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# Piece: https://leibnizlegible.com/api/records/59899/text
# Catalogue record: 59899 — [Aufz. zur Reduktion irrationaler Ausdrücke]
# Incipit: videtur detectum hic quod non procedat talis analysis
# Date: 1698
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=59899&katnr=57602
# Work: https://leibnizlegible.com/work/00068197
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2a
# Folios: Bl. 24–30 (from the shelfmark LH 35, 13, 2a Bl. 24-30): 14 pages, canvases 47–60, page ids 00068197:0047 to 00068197:0060
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068197&page=47
# Lines: 1,220 recognised on 13 of 14 pages, mean confidence 0.78
# 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.
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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 24r — 00068197:0047
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 25v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000047.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 100 lines, status machine
 mo cedit iatius mndr sis
quaritur  Chytha, hy +. hb, hy +. hc) aessumatur Thyaha) L l.H + s. CZ t t CEy atib) L n D n t Z t Hipu 1c. A q  0t 
ntitu t + ff0G
(2) pmmoDtun DE
+hb EE - hb C
(2) pahh 42 +g9C4
L.he GG- hc04
+2hftur VOG nitres ipsius 2E+02+ 24 10 valores valores ita dven, ut primus 
= 2ha 2 10
+2mn D 1CG
+ 2h92, 102
=2 hc 8 10G
fiet: munndD qu (a -b) 43 + L (a. b) ma D
et mmunD D+ H qu cab) 4E + ahlc. b) mn D Z
et tertius item secundo; seu extremimedio; fiet ex  irrationalibus aequa
ht m
Ex rationalibus aequatis fiet
hthih+ A = m BDran Z = EE 4 4 +99 8G fb. ct 47
Et in his dluabus aequationibus explicando lonondim
ht et h2 per ag4. valores ex aegy. 3 et 4,
+5  O2 - ffmnO D - ss nn X) = 0
9+ c9 49mm C 3 99na 34 6
abest hic 00 et 88, 07, 48
non licet sunui eodem movo tactar
quodsi in aeq. 7. ponatur ff H(b -a) h 20, evarescet in ea Oe: quin evanescet etiam ZE, itaque residur
diviso per D, ex aeq. 7. fiet mannD + 2 Ca: b) hmnq-fmmO- fsnn E,  et explicando ff per aequ. habebitur
2 + 2 m 8 =mm C 
+ mmun DD + hhqul c - b) 4G
+ 99n (l. C)..
is valor substituatur in aeq. 8 fiet
+2h (C- b) mn D Q(= gmmOD)
iam (1D +gt
golb. dh a. D h
Haec aequatio 12, compendio et generaliter sic exprimatur
489b. chs 
12004Z + 1020D) + 100291 + 11083 + 1101 3( + 10113020
faciamus jam C) = (14) = 21C8 +01D + 200h,: 000, fet  10o Z4
(15) (000. cro. 1700 98 + 000. 000. 10203) + 000. 000. 1II0 ZD  2ro  AbL..
+80o oro. 1820DD +  H10SD
+P ret teonsx +2  pot. rerveo nortf
210. 210. 1602.  + 201. 201. 1002.. + 2. 210. 201. 1e02 - - + 2. 210. 200. 1002h8 + 2. 201. 200. 1002h7 + 200. 200. 1002 hh
000. 210. 1131-.. 40
+ 000. 2811101.. -+- 000. 200. 110Y
4000. 20l. 1011.- +000, 210. 101..
4000. 200. 1011r
siant sam cdeficientes termindum Y Z, 33, XD = b r savit b. 2. 20. . 1002 Z + 2 201 po. 100 tyt 4 200. 1eb 2 b
ob 77 fit pi 210 + 100. pe 210 4 voo 101 2 = 005. 1101-u0v. 1eobi1 200. hin 2ol  sceuri tromi Mer, 
seu 210 (NO- 4. 1209. 1002) -1101, 000: 41002 . 190
Eatem modo ex D) siet sublato
fiet 201  1 (101r- 4. 1020. 1002) = 101, 000. 7. . Capli est telli. ), in qua a unde fit aequatio, in
qua pro 210, 201, substituendo valores, fiet
3. ∞ob. 1110. 1002 + fL11 2- 4. 12v0. 1002, 10112- 4. 1020. 1002)uo  1) 101 F( 1 (1o1 2ete) 110, L ry. 1101. 1311)
X 1 (10n t) 1101)  = 1011. 101 + V (, poi h) 1BU uot II)
Seu sublantis Lllendis I( - 4. nov. 1002, 1011- 4. 1020. 1002)-
10OL. 1011 = 2- 1110. 1002
et quadrundo utrobique (APP2 1013 4. 101. 1020 (1002) - 4. 1011. 1200 (100716. od. 1020( (101. 102+ 4. 110. 120- 4. 101. 191. 1 sed
quam aequationem purgando et adinando firt tandem 4. 1200. 10d8. 1002 4. Hvt ry. 1110. 1101. 10,  y10. 100L
quae aequatio legem justitiae pulchre observat e a es nit   aitem t e
Exphlemus jam numerus hujus aequationis ex aequ. 12 vel 13, compendii autem causa a + h(t. c) vocemus tp, fiet
1200= tb c) lisitie 1020 minua Chtarb) = 99) t mmnnXq ( , 1002 = - tp gomn 89+h b -a vocemque h9
1D LD- 2mahp et 1101 = 2mgghip et 101 2= mm g p um 99 P
et exaeq. 20 fiet Ahtitpmmnuqueupugum, 4mttip myghp mmgop:
Dividitur hac aequatio per thpperta, et nultiplicetur per e, 3C
fiet =ek9 + 4 ep9g = = ahp ee + 4 e999 L Kpg9
ammontulitipoam heb. C) vocemus tik
 4 amoetitip pmmtriq
in hac aegpntione pronbisat pondatur multiplicata per h, loco tp, et h9 ponantur valores
fiet =AheK 99 = ahek hC) + 4 eg90e aegghK =. ( 4heegos- Ahoehk (+ 409999 t ci 19h0 + 99k99 +9G ht
Unde tandem fiet Kg+ + h k kgy = d
Sed K non potis
aliaqui in aeq. 12 omnes tegmini destruuntur, praeter equm sed hos
 iosers HG et fiA V =b 
ergo t
ibidem termini 1002, Hot, loi, seu termini continentes 
et pracepib simunter  D 24 F5  de 8
112DD + 1448Z + 114)8+ 123F+23O+23C)O =0. pro 3 sculatur V, et fiant
e monnt editraria quaed saedible
ntata compentiao
el 
hon hecse tatabitur tatu inter nisi etiam velimus
evanescere termunos DD, fY, YD quo non licet. quod si ratio datur inter Y et 8, ontato
et viter it ti et  sed
R fumatur aeg.
= 20V + 21H: C = 30 v + 3ch: 6ol 4= 40 v + 41h: deh, et fiec moovv
uet tollgatur ex invenictur 30, ut tollatur v, invenintur 31. et in aeqvotione
simparan ad tollendum hv, puperpunt 20, 21, 4C, 41. Endem faracnvo pro
9. facta ex 8 erit similis calcutus, x pro 20, 21 40, E1, nisi quod per
pro 11t et 14 ex. pontur 211, 24. + etC. quam duau auus a
tollatum 20 pos
2. 144. 40. 41 l.B + 144. 41 hh
obB + 144, 00. 4(hV
L. 00 2d 144 60. 21 p
es rdotio
s  p t
113. 00. 35VH + 113.00.9. hin
maju
qviis
NN van ns
ng..90+ he
requamb. 

## Folio 24v — 00068197:0048
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 25r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000048.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 112 lines, status machine
Aequando duos valores ipsius 30, et ordinando secundum 4O et 41, fiet
Compendio 720. 41. 41 + 211. 41. 40 + 202. 40 40,- 0 fet 2C
101. 40. DC
210. 41. 00
) = 4 qu (. . . . . .) - 302. 320. 4040
200. 00. 00
O1. 220. 40
200. 820 - 00. 00
et (ur 320. 41 = Y 311. 311. 40.40 + 210. 319. 00. 80, + 4. 31F. 310. 40. 1 - 4. 302. 310. 40. 40 - et. 301 920 4o
=g.302. 32040. 40) . 311.40
00 1: 2
220 = 144, (11300 + 123. 20) 8 3H1= =. 2. 123. 144. 21 (113. 00- f123. 20) 310 = = 123. 144. 21 (113, 10+1 2 3. 20)
1+311: 40
220. 4 = 
202 = 123. 123144, 21. 21 301 = 123. 123. 114. 21. 21
ergo ubi ad explicationem aequationis  venerimus
300. 60 == 123. 123, 21. 21 (111. 00 = 112. 20) - 123. 144. 21. 21 (17 00 +123. 20)
apparebet dividi omnia posse per 113.004 123.20,
et siet 3. 144. (113. 00 + 123. 20) 41 = A4. 1232 144. 214 123214424200 +4. 123. 144. 21. 40. 00.
(ACCSt. qu. +. 144 (123. 2121 (123. 141.00 +123. 113. 20= 144. 11300 = 144. 123. 20)001) 1.3. 111. 4. 10
CC3. 40 que 12. 14. 4 00 - 4. 15. 141. 14. 21. 40. 00
Pn 2. 144 (13. 10. + 123. 20) 4 =   x o v)-
Et habetur 22 =
copendi enuntiata
123. 114t (2.∞ +00) + V (4 1s etc
ubi evenescit 40 in irrationali si fiat (44- 114 =0
et evanescet 20 in irration ali si fiat 14 -12 = 0
Eodem Modo ex aq. 3. compendio enuotiata, pro; 123, 144, etc. porendo 223, 249.
2. 144. (113.00+123. 20)
223,. 244 (2. 40 +00) + V(4
. 223 etc .. .. . .)
2. 244 (213.00 + 223. 20)
ubi rursus ut evanescat 40 in irrationalifiet 244-314 20
vel ut evanescat 20 in irrationali siet 244 -212 =0
prout scilicet postremo volumus quarere 40, vel 20
nex valorum ipsius  comparationem, ne opus
sit irrationalibus sublatis id quod quaeritur, v. g. 40 (vel 26)
rursus altolli ad quadratum.
vocutum v
praestitisset fortasse ad calculi uniformitatem non ipsas Z 18, C) explicare per D, sed omnes
, O, Q, D explicare per V, ubi fiet 3 = 10V+ uh, etsi in praesenti Casu, fieri posset 10 = 1 et 11= 0
ita calculus erit paulo prolixior quam opus sit, sed major erit regularitas. Si suppleantur
mea quiae
desunt 00, 0d 80,
3d 4 11208 + 12390 = 0
114D4 1346
_) DB = 111 . eBP + 2. M. 10. 1 LV + 101. 11. 11 lih
0 = D) +
122. 2n. 203V + Z. 122. 30. 21 hH + 122.21. D1 1
+ 133. 31
133,30.30. + 2. 133-3031
141. 40. 40 ) 2.99. 70, 41
113. 10-20 7 152. 1021
113 Jars q 11.20
3 D. X0t ,(er 10 
= ∞ v+ h
. 20v + un
(2.344. 4142 + 319. 42. 44)G + (2. 344. 4. 43 + 314. 43. 44)0
31I. 44. 44 + 344. 41. 41 + 314. 41. 41
cu DB + am P Fhar
sit e + Y( e fh
sed hoc juste pro lixioras
res detenurhatur expagm.
+  9900) + 2 f99 =
Si1. 44. 44+ et
(∞8 + nD + hit tiat LLN + Chm) FThan.
m = f4 +90 hhn = p4+
velsi 1. + 16 llpr _ - alrsf. alpgg = (Hggt allrr = 45 glv. seu Alpr + Fgr = Gr Fgyp xlrr
Hinc haum A = lus
Vedd extrahibile, sead
intentionem notum, unsi fiat f Ale L(2, H- al p, 199- alq) = 2f9- 4lX cut alo, gu - plr-  (g  fiet 311. 44 
V =-mh H (= Brum - CH) sed
jam si sit Gy + CD
sequenti
= 30v+311
= 40M
G2
06+ r08
quod non potest
fet qn taic summntur novissimo tensum
non ut ante
+. 1. . 41..
1121 11. 21
148 - 14471, 00 = 17 04
4. 102. 112. 10. 20. 21. 2
4. 120. (2
seu tiet. 2L 2((2v. 1 2, 120. 104
Deu Yl2. 120. 702)
4. 102. 102. 20.
du 122 1b. 2
torter
R(4) 112. 112. 10. 11. 2021
A. 120. 102. H. 11, 20.20
4. 20n2. 10 11. l1. 20
(4. 120. 112 10. 10. 1121
4. 12. 102 11. 20. 20. 21
4. 102 112, 10. 20. 4,213
qu (N. 27 811. 20) = 112. 12- 11. 11. 20. 20 + 10. 10. 21. 21  -2. 112. 112. 10. 11. 20. 2
 vel 10. 21 - 1(. 20 = 0 Sed si hoc verum Z et V
si te pur asus set
plicatio stpit
in ratione constante, contra hypostesin
4. 150117
(A uIoZ. 11
4. 120. 120. 11. II. 10. 10
4 . 102. 102. 21. 21. 20. 28
I. 12. 112C. I1. H. 20. 2
11 112. 112. FB FD. 2. 2
2. 112. 112. 10.1. 20. 21
8. 120. 102. 10. 11. 202]
4. 120. 112. 10. I1. H. 20
8. Iv. 11. 21

## Folio 25r — 00068197:0049
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 24v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000049.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 84 lines, status machine
Resumamus O aequationem et eme paginae primae W1DrI  Z F 14DGF 2D Z HBDO+,  2O= 0
compendio 12 DG + 11390+, 23 ZOvocemus 10DL + 100L6, et fiet 194 ubi Notu = 1124 + 113 et 10 1 2300
142 1488 + 14. 1448D + 114 DD = 112∞5 = 111. 144. D  - 10. 144 25 - 140. 144h5. Itaque hoc dextium litus
debet etiam esse quadratum, sildensi per constantem mutiplicetur. Eyf  9∞ - 4. 10. 144 5  4. 100. 144. h0
114 - 4. 1. 144  114- 4. IV. 144
Quod comparandum est cum 37∞, DG + sest ergo fiet 2. 10 14.  = a et - 4. 100 144 t 6 = si se
1142- 4. 14. 144
Qu (114 - 4. 11.49) seu (4. 114. 14 - 112, )100. h5 = 10. 144. 4t seu si et
 4. 116. 144 hE
(4. 14. 144 - 1142) 1346 = 144, 1125 + 2. 112. 113. C + 13. _C
sed inde sequeretur C et G rationem habere constantem quod non est permissum
Set si initio fecissemus 1 4 4. M. 144 = 0 feret D- -10oho, Grrot, qui debet esse
integer seu 127C delebet esse divisibilis per N2  + 13C
itaqe praestabuit in faltor, intio statim venire ad divisionem, ob aequationem hajus pugicae primam
si faceremus 1447+ 114D = 0, f et 114 - 111. 144 = 0, superesset D = 100h5: 1otu se et ha valor
Quod videanus au fieri possit
tuea I DD + 1448
 fiat Z - b~  lv + mh, :h
quaeritur ergo modus generaliter efficiendi ut x si Z0 sit divisibilis per bZ + ex fint R-  tath, et dibet esse
erit integer, sec 123GC) erit divsibilis per mQ + 1130
i ut dinisbilis per lov + nhv divsibilis per bZ + . v, id est per +blx + lnitum, pud fiet ergo lnv+ Imto
t b m fit. n = b et In = bl+ ch seu bl = bl + ch, quod non licet, nisi sit c = 0
R eoeamus ergo ad aequationem primam hujus paginae, et videamus a ex qua fit
111DD +14489 = 114 58 = 13DC, :, 112D + 1236 = 3103 + 3030 + 300; 000
410. 410. 144 3 +E. 410. 403. 144 18 + 48). 403 144. CC + 2. 410. 400.144 Dh + 2. 403. 4co1 446h- +. 400 qao 2t hil
sit 8 = 410D + 403 C + 40ch et fiet M AB, :000
et fiet 1et 8.3. 800. 111 DD + 000. 00d. 113
000. 40. 114 DV + 000. 403. 114. 55
+ 000. 400. 1149h
BDD. 112. 310 77 +000.113. 303 9O + 113, 3031C-+000. 112. 300 Dh +000. 123. 300Dh
Redeamus ad aequ. 7 et 8 expressas compendiose, et tollamus simul 44 et 04
000. 123. 310
+ M. 59 +1 YF+ ut ) f + n2 5+ 13D 138O + 12380 =0 2
2D + 14 O8 + 14Y + 1129P + 1130,: 144
fint 144. 223 = 13. 244
ad tollendum OG
seu exsticando 144, 244, 133 229 Ex 24
+21+ D +21282+ 21330,: 21
2a. g2. N. 98 +344. γ1300 + (2314. 4. +525. 11. 4) 04 seu ex a4. 7 t 8 fiet t ( b. h (b - a) (H +hib. 1), g9 (g9. + h (b.c)) = h (b - c) (99 + b - c) h) f. f (5f ilb. - a) h)
seu  100. 144
11a - 4. 14. 144
+= 
=: 212 W + 223 C 4
pel (b- a) 0g = (b- c) ff et ex cgg. 7 et d compositis prodibil L 114. 2447+ 112. 244Q + 11324C + 122. 147
incompendio vocitur  42 F +1 + 42C + 43C; 4 - 144. 2141 - 149 312 8 144. 21C - 144. 2227
 Hinc valor ipsius . substituatur in valorius aq in aequ n13 + 19+ XF + 1 B Y + 1I2 D  + 1330 + 130 =0
en compendio 31. DB + 214 8P+ 3147P+ 312 D9 + 31380 + 34907 = 0
24. 5 22 . 2138. +223
372. 41. H.. +344. 92. 128G + 344. 43. 438(57 F2. 34 45 41. 1  C + 2344. 91. 4310 +2 344. tr. 170g
114- 4. 11I. 144
314 414 311, +
1. 323. 47 44..
 Videamue quibur casibus efficere lieat, sit  Y(22G+ 59300 + 5. 23 8D) extiribilis, posito d et D esse sumplices
H4. 4g. 42
rationales in capitulibus sit b2c; +63(C 460 = 0 seu  = 630 + 69: 62
2.52263.60 -5236062
1(+533.62, 62C0 
=523.69,6300-5236062. Ch
potit . 60 + 32. 63.6 4 5 3, 62 62 - 519. 62. 63
et ee
60 + le tee
 qu (. 2. 522. 63.63 - 5237. 60. 62)
4. (522. 63. 63 +533...)
cur. 60. 80
 522. 63.634533...
fit P = 2. 52267  53 - 523.60. 62, h
522:63163 + 533, b2. 62 - 5 23. 62. 67
fiet qu (2. 522. 69. 67 523,60. 62) = 4. 522. 60. 60, 522. 6363 + 533662. =523,.60. 62
seu qu 63 (4. Su. 6 60 F32f. 62. 65 - 4. sir. 5r2. 60 62. (3, = (1. 52, 65. 60)1 . 522 533. 62. fo - 4. sir. 123. 68. 62
Et divsis omnibus per 52. 60 fiet (525 - 4. 522.533)62 = 4. 522. 523. 63 + 4. 522,523. 60 = 0
 4. 522.523, 63-60.
c
 523.523 - 522.533
seu Sc= 4. 122. 527. 63 - 4. 52 2. 523. 60
Alliter 12027 + 102 vV + 112 3V aequandum quadratae in integris simelicibus capitalibus sit  00 Z = L 1X + 10h 0ov - 21C + 20
ite ZZ + 10 12 523 - et 2. 533
fiet 120. H. IIXX + 120. 2. 11. 10Xh + 120. 10.10 hh, jam ut 30XXF31 Lx +3Chli  sic aequesur quadato, fiet
102. 2. 21 un + 102. 2. 21. 20... + 102. 2c. 20 
u nn iami v.  
303. XX + 30, 21 hX + 9 31 3 h = 4 3. 3ebl - 30. 32hh
30. zoxX + 30. 7C hX + 30. 3 chlr et AL 31. 31hh = 30. 3eth
 nopod  tet
522.60. 60 hk
302. 69. 8 1 t se3. 629

## Folio 25v — 00068197:0050
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 24r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000050.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 92 lines, status machine
po aequ. 7. compendro scrilatur W11DD + 144 4Q + 1+ D Q + 112D8 + 113DC) + 12380 2 0
2u DD + 244 4 8 f etc
Collamus utrobique 8Q, et videamus an simul tolli possiti sC, ita enim residuum poterit dividi per D et habebitur valor ipsum
q integer simplex. Ex 9 fint - 8 = 111DD + 114 D_ + etc.: 144 = (ex 10) = +211∞) + 21434 + etc. ,: 244
et faciendo 1230244 = (12) = 223. 144, evanescet C in reducta eq. 1, et fiet divisione institula per D
- 144211. = 144. 214.  144. 212 8 = 44213
prodit f (ff + (b- at). (- c) h (99 + 1b - 0)h), = 99 (39H (b - 9) h). (b -a) h (ff + b - a 2), seu fFCb -c) = 9g(b - a)
22g hx- f = he 144
(5 Compendio b a sit = e bac sit = K gg f +(b - a) A =  sund gg Ib C) H = trq fit 144 +h et 24+ ft q
114 = 2hemm 214 = -2hkmn 123 = fstip 22 = gglq
11I = 211 = mmun +13= -ffum 112 = -5fnn
tg nm gnn
gg = f(H: e) 99 + b- c) h = HfF K (s Fhe): C et Z = K hf: C. Ergo 244 = hhKKS: e
223ifKktip: ee
23 = - ffmmK: C
212 = -ffnnK: C
Kk- eC = (b6) + cc - 2bc ( bb) = aa + 2ba = breviter) t(5 h)
mmnn 1 V: C) D Ac = 2lmn tetikK F 8
+Tlikmn litie
seu mranntz D + 2h mHKe (C- f) 
Hun trhk(K: e) 
tinhurp..
1C 244- 144 = hh (Kk - eC): C
-frum tihk flk: C)
+tctipe Aun)  + HmmK (C - K) O
et quia s) divisum per ek dat = (C + K) quod compendis do cabumus
=pd + H Yspe C
 2tenin F
Ammk ltiq
etk = b- a + b- c2 2b- a -c = 5
+snC.. qui vator substituendis est in aquatione 9 explicata,
mmun DD + hhop q 2 - 2 hema D Z - fnn D  - ffmm DC + Htip 40 (0
set ptke A DD (= AKKC fFmmu +) DQ (-aL K eff mnty DO + 4h kK exumun ZC
+smeddDD =2mndkff
+KKF DG
tfm 0
- 2Fmnnsfdk _.. + 2VKK fEmmnn.. 18,
TeKoni DD (F AKK esfmmi). (+ AkK effmun) u
si jam in aju e8 liceret  esea ut canes
co stegentio 18 per compendium exet scibum  
cam autem in rem fiet 2 tif Kffmmun (FF + 2he) = 0 seu f + che X 0 seuf = 2h (a- b)
Ergo ex aequ 15. 4g. 19 pe 14 per iy fiet 7h (a - l) (b C)  9gtb. a) seu 9g  h (C - b)
qui consensu valsuum f et g, spem facit ob concinnitatem
lmteemmim
AKKemnnn
auemmun
EKKemmun
H + 2he = hp+ pC
Non autum licuiset in hoc calculo facene idie Y =0 in aeq. 17 vel. tollere seunt BR et D C intaeqius vat Z ita eantu nan
in q licuisset ope  et 10 simul, vel ope 7 ut 8 simul tollere et8 et O. sed vaus fuisse
jam instituimus paus prima, et reperimus ducere cio non permissa, aut certe nomfaciti pirmit ergo peculi calculor quem
aequatio 1B compendio ila exprimatur i _ L 20.
2000C +020DD +(C02 EQ HLIDOD + 1,OIDG HO1∞
t sit 000D = 210( + 201G + 200h. Eoque vadore substitut
prodibit aequatio canonica pro datis. 4. 1200 1D29. 1002 + Vd tet H2110. 2101. 2011 = _ 2110.1002 + 2101. 1020 + 2011.120
fMnn 2101 = 2KKfmnRLN, ubi hN - f+ :tie
et 2011 = -2rdkffmmn quos valores substituendo in aequatione canonica, fiet
YK fErCE hh= ArrSSKKfLMENEKKfEMurhH = ArrSdHKF m ntt Hk fEnt, ama ktismu rtikfhi, atif nuretlinmntlia
dividatur hac aequatio per rrt F 2m BnS. prodit f hhA + 2rhICDS 5 = Y5554 + T8SF4 + AhtNNCM, et pro h substituendo valorem, fit
fahit  +2Y8 f F 4rsshess = (2r58f 4 (atit o ft stst + atihs hhee + ahtia heff, et dividendo per 4he, fit
r6Sff = s eA F hhs f. Est autem thx = 4KHC + rds - +etd ego (rss  = 4Ht  + rodhe - aCKShe. + 2t t e rdd p. t et S
quae aequatio dividi potest per e. in reliquo pro tr subsituatur valor A+he, fiat 0 = 4hHHt _SSt5 + tr ode - 4hR KS + AKKIF - 4KSf
de 2K - d = 2b- 2C - C: k = 2b - 2C - b + a - b+ C = a -c. 4HK +d AHS Haec aequatio dividi potest per 4Fk + 58 - 4kd
 seu per qu (a-c), et fit ff +he = 0. quod sper paginam primam) non licet
 sufficere, ut t, 2, +  t C, D, e sint intigria simplices, sed , 
) et F modo ss sic explicatur, ut utruque in iis coefficientes psirs d
illud adhuc considerandum superest,
sed sufficere ut Q, et qui ab eo pendent 7 s sint rationales
coincidunt nemp ud sit quia gnes erit gg ss, et hil quia jaca hebet esse hHt he = tit
si sit  = f et n -0 fieri potest ut coincidant
si esset b f( + he = 0, fieiet munus- 2hemnt +homn6 + henne
ggt hki -o fieiet mmn) - 2hk mu + htinmo + hk nn 8 X
et tollendo VB sit  E t h (C- K),  - nn Z = nn O + n C = 0 sed ita sit mmnns
+ 11344 D8  
13. 244.
4 172
Generalissima ratio sine omni suppositione soret, ut terenus , ft  per aeq. 9 et 10. fuit 4= 22F 
(144. 214 = 114. 244) 5
Videamus an liceat s+ h(b. a) QG  (Hi +b- a) h) t Hm BD + fpht ED aequade quadrato, ita enim ceque s fiet extrahita
icu licet sit 178 + m O4 + a ODABCFD. siat Z- YC +5Q + LS, sist lde+ lsier LitB +. 1150gt 2LHtOGlst ED +mOEm CD1YE fiat m-Alrs n=- Elet. 
et o= - 4 l5t. et res succedit, hine autem quaerendae Y, S, t, quae hateri possint fit ns = - m: gl rE = - n: 4, st = - p: AC irgu t - n - n: aEr = p: _ 4(5
Erus n5 =rp, seu S = rp. H. sed runus f= -m: AY, fiet rep aL al rrp = nm. seu ry= -va ma: 41p Et eodem modo B -- Ip: alm et HL - - np. 4m
juniiam vioeorem nonsumdere, seu ad CC; F, inio jurendeo rem non
a qu(C + +4) a fri sic disse tutam inter duo quadrata, nn frpomia, in quibus quadeato nominum
posse contnere tres  biniones sed tantum duas ita hic opyseset tollere coe, quid esset recidenem priora. Sed si ope duar aegu.) et 8 vola
et fiit mndt bib. a) (70+sC _tD) = il. r.
procedere seu non posse
Ed ita duae prodeunt aequatione in quibus C) C D inteo sela servant
Sit _ 140bI9 L 2P L 1301 3ZF 1X2075V + 1202 y) 2 + 1211y v Et 1121 9 1Z
aliquet ex dinionibus, lv.g. 1Q notent talis haber aequalio, quae sic fiat extraribiis
+112yVIZ + 1022 vrZ Z. 20 fiat v = 17 FukB,: ~ 2= 2oy + 2h, : on hablnu 9t, 93, 37, 5,3) attoque 414. quique, sed literassoturo 4
dais geneitatem inde cam determinatur atius quod non est licilum

## Folio 26r — 00068197:0051
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000051.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 89 lines, status machine
hy + ah =
h y + ch =
hy+ bh =
8024
+815g1
nal.
pp04 + 2pr4Vo:
1Y47
ff OD +gg DE P ahOZ- ah
2
sp O + rr22 - bhC 4 - bl. 44
un OD + nn D e - choq - ch 44
e + 2810
04 + 24104
2f9) 104  hy, 04 + 24107
-rahZ.. . .
++
+2pr 410G = hy, 04 + 28109
= 26h8....
24
+2mn D 104 = hy, 09
= 2chZ.
u prima 2dae, et 3tia 2dae
Et primum quidem ob irrationales fiet 2 = fgnt ( - a)h Z,: Apt
et rursus 2 = maD + (B. D) h Z,: fpY
quos dios valores aequando fiet
h8 = (fg = mn) D B: a -c, quem vatorem subistine e in altertro valore ipsuis 2 fiet
2+ = f9D +  f9D (a -C) + fl () mD + (b - a) (f g - mn) D,:, 2pr (a -C)
seu 2 = 59 b- c) + mn (a - b), D: k (a- c) Et ita aequavimus terminos
irrationales.
Supersunt aequandi rationales
ompendo sit et Z = (t: H) ED et 2 -(: H) 8 fss at) k pp - bh - X L ud d pt
elis  potit
L = fgumn, :, anc W0 = h5y (V C) uin (an b) 1: pr (a - C)
fiet fGhh OD + odhh D = aR 0 = ahttDB Xt PQ + rr Wlo DD - bh t LDD = mnhli D muhiti D B- ck Oq. ch H. B
nam deprimerestur, omniaper
si aequ tolleremus OE, faciendo = at = exH = - ch, facilius esset quaesitum sed hoc non licet, neque enim equare
licet a et c illudi patet si saltem unum aequemus, v.g. ex H ipsi - ati, seu si faciamus  a, vel
unam saltem procedere depressionem siet si et erit nempe faciendo pp. bh (6h=)- ato
et ex aeq. 1. fret ffhhO + gghhC = ahetD = rr = blittB Compendio
posito  rrvio fla b)het: gotu
e
D= ) (. h) D (: h) O. Hunc vatorem substituamus in aeq. 2 quae
poro qui valo substituendus est in ea sed ipsam aeq. 2. prius  fractemus, quia a
in ea rrioro =bhet +chet sit Q  (3B siet e = atio e + h(3 D) mmOD + rihDQ, ubi pro
substituatur valor ex aeg. 6. fint. 6 (C - a) t. = (C - a) h1b Ct + hh BD7= mmhOD + A C DD= nnO
sed si ex hac aequatione nelimus soes
habetram cd.  per constaideia palet si redeas an
d aequationes 1 et 2, iutlato enim , utique restat velatio inter C relatio in qua B et 5)
D
(  A AD + (A h3. t
higc datur
servunt inter2 (G leges homoneneorum
sit
itaque adhuc unarius foret litera ori varabili praeter O, G, 3, 2, 
lign ab - 106  gg1 2f toq int videdum an ora destini possint, et
quid si divosor foret t+
si quod
fiet (e a) -minh + nnt pro C Dtottendo
(~3) in
quantum tactamus video,
nil obstat quin res succedat
sed oue
C(e) LV =0 
et quia ale esteh C, nec  -0
se si s,  VS tont fdo quod videtur litere ita numerator valeris iplius cynta, h est
quadratus rai onalis; sed numenutor et, quadratus irrationato
ob  = 0, tint exarque (- a) = mm, seu per 7 fiet de (e- a) rrulo 4 (c- a) (a -b h Et = hggmm
pret  qu( hisglb. 9 +hnn sa b) t pove sa- b) qu( sg- muh ap (c a) hggum, set dt L= C fit prgininm (u l) rpp ta. b) plos nas- B (en as A u olis
sit 49: nn = C hqu (a) pp gla- b) (a- b) h + po): pp (c- a
osihh B nng =o fiet irvno calt sut + (C- b) hEL nurriono - (a- b) htt nn 20 4 seu fiet
9 nn) rrioio (C- b, g9) d (a- b, nndatE = 0, et explicand W et t, fiet
9d. nn) qu (hfgb c) maCa b) + CC  (Gtumgghlmm nn qu d- b) nuhitmmungu (a. b) + C- b) gotimmun (a. 1) untimata
seu fic: aghe qu (a -b)-nnt qu a. b) A(C = b) ggpp - (a - b) napp = o seu gg: nn = h quia b) + (a -b) pp H. h qu (a n b) + (C L b) pp
seu
qquos duos ipsia aana valors aequando fiit (a -b) h + pP, : pP (- 4) = (a- b) h + pP, : h qu (a- b) + (C - b) pp
fiet pp (c a) = h qu ( a - b) + (C B) pp seu fiet pp  M qu a. 1): Ia b) seb a) seu pp = hbt a), sed ita fint q9: 1 = 0
contra intentum, nam ut eit sustulimus f, seu feli  et etium 9 = 0 quod non licet
Cogitari ergo de modo quo unam athha
non solum ontinebit A CQ, sad etiam (CD et 1P 
Terarum, et praetera nod ex producto tiam
ita habemus indeterminatis C, D. 2. Ex h, E, 4 - 2od + d
29. et (cyt cb) - 2 19 Y0Z et Hg r bst n tB dri tDC
41107
2+ 104
Ll
 + 28104
nn ODFnuDQ F 2m  
se

## Folio 26v — 00068197:0052
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000052.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 112 lines, status machine
fidt hu, i (Et 1eG) = Hb. Lt 1 f5 04- an  alie 4 e 
= ma CD +unD - bh g- blq + m D L0
 2bh:
= hl 24 + 99C 4 - ch4q- chez 4 rtue  1B
ob rmtionalos fiat th lih P ali4Z HH. abi) 0q . 
Aequitur medius valor,  prico quam
lentio; et ob irrationales fret
ee GC ph (a. b) Y+mn D; hf
1b 
pee bhC G- bh 712 lli 2 1 G. cli 44 + Bg- si  4
t.
ut ex aeq. 3 explicando si, Kofilqu Ca. b) 8Y + mnumi 3D + 2 (a B) trn QD
siet ( ah +5 74
= h (C b) Q + mnD,: h9
oberrationales fine
e
c
fas G ul qule. Gylx  t nuna DB +ih E. b) an 4B Aga (9gt b. e hq = mingus et ungude
(t epaega explicano 3
s in aeg. 5 si facias ff +(ba a h = o, sem f = (a- b) h, ermescant in ea co et 4, et rebidnisdiates per D, 
cb c)h 99 8
siet m  mnun D + 2h (a- b) m xQ nmfs0 + nutt G, vel mmt 3 + vnd z -mm  = E
Eodem modo in aeg. 6, si facias g9 + (b. c =0 evanescent in ea cQ et YL, et residuis divisis pls D
tiet mmnnpA + 2h (c- b) mn q LtumggC + nnggE
Et opi aeqq. 8 et 10 tollendo mmnn D, fiet - 2h (a - 1) mn Z + mn hh. 1) 0 + nn (a - b) G = - 2 h (c. 1) ma tmml . b) ex ul. 1)8
et tirt -2mn G + mnC + un C =0, quod dectu in a = b + b. c, seu in a - c producit aeq. 11 sed di splicat
quod il non inveniuntur.  C) et Q et e intervendentibus  t a, b, c..diiCi.
Itaque praestabit redire ad aegus. 5 et b. et explicae caetera per D. purg. sit X 8 - 30DY2 et C = RCDF 4 et G - 30D +1
lita opus est aeqq. 6. ad tolla ad AA, X, et  D, _D, et s in utraque aequatione, habemus autem in eam rem
hibitrarias 10, 11, 20, 21, 30, 3, F, 9 m. n. Tentemus tamen prius viam conpendiosiorem, et valorem ipsius e inventum 18
Licerit
Substituamus in coqu. 5, pat hhguscabi sq + mmani DB + 2h (c. b) ma 4  mmgu C) At gu (99 b.) ) 2m BC . gy lgge (b. esu mm Cd C o
(1. C)t c9 ..aggnim D apragin YD
3
+ mn vg (gu Xb. 9) t) 9)
h
Comprndii sit . 1200 GL. + 1O7BH 10B2SCL IIO ZP +140,8+ 101D6=20 (tmnungu B)
sint 3240+Zh et () = 304 + 3ih, Et ex 4eg. 13 fict 120078. + 1020. 20. 2098 17 1020, 2d. 21hG + 1020. 21. ahh 4
Ubi H aequando nihilo coefficientes ipsorum
e
L
+∞2 3030..2. 1002. 30.31 ... + 1002.31. 31 th
+1110. 20. 1 21110. 21.
IIN 30 , +1101. 31
1011 20. 30,-1011. 20.31 .. +1011. 21. 31hh
1011. 21. 30 1
Sed quia is calculus paulo prolixior ideo redeundo ad aeq. 13, faciamus O = 210 4 + 201 D + 200h,: 000
0f.8) -7 210 - 1002. - + 20, 2. 1002D7+ 2. 20. 201 1002 - . + 2. 40 2001002. (  + 2. 201. 2902 D h, 200 L 602 it
et ex aeq. 13 firt 000: 12008 + 800. 1020D7 + 9002. 111087
O4 20. 1101 ..
000. 201 .1101. 300, 200. 110. 1
+000. 200. 1011Jh
uti tollere spaptet tres ter minos summos. fit ob X4 210 4 00e. 101, 210. + 0002. 1105 = 0015. 1101 - 4, 1200. 100Z. OV0
l s
800. 241. 1011 DP +000210.1011..
1002
4. 10021∞o fiss 4. 1002
adeaque 210 - ( (1101 - 4. 1200. 1002)  110I, 00b et eodem nodo (20  (10112 - 4. 1020. 1002) F 1041,   1 002
superest ZD et fit 201 = 000. 210. 101 + 000= 11107:, 2. 20. 1002 + 000. 1101
seu =2 L 000. 1011 (+ (1101- 4. 1200. 102)  1101) + 0c 0. 1110
(2, 2. 1002
201 =
O (105 - 4. 1200. 1002) - RCoc. 101 + 960- 1101
vuo
(25)
(2d L (101, - 4. 1020. 1uo2)  1981 7 2.1002
1011 +(110 - 4. 1200. 1002) f 1011 . 101 - + 2. 1002. 1110 == L (1C11- 4. 1020. 1002) 1V11, V1101- 4. 1200. 1002)
Emotandem 1aH H (ctO H02).
et firt - 1011. 1101 + 2. 1002. 1110=- 1(1012= 4 r1020. 1002, 1101 - 4. 1200. 1002)
quae aequatio explicatis significatio nobus pumerorum patiticorum, praestanda est ope residuarum, arbitrariarum m n, g
et ex aeq. 18 sublatis 2Z, DD, 4) sperest 2. 20. 200. 1092 CB + 2. 20: 1002Y + 200. 1002 h0
000. 700 H1C1I 
undem redacta sunt ad  per valores simplicissimos
en
(1611. n. 1101.  + 4, 1002. 110  4, Per. 1101. 11O (122) = (1011. 1491) - A. 1020 (1002) 101 - 4. 1200 (002) 1011 +16. guor 1200. lest
iaque superest explicatio numeroum fictitiorum
Seu A, 1002. 111O _ 18H. 1101. 1110 = 4. 1200. 1020. 1602 quae aequatio servat leges justitice et potest inservire pro lanone
 1020. 1101
e 1200. 10112
astenus vene nor culculi
ll
se nerali, aequationis trium literadum legis homogineorum inter se servantium
reducendae per mtires simpli seu reddenda identicae per valoces simplicissi
5
duarum ex his ex tertia 
1200 = heb. 9 (hb. 9 +a0) 1020 = mump C h Ca b). B) toot  mngs (9or hsb - c)
h (a- b
quia g9 +. (b- C) h. tam saepe occurit
111O = 2ma (9gth cb. c)) HIOC =mng Cgatib. Gh) _ 1011 = umgque (gg + b. d6)
ideo compenvia vecetur hp et Hba
(29)
ha. 6)h
Et exaeq. 28 fiet: ahm sipinta  ahemingu hxmmnntilipp hCa.b)
munt esthe man hegomnunaF ahena muhe mngghemmgglp = ahenn tb. c hpmnanche go) napetip
(bnc)
Mhib. e)hp mgh
vil alegninp + h ccg aen p - ke genenpp +
(et divisrs onnibus per nggm n pp, sic et mtu tistu
Thecgsque porco pro hp - is(b. C) reponento g sit - a
et  contrari, fietque -3nee + egg - 2he ( 4+
 htegp=-gheegmn p aie(.  ma pp
s eheegu -eg + h( b. cg ep L rt eg ep = - 4thev (-c) + 4he (b.) 99
)u +2egt = 2 helt- 1)gg = 0 quae divivi potstael g
et explicande e per a- l, fiet
hra + 6hab - 3hbb + agque - bg9 t B ∞ gb C Et c eticb B lb + hcc = 2hbc = 0
gg: h = +  ua + 3a
unde tandem fiet
ct 40b 108+ aub.  sed clculus resumendus
2eng99F 2eggti(b d
acogtsb 
siet
= 17) =0

## Folio 27r — 00068197:0053
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 28v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000053.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 85 lines, status machine
entemualiter equationem 60. 6a (20c - 46l)  20. 40a (:0 - 40) = 0  et fingagus esse 20 - 40, a = 4, 20c- 40
fiet 20, a 4 C = 40, a t b, f. et 40 = 20, a 4 c, :, a 4 b et 60 660 = 12040 = 20 H(a + C, : a +b)
sit 4C + et 20 =  a - b, et 40a == a = c enit 460 (a - b, a -c), et cum in Coniccus curva, cui
haec destinamus sunt b et c quantitates rigativa hinc. a a - b, a-c est quantitas affrrmatioa
ademue 60 est possibilis
Est ergo 20- 40, a =. 20c- 40b
20- E. 47a = 40 c - b  = 20c- 40l
Aitera aequatio resolvenda est 20. 30 (20 - 2. 40) a + 2.50. 60 (20c - 40b) - 30. 60. 806+
a (a - b, 30 (2C- a - b) L50a (a - b, a -) (c:- b) _30H (a - b, a - c)
L2.30- 40,
0 seu ana
et
30, 2ac = aa adrb t ab t lb + a  t 30, aa 2C - aaa C a ab ab2c + aba fabl + aba- abc Cbba tb
30 (f(a - b, 2ac- aa ba + 50 ad (a - b, a - C) (cub). fit 30 = dec- b Cr aa -): a
et 50 = +2ac -aa bce raa = ab): aa
Certia Aequalio resolvenda est
12. 40 (2  40) dat 2. 56. 60 a (20c- 40b) +1 260. 60l = 0
seu 12 (a = cg 2 2b + a + C) +rpa ab - ac +b) 12 (a -C, a - b) C  + 2.56+ (a- b, a - c) (c - b)
seu 1F 12 (a -c) (2 ab taa + aC + ac - b C) + 2.56 (c - b) H(a b, a - c)
fiat: 12 = 21b-c) Naa- ab): a et est  t
204 +b 1 et
  et 40 - 20 = 20) Eca +b = 1 = 20, E= 20a - b s. F C = E0a - b p = A, srol -t0- 20 - 20 s
0a+
20a +b
 fiet c = 20a - b = anac e 20a  F bcd  fiit 20 = AC - b + bcA, 1, 4) +4
i 60 fingereter  = 20 = 40 fieret  20 - 40, 20c = 4ob = 20. ~ a
seu 20. 20c + 40. 40b 220.40 b. = 0  sit C = VY et b = - β3β, et fuast
=20.4u c 1+
aoVY 20. 40 aa = 0, fiat ββ + V7 - aa = PB)
a  LC + y sin aa = CCFYY + 2B) et a B   et B 2ost +). 
A po
potest suma 20B et (40 - Y
 et 20 + 40 β3 a  X 1.) 7.   
 et fect 6o  β- V esse g+
Ex aequa pet d . 30 (+B 30 +B (+ B- 3) p B +)y) + (3(3 2B) + YY) eB
fiet a = + (8 4)
+
et 60  et (+) + 20) (+)  f 20β = 0 et fit.
20 = Et fier potest 20- (4) + B et F
 20 = (+) 6 + 40 -(1)) et 60 -() B(+)
seu 60 = (+) a.
Ex aeg. 2 siel fe  (o A 2. 4) t (p.) -(4) 2u 5a (20
30F20 (20- 2. 40) a ∞ a Bβ) (+) 2. 50 (20) 1 + 40 (B)
P. e a -) B F (1+)
seu  BB- 2β)8+ B 1 (4. ) 2. 3( (+) eY) (4 0)BP) =  quae aequitio d di ptest per β3-)
et fit 30 a 2β (+)  (4, 2.50(), =d seu fieripotest 30 = +4) et 50 = a 
30a.
Resumamus in aequ. 1. sumsimu 60 = 3 (20 -40). firt 60 20c et aC = -yy et ab = = ββ, et ex aeq. 1. fis
22to-40)  (. 20y + 40 (3β) +20. 40 aa =0 p 1 seu, 20. 2014 4. 40 338 20. 40 (3= 20. 40 a a: N
F20.40
Ergo fiat β(+7 3 + aal.s  = + 2β5 (+) q al. D +B +7) + ux
et ficret ex huraequrtine + β +7 fjaled) =   et ex pusic rd   avito ( C =  (( 20 4, + p)  a t 2)
Naplical facire
An sis 2au  20.207 140 403 4 ) = 12 40 ) +. 20  20. 40 (Y + 20. 4ol B 1F p) + 20 +0 a4
3
2. 20. 41
si  admiteremus imginacta posset, poin 60 = 1 (20 40). et ex aequ.d. f. Eet A (20. 40) C o s
seu  aa: A A = 20. 207)+ 40. 40 33 = 20. 40 B(3 - 20.90)Y.   qu et peret 4120 )40β = 0
s poneremus 20. 40 (3(+.). 1. t aa =. (+1) 2. L9. 4 (. BV. sed ita finit aa: ad quan
tX
Faciumque re 50 = R. 20. 40 fiet A L. 20. 90 ( - 20)) + to (()  0 - 10 = 40 - 20, ab
 re titus negativa, quod non licet
Si fiit 6060 = X  20. qua, fiet. A (ao C qu1)  t Eo - 4) a u = 2017 + 40 β( = (40- 20) aq: X
sed non video aliquid commodi inc nasci
siet 4u  40 (3B - a4) = 20 (V7 - a4)
fiet 4 W)Y = = 20 βB
et qup
set hoc ut prus. Finganm aa:  = c BB+VY
et potest esse 40 = β3B et 20 =VY sit A = 1 fiet
ob - 20)1 +4o(d B = (10 - 20) aa, fit p 7 = ( - y, (B 3 + V7
20c - 4cb =   ((3 - p+ + (+ = aa (2- y7)
60. 60 = β31)  Ergo 60 = = β)
Itis positis in aq. 2. siit 30 fad (-s (  )B 30(e  ( ) (  -  ( p e 1 ) ((3) qu. 50. st ( y 
it  ut patet
fiat 30 = + . 2βa a (BB )7) et 50 =  aa B3) - y5
Et in aeg. 3 tint 12(2YBC Bna BPIY WVH) + 2. 563) (C X - y47 = 0
37
seu 12(2 Bp y + (5) aB - p6- (Y  BB p ) + 2.56 (B 
fit 12 = + 2 7 (pa pa5 et 56 = fBY) (BC+ 7Y) - β 4 seu  12  4 2ffer
12 = 4 2aa) (1). () et 56 = (i) Vaa = β
 os a iactun ti  adet iot is e u  10. 60.  
 30no, gdiditer. im eaius li tandeus avenduu lduitum et ne siet v 20. b0  i a
ac
X
h.

## Folio 27v — 00068197:0054
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 28r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000054.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 104 lines, status machine
gestructis Z, vZ2
en +2, 12. 20, 30aaVZ
20. 117- 2. 12. 30. 40 aavZ
120 D 22. 13.50. 60 aC YZ
P2. 215056 aC YZ
30. 5obo bl 2-240 50.5F ab VZ
et 24 40(, 50. 50 =
20c - 40b, 50. 50 =
1.7Z, costant
+ 12. 12. 4daa a2
+2. 12.56C aa CZ
d. 5656aacZ
Adegeaan
+12.5656aC
 40. 56. 56 na bL. 4d56. A_
+2.1450. 50 C t 1250acvV
30. 16 5b a abv 20. 50 56b 13
a e V VVet P, ut partes ipsus VZ inter se
-30 (4030 + 50. 60), WvZ. jam ses 50 = 2. 20. 42C - 20. 20 ac _ - 40. 40 ab , 20. 30
5o Z. 230. 56 60 abVZ
atur (20- 40) 5C
3 5b
perulai pu  et B, uret ho, voaos, noit-a
.8. 40 hacc- 429 acc - q. 20 40 dgbF
FHoae 2.20. 4 aabc +40 aabl
15 qu (ob - 20C) (20 (1e), (29
ii itq
4.264
earitur
2qu (40b - 20C) + (20. 41, 20- a01a :, 4ob. 20c)
28.20.  (rec- 40b
 20-40
420.40 a ct - (9. 20. 40 a
41 20Y ac
-720.40 aabc +C
+ 4 qu (aob- 20c), (2oc + 40b), 20 +40
50 fit 207 - 49 = 701
2. 20-40 = 72
+2. 30a40. 40,2. 20. 4
4qu (. ck.
ceetu
20. 30. 30 ( v. 40 40D + 20. 3
2b. 40 )lbb o. 18. 1. 1
F20. 30. 3002040. 8. 20 abc- 20. 30. 30. 20.40. 82040 abc v
=20. 30. 30. 20. 4. 420. 20 ac a20 p. 10. 20. 140. 4 20. 20 a6
14 4qu....
+0ba 20c = 71a. fit 60 = +(20. 40. 30271)
2. 40-20 = 43 73
50 - 20. 30 7 a + 30. 60b, : 2, 60. 71a
56= 12. 40. 72a + 12,6060C, : 2. 60. 71 a
2 (20c 406)50e 2d. m sa
vel 50 = 4203,0. 30, +20. 20, 2. 20. 40 7  20. 46. 70b, :71.
20 40 20 20 90. +0 b
20 20. qub. - 2. 20. 4 b
=283C 4. 2.20.40. 2C
50 = 29 q 30, 20. 20 = 2. 20. 40a = 60. 60b 3 :, 260 (400
Sin 50 = 203, 20 (240-20) ac - 40. 40 ab, E 2. 60. 71. 712 a c
5 = 12, 2. 20. qua - 40. 40 a + p. 60 b0c,: 2. 69. na = 12  2. 40. 40ll. = 4040. hol +
Zoq0aore (440 40 20aC
seu 6- H 40, A0 (2. 20- 40) a- 40. 40 al, 2. b0s 31. 1aa . 2. 60. 71. 11 aa
2. 20. 40-20. 20 a C -440. qrab
60.60 = +040 ac = 25. 10 1la. )
220.204 (C. b) ac = 2. 20 foac,: 2. 20 H. b) (20c + 22 9+ Cc 11) b
2. aevac = 2 ae Vab s,∞ re : 1)
sunqu
2. avaa, Gbc +. H(C
=Eora = 2. 20. 201 sc: 6) a 20 it 10
2. 20 ), 20 cc = 2. 20 fbc 4 bc
ib 20 +
Y2. 20do ad tc. b)- 10 20ae- (20no a d; u
50 = 1a
2, 20. 1 ca: 6), 20, CF
2. 20. 20 HC: b) a6, - 20. 20 a(] - 20. 20a
2, 20 fa : b), et b0 = qubc, 20
56 =0 seu differunt signis  et 3
a, sb=ve ss et adei2
divellantur ut fecimus
quo = Ya3 b - X Et - X2 - = qu = 3o X6 - 3_2 X B+ x6 -, X9 + 3 XX 9 - 9 et ,  1)
vroa qetrs
X- 3 = h ∞1013 = h +X
quo = (126 - Citu g  t
si scilicet aeg. 1. _ seu 10.90 (20. ex). + 6v. 60(90c 4 b)
la, qu (∞ - Y
412. 12. 30 aa av
20435, (20c 40b)
H N 20 30
enbiaaditat
204 +v. +o
17. 3020 4 30 a
P2. 3016. 40. 402020ac
Et 2. 40020s
4u = 20+7c 
20+. 40 = 7X d 2.30 -40 1- 7
326-40 - 2.40+10 ∞3.70
X 3. 10+7522. 2040-
 75.3, 1d= 2. 20 -4. 40
3o co o goqu
nde 
2020= 40. qub
60. qu (40b - 20c)
+2. 1. 10 b6d, X0. 40
= 20. 30.30a
qu setb
ra, qu (Se. 16)

## Folio 28r — 00068197:0055
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 27v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000055.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 94 lines, status machine
∞ OD in OQ in D4 = 00D) P
sit ax = CD et ax + ab = O et _x +ac = DC
finte On = C2 - ab = 52 - ac et B = _ a (b - 5) ab + OD: 0 = ac + C),: 1 sit abD + BV -acCteoF
faciamuse C = 1P + Na, a :, 30V + 31a  et D = 2.V + Va, a:, ZoV + 31 a
fill - 10. 100V + 2 10. 1aV + 1. 1aa, 20V+ 2a, a = +W. v. 20, V + 2. 18. 1. 20aaBP + 11.11. 20a B + 1(11. 21 a
=+30 pV 4, 10 2 av 21. 2 aa, 1oV AIa, a = - po 2020  - 2.
on - 2.11. 20. 2 3∞ - 11. 21. 21a7
= 30. 30 vrt 2.30. 3lav + 31. 31 aa, 20v + 21a, - aa b = - 20. 30. 70 aB - 2. 20. g a. 31 aa vv = 20. 31. 31 a
21. 20. 20 aavV - 2. 27. 3031 ar
ut destruatur V, fit 30. 30 = - l0. 20 (10 -20): (10-20) Ergo 30. 30 = 10. 20 et parifua 3HH. 21
et pari jure ut destruatur V, fiet 31. 31 = = 11. 21, Ope prioris tollatur _19ope posterioris tollatur . 11
pyt 16 2 - 30. 10: 20 in aequationibus praedituris, ut sint  coefficient ipsius v, et ipsius V, aequales nihilo. Et in duobus
istis coefficientibus _ etiam est asenomia, ut scilicet unusfiat ex altero per mulando 10, 20, 30, cum 1, 21, 31
itaque habita unius prostatio destructione, facili habetur et alterius destructio
2. 30. 30. 1L. 20 . 307 2 + 2. 30130. 31. 31. 20 11. 20. 20 _-2. 2051 + 1130. 30- 2 20 3031 14191
seu +2. H. H. 2020. 50.30 + 30.31. 3] 2 20 20. 30. 30. 31. 3 t 11. 1. 201 + 2. 11. 20. 3831-1 1. 2020303. + . 1. 20. 7 X p
peli ando 10 et Ul.fito - 4 2. 130. 30. 21 31. 20  + 38. 21 + 2. 3010. 20. 21 + 31. 31. 20. 20 2, 30. 4 - 3 1. 113070 - 2. 20 3031 - 1. 2130, 20
Iir Verulicando
10 et 21 fiet
seuo 2) 28. 30. 31 + 307 21 21 4 (2s. 35. 20. 25 +  25t. 31. 31 - 2. 20. 21. 38.31C 20. 20. 3050. 71 31) 2. 28. 1 30. 3(=20 26. 21 21302
12 (202 1  36)  - 2. 31 (20.30, 20 496) + 30.30 (20 +30) = 0 et divisis omnibus per 20. 20 + 30. 30
ep  destmendo,
Et prorsis eadem prodibit aequatio ex v destruendo
21 sed hoc non licet, nam ita fiet D quantitas constans. Quin et si conjungamus
31.20 = 30 = 0. seu 31: 21 = 30. 20
nc huic aequationi binas priores 30.30 = - 10. 20 et 31. 31  = - 11. 21. fint 31. 31: 30. 30 = 11 21: 10. 20
sed 21: 20 = 31: 30 Ergo fiet 31. 31: 30. 30 = 11: 10, 31: 20 et fiit 31: 30 = 11: 10, 1el 31: 11 = 30: 10. Ergo
etiam C desineret esse constans. Haec ergo via non procedit.
Ergo pro explicandis (3 ut 9 assumamius literas binas: C = DaV+ 2+ 12a
v + B0Z + 56a
fiec + ()3D = + l0. 1ovV + 20. 2572 + 2. 10. 20vZ F 2. 19. 12 ba + 2. 25. 12 Z a +12. 1 aa, 300+40Z + 3+a 3f a
Fiet (- 99) = 4301oVMt 40. 407 Z+ 3. 30 40v2 + 2. 303qua + 2. 90 347a + 34. 34aa, 10V + 40 Z +12a, - a
O= +acO = + 50 50 vn + 60. 63 77, + 2. 50. 60pZ + 2. 50. 56Va +2. 00. 56 Za + 55. 56aa, 18v +203fna, + aac
l- abD = 4 50. 5ov P. + 60. 6077 + 3. 50. 60VZ + 2.5056va + 2.6056 Za + 56. 56aa, 30 + 402 +3 4a, = aab
Hic tentemus tollere Z,  Z3, VZ2, aZ2,
+20. 20. 4∞a = 40. 40. 20
d346. 40. 
O2 1210. 20. 40 a
+ 60. 60. 20c - 60. 60. 40b-0
265 50. 10
D. = 30v+407+3+ a
a 500 +60Z+56 a
A2. 30. 40. 20) +250.60. 205
Unie 40. 40 - 20. 20C
etE 260=20 laib
60= 40_c o 10- 20l(.
20. .  1660 (roc 40b)
- 60. 60.3076 = p 20. 39 (20- 2.40) .10. 40 (40 -2. 20) a
- 10. 40. 127 +50.60. 12 7c
- 2. 40. 34. 20. +2.60.56. 205
-2.50. (D. 40) 7 60. 60 (10c - 30b) 4 3. 50. 60 (20c - 40b)d
= 60. 60. 34 7b 29 L20. 34 a β2- 2. 40) 460. 60 (12C- 346)
=3. b0. 5h. 40d + 12. 40 a (2. 20 40) 2. b0. 5b (20C- 406)
Ex his tribiis aequationibus prestitis, haberi videtur valo. ipsius Z rationaliter per v
+20. 20. 34a
in aequationibus prima et ultima, absunt 10 et 30, quae possunt reservari mediae
Z 11. 20. 42 toa
9o. C 20 toa I prima et secunda absunt 12, 34, s quae possunt reservari ultimae
Aqui sut (60
sed in  nihil ab est secundae et tertiae, adeoque prima nullas proprias habet.
20. 40. 40a +60. 60. 2c  media fit 30 = 10. 40 (40- 2.20) + 2. 50. 60 (40b = 20c) = 10. 60. 60C,:, 20. 204 - 2. 20.40 a - 60. 60 b
dicidinosset in
60. fol, 20 10a =
Sidil f. ret =C
X ultima fit 34 = 12. a (40 - 2. 20) + 2. b0. 56 (40b - 20c) = 12. 60. 60C3:, 20. 30a - 2 0 +o a - 60. 60 b
nhis pro b0 substituatur valor ex aequ. 1. qui est 60.60 = 20. 40 a (20 -40):, 40b - 20c
20. 40 20a 460 sid indeo trevis procedi quaerendo solas 50 50 ex media, 5b ex ultima, 60 ex prima equatione
 doibe = +  fit ergo ex media 50 = 20. 30 (20- 2. 40) a + 10. 40 (2. 20 - 40) a + 60. 60. (0c 20b), : , 2. 60 (40b = 20c)
Aru saibe - 14
si firiol 20- qu, a m oc adbd et ex ultima 56 = 20. 34 (20- 2. 4) a + 12.40 (2. 20 -40)a + 60. 60 (12c. 346)3 :, 2.60 (40b = 20c)
oc fatei quia ergo tot literae sp superfluunt, videamus an liceat sevare in numeralonbus
sit roa + 3= qota + b) prima audem et 60. 60 = 20. 40 (20 -40)a :, 40b = 20c
2020, (a +C:, a + b) ipsorum C et D solas 10 et 40, et destruere 20, 12, 30,. 34, fiet
e40 = 241 + c): (a + b
et 2C. 40 = 4 60.50
(sed ita C fieret
et 60 fiet etiam d
50 = 10. 40. 40a + 60.60. 10C, :, 2. 40. 60b et 5b= pd ita fit et 56 =0
itaque ante omnia conservare demius 20 et 40, ita stabil 60: nonplicet simul rejicere 10, 12, 30. 34, sed
videmur posse tuto rejicere 10 et 34, fiet 50 = 20. 30 (20- 2. 494 - 3060. 60b,:, 2.60 (40b 20c)
t inisii sit es ti, et 56 = 12. 40 (2. 20.40) at 12. 60. 60C, :, 260 (20b - 20c) et 60.60 = 20. 40 (20- 40)a:, ob. 20
ita tpplicaro bosit 50 = 20. 30 (29 -2. 40) (40b - 20c) , t 12.29 4u ( quiae - 30. 20. 40 (20-40) Ab, : , 2 qu (tob- 201) 60 
1 11
seu 50 = (20. 20. 304065 () 20. 30. 90. 40 (b (= 20. 25. 30. 40 ab t 20. 30. 40. 4 cab) + 20 30 (20 r2. 40) fC, : 2. u( goba
et 56 = 12. 40 (2. 20 - 40) (Geb - 20c) _a + 12. 20.90 (20 -40) ,: 2 qu (40b- 20c)604 seu
SH = 12. 40 (2. 20- 40) 40F -12. 40 (2 20. 20 12. 40. 4020 a  fF 12 20. 40. 20d C 1. 2040. 40 4D;, 2qu (gob- 201) 604
fit ergo 50 = 2 30 20 (2.40 -20) aC = 40. 40 ab, 262 2 qu (40b - 205) 2 1 (20. 40, 20- 40, a :, 40b - 26c)
et 56 = 12 +0 ( 20. 42) (1ob. 2oc t 2 (20 (2. 20 - 40) ab - 20. 204c, 12. 40: 2 qu (90b - 20c) 1 (20. 40, 20- 40, a:, 40b - 20c)
dea: si esset 60=(0, fieret ex 1, 204 20=40
 ati adtuo unb acehonao o peletaee   disi o oa t
u te tius pari tndo dun lict lactre ab 40 ito pec lalea  s d 250 simal uee lace totu
 3A; vot 12 et 30 4
o rondorai u
H1 = 26b H. b) lbc -20 C guat)

## Folio 28v — 00068197:0056
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 27r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000056.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 81 lines, status machine
A X, X. XV, y
AE, 9
BD, h
vy: Zlh = AVE: ACE = AYE: L99. Ergo gu = hoh, gX - XX.
Hinc si fiat rg hh fiet yy = rX - =XX. porro cum h, sit minor
quam 9. ser h: q ratiominoris erit et r. 9 ratio ad huc
itaqueY potest esse una ex upplicatis lipseos. sit FR,. fiet r = YX - XXX,
hine quaeratur X fiet   qx 74 -gr = XX - gX. seu  39. X-
C494. ) 1d = (t  H) = CF  (L49 i hh) = CF = ( FH
Cum ergo quadratum ab FC sit differentia quadraturum FB etBC, vel
Quadratorum ab FB, et ab , erit FB, L9. Ergo F est focus
 CY _ est =9 - X Vocetur Z, et fiet Lx erit X = Lg -7 et, et ex
gy = Lrq= et
minoris quir r: q = hh. 94
Et latus rectum est ordinatim applicatu ad axem in foco
2(+ E 42) - YZq = = r9 - X7 =  tt - hiti
qu
si est esset B, ei Ex masto cpseos F agatur nolmatis in aer cd jngatum Bd normalis  et occurens in V
et x CV vocetur v erit D - 2h - y, et feret  a Lhh - hv + bD = gy = Llit - hhZZ. et fiet Z2 = 4a v- 4qvD
plu 22= h9v - 2 W. sed prostat uti literii y et Z, ubi coordinatae i agibus ve
sumuntur, et a absunt simplices lariata. Poro Aris compug atus empue mineestidiveto e eutu an
quaeritur fa tagay + dz d2) seu Elementum curvae Ellipseos, fiet: est autem ydy = -y dZ
et duys dy =  he yque deder ,: 12hb - sitqque, adeoque dpdus + traz - 1 2dq, bili 23 1 - 12 2
itaque summore oportt AZ H 99- ∞e22,: 99. 94
ht yn =gaco et  et 7d7 = 2aded t d=
= 9122
 dydus +, drah = + g- eez 2g
Aadeo:, (3 laco s dz
d2d
F agged, aco Adesque de = a dco, ap ace + q 9972
A Et fiet dn = a du C1G4 4 raeeto
Y(aco, 99 aecco, +q aqqco
ta
az au
 itaque extrchibilis aedtenda est quadatice formula aco acoco-=  aqqeo-  vel ducendo invicem
quqqo.
4qued +qBac, quod totum dividamus per ee qq, tint C-44 (eergqu) nacoco +
Qarqus coco
Uhius sermulae vallies sunt acal, acd- 4 99, aed -9. ee 60, h et cD,  ibeC
Ia Hypert, A5, 9, CA, G latus rectum Y. sit tih = ce AX g. X, y
19 = ry -Ixa LYyy
yu = rX + YXX. sit Z - L9+X se fet - 24
+F
seu yy = XZ2 - L Yaseu gy = tih 2 + gtk
jungatur AB, ejque paractia dualu aXG occurrer ipsi  axi conpigato in e exsi
erit X Y, seu y, medi proportionalis inter DG et GB, seu inter summam
disferentiam ipsatum
itaq generaliter yy=
et
CB, et CG. idque tam in Ayibola quam in Ellessi reum es
22Lth,. ubi in Hy perbola
ydy = + tih 7dZ et dZdZ + dijdus = d2d2, Z2 +th F  B F, in Cllipsi minas est-.
seu d ez t ayaus - drc d N = p A  a q + ee 2,:, + 4922 + B L 90 7 4 223:,  nt
Ergo generalis haec est construssio pro Ellipsi et Hexperbo lice curveatemento, de
Z + fil
aut sot
s mbiguitate in valore ipsius
n
e in Ellipsi es distantia focorum quod sic ostendo; FB qu =- BCqu = FC qu, ident est quod 294 - L hh = FC qu.
 Ergo oC = 49 - hh= AFC qu = FO. qu.
ad quaeramus distuntiam focorum generaliter, tam pro Ellipsi quam pro Hyperbola
in aequatione si sit generli gy + ln q2 +  lh, sit y= LY, siet  vr + hh d Z ( hh. sec. 1Y = h: 49,
et fiit Zhh = + 22 + 419, seu 47Z = 49 F hh. Poro  27 = bis CF _  = FC
Ero FOqu = 49 + hn = ee, seu C = FC. itaque in . Hyperbola non minus quam in Ellipsi
e = YCga =+ lin) est distantia focorum
ape directo, seu latere transverso
Haec in Ellipsi est minor, in Hyperbola major
aa : 7 = 9 fi de = - a ly:    d8 d 5  de 1e = des s (a + 84):  B 3.t 2Z = A at, fiet X- f ak
et dus idh L  et ft s(d 9des + d3 de) = d dX sC at + aaxK, - .a  d(at tate)
eK
quod aequandum   Rk tatu = KK WV fit ( siet a + akk = k10c0
1
RK. = K. WW +  . W0
Zl -aa. = qua) W- 4 Ergo rest redit eo
ut. 15 - 4 a aequetur quadinto quo sit si aZ + ak  qu Caa + K K) suit aK AA = a M L XK  aa
44  4. sit 19: e = f et sit  4 = 3 a f - 4 fint 99+ f = 25
H= Apura +.
sin H(2
+ ff.) =. 5
Sit +hhf9e - ee
11

## Folio 29r — 00068197:0057
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 30v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000057.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 75 lines, status machine
OD in OE in 3( = 00DDG Esto ax = O D, aX = bb = 09 et aX - cc = DG
erit productum ex his tribus ax, ax- bb, ax- cc aequale quadrato.
Quia ax = OD erit OD - bb = O∞, seu _ = 1B - bb,: O  Ergo O0D + bbD = ODD+c
sit O =
et 0D- cC = DG seu D =OD -CC3: D vel + 6OD - CDDa +bbD
207 + 12a
et 2
= 300 +40Z
a
602+56a
602+sba
Aoet +(3O D = 20. 20ZZ + 2. 20, 12 aZ + 12. 12aa, 30v + 40Z, +aa
a
Fiet o= -(9DD = 40. 4072 + 2. 40. 30 Zv  40. tovB, 12a + 20Z,=aa
+ bb D = 60. 6027 +2. 60.56aZ +5656aa, 30v + 40Z, + bl
-cc0 = 60.6022 +2, 60.56 a7, + 56.56aa, 124 +20Z7 = cc
in hac equatione destruamus terminos Z3 VZZ et aZZ, ut solum restet Z
fundamentalis
Habebimus tres aequationes destructitias
O= 20. 20. 40 aa = 20. 40.40 aa + 60. 60.40 bb - 60. 60. 20Cc
(1
0= 20. 20. 30aa - 320. 40. 30 aa + 60,60. 30 bb
2
02) 220. 40. 12 aa - 40. 40. 12a a + 2. 40. 56. 60 bb - 12. 60 60cC. 2, 20. 5bboe, aZZ
Ex aequ. 1. fiet 60.60 (40bb -20CC) = 20. 40 aa (40 -20)
60. 60bb = 20 aa (2. 40 -20)
2.56.60 (40bb -20cC) = 12, 60 60cC +40. 40 4a - 2. 20.40aa
2
Aequando binos valores ipsins 60. 60 ex agq. 1 et 2 firt 40 bb (40- 29 = (2. 40-20)b0bb- 20cc)
3
seu (40. 40bb) (20.40bb) = (2340. 40bb +20. 20cC- 2. 40. 901c (20. 40b6)
unde siet 40. 40 - 2. 40 20 eC + 20. 20Ct = 20. 20c- 20. 20 bbcc
62
tandemque 40:20 = cc  + C Y (cC - bb),: bb. compendi,
causa  HCC- bb) vocetur f, et fiet 40, 20 = c (C+): bb
Et licebit facere 40 = c (c+Y) et 20 = bb. Et hos valores
substituendo in aequ. 2, fiet 60. 60 = aa (2cc +  2Cf = bb) vel
60. 60 = aa, cc _ + ff + 2cf adeoque tandem 60 = + a (c + f)
Superest aequatio tertia, in qua substituedi sunt valores numerorus
Inventorum 20, 40, 60, supersuntque quaerendi 12 et 56. itaque ex eq. 3.
fiee (onsidero autem quia 60 et 40, habent communem divisorem C+ f, posse
inter explicandum aequationem 3, simul fieri hanc depressionem, et fiet
12. 56a bbc ( + 2. 56abCf ((2. 56 abb C0) = 12, aa C t aacf t ec aac +ccaaf. raalbc
vel + scab 56 bb f = 12 a   (a Cc - bb) =) aufs + a ef, et
divisione facta per f, fiet tandem +56 bb = 12a (f+C). Unde fieri
Quod si faceremus
potest 12 = =bb et 56 = a (f + c) = 60.
F esse + fieret ut 56 =60, ita 12 = 20, quod non licet nam fieret
d: a =  itaque debet esse = idem quod - et fiet 12 = - bb. Sed
Hlabemas ergo valres intentos, prore nata setise simplices
20 dbc 122 - bb e se = a Cets) = 44 =C e
sed ita fiet 60 = = a (c+ f) adeoque 60 = - 50
Et ex
sed ita fit e=-
si omissa divulsione  consem u ratirem 40: 25 seu fiassemus 40= 20, ce tcf b9
a fint bb Z - bba
ZSolba f)
videamus an itam eadem difficultas occurrat, et ex atr et ex 2 fiet 60. 60b ea
-569 = 5ba
2.X 20g onoa - 28. 20 A2. 20. 20 (CC + fod:bb) seu 60. 60bb: 20 20ap =
bb - bb + 2cc + 2c. f,  seu cA H + 2tf et +60. b, 20 a = c + f et 60 = Za, C+F: bb
CC = 12, aa cccest aacc fstraacces tec ccaace + ccaqet rceaaet 
Hos valaes o et 60, substituam in cer. 3. eoanescet in ex 20, et fiet
Se sbt = 12, 2 au C  raa aace (a- bb7 =) 2aacff+ Laiff + (2.a a CC= bb) est
25oabbC
sbacbbi Hets
urau
pato
etragus cs
 aees doseets us adt nast in P ti t ena
 l constanti
b
Aquod nescio
au liceat.
Quorsi

## Folio 29v — 00068197:0058
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 30r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000058.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 77 lines, status machine
rit
+BDn ut 3r 30.3obbrB + 2. ZobbrrC BZ Hbbcc ZZ
4110 -. -  - - . - + cebbr rr aa + Z. cc bbr ra a Z + Ccbb (  H. ZZ
_bb
 OdD5
sit r= c+f
Videamus, tamen an non liceat facere e) constantem magno utique alculi compendio
Ergi Si esset C= obix, ex aequatione fundumeritati fielet
do
=aat1, 40
sit C = m et  D = fiet mDD - bb D + cem = 0
sit jam D) = 30 aV + 40 aZ:, 60Z+ 56a, Et fiet
FM)D.. . . . . 30. 30 aavV + 2. 387aa v Z + 40. 40 m aa Z3
= mm 5)
abb13...- 3160 .. ..  30. 60 (b6 a v7- +0. 60 5bb a7Z. = 30. 56, b 7. aaP - 40. foz n n 1442
MM
Enm,
26b)
+cem
+ 5b. 5b cem aa + 60. 60 ccm Z2 +2. basbemaz + 1. bo sbcem az 
Et fiat 40 = c + 60 = a. Ergo, oCCaa  = a a Cbb)  aacC C) =0
 fit ox 2uacem - uac hmm
Ergo exito. 4omaa
o. siqbt za profilo-
+6060 cem
tccaa f aacce
2aacCf
L= ac
F ad, c bb,
caycets
Quia bbenim = 2Cm, hirc tota aequatio diridi potest per m
desructa tma
atb, a b = Ba - b1, c
Tollitur e sublato termino aZZ, reperio toli etiam terminum aazue, num
nam fit.  Rc. 56. 2cm aa Z + 2a. 5b. cem aZ=c
itaque tandem ex aequatione nosta fiet imo tllitur etiam VZ
Itaque calculus tam facilis non procedit ad scopum
30. 3 a ox
Videamus an evitare incommodumliceat, non assumendo 40 et 60,
ut fecimus, sed quaerendo, (compendio sit bb +m = ag, et aequatio ad destruendu
nimum
terminum Z3 est 40. 40 aa m - 40. 60aag + 60. 60 cem = 0. Et si
Hinc 40. 40  40. 609 + 60. 6099 - 60 60gga6060a am n ce
mm
qa nm
seu 40: 60 =   (_ aagq- mmcc): am
40. 40 aa - 40a60a9 + 60. 6aagg- 60 60 aague  60 60cc mm
amm
amm
fiet 40: 60 = ag + sCaagg - 4 cemm), : 2 am
si faceremus ag = 2cm, se Haaggs dumm) =0 recideremus in priora
et fieret 40: 60 = ag: 2am = 2Cm: 2am = c: a. Sed si facerenus
ag = - uem fieret 40: 60 = - 2c m: 2an et posset esse 40 = = c et 60 = a
et fieret bb + mm = - 2em et mm t rem  ec bb et nim + 2cm + cc = cc = bb
et m = H(Cc - bb) = C seu M = f-c. Videamus jam an etiam hoc modo
destruantur iqe et a7, Ut vereor ne sit idem. ll e pro aZ f  2Cm Ca +2aCCm =
 et pro vZ vZ, fit -2. 3oema +30a rem. Ergo relicta adhuc m, faciamus
Faciamus
40:65 a ag + (aagg- +cemm) e: et 60= 2am,
vel 40 f0 - bb +mm + YCb + m + abbum - q ce mm)
bb +mm = 2mn, fial 40 = n r. 40 :60 = n + Vbin- cc): a
nn-ec
Et fit aZ, = + Nan -c4)) 2mn F a ccm. = ram_.  e _nlun - c)
Haec stare possunt
=a 2mn = 2am, dena - cC)
ut Quia r est arbitraria, modo apta, faciamus nn = bb + cc, fiet m = n + C et 60= a  - nt B
aZ, +2 (n + + (un - c))ma
mm - 2mn + nn = nn - bb fit m = n + senn - bb)
Vemamus ad nostram aequationem et his valoribus pro 60 et g0 substitutis videamus an destruatur 22.
o= nn M H2nomixbbmaa au-a (-armub) Laaccm fit bb FcC = nin, ut oportet
no mntbb ponendo 2m, fiat 40. quaa  =240. boAN + 60. bocc fit 40 40 : 60 = n + Y(pn - cc),: a
seu m = p + C
seu 240: 60 =. n + b,: a, sen 40 = n + b et 60 =a, ut prius et  quia mm +bb = 2MN, A sit m = n + Y Cnumbo
Repotaums quraes
un FEkb +bb 2 nF X
nten
0. = 2

## Folio 30r — 00068197:0059
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 29v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000059.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 115 lines, status machine
Nunc ergo ad aequationem nostram primariam redeamus, ibique ualores repertos
ipsorum 40, 60, et m substituamus, reperiemus autem omnia posse dividi per am
Et sit ex ea 30. 30 avv 
O= 3030 av0 + 2. 30ap L Z - 2. 20. 56n aY - 2. 56  f ) t. aZ
et fint Z = 30. 30VV - 2. 30.56nV + 55. 56CC, :, 2. 30b0. - 2. 560 (n +b)
Et 30 = sC = 2r, firt  ∞
A+ Er 4
= vV = 2nV rcC
-bv + DCn + b)
ipsias Z, fiet
2gu(rt ) B
sb o a alui + (n +b) CC
ubi ulique evarescit Y, et faciendo p +b = p. et n. b-9 pV + 2 b n) V
utmunte sic lompendio b + n + b sit pet nab situq. En firt D 
lh.
st
ut m = n +C et n = YCbbAce)
_, ibum D =-
+2r
2 E fit v DxbuY L 2YbpB
jam C = m D - ab, : m = mB - cc, : D
vv - 
= Apr VV EBAB p
rvo I CC
A 2tbD + 2rbp
= bb v + 2bbq v
+
 2bp N. ton +2ce
= bbcc
an C
2bp
quo = 2qu + pcc
sid redeamus ad aequationem mDD - bbD - mm D + mec = 0 vel ob mm En
mvb = 2m V + cem
+ 2m bP
sint, AD = 2nD + cc =
qpqu + 2 pcc v
.
tbn
 a(pju =qp cCv  pp C
+2n 2p9V. - 2n pcC VV
tant -qurqpp .. +2n 2pKC
=2ncc_ on LEnCC
= 2n 2bp 3
ob
+ ce VD = 2CC4
av
pa ccip.
 ACCH
e
Ex v sis abb abb t Abn + sum Hoba CG + qon Alt C abidC tud- qtn Lbb c
Ex v sit 2nb (bB-Bbin =) C) c) + 2nb A cc= 0 suvedit
Bg esse nn - bb = ce, unde lotis termitus divi di potest per ac
Ex V diviso per p, fit nn Z_nbb + 8 anf _nb - 4cC = 0 succedit
et D fore constantem; idque praevideri poterat ex natura aequationis
nt -cc . debet ex identica depue hendi inventis valoribus ipsorum Teretur t 2
n v, ut nonaetit Conorutamus, novandum est q+ p es = 2n
 tecai
Fqu
+ mvo - 2mpV + mpcc Am on = Emp V + mpcc
+ sem
aee.
tepe.
2nqv
ten
mun
 2
+56. 5baCC
D2brwV  2r(t) V
(2tb) rWG= 2 (ntb) u V + (n +b) reC
Arv- 2ArnV + arcc
82rbV t2urb (ntb)
+pcc
topec
bo - 24  Esp
3 = Z: 2
tit mhrt - bb, D
= in E- Cc
nt-bl in GbEH t. Z
in hk bbHE- inti - nCc
42 =
-2nt
Encepcc
F2nEopipq.  2n
+ qccbp
= CC24
-45
Hoc
aequatio
debet esse
identica
per V, abcunque. sscondant si sint vjusdem gradus
Sed video dandem totum ioc nape
rao uatet
redigi
et quidem
deber debere et ipsius e binos hos valores seddi posse identicos it est si sut op
et et habere eundem denominatorem et numerptorem, vel saltem divisionibus. t dibersi
taeu maniteste petent in nostra aeg. N, quae cum sit identica, debet etiam in frrma fractionis ilpalica ostendi posse
per constantes, si sint ntrobique ejusdem gradus; v per formam rariantem
at ide
 inqrdec  pie 
ilt od dls
Htine _ cum sint
imus et nus ternim
iet cem + embp. :M = Bcc: q seu ccD + 2bpq = piC. quia pu = Ct
ejulda grodus dividatur aumeratur verter per m, et simiter per. 
ailet rem contingere et
e
s  ttrae
b.  sed hi duo denominatores sper constantes
ix itatu abre qurote utu sestaorm
identicae, nam aequatio
Nempen
asiore

## Folio 30v — 00068197:0060
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 29r); this page is its left half.
# No recognised text on this page (recognized).