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# Piece: https://leibnizlegible.com/api/records/12269/text
# Catalogue record: 12269 — [Mathemat. Aufz.]
# Incipit: sit divisor x, numeri naturales sint n
# Date: 1693
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=12269&katnr=39154
# Work: https://leibnizlegible.com/work/00068255
# Title: Leibniz-Handschriften zur Mathematik LH 35, 3 B 5
# Folios: Bl. 22 (from the shelfmark LH 35, 3 B 5 Bl. 22): 2 pages, canvases 43–44, page ids 00068255:0043 to 00068255:0044
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068255&page=43
# Lines: 133 recognised on 2 of 2 pages, mean confidence 0.85
# 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 22r — 00068255:0043
# Source image: https://digitale-sammlungen.gwlb.de/content/00068255/jpgs/default/00000043.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 67 lines, status machine
sint sit divisor X, numeri naturales sint n
nix residui, sunt n usque ad ∞, deinde sunt n usque ad 2X, et iba porro
Nempe sit n= Y3Fh A fX +Y, Ergo residui sunt omnes ordine rmdinores n
nn: X Residui sit. n = sR +Y fiet A nn = GEXX + 2fXY+ rY
ex nniK, sidem qui residii ex rriX
feErunt residui ux, seu residui  ex quadratis resionis
seu residui quadratorum, sunt residui ex quadratis sim resituis
Eodem modo residui ex nA3: x erunt residui ex Y: X,
simplicium.
seu ri idemque est in altioribus.
Adeoque residum arborum
sunt reficari ex potentia est residuus ex potentia residui
lateris
ex
 Marente eodem divisore, et ateribus existentibus
ex simplice ex latere
numeris naturalibus; residnorum periodus cait numer
rum naturala cost constabil ex numeris naturalibus
divisore minoribus. Hi vocentur m, fiet
posito C esse complementum eorum ad x
cum ergo m sunt residuu en. X, erunt utique
M = X -C
Residui ex nn: X
euut Residui ex mm: X
n: X
n: X et
Mm: X MmX
CU.
t EXC+
sed si n sint progressionis gevium arithmeticae
seu Rasioniex 
etiam residui ex mm:x erunt progressionis arithnetrae
 sequens m
quod sic ostendo sit M = A - C, tunc
n) erit pro X. C+I. Et nm = XX - 2XC + CC
d
et (mn) = XX = 2K C + 2X + CC - 2C +
seu C
sit (M) m+ 1 emul resi et nitu (M) (n). X erit mm+mHI;
sit um = 
4 3
b X +t
et residuns ex (mm) crib residuus
115 11 p = 3 H2 h- 1
C
ne
1
e 10
16 25 36 49 64 81
3
4
5 9
100
quadhatorum seduis
adtantur differentiae
respuuu
quadrati seque
12 10 10 12
Ep. gr. 3 est resiouns quadrati 16, differentia quadrati ib a
equente est g. Ergo, 8 +) dabit 12 cujus residuns a (3 est 12.
simciliter 12 est residuns quadrati 1b, cui addatur differentia
a quadrato sequente quae est II, it 12 + 1 est 33, inde tollendo 13 est
13 restat 10. Et ita porro.
itaque qu Residni vel quod eodem
Regulisro jonoo
quadratorum
resiinus euit

## Folio 22v — 00068255:0044
# Source image: https://digitale-sammlungen.gwlb.de/content/00068255/jpgs/default/00000044.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 66 lines, status machine
s
redit,  ad residuum a quadrato addatur residuuam
a differentia, inde residium erit residuum quadrati sequens
Resodui differentiarum sunt residui numelom propesionis
metricae arithmelicae, nempernum nam differnliae
quad ratorum sunt numen impares. Haec regula est
generalis pro omnibus potentus, ad residuum a quad potentia
addatur residuum differentiae a potentia sequente 
habebitur residuum a potentia sequente. Hinc psidui
Resdm cuiuso generaliter residui curdusvis sriei, sunt
residur, ex summis residuorum a differentiis
sit scires
differentiae
a b c d
pdm aferminis
Res
ven
qet
mn
Resiou a differentiis r
Resid ab ma f = n
resid ab
resid ab
e p
Fdnes a b c d e
h K
differentiae
Rotioniprici
+ d
l m n p
Retidinidifferentiam Y 5 + V
Resid ab arg - resid a b quia a + =
Refid. ab d +K = refid. ab C) quia e = resio a d+K.
Refid ab Resid a dt  d_d +k = resd.. R(dFK) = _ R (Rd + RK) =
tr. a b cde
n n P 
D+ V)  p = R(C+ h) = R(rCt rh) = R (n + F)
Ergo 9= R(n + + +0) et eode
R
modo
pergendo, patet fore A(l+ r+ 5 + F +0) =c
e
 plama differentia sit aequalis primo
RpeY.
l
X
Rdis l Y 54V
ddiff
25 g 2h 2k
eum residuo terminiprimi
derminu cum scilicet incipitura
Rresiduum etagregato a residuis disfedntam Agroat numerorum
d Eut Residuns ultimi termini inserce
progressiones nirthmetica. summa residuorum est residnum a multis
trmitum primm progressionis.
No
ifferentiae constantis aucto per
4 9 16 25 36. 49
222 2
Resid. a  1 est resi. a 2. 6+1
Resid ab
Dere
2. 5+ 1
2.4 + 1
ere
Resid. à 49 = Les e P13 + R. 11 + R9 + etc)