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# Piece: https://leibnizlegible.com/api/records/60572/text
# Catalogue record: 60572 — [Aufz. zu Gleichungen]
# Incipit: Sit Formula x^5 + 11x^4 + 12x³ + 13x² + 14x +15
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=60572&katnr=57470
# Work: https://leibnizlegible.com/work/00068196
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folios: Bl. 220–221 (from the shelfmark LH 35, 13, 1 Bl. 220-221): 4 pages, canvases 435–438, page ids 00068196:0435 to 00068196:0438
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=435
# Lines: 164 recognised on 3 of 4 pages, mean confidence 0.62
# 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 220r — 00068196:0435
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 221v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000435.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 66 lines, status machine
20N+21a 3
st Formula X L 1X + n X + 13 X + 1aX + 15. si fiat X = 30 y + 3i a = 3
O3+ 1O D + nOD F BODD+ 140D+ 15D
O = 20y+ 5.20.214Y + 10. 28. 218 4+1028. 214yy + 5. 20. 21 ay +2Ya
D 42030 +4. 28.230a + 6,10. 2. ua + g. 20. 2 3a + 1. 21. 30a
+1128. 31 a +4. 20. 2. 31a
O= 12030 + 33 20. 230 a + 3. 21021 30a
+721364
+2.283. 3031 a + 6(26. 113 31 au + 610. 25. 30.31a + 22P. 30. 31a 
2th∞. ZPaa +3. 20. 4. 38 + 3.202l. 3ra +21331 a
(D= 1. 20 30+ 3. 8 2B. ea + 5t28. 30. 3 aa 4 6hroal 20. u2 + 2. 20. 21. 31  y2
2.20. 21. 38a + 6. 70. 2136 31 an + 3de, 30. 31 a +3721. 3031  f1. 31 a
221. 30 aa
ed tios colius
+6. 20. 21. 312 49. 26. 2. 31 a + 1, 21. 31a
1. 31a
()D = 1. 192030+42038. 31 . 1, .1. 30. 2 a + 42030 3P2, + 1. 14. 20. 3 a
et 4 14. 21. 3031
+5. 30. 3 a + 15. 314
Manifestum est ex formula sit explicata posse tolli Terminum primum et ultimum
N = 1390 15. 53d sl a tro. 36. 31 a 4 B 20
licat per aequationes ad summum ansurgentes ad gradum formutae, sed an liceat simul
tollere terminum primum et secundum  est dubilari potest. Aeu antem
facilius agnoscemus  considerando formulam ut productam ex radicibus X +21, X+ 3t, X+ 42,
X+5I, X +bI etc. ponamu  Esto X= 90Z + 9134, 80Z+ 81, et fient radice srumerat
ete, divis per ( (80Z + 81). posito
in p tot esse unitates, quot formula
data habet gradus seu radices
90Z + gia quaZ + _a, qua Z + g1a
802 Z481.21 80.21Z PI.31 80. 41Z 8t. 41 1
Hic palet posito non adesse nisi unam radicem, non licere simal tullere gradum
 primum et secundum, nam fiecet simul qua +. ∞. 2 = 0 et 9ia +81. 21 = 0
ergo fieret qu. 80 = 4igC, quod non licet, nam vel  fieret
ex  Z XIa = Z+ (91291), 5, ZF 91: 98 seu Xiq = Z BL ) 51, Z- (21: a)
unde vel fint X = a, quod ab contra hypothesn bel 2 - (212 a) = 0, quo
Cna non est constats
casu evanescet Z a X in aequatine X: a = 9u2+91, a:, 80Z + 8I; quod
est contra intentum, desideratur enim explicatio ipsius X.
Urmamus ad formulam secundi gradus seu duarum radicum, ibi ut fiat
primu terminus aequalis nihilo, oportet in aliqua radicum ipsius Z coefficientem esse
nihil ut si sit qua +90. 21 = 0 seu qu290 = _ - 1: a. quod si jam
Secundus terminus  post 90a+ 80.2 =0 est solum 90a180.31, 91 a + 8. 21
qui ut destruatur altemtium eorum esse oporset aequale nihilo, sed nentrum licet,
non primumque nam fieret 21=31 Montra hypothesin renisi cum sunt duae
radices aequeles, tunc enim sublato termino summo poterit _t, tolli secundus.) v
non alterum licet, nam  nenpe ola +81. Et, nam fieret 91: 81 = -21 ia = 90: 80,
in Formula 3Hi gradus seu trium radicum, manente 90a +8021 =0 pro
quod non licet ob rationem dictam
sublatione termini primi, in secundo non petest esse nisi 
904 + 80. 31, 90 a +8041, 91 a +81. 21 =0 ubi quodvis aliqua extribus
pro ducentibus debet esse nihilo aequalis, sed nisi duae sent radices aequales
non potest esse g0 a + 80.31 =0, vel 10a + 80. 41 =, alioqui vel 31 vel 41
fieret = 21 sed nec licet facere 91at 81. 21 = 0 nam fieret ursus 91. 81290
In Fonnula quarti gradus seu quataor radicum mantute ut ante
14+ 80. 21 = D pro primo termino, habebimus. secundi termi nn coeffi
cientem, fieri ex tibus coefficicute ipsius Z ex reliquis radiabus,
 quatuor radicibus multiplicando per conca +81. 21. Ergo tursus
sed n horum fater  coefficientium licet facere arqualem nihilo
nisi ad sint duae radices aequales; nec qua + 81. 21=0, nisi
quia fieret 9)281 = 90: 80 quod non licet aut non prodest
nunquam licet simul tollere
usse
Ergob 2C
ex formula Terminum
primum 8 secundum, iisitctabea
is ratres oqpates

## Folio 220v — 00068196:0436
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 221r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000436.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 97 lines, status machine
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## Folio 221r — 00068196:0437
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 220v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000437.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 1 line, status machine
2

## Folio 221v — 00068196:0438
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 220r); this page is its left half.
# No recognised text on this page (recognized).