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# Piece: https://leibnizlegible.com/api/records/59325/text
# Catalogue record: 59325 — [Aufz. zur Reduktion irrationaler Ausdrücke auf rationale]
# Incipit: x, x+b, x+c aeqv. [quadra]to.
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=59325&katnr=57521
# Work: https://leibnizlegible.com/work/00068196
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folios: Bl. 322 (from the shelfmark LH 35, 13, 1 Bl. 322): 2 pages, canvases 639–640, page ids 00068196:0639 to 00068196:0640
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=639
# Lines: 129 recognised on 2 of 2 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.
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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 322r — 00068196:0639
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000639.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 84 lines, status machine
X, XYb, XC aequ. Oto. fiat X quadratus, OV, eriet vv Dab, et XXBac quadratio quadrati multiplicat
per communem seu vobab = O0, et Wdac = ED) Ergo VaYab, v- tab  008. Ergo
An. vv- ab et vo ac possunt habere communem divisorem in indefinitis.
m  Ce v= 12 Hlm + 113 In + 13 Imn = 100a vv - 12tm + 2. 112. 13l Hun + 2. 100  2
O- 210 fl+ 220 +m + 230 Lan  O0 = 21ia F  am 8= 412 Ulm + +13 Iln + 423 imnt quod
Et
ana namsm
O0g = 210: 412 al Hm + 230, 412 an AImt 2. 210220. qu0 a Vm
ona
visvot
D = 3 10 val + 320 Jam +330 tan
+ 2. 24d. 30413. an sIm +2. 210. 230a 423 an IIm 12. 210. 220 atm H2. alm + 210. 400 aa l
ita OO Z fit homogeneum ipsi ve +ab
= (avv + aab =) 112 alm + aab + a 12 1 al u 2. 113. 112n Hlm
Sed aliter brevius aa, v - ab, vo- ac = Dto. = O0Y. 4N
vo =  12 lm + 2. 11. 1 m Sn
X = 2p + 212 SIn
8= 2t ala + 2. 31. 3 a sin OY -alp 20 3jal P + 212 3 al YIn
v= idlm + 1 Ymn
O = 31 Yal + 22 Van
D= 41 val + 42 tan
n
20ge anp
212.Ean..
2. 313222alM 2. 20. 31 3rap..
p=4 al +2. 41. 42a n 95 = 20. 4+ al +212. 4 al sen
42 an
20.qu’ aup
22. qu an
cums ob OB8 ∞ avv = ab, fiat ualm + 1ann -2a b = 20. 3i a lr22. valu 2. 11. 92. 12 alni 2. 20. 41. 4ap..
(2, 203r anse
Iius  ob YD) ∞ arv -aac firt ualon +CEamn - aac = 20 4i alp + 2. 31. 42 21 2aln
20. 4 anp.
et firt 2 (131. 32. 212  -9. 41. g2. 212) aln + aal,:, 20a (a- 3P)B + 20 (42-3 n
et m2 20a 7 3)Ip + 2442+35) np + a (2. 22 a (H. 42 + 3132) t aab taac s:, 2a 2a ( + 12 n)
Exirratios ob OC) X ∞ avv- aab fiet 2a 2. 11. 12 a m = 212. 35 al + 212. 32 dn + 2. 20. 31.32 Ap
aulibus  ob 00Y ∞x app -aab fint 2. 11. 12 m = 213. 4( l + 212. 45 n + 2. 20. 4 42 p
unde fit  p 212 (4 2 2() l + 212(42 32)n, :, 2. 20. (31. 32- 41. 42). Mosque ipsius P valores aequando inter
se fit 212a(i - 31)1 +(~. 32)1)= 4. 212 (1.32 - 41. 43. aln +, 31. 32 - 41 42) aalb- c) unde + et a interventa onstantium ex se invicem
habentur. Aursus ex Locu valoris ipsius in prolixioris sed homoeoptati id aeq. 4, potest ex aequ. 1. dici duci talis ME 2 20. 3i alp +20. 32 anp + 2. 3(. 32. 212. alnt aabs:
Nal. +12 an, et pro p subsis substituendo valorem ex 17. fit, 20. 2F 35 203( a l212 (4127 L+25. 3( a l 212 (426) n +. 25. 35 an 212 C39) 1 a1sti 31) n
m.
Rarsus et 6 et 7 fit
+2. 25 (31. 3e 41. 42) 2. 31. 32. 212aln + 2. 36 (31B2. 41. 42) dab, :, 220(31. 32-41. 92) 11’ al + 2. 70. 31. 32 414)’an
226 (3L. 31 -41. 42) 2. 11. 12 m= 20 (31. 32(9142) 212. 41l + 20(31. 31 (41. 92) 212. 42n + 20. 41. 42. 212 (45 3() l +20. 4. 42. 2n () 31)n. Ergu ex
ei seu 50 (31. 32=91. 42) 2. 1. 12m = 212 (31. 32. 41 -41.42. 31)l + 212 (31. 32. 42- 41.92. 35)n. Et ex aequ. 10 fit m = 42. 3(453 ) All 1
4212. 32 (4- 32) ann + 212 (31. 42 + 35. 45+ 2. 31.35-4. 313241. 42) aln +2(31 30-41.42) fab, :, 2(31. 32. 41. 42) (1r l + 12n). unde
ex aegus. 12 et 13 duos ipsius in valores aequando fit (31.32. 452- 41. 4332 pi Il + (31. 32. 45- 41. 42. 32) 12 nn  = 3F (N = 31) 1l. tl + 32 (4 - 3 r) uN nM
 in qua rursus est aequatio inter l et n
sed malim post aequo e omnia relumere taistituendo calculum secundum homooptosin, quis comprobationi servit,
+(35. 45432. 41+2. 35. 30-4. 3). 82. 41. 92 Y 1. 12 In + (31. 32-9142) 1n ab
Ex5 et 6 est 4. V1. 12m2 = (9). 212 (31+ 45)1 + 212 (31+ 42)n + 2. 20 (31. 32 + 41. 42)p et pro p ex aequ. 7 in hac sbsti
tuendo valoremfit m = (10) - (31. 32 - 41. 42) 212 (3eta C) ( (31. 32 - 41. 42) 21265 + 42M,:, 4. 11. 12 (31. 32- 4142) seu destrndis destmendis
in CaH3 (g2. 3C) F 31.72. 41 (=41. 42. 45) - 41. 42. 31) ( ((3l 32. 32 2+ 3132. 2 L 91 42 3V (AI 42 a. ) n, , taumeisit
(31. 32 +41. 42) 213 (41 3e) +(31321442) 212 (11 30)
(.31. 32. 3(7 31. 32. 4C + 41. 42. 411) = 41. 42. 312). (. 31.37. 312 31132. 42 = 41. 42. 3r f 41. 42. 4
seu m =12) = 231. 32 (45 31)C 212, 31. 41 (32.41 - 31. 42) l + 212, 32. 42 (31. 42 - 32. 41)n., :, 2. 11. 12 (31. 32-41. 42
seu m (3 212. (32. 41 - 31. 42) ( (31. 41 l - 32. 42n) p: B 2. 11. 12 (31. 32,-41. 42)
sed ex aeq. 4
quaeramus rursus valorem ipsius M, explicando, in e ea p, per aeg. 7. quispl simpliciorem quam 3. sufficitus atturata
alterutia ex ipsis, quia ipsae 3 et 7 jum sunt aequata. Hinc ex aeg. 4 per orqu. 7. fiet:
M = (14) = 20. (41+9,2). 212. (4,= 3,) l + 20. (4143(). 212 (932-3 5) In + dab t9) 2(2(. 52- 41. 42),:, 4 ( l + 1 2n) (3132- 41. 42)
20 (42435) 212(91- 314) In
20 (42 4921) 212 (40. 38) nn 4q (31. 32 - 4142) ae B1 3et a 42) b2
+4iqu Ar. 32+3qeL 35. 32
uti coeficirus ibsius in est 4
4+4. 31 37- 41. 4
Seu m. (15) -20. 212. A 1=- 3) I + 20. 212 (421 -321) nn +  (31. 35 -41. 42) In + 2a (brefo1 32. n a2), 1 a (u tri a) (31 gr. X1 42)
vel m = (15) = 20. 21 (2) aPL∞ 42 n +3r l + 32n) + 2a (b7C) (31. 31 - 41 42): 4 (l +ien) (31. 32- 4142)
quos renos
qualores 13 et 16 aequando sit 2. 211 (32. 41- 31. 42) , 11. 31. 2 l = 12. 32. 42 nn +. 122. 3, 41 ln = 20. 22. N.  (17) ce qual.
20. 212, 11012. [2(411 - 421) + 20. 212. 11. 12, (1351 -3En) + 2. 11. 12 (31. 31-41. 42), a (b+C); Haec rursus est
leziarn
aequatio ad l et n quae debet identificari prion nempe aequationi 8, ut saltem habeant  unam
hadicem communem, itque idque obtinendum est per arbitrarias, quod si efficiatur habebitur solutio desidenta
dempe ope quatumr collatitiarum aequationum invenimetis
m relationem inter L et n, et deuique
de
 aequ. 7. valor m per aequ. 13.
relatiom inter C et n, per aeq. 8 vel 17 quae i ficts et denique re atio dater constantes per
e
alor
quandum relationem inter constantes. Et adeb
ea conidentiationem. Hoc ergo resumendum et absolvendum

## Folio 322v — 00068196:0640
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000640.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 45 lines, status machine
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