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# Piece: https://leibnizlegible.com/api/records/27988/text
# Catalogue record: 27988 — [De resolutione aequationis quinti gradus]
# Incipit: Tentemus compendiosam ni fallor resolutionem Aequationis
# Date: 1678
# Other printings: Knobloch: Beginn der Determinantentheorie, 1980, S. 170-172
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=27988&katnr=56675
# Work: https://leibnizlegible.com/work/00068220
# Title: Leibniz-Handschriften zur Mathematik LH 35, 3 A 7
# Folios: Bl. 2 (from the shelfmark LH 35, 3 A 7 Bl. 2.4): 2 pages, canvases 3–4, page ids 00068220:0003 to 00068220:0004
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068220&page=3
# Lines: 134 recognised on 2 of 2 pages, mean confidence 0.83
# 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.
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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 2r — 00068220:0003
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 4v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068220/jpgs/default/00000003.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 65 lines, status machine
entemus compendiosam in sallor resolutionem Aequationis
jam ante multos annoz.
quinti gradus. Inveni alibi si sit
257Z m aqu. 0. Sore
Zaqu. S: M + S: mo - po + Y a m - s: mo po 
Quodsi ergo alias aequationes surdesolidas ad nanc arte aliqua reducere potetimus
utique et ipsas resolvemus. Cum autem aequatio urdesolida habeat quinque terminos
praeter summum, opus erit (summo semper eodem mantnte) arbitrariis lideris quinque
adsunt autem jam duae l. et m. ergo tantum ad huc desiderantur tres.
sit data aequatio v+g Primum repleamus omnes terminos aequationis 1.
ponendo: Zaequ X + et habebimus X + 5MX + 10n X + 5O n X + 5n + 1
(4)
- 5 l X - 15ln X - 1lX - sInaequ. 0.
et habeti potent X est enim xugo Z- n. Habetur autem Z per aequ. 2. +5V X +5I
Sit jam aequalio data. y49y + ry + sytt aequ. 0. Quae reducenda est ad hanc formam,
ut comparari possit cum aequatione 4. Assumatur aliqua aequatio unam habens.
cum data radicem communem y + by +cyfx aequ. 0 ita enim impletur numerus
quinque arbitrariarum ad comparationem necessariarum, , quae sunt ln mnab, c, ut e
Et quaerendo communem divisorem aequationum 6 et 7. necese est
divisiones semper instituendo divisorum per residuos, post ultimam divisionem
residuum esse nullum, inde habetur in fine equatio in qua abest, y ideo
res eo redit, uet ope duarum aequationum 6 et 7. Follatur litera y. Quemadmodum
et animadvertit. Huddenius; fiet y aequ - 4y - r.y - sy - + (per aeg. 6) et y aequ. by. 9 Xy
per aequ. 7.) Et aequando valores ex aequ. 8 et 9.  Habebitur equalio nova (
ubi y assurget a tantum ad gradum quartum, quam scribere necesse non est. si jam
aequata multiplicetur per by  coincident ter ut fiat aequatio 11. coincident aequatione
10
1D et H, et Termini summi aequationum 10 et 11. et fiet: by aequ. +97 +r y + t per 8 et 9.
et rursus byt aequ. - bby - bc y - bxy. Et ex aequationibus 10 et U. fiet aequatio
nova 3tii gradus: bb y aequ. +bc y + bx y quod si jum Aequalio 7 etiam
t.
(13)
multiplicetur per - bb +C - 9, habebitur etiam , b7 aequ. +  + B yt 4 bc y + bbx
9
 169. +c9. + qX
LX:
quos duos valores inter se aequando,habebitur aequatio
ubi y tantum assurgit ad secundum gradum, nempe: 1b y aequ.-bc
2bc
Quam Aequationem 14 multiplicando per y  et Aequationem 7mul. +b4.
ex:
1
cq
+
tiplicando per b - 2bc + b9 -Y +X habebimus duos valores ipsius
-bx aequ
+ CX
= qX
b3- 2bc + bg - r +X. y. quos duos valores inter se aequando fiet . bet bX + C c9 + + b + cx. q Y
aequ - b + 2bc - b9 + r -X. by +c9 + X et fiet 
brevior rat est ex Y aeg. 6 027 faciendi
novam ine X. si amba per abbitrarimuti,
plicentur pert liteam Ay By, Cy etc. haec
per Ly,  M3, P5, et. quantum  et
brgignum addantur provementia, donec
salis sit arbitvariaum
 - bE y - bx aequ. 0.
2c
A
1
+ 2bc2 S. +2bCX
bcq
t:
9
Ex

## Folio 2v — 00068220:0004
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 4r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068220/jpgs/default/00000004.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 69 lines, status machine
resumamus: a r 29y + ry + Sy + t aequ 0. cy b5 + cy.+ d aequ. 0
Multiplicando per crucem et componendo: eqy + ery r esys + et aequ ab y + ac y + ad y
et rursus:  + abbus + abcy + abdy
Resumamus. 10y + 1 y + n y 4 1y 4 149+15 aequ. 0. et 20y + uy + 219 +23 equ.0
+ ae
et multiplicando per crucem atque componendo:
+m2o 3 +12. 2o y+ 13. 20 Yy + 14. 20Y, + 15. 20 aqu. 9.   H 1 on y + 10ns y 
- 10. 21
(0
= 10. 21.  10. 23. cau cs
+ 12. 20. 20 y + 13. 20. 20 y + 14. 2099Y + 15. 20. 20. cqu 0
Et aequ. 7. et 8 rursus componendo fiet:
= 10. 20.2d. - 10, 20. 23
- 11. 20. 21 - - 11. 20.22. - 11. 20.23
+ 10. 21. 21 1 + 10. 21. 22. + 10. 21. 23.
Sed et puo 10, 11, 12, 19, 14, 1 20, 21, 22, 23  invertendo terminos, fit:
scribendo.
(10)
15, 14, 13, 12, 11, 10 33, 22, 21, 20
10. 23. 23 y + 11. 23. 23 y + 15. d. 15
=14. 20. 23.
= 15. 21. 23
- 14. 21. 23 ∞14. 22. 23
+ 15. 20. 22: +. 15. 21. 22 + 15. 22. 22
lnde componendo cum aqug c aequationesu cum aeq. 3. fit
F 13. 20. 30401- + 14. 20. 10. 20y +15. 20. 20. 20
10.10
-11. 20. 
et
= 11. 20, 20, 24
+10. 20. 21. 22.
+ 10. 2D,21. 23.
+ 10, 20. 21. 22. 410, 20. 28. 22. + 10, 253, 22. 23
= 12 20. 20. 21. - 1
a 11.
4 1. 20. 21. 22. + 11. 20. 21. 22
2. 4. 21. = 10, 21. 21. 22 = 10, 21. 21. 23
+ 10. 23.2323y = 11. 23 23. 23y + 12. 23. 23. 23 9
et componendo aequ. 10. cum aequ. 3 fit:
Et rursus scibendo oppositus.
aeg. 0
1- 20, 23. 2
- 14. 2. 23. 23.
+15. 20, 12. 23. 14, 21. 23, 23
213. 21. 23. 23. 415. 21, E2. 23
PX 20. 23. 23
+1g. 2021. 23
15: 20, 22. 22. +15. 21. 21. 23. - 13, 22, 23. 23
9. 21. 22. 23. 4 15, 21. 22.83
-15. 21, 22. 22. + 14, 22. 22. 23
= 15. 22.22. 22
+ 14. 20. 22. 23
(12)
89. 9.
(12
et Ly + My + N ry. 0.
Et jungendo aequationes H et 12
compendio scrbamus C A st + By + C aequ. _. 0.
(ubi nota C et  et A et N. et B et M, esse _ sibi invicem inversas
+BLy + CL aequ 0. Ergo et: .ct y +
N
eN
quas inter se aequando fit:
Ergo fiet denque  = t
-AM y - AN
(15
CM 2BL aeq. 0
+BN = AM
go