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# Piece: https://leibnizlegible.com/api/records/59865/text
# Catalogue record: 59865 — [Aufz. zu Gleichungen 4. Grades, zur Integration irrationaler Integranden]
# Incipit: v = 112√(lm) + 123√(nm) v^4 + afvv + a³g debet esse aequalis quadrato
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=59865&katnr=57576
# Work: https://leibnizlegible.com/work/00068196
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folios: Bl. 423–424 (from the shelfmark LH 35, 13, 1 Bl. 423-424): 4 pages, canvases 841–844, page ids 00068196:0841 to 00068196:0844
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=841
# Lines: 241 recognised on 4 of 4 pages, mean confidence 0.73
# 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 423r — 00068196:0841
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 424v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000841.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 2 lines, status machine
Hocrtutius
LH ∞CV, XII, 1

## Folio 423v — 00068196:0842
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 424r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000842.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 109 lines, status machine
v-12 sm + estum v + af vB + ag dalot esse aequalis quadrato cujus latus ap + GsIm
a-
+udUlmm
+ 123F un MN
rafvo. +
aLm
+ 4. 112. 123 lmm In C
4. 12. 123n MM VIn
+ 2. 112. 123 af M rin
+2. 12. 1233 a mm.
et m Umm +612 us Enmm +1Eaflm + ag = aa pp + En 49
129 inmm
125 afn M
ubi substituendo valorem ipsius B, siet
4. 1. 125 np  2. n usasnm
Hac aequatione divisa per ln, fint comprndio 9 u ar 49 =  ideterminatis,
299= ar quar- 4c5). Et quia q debet esse ration.
et
eb etiam 29q erit rationalis in ipsis indeterminatis; adeoque   Caarr -44 5) debet esse quantitas rationalis,
quidem talis, ut ea addita vel subtracta respecto ar si proveniat extrahibile, ut tandem in rationalibus
determinatis habeatur q. Porro l, m, n debent esse quantitates rationale
m, et si
n
implae, et ideo
im quarta
et n possunt integre ltohalitar e ica possent L m, n, tamen res e  praestito res reduci posset ex
etiam l, n rationaliter integre simple explicari possent per n, quippe per quam X sit explicari so test
Hinc posset fieri l= 10M + n a et n = 3o + 31 a et ln = 10. 3o mm 4 10. 2, a M + 11 31a a. 
in mm = 30. 30m + 2.30. 31 a m + 31. 31 aamm
1p = 10.38 m X 18. 3Bam + 3.
ArOu. 10m 4, 12. 10. uan + 1t umamm
ar.
13. 30 n, +2. 125. 30. 31 a M 3 + 125 F. 31 31 aanim
612 nJ. 1930)  6.1E. 13. 10. 31 ar3 +B. 12 125. 11. 31 aa mim
16. 12. 25. 1. 30 an . 10 a f rmmt 112. 11 a F m
m qerolia pus
+12330. af mm + 125. 31 f m
2. b. 40. 0. 31 a m 3. + 0.l1.30. 3 aamn A 11. 31s1 a
(4. MEAS. Eo M + 4 1 .19. 1oIam Z. 1E M J. 16 1 SIaaN + 1211E. n5. 1011. 51am + A. M. 13. 13 31 atm
gtens odon
i l hs o s ari +n. 16. 1 . 10. 3oaa n 6 4 4 l. 15. 13. 31 a m r
. l l 16 1 1 15, to ti cum 24 4 112, 123, 10. l1, 30, 31; cum 123,12, 30 31, 10, 1
ue 1n35 6 o  d. f. se toe ta estatit
 12 pnnum 123 afn M
6. 1rE. 13 Cnmm
xrationaliter integre simple explicari pet
assumeretur velut X, per quam
ln = 1830m + 102
niuion hon aam 1n S. N. Di _n
+8. 1. 2f. 112. 30 aa M
14. 1 a8. 16. d0 af M + G. 12. 2J. 1011.30a f m 5 + 4. 1 23 pi. 3oafm L + 4. A2. 18. 12. 5. f M
+16.12E113F 10 HZahtaamB
+a 1E. 1232 16.31 a f M + 18. 112. 125. 10. 11. 31 ∞ f 1
idem rarsus, nisi quod permutari debent a, 123, 10, l, 30 3l; cum 123, 112, 30, 31, 0, 11
+qrapq sn
46. 112. 12s bn MM
d
12 aF R M
0 = + a app
+9q ln
8.12.5 n m adeoque
2. 10. 11. 30.30 a + 1010.  mm
H. 11.
+2. 10. 11.30. 31.
Un = 10. 30 m 4246. 11. D aM = 1I. 30aM
+1S. 3F amm t2 1O. 11314 n  17
I = 1030 u + 2.1030. 3, am + 103i a m +. 11.3 a3
11. 302 amm t 2. 11. 30. 31a M
saciamus _ q 10m
(112. 123. 130 aM
un -1o ma tu. spsem esiaa
ln.
Compendio fiat a5 = 60 n + 6) m + b2 aam + 63a m + 64 a m +65 a m + b6 L m + 67 a m + 68 a
comprudio fiat = sem 40
a3Y = 50 M  + s1 a M + 52 am + 530m + 54a
So M E+ 2. Go. sta m + 2. soma ni sr soss aur2. so. _ an
5ISI. a m +2. Sus2an t2 31. 53 an f. 251 59 am
75252 a n + 2. . 53. a m + 252. 59 am
153. 53a m L. B. 5iam + 45ta
compendio fiat
ar=
(1E Im + 125n p. + aB) Lmm F. 2. 12. 115 lnmm +2. 12 ar lm F aarr Svestmm +2) + 4 u2. us si un t 1e emtag. 149
es arntr
ap= 1 lm + 123 um tar = 2113. 1g lmm + 112. 123 fm-: 9 seu uiti
e
velmqus 1 . (2. sa nnon.) 2t. rs Cmma + vr. 13 asm brmf.eti aa)
B. p3rEL
sit q tm:a et r= sin t adlage, et prodtunt hae duae aequatores: 2.B stnter 2M a a iaq p 1s snop  sparus    ii la mi itus u ag. betftiaa)
a2 t: glimm, + (t: a) smyl a(t. a) a sag= 2. i. 13 lntr turrs apt 2. 135 snmgs + 2. n ap
13 t. a) tm
Hiac duae aequationes ad m.
2. 112. 10. 50 m + 2. 15. 12 5iam + 2 13 ustaam
190
50. 50 H + 2. 5a51 am
A. s.Soion en
12. 9
n osen
indaltiin  nia taq a
saaco
cis te inor
125 a Ncap. 314
+ratag st a
ii
et s = 10. 5omm 1. stam 4 nsiaa  sa = 30. 50 min + 30 siam + 31. 51 a
An sieprostel. duae aequalionis sunt 2. ir arlm 4+ aarr = 9 102. tigrlarum + 112 astem + a9- b  44
et n= 30m + 31a
S=om 151a. fint A5s n = se  resanitsian
et aegla Farq= 2. ud 2glum rur rasm sed ita encul g aqque pant i  tistius etc l m
t purata e fin sm  t sit anen tme ne    naect atm

## Folio 424r — 00068196:0843
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 423v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000843.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 67 lines, status machine
9. 10 XX + 12a f. raa ∞ ∞, _ 25 XX + 2. 20. 1 f + 21. 21aa fiet B20 = l9. 10 21. - 19. 1. ergo
2. 20, 21 = 9. 11. su  2. 4(10. 12 = 11.
9. 102 t g. 19  + 9.   + g. W X + 9. 13 X + 9.14 a ∞  C 16X (2) (2o XX + 21 aX + 22) - B
9. I2aCXV
m si umm mnneactm ra) aqq = sa euillut anm  i s 1)  20 X + 2.2o nX t 2. 20 vaarx ger.
21. 21 caxX + 2. 21. 22a X =+ 232
fiet 20 19. 10 22 = 19. 14  21 = 9. 11 : 2 19. 10. - 9. 137. 19 9
et firt 13 _o = 11 s14 seu 13: 11 = 44: 10)
denque ob terminum medium 2. 20. 22 + 21. 21 = 91 2. fiet 2. fS10. 144 + f. 11. 13: (1.0. 14) = 5. 12. seu fint
2. 10. 14 = 12 110. 14 + 11. 13 = 0 seu 12 = 2.10. 14 + 11. 13,: 110.14.
9. 10 X + 9. na X + 9. 12 aa X + 9. 13aX+ 9. 14a X + 9. 5a X + 9.16a  (2) (20X + 21aXX + 22aA X + 25)
∞ 20 X + 2. 20. 21 aX +2.20. 22 a X L2 20. 232 X L 220 Ha +2.
XX
+21. 21 acX +2. 2]. 22 a X +2. 21. 23a
e
+ 22. a
20 = f9. 10 23 = 19. 16
21 = 9. 11 . 21910. 22- 9.15  6: 2 (9.16
.f15 H(lo: 16) + f. 11. 1. (4. 10) = g. 12 2. f. 11 (. 2. 19 10) 1916) 1 99 15 15: (4. 9. 16) - 4. 14
2. V9. 1o E. f o. 16) 1 5 Y. g. 1. 1 : (. 9. 10) = g. 12
Bgo (E) (dum . 15: (E. Lg 16) + g. g. 11. 11 : (4. 410) = g. 12 seu 4.10 Ho 15 F e 15 Mo. 16) + 1. 11 (.: 1to) - 12 sen
. g. 16) L. Hf. 6). 9. 11: (2 14. 10) + gtg. 15. 15. : (4. 415) = f. 1 4 seu P. 11H. 6. 109) + 13. 15. (4deu
410. 
220 2. 9110.16 + 7. g.f. 11. 15 _ t: 4 1 (. 9. 1016) = 8. 134. 10.16 + 11. 15, : 2 110165 = 13
a
9.10 X + g. 11aX g. 12a a X - + 9. 132X + 914 7 + 9. 154 + 9. 16aX + 9. 17ax +9.1 8a 
F _ 20 X T + 21ax2+ 2WXX + 23( + 24a ∞ F2..
∞ 2s X praactax + 22oaH + gronsat + 2 20. 24
20 = 19.10
24 = 9.18
e
+2,5
8. 10. 910  n) e. 4
se 8. 9. 10.s9. 18 4.
24 t0 21 = 9. 11. fr. 19. 0) e  2gas, et = 19. 18 - AB Ed 9. 17. Wd ara p2 - 9 9 (1. 11. 4 (940)3 : L. +f. 10
+ 21. 21
+2. 21. 23
+2. 21. 22
et pursus ar 9. 16 9. 9 (1717: 4 (9 48)):. f. +g. 18
seu 22 = 9. 12. -9. 9. (11. 11. : 4 (9, 10)): 2. squo et rrsus 22 =9. 16 - 9. 9. (17. 17: 4 (9. 18)), : 2 +9. 18
23 = 9. 17: 2. Yg. 18)
seu  unde g. 12. 4. g. 10  1g. 18 - ggy. 11. 11. 7 fg. 18 = 8. 16. 4. f. 18. 4. 9. 10. 714. 10 - f. . 17. 17 4. f. 0. f ff. 10
seu 4 1.10. 18 - 11. 11. 18 s1g = 4. 1 16. 10. 18 110 = 17. 17. 10S10 seu 18
12. 22 =
18.16. 6B - 17.17
4. f.9. 10. 12 2 9.9. 11. 11 4. g. 9-16.18- f,.9. 17. 17 durs ralores invicem aultiptrando
8. f. 1097 (9. 10)
 +L(. 18)
cb2
+ 4. 9, 9. 10. 12.16.18 -
2.. Hr0. 18 F2. 13. 9. 11. 17: 4 10. 18)
28.G. Siditro  . 4. . f. 1. 17 iaa + () f. f. 10. 1216198- 4. f. f10. 12. 171) C + . f. 11.1. 1717
10
= 4. g, G18. 16. 11. 11
64.  10. 18 110. 18
terminis X et X fiet, ex prire 2 2. 9. fo. 18Y 2. 4912: (2. (9. 10) 2.
ex priore _2 ga X. H. 16) 9. 17. Plilg 18) + ff. 9. 1( L. 19 16) 9. 12 - f. 9. 1 (. L. 1919). 9. fB. 11 11. (. 4. 9. 10)7. 289, 10
seu _8. B. 9. 10. 17, + 4. f u1. (12 g. 16. 19. 13 - A. 11. 1 923 8) . 8. f10 s 911 8 = f 13
Ex posteriore f. A9. 18.) 9. u illaue) + (. 9. 17 (chAg. 15) 9. 16 - f. 9, 7 ( fl9.19) 9. g. (17. 17(. 4. 818) 2lg. 18.
8. c. 1.18 1918) + 4. 9. 16. 17, 18 19. 10) - 5.4. 173 1(9. 10) 3: 8. 18. 8. 1910 = 815
A. 9. 1
quae multiplicetur per   8. 9. 18. 5g. 10. fiet
Hinc
oterinomer
a
+12
+ 1E.123. 10. 3042 3 1 ~ AL. 13. 103  es 1 + 12. 125. 13 a7 M

## Folio 424v — 00068196:0844
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 423r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000844.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 63 lines, status machine
51 4- 7om ex aqu l9 In q etc. quae est C pagnae aute praeced. Liet:
 H (4. 112 125= ) Umm, + (4.12. 125 + 15. , 5) astm 4 112. 125 aa f = 0
(4. 112. 12 F uum (4. 112. 124 4123278) afnm 70 a9
(8. m Fmns) en mon compendie Z. c2Umm + rilumm + 21 as lm + 2004 -0
sit undcaueu -70. 1. 2) ) (ame n o5) = 8 12 1 60, 5. 1. 123 +7 st
et A 2m +6oum, + diaa golma +  gonm + _.
Emum
12olm +60 nin P 4 31ad
e
e
2212.60 lmint
o filitu
 
tende 2197 - 212 50 + 22. 30) (2olm +60 n 
se
31 a 
281a. 2il
Gia. 17 N
eraa
a via (at 60. 2oc Drachile    vi a (291 bens. 3  B1 2a)
ma pes sot eg)a leren homoeplilam, ateiqi pot sol dutrell 213 l  anasuer
Be
veita ocipid ad servanda
a0 6. (21. 20- e1. 21)
1 2. (2 1. 00- 81 291)
ruat ) =Biaa (anro- dian) hoc = 20uaa
60n
181 241) - 81 aa (231. 6a - 81. 232))
2231 an
+, 201es tem ) vrat
2 pl .  giaanm et fE 2ias - 212. Biaa = 23160 af - 131 81aa
et ha
20
+81. 281 6d Eupet
iaculosegum
Eruno
Hin ult ii orire
l = 10m + 11a n = 30m + 31a
Zatro. = 10. romm + 11. 2B Nm t g. aa sed hiet
Tolaa, 30. b2 mt 315dan
b +2
tit ad nas
pa
18 - - 7 2 = 10. ll = 16 pt 1 .  
et video non polle evitare quin fiat, 81  etiam
ob caeterii terrminis per
pu asalibet vdncalunda
de suma in ergo aequ. O et ibi explia ae agante. Et video latus sinist dexerum divisum prius
nda
 divisio
us (eat B1 =0 - 0
1h.
am
bis
in
drodi polle per qu Et quotiens per num rursus cumalipeicatus debet aequar laberi sinistro per aequades per qu divi
pr
(7ol. 4 81a
tot +0ia
it  21260l
arum enrela
21 = 22 be + 21 2
 3 a semm ia 1