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# Piece: https://leibnizlegible.com/api/records/60616/text
# Catalogue record: 60616 — [Aufz. zu Gleichungen 4. Grades, zur Integration irrationaler Integranden]
# Incipit: Productum ex xx + bx + ac in xx + mx + an, aequari debet quadrato ad u
# Date: 1690
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=60616&katnr=57532
# Work: https://leibnizlegible.com/work/00068196
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folios: Bl. 343–344 (from the shelfmark LH 35, 13, 1 Bl. 343-344): 4 pages, canvases 681–684, page ids 00068196:0681 to 00068196:0684
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=681
# Lines: 241 recognised on 4 of 4 pages, mean confidence 0.70
# 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 343r — 00068196:0681
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 344v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000681.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 18 lines, status machine
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## Folio 343v — 00068196:0682
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 344r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000682.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 97 lines, status machine
Productum ex Xx abatac in ex +mxtan aequatii debet quadrato ad usum Tecragonisticum
Sit  Xx+bX + ac =00D et Axt mxtan  D28 tipt Lxnrburac, aermut ac = 0
facimus X= (, 2 + 1t6 + 148 + s DZ nott  +130rh6 + vosq6, + 2 C  rotg rro  go sq 16ol 4 1906
22 6och + br0 1y +63016
e O= 510 sq + fro su +. 53016
BC = 100r 2 20B +2) 6= ∞ +31 N= 400 + 4 2 =  oDV + 001
nn- h4 +26 124 (+ 29 1h5 (a2 q 146
bx = bse + b -. + b..
Habemus literas 10, 20, 30  1, 2, 31i n0, 19, 2H, 5 10, 0, 530; 40, ocv 11, 001
in summa literas  19
aquaues seop, prumal. bblu
aberuus
9paratitias ad
1 limatculi th
00D Eususq e msa eqs 13 5o. co1ro f hb sn songo h GS pra B, r0, e   10. 00d
520. 520 130 3 .. 520. S. 230 2
pro 0, 10, 20, 30; 510, 520, 530; 40, 000
325027 siaso1o
32. 530 78 Bos3o t.. 51. 530 130 8. . 530. 520. 2308..
2hamenteeste srgob. . (1510. 8o2go2, ..  2 3u po17ot . . ( pro e, 11 21, 319 610, 5B 630, 41. 001
2. s1o. Sgp. 19 l7 2. 520. sto. 1406. . 2. 520. 5301204.. 2. 50.530 10t..
2 5v sp 2e  su 7 . . 2. 810 530. 4 . . l 2. 520510 7 ..
610,620, 630.
i Ad OOG pro irrationali it 3 ob v, fit , 3s3o +2. 40 Hb =. 10. 120. 510 + 20. 128. 520 + 30.120. sy0 +2. 7021D. 510 5)at 2. 30170 520. 170 , 00B. sior eo
Eodem modo coincidentiabuatur axt HXtan, et D9G.
pro irrationali eti4 obv 120= 20 +2. ∞0+ AB 3. 20, 24g., 1. (.  . 10 510, 20 (. si0 vuip (1D aor, 5i . s3o . 7. 20. 510. 530
sin 120 = D 2. 30+ 2o
2. 10+2. 40 = 2. 10, 130, 510.520 = 10. 230.510 -20. 230.526- 30. 2385, 1 -2. 005. 530. 52,: . 10. 510. 530
 Ex amobus posterioribus valoribus insius o  130 = tE  Msso = 2 2o 3o. Aru. so +   B  asiB 3 a 138  d    0
pro irrationali 1t2 obv 1252
Seu 18. t20
aaoer
22. 1042. 40 + 2. 000. 570.530 (910 270. 510 + 20. 2. 12o +30230. 57D1. 2,20. 520 (579 5 110 o. u  3t  ) (  1B VaHLC530
1302 +2. 40 + 2. 40 - 2. 000. 540 530 (- 2. 20. 230 310. 52d, P210. 510626)
o l. rao F 1, 000. 520. Ggu + 19, B0. B250 nome 105IS,  10. 310t 2OSIO 3053D)Pi amas g0. 210 Sar
fiet
 t. 40 210SOZ 4 o 1 90. 520530 r
quos duos valores ipsia 130 ae quando inter se
-21 +2. 40. 2. 20s +4. 0oe 20521350 na  o  Wost dr305o +2  250. 
+2. 3052. 90. 210 sie. Ssu - 4. vo se stoisu s  946.38. 238. 513. 5,  vniresi
=1.10-290. 42 00o. 5 20. 530. 18. 5167205 5a3 +290. 
do
2oge seige - 20. 528
+4. 10. 20. 510 4 4.o. 40. 510 - 4. 00o 310. 2530, = 520, 10. 51a1 30 51du 3S
+ 2 20 +240 - R oe 10.5
 5ro-4. L 45 152P, 185320.3130
vio 2. 40 + 5. 000. 51050, 15 1, 10 s 120 5id 30yse
+2. 20 +2. 40 = 2.B00. 510. 530, - 2. 10. 510.520, 10. 510 + 10. 510 - 30. 530
-2.10 - 2. 46 + 2.000. 520. 572, = 2. 205243 p10. 51d + 20. 32 - 30 530
+2. 2 + 2. 40 = 2000. 5I0. SIB, 2. 18. 510. 530, 10. 510 - 20, 520 4 30. 54
L. 10. j19) 520, 10go (po gui o 11e
onmodi +oofeso rronrosio i
=210-2. 40 4 20000. G203 30, 1d. 5d4 40. 2. 184 538 +30 (39+ 182. 510 H 7B. 30. 5t0. 130 F. 2s1o. 20.30. 510. 520. 530
  .on0. 510. 120, a 10. 510 4 0. 310 to  . 510.
+4. 1020 520. seo,
= 4. 10. 40. 510.520,
74. 10. 000. 510. 516. 530,)
(4 10. 20. 520. 520,
+4. 20. 40. 520.5201
= 4. 20. ∞0. 528. 530)
= X 10, 30.510. 5309.
- 4. 10. 40. 510. 530
( 4. 10. 000. 510, 140, 5303)
F 2. 10
+ 2.40
 = 2. 00, 520. 530
1. 18 sie _ naste  Hosi
 trose miniotusi ex di
seubreviu
eno i B. 10. 900. 515.: 520. 530.c.
+2. 30-2. 40 + 2   13, 2. 10, 510, 520 +2. 10+2. 40 - 2000. 320.530 2. 20. 510. 520 10. 510 +20.520 -30. 53
2.30- 240 + 3 2.10. 310.530 H   210 +2 - rm. 1B,30 13413. 20 gyt 3051
8. 10,000. 510. 520.530
30. 51 2 ,6. 5 7 328. 510
d l + decies 1020.30.7M 350
in summa
aequationes
F 3. 1030. 519. 530
20.30. 5V. 530
+20. 381616. 510
130. 5d( + 18. 29. 51. 510
Crom 52o, 20. 51d -  3id, +2 10. . Su. Sod
38S B. Eiroabis reluum eon nescit
(vono ud  so o nes oron to st
(4. 1030. 220. 510. 550  (on Wex 10. 319 (130. 53d
+(.520 1840. 30. 510. 530
438. 530 4+18. 20.517.520
10. 26. 310. Se
=20.30, 526, 536
420. 30. 626. 530
+10. 510 - 18. 30. 515. 530
- 10. 30, 510.538
= 10. 510220,526
+Lecies 10. 20. 30. 510. 520. 530
uos ut his calau it bay tia h itbea si hilese odem modo
) P10. 518) - 13. 30. 510. 51 (24)
20. 5105 - 10. 36. 1d. 538 = 230

## Folio 344r — 00068196:0683
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 343v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000683.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 99 lines, status machine
o= 91+IIa +21b +31C ex ∞4. 15
0= 02+12 a 122b+32C ex aeg. 16
o= 03+13a+29b +33C ex eg. 17
l
1= 2 13 =h
O1 = 2. 16. 520 - 2. 30 - 240 11 = 318 15204 5a. 130 2 = 2. 30. 510. 530
O2= 2.13.530 -220 -2. 10 12=220. 520.530
63- 2. 552 6. 630-2. 10 - 2.40 19, -2. 10.510530
13. 2i 3i
31 = 2. 30.510.550
22~10.516+ 20. 520+ 39336 32=2. 20, 510. 520
23= 2. 10. 510. 520
ubi oanes simplices u termini non nisi n
33 =F0, 510 + 20. 520 + 30. 538
Obi omnes litere velorem habent auius membri preter eas quas
et 11, 22, 33 sunt copositi ergo earum
impediantur et, et, ot, et
ubi valore, ipsarum 01, 62, 83,
ex solicatio differatur, caeteris ergo explicatis fiet
= 8.10.20 3. 90. 520. 170)
+ 4. 10. 30. 510. 530, 10. 510∞. 520+ 30.130
+4. 10. 20, 510. 520 10. 1d1 20. 520830. 53d
61. 10, 2030. 90. 52a538
3(. 10. 510  10. 5id 138. 5d
  20 qu. 3rd. Be e i sio s
+4, 18  5105, E3OH -4. 600510,525gC
3.40, S10. 52M = 4ba. 510. 50. 2
520 53C) 49  360 5IB. 520 5d)
=4.40+ 10. 51d 12.1020) f2, 000. 20. 570R
L2ionosio Sib tEtoge uis E ro2sosio 3 6.
18 20. ga to. 310. 10) . q uno sed su
403. H. L2= =420. 30528536 13
- 03. 12. 21 = - 4. 2030520.530. 03
e1. r2sso = id d peio nd
a
-13 2231
119. 20. 519, 5210,
3.18. 30. 310. 530
3. 10. 25. 510,530
3. 20. 30. 520. 53
(3. 20. 30 520. 536
= - 14, 1B 2c. 515. 528, Gorsti  2 Sd 4 3aso 10. 510
= 418. 1. 518. 530)0. pi t 1e s9 130 330
itaque tandem fiet:
 250
2( 230 =
elo aeeu acentens instirioe, ut ro cete
e dem modo tractentur
Unte simliter habentur  120 puo,  0 eto porntado 1010
-20. 30. 31b. 530
Calculatio hujus paginae etsi non minus prolixa quam paecedentis tamen in eo praeferendus est
= 20. 3. 52, 530
Nominator autem in onnibus his valoribus est idem
ubiquse in medio quod ut ubique in media operatione leges justitiae servantur, quae
paginae non ubique apparent, etsi postremo sub exitum rursus apparere debeant
unde etiam in etror ibi admissus, quem aliquotres  licet respiciens non deprehendi
Eodem modo ex irationalibus ob V habentur valores ipsarum, 120, 130, 230, tantum pro 10, 20, 30 po-
et si subesse aliquem  satis viderem
neudo 11, 21, 31. et pro 240 penendo  2. 4 + b; et pro 000, ponendo 001. dit ita fuot
2.+ 41+. Ll. =00. 510. 52b, 4. H. 510, 5. 30 11. 510 = 21. 5W 141. 238
26 222 27 41r 2 b 001. 51a 530, 14. 1L. SIa 10, 1131
(24) 11 + 40 L. b = a1. 5pl 520, 23 h  e s1B sEC + 21. 1. 3, 5. 5) 5 - 2. 2. 31528. std
5eo 4317. 53
si
230= p 510 = 11. 1. 510. 51 + deces 1. 21. 31515. 520. 330, =. 1251
+23546 = II. 25. 510. 520
pro 2p. 3( ponnt + 31. 535 - Ho 31. 518. 630
= H. 35 510. 510
= 21. 31. 520. 530
Et 15d pro ei ospuou
jam bines valores ipsarum 120, 130, 230 aequando interso oportet aequare inter se
- 21. 31. 528. 570
120= = 1200, 1230 13 =1300: 1230
et 230 2 2301 2 1231 et 2 1202 1201 : 1231 et 130 2 1301 : 131. Et pet
2300 = 1230
causa sib ex aeq. 21 fint 290 =
quoniam licet novam facere arbilravam latentem in ipsis. xab et i X b, ac, in, an latentem
1230: 1231 2 2300 : 2301 2 1300: 1301 = 1400: 1201
arit forta liceset facere hy pothaesiu servientem ad compendiam calculi, ut si h = 0 = 11= 22 = 33, licet
id nis facial incidee in imaginarias.
ad o lu 170 cano
+G. 10 20 0. 5ub. 110
+ I1it2: BC
+ol. 1223 L 4. 10. 20. 510. 520. 530. 01
-ol. 13. 22 = ∞ - 2. 10. 510. 538. O1.
=62. 11. 23 = = 2 10 510. 52
+02. 13. 21 = + 4. 10.30. 510.520, 538. 92
= 10. 30 520 3t0, oi aap ite 1 0 pso (101 10)4 20 17d 430 530
20 + 40 - 660. 51 530, 4. 10. 510. 520, 110. 510 + 20. 520 - 30. 530
410. 510 + 28.528+ 38. 530
10 4 40 - 000. 520. 530, 2, EF, vo s eitiso ge. Sid. 19o - 7. 20 30. 528. 36
+ 10. 51 - 10. 10. 51. 52 L(, decies 10. 20. 30. 510. 10 538 = 1210
-10. 20. 51.520
= 16.30. 510. 530
= 10. 38. 510. 530
A30 + 40 -000. 510. 520, 4. 10, 510,530, +r0. 510 - 20. 520 + 30.530
calculo prioris
= (+8. 10. 2030. 518,520, 530
=+8, 10. 20.30. 510.528. 538

## Folio 344v — 00068196:0684
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 343r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000684.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 27 lines, status machine
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