias per panciores
i eexesolis d relra plot
u artes 20 levera habendiae solum
ere 1r23. 125. 23) 4 1712 23511, =0
ease nos sequi electiones, qua idem bis s
aq. 5 fel mru et 8 hit 1  127 seu
etsi non nisi tres fiunt i mobis hypotheles 5 F aequationes
mo)  +235. 1273:9 125. 237 seu unde 125. 237
go si duo valores ipsius ipsius 1723: 1712
no) 1723 = (712 sive 1723= (4(+)) 1712 ad eoque fit
t
Evolvaum
unide colligas revera obeng. 5 et 6.  non n
235. 125.237 =0 seu fier
amen non prorsus constat. plrcet ad hac
(+N 115. 237 seu  235. 127 - (3C=B 125. 237
multiplicentur invicem fit 1723: 1712=
determinationem, quoniam, ing : 1712  sam tum habetur. taque sola hini prodit aec
jam patet sequitur ex. 12-0 vl et 7123 =0
lesolvere aequuios t ) s 
((4)) ∞ +. fiet (urs. 235 r27. 17 (4)125. 235. 127 237 = (2) 235. 127 H225 27 +() 15. 237
quia sic inciditur in aequ. g. utroge
sin fiat (3C+)) ∞ 4 fiet (2)(125.235. 127() 235, 127. 237 +(2) 1252. 237. 127B (2)
sinc acamus, patet plucet
125. 237 (125. 127 -235, 237)70 seu 235. 127 + 125. 237
etiam patet alterutram aequationem hic habitam 5 aut 6 articuli hujus, salis facere aequationi
uiam nunc omnibus
275. 127 (125, 127 = 235. 237)
∞ + fit ( 2 no) (4)) x+sin
modo aequationis i0 sensu e si()
Y. Z Natisfecimus, ut recollatio
necollectis omnitus auc contoribus, inventis satisfuciamus etiam duplici valori ipsarum
aequationisus ad destruendum irutionales evitandaue
nib ipsis 512, si3, 523,  quam  in 7(12, 713, 723
 ps sur atu u e bisatur alimi prart osfuntars
2IL, 9∞ 213, 223, inventi quae latent u vo

## Folio 70v — 00068255:0140
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 67r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068255/jpgs/default/00000140.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 19 lines, status machine
Haec ad aequ. 1 artic. 29.
22
2352 1277. 1
A et
1H
315. 12 X. 125)
125. 235, 127 - 2377 = 125 - 125. 235. 127+ (1257. 235 (23a) 2 (1257 235. 127 (237) pro
127. 237, 125 - 235 = 129 (237) 125 125 (372 235t - 2. 127. 237, 125. 235
125, 235, 117- 237
127. 197, 125= 235
1352. 127 (153) 235, 125. 1252
235 127: 125
125+ 635d 127 + 2350127t - 2. 125. 238127+
125+ 127 + 235. 127 - 2. 1252. 235, 12,
125+235. 12.st + 235627 - 2. 125. 238. 127. 
127. 235, 125 -235 = 125 235. 125 +2356 1274 = 2. 125. 235t. 12.  =
ergo 125. 235. 127+ - 237 = 127. 237, 125 -2352 C
1254. 127
125127+ 235. 127-2. 125. 115. 1272

## Folio 71r — 00068255:0141
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 66v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068255/jpgs/default/00000141.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 118 lines, status machine
In aequ. 1. aitenl praecedentis 16 poterit expliveri U1213. 1L313. atbu Seu W2 61B in 127. q = W121a5 31E + 22. 1. S1
Ergo H213. 1231. ADCL 112. 123: 113 = 112. 125 . 513 + 1223. A ~. 13. 513 + 112. 1B. 512. 523. uside explicando fit
M 125 38 - (12. 123 bb 2. 
Mo3. v2. 123, 513 = V quh qu e. 12) b. it )
adeoque tandem u2Bh. 12313 = ubb. 24. q, BC. 16h + g, 1152
posu asu satorn qegit 
g33us iht Bhh
 Ahl. 9Y (-29hater. 123, 6. E
lit f. 28. in trillnst _ sit b  ( y . ur 1es Ul mol:  t 192) N au. D8
A, 162+ +. VY
sinricin nud 2 2t 3 ila omestandem irrationales ex calcude sunt sublatae
sed absolvamus penitus aegu. 1 et fiet
Gat  M no s s V 2h+, ihaf = Eh + fa9 et fil (Em) me n2. 123,. 512. 513, 2h -f
rgo ob et ex 9 et 3 fiet
FB(12. h23. 52. 123 = atgg: (11 123. sn 525 ta 9)
An sic potius sit s ut e artie 19 aeq.  vel s ut succedat extractio fiat 2h+f = (guo) 13. 523, te  d: 
(ob en artic. 14 aequoo, ut sucxat exi
i 2h +5,  g9 :tfiol f. h
123 1 (13tio) (11223h = fh+. 5) 49
ergo ex aeq. 3. per y et 10 fiet 11223h = (4mo) 112. 123. 512. 523 ad: R, bc
Et duos ipsius h valores aequando   et i 
siet 112. 523 2ibb) + 123. 51 (CC) +. 1. 123. S12. Sig ((. 6y 4 ocC)) = 4 H 13 f12513 (D. bc
fiet 112. 528 (1: b6) + 123. 512( : cc) + 112. 123. 51 2 525 (1 : b6) + (1: Cc)) = 4. 112. 123. 512. 323(1: bc) Unde
faciendo H2. 123. 51 seg ut jam determinavimus oportet
ras nec quicquam h
set si  24. videtur hac aequalit
auluusus heris
112. 523 (1.6) - 11, 312 (1. C) = salua no vi b. b) e
facere aequalem quadrat
licentur in ea f et h
ad quod postea recunitur
et (10mo) 123. 512 = 15. 2312: a. Ergo
ergo praestat statum facere ob aeq.C
(1Imo) 12. 129. 512. 527 = = ( 152. 14232. 2312: aa et 1 - 12. 123, 512. 523 = (12 mo) (1B 15. 1223. 2 212: a
sit ergo (9no). 112. 523 =  e ex 415. 12523: a
exaoqs et 51223 = (13t0) 15. 18237: a + 15. 332.a et 11223 = (14. to) 15. 13523( a) - 15. 23s1 2L. a)
iter se percuntant officia, et cum i 51223 prius contineret + et =, sed 11223 tantum +, nunc fit contra
Ergo in aeq. 4, retinendo 112. 123.
513: 113 = (16t0) -(15 : 4) 1825. 23str (= (15: a) (1825 = 2312)), - 2 4 (1225 + 2351 2 , ., 12523. 241 2)
fiet f: h -(1560)  12 5t + 2 32, (152) : 12523. 2.3 = (1514)
(17 mo) 513: 113 = (15: a) 1823. 2342. (12523 623)2): aS B1) 112. 123 (12(23 + 23)12)
suppono autem 5_2 et 52
513: 113 Lae quationis 4 sic
Sed per aeg. 15 et 5 - at ot = (19 ab) 12 .2 HE :, 13  B2
Ergo ix 18 per 19 et 9 fit 12. 513;, 113. 512 = (15∞4) 12523. 122Z = 2212 :, 1sL 3. 2g2 (15: a) (12t5 - 2342)
inveniri per 112, 113, 15ca, et r2z et 2712
112. 513, :, 113. 512 = 112. 523, (f = 2h, : h): 112t23
123. 512, f - 2h, : 11223 = (per 10) -(3: a). 2412 (1323 Lyp2. 1223. 2252). (15. aB, 122 = 23E)
112. 513, 1, 113. 512 = (20m0) (12523 : 2392), 1223 - 22423 :, 1243 + 2312
(2205) 123. 513,:, 113. 513 = (2212 12) 2 22 s s
123. 513:, 113. 513 = (21mo)
ii
Fajesi) et 31223  . a) n  (37 : a), 12725 - 23712
  t 
aa et 71225 =(41) (7: af, 12725 + 2371E
L 3f. 1272551533(. 2123 - 23712): (12723 + 2371331253, seu
312. 713:, 313. 712 = (3mo) (12723: 23712), 12723-23712, :, 12723 + 23712 et 323. 713;, 313. 723 -(800) 23712. 127 25) 23712- 12723, :, 25712+ 127 23
et (2do), 373. 712 = - 37. 23712. a et 312. 313. 712. 725
A     apiai ure it
et 713. 313 = (6to) -(37.
aeq. articuli 12 designaitae articuso 1
nat = (5to) 0. quae ne habeatur v: Z divellenda est in duas, et fiet
163 Hinc jam _ at quia per aeg. 14 satis fecimus extra ctroer in. et pari jare 312. 7126 + 323
L51223 YEptC13. 1213 3Z 4 7 122N = (4. to
fint + 12 Si. Amarev + iy 123. pa  - s, tA B nde l) Msir, 1, 3nop2 = 129. 523,: , 321. 723
in   3 1)o et
non
6, 7, 8 duae sufficiuct cum ter
B_AA, :, 721C)C= ubi par  per ag, 11 articuli 14, et autre. qu. 3 artic. 15. apparet utrobique
M2. St. _, :, 312. 7 12BL sra AC:, nBA  Bpu) 12 1. 52 3, id) 2 H  a q
V9( 1121612. 323
123, 523. 312. 712 quod institiariam multiplicat
onem et fet: 5 ed) n. SC2. 12517 13. 15. 125
123. 523. 3I2. 712. nt antem ae quatio
seu 113. 523, :, 5tG. 723 = (1mo) +) 5M, :, on3 H = 112. 512 s’ podit au
2  im. 527d. b_) = 127. 512 tCC)
h- f = (0 mo) 123. 512 ad: E
feliniter exenit extractio
cc. ih. 33 t  s et C 8. y. Jnda  aa b.B, B
et fit (u2 23 (l mm) 16)  2h +, D.
pari pre 31213. 3231+)X
Th 112. 523(: b) + ,29. 5121d: C2)
rachonem
Nita redimus ad eq.t
rst
compuno iasrevhrn
t 12. NS. SM. 523 - -1b, +hil. 1. 
Seu (1m0) S1 = Au 2. 123. 512 125 = 11223 112. 123. 512. 527, - 2, ME. N3. 12 S3) Ergo Aa3 esB3. M. 13. 13 = bb. ih. q Eh.  ist  - 4thr 9- +h r  chi q -  n.  pats. it il g.  . q iB
gutrus 2 l122(71227
 AI: n B
i. 12) en 13. sedium
Pit pari pec ex ajo.  alriansous estpis
3. 113: 
 iu
a 61q 513: 113
1122
112713023712 = (10 mo) otu M.
Sed quia in s, continetu
alitiome et pcab
ilimque duae aequaturs. ies. 2Eu (F
peia
ant rediietionems
 ungur rto  besetes
byg. 10 et 12 artituli 10
1527) (323t723) i E 23. 112∞D m (am
3t 2 V2. 123Si, DI2tII
 ar it si-
 H2. 1L 3
quis duo valpres equandi,
surds otro
2123 (123+523) (327
mysore dis utia nm  aspro, st q. L, v  
nnon, mesusat.n) iss
 ters, et exrqoe
32g  is iti n situ ubi 
n ot mua antu ititand aeuaa ad
Eex as per
ome 713

## Folio 71v — 00068255:0142
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 66r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068255/jpgs/default/00000142.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 19 lines, status machine
(B B
in Artieuli 18 aequatione q per errorem putavi festinans esse 11223, 71223 = 51223, 31223
unumque ex alio fceri permuta ndo, sod termitiosque easden LS;  15337 seu berminosque eosdem permutando
manere. sed revera hoc non succedit per se, sive non est hec aequatio dentica, sed succedit
per novam suppositionem, quam reperi esse felicer; et aliquid ende prodir
quod ex ista ipsa hypos hesi daximus. Nsempe
11223. 71223 = 12. 523 + 13. 512, 312. 723- 313. 712 Hec posui esse edem substituamus vilores
S12 23, 31223 = 112. 523 = 123, 512, 312. 723 + 323. 712
indentos Haec pro aequalibus habui per errorem, viitendum tamen an non is succedat ex jam inventis
hac hypothesi assumta, ut ita enor justificet se ipsum, seu fiat felix. substituamus valores ipsorum
5+2, 3 723, 323 fit 1 Ex aequ. 14 articuli 14, et 4 articuli 15, flet.
11223. 71223 = 15.37, 125 = 2352, 127 + 297. Et ex ag. 13 artic. 14 et aequ. 5 artic. 15 similiter fit
51223. 31223 = 15. 37 125+ 235, 1272372 quae aequatiu Ergo fic A 1  1223. 71223 = 31223. 3223. fres
125= 235, 1274235 125+ 235, 127 - 23 p sed per aq. 8 artic. 26 est 125t 2 = 235 1 habemus
Ergo non consentiunt interse sed est nova determinatio, et
praestemus actu ipso
(125. 127 + 125, 237 - 2357127(335. 237) = (125. 127 - 125. 2377 4 235. 127 - 135, 237
et sit (2) 115. 237 = ( (2). 235. 125. sed ita incidimus in aequationem 8 articuli 26. itaque error
est felix

## Folio 72r — 00068255:0143
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 65v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068255/jpgs/default/00000143.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 88 lines, status machine
Extractione igituo fucta ex aequ. articuli 12 sejundum articulm B3, fiet
12. 512V + 129. 523Z= Sir23 ((11223CU - mm.) I. p (1213. 12913)). seu per compendium
H2. S12V + 128. 323Z + 109. 100 a =0. Et pari jure ob 8, p- 0 fit, 312.712D + 323. 723Z + 200. 200 a0
quae duae aequationes debent multiplicata ut debent coincidentiari, ne per eas reperiatur
Y vel Z, vel veZ. et fiet  112. 512. 200 aV = 123,523. 290 a2 100. 100 a 40 quam aequatio
nem jam identicam reddendo, fit 20: 105 3121712. Mp et 208. 100= 323,733,:, n35 Unde
15) Omnia ergo rducta sunt ad aequationes 5 et 6 arsiculi hujus, itemque  3 et 4.
1. 2, 3 g articuli q
vgrte tamen adhuc se imstendia quaedam
et  semper let u3 vel siem T1213, S12B, (1413, 52313 solum
comptendia
tantum extra haec t occurrit 13. 5C3 in valore lationali MI3C iC. 313 + 13OSAZ et 1X5B. 2
qud seli et  ieatoribus P, 
e . et H2B i si3 n sI. sit mBsB = I S) an ergo praestat tollere 113513
quam caetera. Nempe 11213 = 112. 513 + 1γ. 512 et 1213 = 125B- 113.512 ergo addeado
11213 F51213, :2) =. 513 = 123(3 +52313; 2 2. 123 Hinc 123, 112I3 +H3. SnB = 112. 123 Gru2My
1213 - 51213, 2. 512 = 113 = 12X13 - 52313; 2. 523AIc C5. 11213, 52.51213 = 5I2 NN , 5yV
monisin
quod si  velimus tractare eodem modo fiet:
mess meneurs erdussus
1213 = spH + (231 - si5 - 11. S13 = + 3. 523. (27- 195711215 2. 8. m B
4112. 512. 1532 112. 12. 52313)
et Rationalis aequatio destructis irrationalibus erit
utim ( us ib videguns an aequationes  4 articuli 9 possint
u
jam 
Ex aequ 1. s  fiat - 112. 513 + 13. 512-
-312. 7B3 + 313.712
fit 12. S13 + 113. se2 - 571
-123. 513 + 113. 523 - 1323
1312. Ex a9. fit 2, 3 73t313 723 3123
B pro 1, sit  1 a + r2bl. L  0 pro 2 sib ad 2a
129. 511 F 13. 522 - 5 23
pra dsic Matp = 3 pro
pro 4 sit na- nd 21 40
Ex isit na + 12b = 10 (1d + 14).  Et ex 3 est na - 12b = 30 (13d - 140) Erg. 2. na =  ( d 312C) 
deniqu pro 3 sit 4la atrb 3 30
2. 21a = 29 (22d 1249) + te su d un BC)
Et eodem int ex aequ. 2 et 4. fiet Et eoder
2. 12b 2 (0 (1d +14C) - 30 (13d -14 e)
222b   (23d1240) - 48 (3d-24e)
similiter suburendo ab aeq.). sustichendo aeq. 6. fit
et ex aeqq. 2 et A pari jure
Ex 7 et 8 fiet: (12 13,  + f 13, 22 24, 40 23d) d = (2 te 14. + 28 14, + 2 14, - 4e24) e 80
Et eodem mod ex getiosis (2. B n - (3e 1u- renn +. 40 21) d  = (- 1∞3. 1472 4 0, 4. 22 + 28 1412 + 20. 24 12) e
Eiqu ex 11 et 12 fil (10. 1 2) + 30.13 21 - 2a. 1125 - 40 241) (- 1014 2 - 2 19 21 + 20. 24. 12 + 40 24 12)
1apat - 30. 1722 - 20. 1712 , 40 11 n  (- 10 1 4 21 + 30 14 21. + 20. 241 - 40. 4 1)
prasabit Addantur in unum egus  et c2 jam 2+ 2 = sar 123B
11223.51225 (inise
2. 11213. 2313
et deni
Et subtrahendo
=11223.51225, 52312 (l mon)
= 312. 712. 1001. - 323.723. 106.. = 100,. 200ad
etiam  112. 512. 323.723= 123.523. 312. 712 seu 12. 512:, 123.523 = 312.712:, 323. 723
x articuli B3. aq3 et 4 quaecamus 51: 113, et 713: 313. Nempe ex aeg. 1. fit 51 F 4 Scus 5i + 4. 5u s.
4. 1123. 125. 512. 525- 11. 123,. 512 625. 1225. seu
((12 a  mte (4. s 51s
u2. 125. S1C. 325 = 112. 123. C. 312. 512. 123=Z
112, 123
determinatione consequamur
11223, 112. 123( to) 112. 125, 512. 523 H: aa piet
seu (4to). 112. 113. 312. 5(3 = 11225f. ) atih- H
faciamus 4.113. 123. 5123, 523-112. 123. 312. 523. 112257
4. 112. 123. 5p b23 11223 = 112, 123. 513.523f.tah
et fiet SeZ=  (112. 123. n2. 513 0(f. 2H) f seu =)  512. 51Z (f - 24). Et ob G, F-C pari
Mg. 12. 13t 31223 a: +ka -99  et (7mo) 713 = X1. 724g- 219.
turc fiet (670) 312. 323. 312. 123=
dum tamen res pentus ad rationales redacta est quia 12 173, Si2. 523 latent in 1122
112. 123512523(4aa - B) ~ 112. 52374 123. 512,2. 112. 123. 512. 523XA. Unde transponendo
fit. f112. 123, 312. 523ff = (9n0) 112. 525 aa + 123: SIkR = 2. 112. 143 512. sB Hk, ergto
tandem fit f f- 112. 113. 512. 523  ffi12. 523la 4- 123. 512h =4 51223k.
 Si faceremus () aha -Y = 0 seu f= -2R. quia ob eg. 5. non
licet
Ergus faciamus b, aAA- fs, Cd, = M. 523A + 123. 3. ita nulla fit nova determinatio
et x asg. 9 fiet 112. 123. 512. 52317 (112. 523 ab + 123. 512 ab seu =) 11223tb seu per n et 12
un fit B. 123. 512. 52 = (1310) bb, 4aa - f. Ergo res reducta est ad aegq. 5, et, 11 et 13. quod
jam seto com jam bb, ata- s compendio vocemus  B Nam  majus esse debet quam ata seu
d ps  quel. 10 113. 123. 512. 527 est quantitas
au atoremjam ex aequ. 1  sit B = B
112. 525 ab + 112. 52B  ab B) B 112. 515 + (P. lb) 112. 523 + B8. as bb
30 + 4 aabb3: 4habb. seu 12. 523 ub (B 2ab) ( (+ aalbb - β: etb
sergo ex 4 et 18
jam 3β2 b SB. qua et B aabb dabb sbb et B  b fona _, ergo fief
tanb
  a ia oses it siasit e s
sis reduci potest ad rationates
uie art asnse

## Folio 72v — 00068255:0144
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 65r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068255/jpgs/default/00000144.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 2 lines, status machine
RIED WILHELM LEIBNIZ BIBI
EDERSACHSISCHE LANDESBIBLIOTH

## Folio 73r — 00068255:0145
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 64v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068255/jpgs/default/00000145.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 111 lines, status machine
id cnfendo aequabiones 4 et s articuli 8, eum aequationibus 7 et 8 arti culi 7. habetur
quaedam depcressio. Nempe multiplicetur aequatio 7 per 4, et 8 per s, fiet ex priori
multiplicatione Sircgl. 31213. 5123,:, 7121, 3. 14213. 7,223 = 51224 (112+ S12) : 712253(32+7,2). Sed hrl: 1113 -521
et ita in pluribus. Ergo fit: Sin13: 71213 = (1mo) 11223 (112+512) : 31223 (3121712). Et ex posteriore mulbiplicatione nempe y8 per
Cq.5. fit 523131. 32313. 51223,:, 723133. 12913. 71223 = 51an 1 (123+525), : 412233 (33+723) unde factis destructionibus
fiet : 52313372313 = (2d0) 112 23 (1234523): 3129 (323+723). Eropa harum aequasionum 1 et2 articuli praesentis, Et
ex aeqque 4 et sarticuli praecedentis fiet 3ng. Im  t) li13 . 31213 = (3tio) Si2e3 (312+ 712) :72 .3 (112 + 112) et
Et eodem modo labebita
12313 = 32913 =51223 (323+713): 71223 (123+523). His igitur quatuor aequationibus continentur inventae jam
10) Tollamus jam earum opelitums  113, S1g, 213, 21B. idem Explicanto pa quae pertinent ad 113
relationes determinatarum. Comprudio pro aeq. 1. serilamus sirs  M. Explicemus
non sertinent; ex equi fiet _  st nSt, :, 3ec. 73= 313. 712
 seu 12 S3a 113 Sisa ex aeq. 1 fit H~ Seza - 13. ra  3rETIZ. NeL 213 112. 7512
Ex aq sit (200) insua- IB 503a L 313. (13.752 313. 72525. Et (x ag. 3. (C
et (3to) _2 Osa + 1c3. Sra= 312 713. 5712 + 313 712 5712 Et ex q. 4fit (4to)123 sisa tnSga 32  5xyf 313 Y. 57
F112. SB = WX3(A1I. Sez = 51223 et ita porto SuE siiis. at . m. jujis a seinvilem agg 3 et  fieta Auferatur etiam aequ. 1. ab aq. 3 fit V13 sua ( m9 (71 Y51)S3 +(BHCE 11)313. 712
s
et auteratur postremo aequ. 2 ab eg. 4 fit 2. U3 523
3 l 73  In 313 EX ex aeq. 2
t (2d0) 123. 3s. siz - 523. 3323. 118 - 303. 11237 13 713A 313. Et ex a42 ft (otro ) 1. 712. 51g + S12 mu s3 B3,2 MM2 1 X2 31, 3
e.772.535B13. Proquibus
A 7 M. VE S1  Sir. 33i2 114 reC m. 313, preoite sit bmo) 15. 13F1. 513 4 13513 + H. 713 L0,
Et ex eg 4 fit (4to) 123. 7123. 51 n a
 3F 2L53 + 13. 313 + 24. 713 =0 et tib) 31. 715 +32. 813 4 53. 313 + 34.713- 0et(4to)9 1tyr 2B3+43831 17B
 jam earum ese liter 113, 513, 313, 7 13. Et ideo in aegq.  123, 4 articuli praecedentis
Ex Ya 1Daexplicemus terminos ubi 13, compeudiole exprimemus as rubi non est 13. fit (1mo)  H12. 513 G2. 1iga =
Z 312 IN 21 M. 13I 4 1312. 312. 73 -1312. 712. 313 et Bdo 1∞ 1235Ba-523. 113a= 132932713-1325. 33. 313
et (3tio) 112. 513 a + G12. 113a =, s3I2 12 Sg. 5712. 317,71345712. 712.313. E(1) 123. 513a 4 523. 1134 =5733,323713+5723. 73,313
Ex 1 et 3 addendo fit 2. 112. 513 d= (1313+ 571 e (2 (+ 13n)7B+ (+ 512 1312)712. 313. Et subtra
fet ((10), 1a 3 fil 2512. 113a = (64) (5712- 1312) 312.713 4 (571241312)712. 313. Et ex 2 ct 4 fi( ad
2.123.513a (5723+1323) 323. 713 4 (5723 - 1323) X23. 313. Et gubtrahendo 2 a 4 fil 2. 523. Ha =(657 23-1323 )123. 713 H(51234323) 725. 313.
ubi unde ex aequis et 7 fit (5712+ 1312). 123. 312. 713 4 (5712- 1312) 123.712. 313 = (9no) 5723 + 1323) 112.323.713+ (5725.1324)112.723. 313
Unde fiet (1b ino) (1323 - 5723) 112. 723  - (1312∞ 5712) 123. 712( 6., ((13224, 5723) 42. 323 - (1312157 12) 123 31 = 713: 313
Et ex ggg.  et 8 6 et 8 fit (712 - 13n) 523. 312. 7(34 (5712F1312) 523.71. 315 (5725, 1323)512. 323. 71345723+ 123)512.723.313
se unde fie (12mo)  (1323+9723) 512. 723 -(1312+5712)523. I,: (1323-5723) e2 313 - (13. 2 5 n2)523312 = 713: 313
iea unde semiliter pro 1312 et semilibus ponendo i habetur 513. 313. Et notaridum ut ex duobus valoribu ipsi rviin
513: 313 fieri novam Ane 2. 13 
1+ sed fortusse brevique propedilicebit, refinendo H3I et similes, nempe ex articuli 9 aequat nbus 1,23, 4
sis set 13a i fima) 1 Et B1. S et S23Ca = r 1325. 113B et tu) Mra = 571 3113 et dem
Ato) 1231hA  5729. 32313. E 9. sed ex articuli 10 aequatio 1,r 3, 4 alies fabrica vimus vable
D  et d destructis jam irrationalibus in Aquatione articuli restat sequeas aequatio
12 2. smepsues Gt4
mm
, 1323. 11213 -51223. 512B3 - (a + b( lan)(a- lisl- n
123. S12. IIZ. 1121y12313. - 123. 322. L12a. 12313. 123137
Btam abL bm = Lametbl = (M2. 523+ 13, 4)  (In 527 11 4. 51) (1. - insns
2 112. 512. 118. 529 4 2. H2. 512. 173. 513 == 2. 112.512. 12313. Ergo coefficiens ipsiu ev
est - 112. SiZ. 11213, 12313 et coefficiens ipsius ZZ est - 125. 5233 12313. 11213
ipsius vZ duo quae per L conpanximus, sunt 10213o 12313. 11225
jam in coefficiente
Ex reliquis tubus nemeris
doefficientis il si22(II2. 512. S1323I + 123. 523: 512131 + 113. 513 512281). quae explicando
de bos ae dad
pro quibus substituendo valores, ex aequationibus 4 et S articuli 6, et aeque 1. et 2 articulia 7, prodibit aequatio (M. 51, S1223. W SEC. SIB231 = u2 H 2h. 5+32(+13. 
ubi interpretanda 3, B. t per a9. 4. 5.6 et quide
+123 52L5I231923, 1133372
L123. 512) 1272515
8. 512. l1213. I122Z V = 523, lgg3. 71223 Z = 312. 3121B. Suues V. + 31. 32313. Si23 Z =)- 112. 1.213 fin2, Vel13. 12sis. In23Z + 3ahug siu, . 79tus. Sicese) et edem modys 123. 523. 51227. (2 131 9
plura inveniamus.
71223 (G SI2) N3. s 3B. 712
 512Vyt _. SBPH + a3. 513 VZ + 100p50ot-er mon =6
=112. S12. H1223,. Iu5. 1113 (B. 1p112. 513. I12rZ. 112. 12313.
+ Ig. S13. 1123. 51213. 31123.
por coefficiente ipsius yv. vel Z
obso vetur infra artia
S12 23 (912(712) Ar. 523, 327 a siZ,
aequationes per A2B et Aimilibus Nam addendo E et 3 fit SiB.
 (1t0) W1 tsyn in (aF
f112. 512. 5123. 11212. 5213VB + 112. 512. 31223. 11323,5B23V.
+n3. 523. 51n3. 1117 51213.
= ab+ am +blm
G(Sie 75IB23.7 4 (5, 23F1325) 313 723
Hini ex aequ. I. fit (imo) nc. 3312. Sih- Si. 3312. 13
pro quibus breviter scribama
13= l p. 113+1V.
ysPh
valorem Y) sic scribertuus, 114(3. 5121) v 4 11323. 5143Z,:, 81223. 5I223. Inde fint
ft 123 hg3mi=Ds
u2. 522 3.. 723 = 13 sn. nq addantur aequ, h et 4 fiet 2. 12 5a = (6to) (572317525) 3C3. 713 + (3, 23, 15 a1) 313 223. Aeferantur etiam
 ex a
Andantur aegu. 1 et 3. firt L. iZ sca (to) (S +Y51) M sA ((5)(2 7612) 313 n Semiliter
Qfiet 17.S1  Ec, Se, ., SIE h segit= 5ri.  lt ex aq. + fiit 1 sp 113. 5e5, , 32g H3. D
= Hll bitur ∞, D- 0, tantum pra 1 vet 5) et compendiose scribe
s
63523. 52312. 513171
(uddatur 11213. 1233. 1225 ot1 12. 92 I4123. 513 f,112. 525
113. 512) VIB. 523129. 51
 i
51223. 112. Si2. 51323I = 112. 517+ 12 5_)NZ. 523
1 ssu). 125. 51 lu e
Su. S12. p2. 52Z. 115. 52t X
12. 5Ic112. 529. 17551
ur. st2. 127. 512. 17
(+ 123. 523. 129. 312. 113. 51
(∞IL. 523. 123. 512. II2. 51
172 . . S2MB. Se
4 123. 52. 112. 58. 115.51
B.Z. 123 513. 525 (112. 175. 5. 513 + AE 13 512 siz s
ot pS. 113. 513. 233  12. 117. 125. 512. 513. 523 eU9  
.f 115.13. 123. 512. 513. 35  113. 125(. 31B. 113 2. 112. 11) 14. 5 512. 57  
127 12. 1132. 512. 523363. 125.113. 532 in 113(123. 512.52
idat msu
an  fi  sii in in sr hnuns ns ffshtun ns saisis
(VS1  1123. 5253, H3 Se5IUI3. s13 L 2 1ssi (L113 +2 1. n3. 523 513
ues su t E r.tet ( 
s i
seu Bsmontus  A3 3t +
is
est aest r
11235
Slohis 1 127

## Folio 73v — 00068255:0146
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 64r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068255/jpgs/default/00000146.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 43 lines, status machine
(10) Has 4 aequationes articuli praecedentis aliter sexprimemus, est autem 1121 = 12. 513+ 113. 51 2.
et nyis 13 83+ 1g 323 et sit 12 5=
.sit
AD 123 = a ab 133 = c+d 131215  a- b 533 = c- d
122. 51=C
31213 =+6) sit 112. 513 = a  134612 = b 312. 713 =c
123. 513 = (a) 1. 113. 523 =(6)
fiet 11213 = a + b 13(3 = (a) +16)
323. 713 =()
4 323 =8 et fiet
31213 = e+ d 32913 = (C)H(d)
S12I = a b.  52313 = (a) - 1(6)  X213  c - d 71313 = (C) - (d)
313, 712 = d
313. 723 = (d)
Hinc ex . Articuli 9 aequatione 1. sit (a - b,: cud = (imo) 112: 312 a
S (1) -6), 51 ( (3) = (2d0 123: 323
a + b 3:, etd = 3tio) 312: 712
(4 (b), : (1 (d) = ato = 523: 723
ex ifit eud = aub, 3t. 12 cx3 fit E+d = atb, 712. 512. Ergo 2C = (6t) sa b) p12: 12) + (a + b(712 :512) Et st
et ad = Cto) (b- a) β312. 112) + (a +b) (71  (612) et pari jure ex 2 et 4 fit 2(C) -. (mo) ((2) - (6)) 383. 123 + (a)+ b)) (23: 573)
et 210) _ = (8B0) ((6) - (a)) (377 2122) f(a) + (b) ) (723: 523) jam c. 312 = 713 = ( (C): 323. Ergo 323C 311C)
Ergo ex 5 et Y fit, (a, b) (3e2. 323: 11)  (a+b) ( (71292Z: 512) = ((a) = (6)) (312. 323 : 123) + (() +(b)) (312. 7231523).
Et eodem modo 723 d= 7121d) et fiet ((b- a) (312. 7 3. 12) et (a + b) (712. 723): 512 = (19mo) (6) (a)). _ 72 3 A(1 +(b )) (Ep2. 313. t
et ex aequ. _ et 10 et 12 poterunt inveniri a et b
compendii causa scribumi oe  ex  bgo + (a+ b1 = a (a. b) 1 3 + (at 372
pro aequ  b7 331  (476) 73s (a -b) e + (a+b) = (13tro) ((a) = (6) 9 + ((a) + (l)) h
et pro aequa (b- a)  + (atb)  = (gto) (1b) - (2)) 7 + (a + (b)) 12
agus 1y milliplicetur per E, et ag. 14 per e, fa et addantur un anjum fiet
sambE) + (t6) 52 ((a) - (b)) gC ut (a) +67) _E  in conjungantur jam a et () item b et (
e
b et fiet
tex 19 firt +a e + af (a)o - (a h  Ab e = bf - 1b) 9+ (b) h jam (a) = ~r 123a1112. 
(b- a) e2 + (a1b) g (B) (1) Ve + ((a) + ()) it
ex 14 fint = aE + a∞ + (a)V = (a) 1 = - bE - b 8 + (b) γ + (b) 
(6) 18) 523b : 512
ergo ar ob 17 et 18 etraque aequatio, is et 16 multiplicetia per 112. 512, et pro (a) et b) substituatur vald
fietque: +12. S1 a + 12. 512fa = 123. 51Z Ga= 123. 512ha L) 112. 512 ed = 113. 512 fb - 112. 5239b + 112. 523Ab
112gG. Sit 343. 71L. 123, (7a7 13: 523 1(20m0)
 tC Ea L m. SiGa + 122. 3ILya = 13. SIEBa,   12. S12Eb, - i. sm gb + 112. 523Y b + 112. 523, 1b
112. 7:1IL
unde ex agurg fit (129. gt2. 312. 313) 4. 112. 323. H ( Gr EsBir. 125, 12S 5 t2. 312. 723, a   5 2. 523, 312. 729 -- 1121513 37 p , 1. 515. 142. 3 + 12. 123 53, 312. 723, b
tex
Hic suspectum mili erois, quod 23 sepius reperitur in mentris harum duaum ag9. quam