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# Piece: https://leibnizlegible.com/api/records/43293/text
# Catalogue record: 43293 — [Aufz. zur Zahlentheorie]
# Incipit: 1a+0b 0a+1b 2a+0b 1a+1b 0a+2b Ut qvilibet horum sit par, sufficit a
# Date: 1680
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=43293&katnr=39556
# Work: https://leibnizlegible.com/work/00068195
# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 2
# Folios: Bl. 8 (from the shelfmark LH 35, 12, 2 Bl. 8): 2 pages, canvases 15–16, page ids 00068195:0015 to 00068195:0016
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068195&page=15
# Lines: 54 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.
# 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 8r — 00068195:0015
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000015.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 53 lines, status machine
1.8
1.
D2
a2b ab
210 BH () 2+8 1t1 o+2
5.B 4. 1 32 
1a+0b. 0a + 1b
a 
quilibet
2410b 1a+ 1b 0a + 2b ut qusedlibet houm sit par oportet
Ut quibbet horum sit par oporter
asut
Ut quilibet horum sit ternaria quinon iea
esse ternarios, quinan
ut quilibet horum sit quaternanus opus est a et b esse pares
ut, quolibet horum sit senarius, oportet q et b esse temarios 
ut quilibe horum sit octonatias oportet a et b esse quaternanos
3a + ob 24+ 1b + 1a + 2b + 0a + 3b
ut qudibet horum sit temprius, oporteverit 2a + 16 = 3C seu b = 3C- 20
et 1a + 26 = 3d seu a = 3d - 2b. seu 2b = a - 3d adeoque
a et b veb a et d vel simul pares vel simul impares. Porro
et simeliter b et C. porro a xt valorem ex posterior
substituendo in priors fit, b = 3c -6d - 4b, seu
eat 5b = 3C -6d
cubetur, quaternetur etc
e
X +Xx, si quaedretur, quae vi quibis terminus erit gradus paris
X + X
  xd si cabetur fit X.. + x X + XI. X + X. X. sunt
sunt omnes termini potentiae ternariae
 H si sit aequatio X  pX  cubetur
Aerx, si quan X +X si quaternetur prodit 20. 15H 10+2 513
qui omnes sunt quaternarii
X+ X cabando fint P 9+, 6+2 3+3, et ita seuper par
. X +X si quadaetur, prodit R 8. 4+1
X +  = x ax X + 1 = X
X LX
20
aequalio ubique gradus quinarii hoc modo
X L, XN + X, X, si Insetur, prodeunt
m
Generaliter X + X, X si etur prodibit potentia
1
formula cujus omnes potentiae sint gradus (n n. ani
cutanose
fiet X X 4pX + p X = q + 29r X+ re 24YXX + 
Hinc ergo
+ X, X
X
Xn + F Iut etur
omnes termini in productae aequatione erunt potentiae nariae
+, X = x + X R al omnes quinarius, 
sit -  sit E V+ X X= X 

## Folio 8v — 00068195:0016
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000016.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 1 line, status machine
aotecte ravorsora