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# Piece: https://leibnizlegible.com/api/records/52786/text
# Catalogue record: 52786 — [Overbeck]: De Cyclis Diversis ad unam Seriem Progressionis, qvorum unus Cyclus est alterius Multiplus.
# Incipit: Locus 0.1.2.3.4.5.6.7.8.9.10. ...
# Date: 1715
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=52786&katnr=36911
# Work: https://leibnizlegible.com/work/DE-611-HS-861255
# Title: LBr. 705 · Overbeck
# Folios: Bl. 81–82 (from the shelfmark LBr 705 Bl. 81-82): 4 pages, canvases 161–164, page ids DE-611-HS-861255:0161 to DE-611-HS-861255:0164
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=DE-611-HS-861255&page=161
# Lines: 83 recognised on 4 of 4 pages, mean confidence 0.87
# 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 81r — DE-611-HS-861255:0161
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 82v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-861255/jpgs/default/00000161.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 1 line, status machine
:

## Folio 81v — DE-611-HS-861255:0162
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 82r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-861255/jpgs/default/00000162.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 72 lines, status machine
Residua Progr: Bimalis
Cycli 11.) 1 Z. 4. 8. 5, 10. 9, H. 3, 6. 1.-
Cyeli 7) 1. 2. 4. A A. - - 1. - A. e
Cydi 77) 1. 2. 4. B. 16. 32. 64. 51. 25 50. 23, 4b. 15, 30, 60. 43 9. 18. 36.72.
Cycli 88) 1. 2. 4. 8. 16. 32. 64. 40. 80. 72. 56. 24, 48. 8. etc:
Cyclig9s) 1. 2. 4. 8. 16. 32. 64. 29. 18. 17. 34 68. 37. 74. 49. -
Cyeli 13) 1 2. 4. 8. 3. 6. -1. -2. -4.-8. -3. -6 A.
Cyli 39) 1. 2. 4. 8. 16. -7. -14. H. 22. 5, 10. 20. A.
Cycli(21) 1. 2. 4, 8. 16 11. 2. -
Cychi61) R. Z. 4. 8. 16. 32.-1. 2. -4-8-16-32. a.
Cycli(91) 1. 2. 4. 8. 16. 32. 64. 37. 7457. 23, 46. A.
CyclilII7) 1. 2. 4. 8. 16. 32. 64. H1. 22. 94. 88. 59. A.
Cydi (9) 1. 2. 4. 8. 16. 32. 55. D9, 11. 22. -5. -10 - 20 +9. 18. 36. 23.-3.-6 -12 -24, 1.
Malticlod
s Residuu habet a. ibi quoque illius simiplans habet, Rasid. 1
Nam quia integra, sunt eadem, 3pone Egilum Simplantem = c. Multiplantem = gc.
Ergo ubi Multiplans habuit Resid. 1. ibi integrum fuit Zgc. + d.
sed ubi integrum est Z9c + n. ibi etiam Cyolus c. farit Resicuum 1.
Obs. 2. Et quia Cyclus 77. habet duos simplantes 1. _ 7: Non poterit Cyclg 77. facere Refiduum 1.
Obs. 3. Locus ubi primo, recurrit, Residuum 1. metitur omnes locox ubi postea recurrit Refil. 1.
unupro erie faciunt Resid. 1.
Recurrut enim ibi initia Periodorum aequalium sicut ulna metitur omnes longitudi.
nes per ulnam (ceu per aequalem perrodum numeratas)
Obs. 4. Ergo ubi in Simplante recurrit, primum Resid. 1. istelocus metietur omnes locorum numeros
ubi in egilo suo Multiplante recurrit.
Obs. 5. In Primitivo Cyclo unitas recurrit proximo loco ante numerum Cycti seus numerus Cplit
ut Cyclus 7. habet Unitate recunrentem loco serto, e
ubi
Obf. 6. Et quia 77. utriusque Multiplus est per obs. 4. non potenit 77. habere Resid. 1
alio leco quam tali quem et
11 loco decimo
6. et 10 metiuntur
Horum autem Communis divilduus minimus est 30. Ergo 77. habet Residuum. Loci
REGUL A pro Compositis Numeris
Obs7. Resolvatur cempositus in suos Primitivos Commultiplicantes
Singuli Primitivi illi minuantur Unitate propter obs.
Communis Dividuus Minimus Numerorum ita Minutorum est Locus ubi in
Numero Composito Unitas Residua recurrere debet
Sed fieri tamen potest ut alio loco praecedente ja fuerit Unitas Residi
EGULA pro Qvadratis et Cubis.
Ob. 8. Si Egolus fuerit Quadratus Numerus. 
no
Loco X, X - 1.
Residuam Unitatem habebit
ut Cyclus 49. Unitatem reducit loco4
Cubus, autem loco (X, X, X - 1.)
Oblsi Cycus sit triplus Primitivi, Residuum egus 1. est eo loco ubi primitious ipse facit Resid. 1
Obs. 10
Talis sit 3p. e Primitivus p.
a
quamvis per Accidens adem ju fiat loco dimudiore
Part Fit ergo 3p. ex 3.
p - 2.
minuendo Unitate fiut 2.
est p - 2.. (nam p- 1. semper par est, quia primitivus p. ubique impari
P
et utrumue
Horum Communis dividuus Minimus
Quare loco p- 2. recurrit Unitas in triplo primitivi per ipsam Regula uod est propos
Nullus Cyclus facit majorem Periodum ipso Egcli Numero
quam membroni
ut Cyclus 45. non facit majorem Perodum Residuorum
uumeno
Si plara dentur Residua necesse foret aut plures quam 45 numenvr dari ipso
45 minores quod absurdum, aut idem Residuum in eadem Periodo bis necunrere
i
quod etiam absurdum.... 
De Cyclis Diversis ad unam seriem. Progressionis, quorum unus Cyclus est alterius Multiplus
57. 57
37
29:5
-2. -4. -8. 16-

## Folio 82r — DE-611-HS-861255:0163
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 81v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-861255/jpgs/default/00000163.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 7 lines, status machine
Locus 8. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15, 16. 17. 18, 19. 20. 21, 22. 23. 24. 25. 26. 27. 26. 29. 30. . s2. 33. 34, 31. 38. 31. 38. 39. 40. 41. 41. 43. 44. 43. 40. 47. 48. 49. 30. 31. 52. 3 3. 54. 33. 3655. 36.59. 60. 61. 62. 63.64. 65.66,67. 66. 69, 70.
Cyclisiq) 1. 2. 4. 8. 16, 32. 64. -15. - 30. -60 roz. 46. 92. 41. 82. 21. 42. 84. 25, 50. 100. 57. - 29 58127. 54, 108. 73. 3. 6. 12 21. 48. ∞. 49, 98. 53. 106. 69. -5. 10 -20. -40. -00 -17. 34 63. 17 14. 28. 56 112. 81. 19. 38. 76. 9. 18. 36. 7Z. 1.
fac. Loco 147
Residun1
u 43) 1. 2. 4. 6. 16. 32. 64. 128. 256,169,. -5. - 10. -20 - 40. 80. 160 133. 46. 92. 132. 254 58. 100. 200. 57, 114. 128, 13, 126. 109, 48. 93, 186. 29. 58. 116. 232, 121. 242, 41, 282, 21. 49. 198 53. 106. 212 81. 162j19. 38.-76. 152,139. 18 155. 31. 62. 124. 245. 153, 37. 74. 148. 296249. 155, 13.16, 152 26. (. 185, 27. 54. 108.
Unitale
45.

## Folio 82v — DE-611-HS-861255:0164
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 81r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-861255/jpgs/default/00000164.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 3 lines, status machine
FRIED WILHELM LEIBNIZ BIBLIC
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HIEDERSACHSISCHE LANDESBIBLIOTHEK