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# Piece: https://leibnizlegible.com/api/records/12270/text
# Catalogue record: 12270 — [Mathemat. Aufz.]
# Incipit: 0 1 2 3 4 5 6 7 8 9 10 11 12 13
# Date: 1693
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=12270&katnr=39155
# Work: https://leibnizlegible.com/work/00068255
# Title: Leibniz-Handschriften zur Mathematik LH 35, 3 B 5
# Folios: Bl. 23 (from the shelfmark LH 35, 3 B 5 Bl. 23): 2 pages, canvases 45–46, page ids 00068255:0045 to 00068255:0046
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068255&page=45
# Lines: 196 recognised on 2 of 2 pages, mean confidence 0.72
# 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 23r — 00068255:0045
# Source image: https://digitale-sammlungen.gwlb.de/content/00068255/jpgs/default/00000045.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 190 lines, status machine
14
sqb
Rosd pn 2.
a
ib
01 234567 39 10 11 12 13
d. 1. 9, 1. 9, 1, 0, 1. 0, 1. 0. 1. 0. 1 quata )0.
O1. 2.3,0. 1. 2.3. 0. 1. 2.3.0. 1
0123456701 2 3 45
012345678910 11 12. 13
9
4
 14 15. 1b 1y _ 19 20 21 295234424925
1’4’g’ 1b 5 36 496781 100 12y144 69 150 us vl.1 57  4 1 00 4 41 487 519 576  tu. st
2 2 4 5 6
00
∞ 10+
0
1 e 1d
0
10 11 a 12
0
10 1010 10 10 b o
4 10r 4
1 4
10
01 4 90 9 4 1 0 1 4 9 0
0
10101
0
 qo
1 4 10141 0 1 4 11 4 1014
9 4 101490 94 1 0 1 49094
120 1 4 9 1b 25 4 17 0 17 4 25 1b 9 4 1 d 149 16254 17 0 17 4251691
640 1 4 9 16. 2536 49 0. 17 36 87 16 41 4 3 0334 M 16 5y 3b 17 0 49 3625 1b 9 4
18 0 1 4 9 16 25 3649 64 81 100 121 16 41 b8 97 03105. 16 57 leo 17 69  3b 8) 16B 4
0
65
61
63
1
10st
4 9 16 25 36 49 64 8i 100 121 6
9 i 3 15 17 c9 4 2
 65
(qui suit 12
quaeruntur residui quadratorum per  divisorum. primi sunt omnes quadrati infra 128 nempe a 0 usque ad 121
inclusive. a proximo quadralo  144, residuus est 16, is notetur. Ejusdem quadrati differentia a proxime sequent bu
est 25, inde detracto 1, manet 24. is numerus etiam notetur. _ Hiuc dedacemus residuos sequentes hoque modo nempe
erut  16 A, 1 16 + 124 +1 16+2. 24 + 4 ) 16+324 +9 (16 + 4. 24 + 16).61 5, 24 125 1. s. 1 6. 6 165724149
qui faciunt 16
quorum differentiae 2471
3. 32+9
105
05
41
4. 31, F 1b
24,15
68
24+3
16 + 1.402)
415 9 2447
16+1
(128)0
33 69
249 24H1 24+ 13
16+ 1958+1
24+33
ar oro mar 24125
2Ary 2tg 2421 24+23
161254 +4
24+35
Aribo 
24437
C
C9
infdiunaturs bmartuns btngnn sibsq. 2ay,q t sbraune (97
113. 964
174396 +9
24127
(6+ Xit albub )16+r )1b+ 18 2+ B. 18 (16 + 19 24 +1919) 16F2029 + 20. 20 (16+21. 24. + 21. 21
36+2. 6114
16
1
4+2.60146
24+39
656164+1 so+ 16414 96
24+41
1gt28
duo secundo assumatur fsiduus primus, auctus sequente impare semel, et primo quadrato 1. pro residuo tertio assumatur residuus primus 128
29
L1 = 10C
2E +1 =
e
2E
Et ita porro donec rursuns eatur ultra diuisorem
1. (oB  = . 2 e atur Rofitui ex quad
l esoy. 2 = 34 essy 100 +1 =b Aequantur residus ex quadratis
Si
2 - 3
 + 1 = 6b = 1 
 di +C = br =
d
I
deu E1 91 = 2 = gE b+ = I 2 = u 1d L = W A
1H2
122
esigagmos  esod  eso =  osit atias
indes
10
il
3+
2
(d + f12)
=
ito
U
: U. P.s i_ = (, 2: I. P.s. H) = + 50 2l: U. P. s3 D qu su H
147
u
1+ nC
ul all 2.
=
143C
-
ui uul caand
quissus esser nos chro DXH.  e5dt
vouosur juns vab untomu onp. 50t
ranguribos sineisga ad toeuinul 13
dua
pund 3
nnon ept. B  ( F 1) eis
2
2
ilust esda pu
no 2 + a + f2 = (uid 5 13
u tD eisaa = tud. egoV
1+a + 2 = ()  es my
A + f. E = 22
u . (15da =
a 2150d
cia) oudi ud
a
45
2a
ubi residui angen crescerent, ultra divisorem, assumendus est pro primo assumendus est qui superest detracto divisore; proresi e
4
L1
3+(3
zo
sicot4 ota
2
t
Residuis ex residuo quocunque aucto per proximum imparem repetitum
et quadrato numeri ropetitionum
Ex. gr. Exgr. Residuns est quadrato 625 est 13, sumatur alius
residuas quicunque ut  Al C 1B proximus impar est 27 fet menum
f 49 + n 2b+ un = 113+ e. 128. qua quad ratus est 625, habetur e,  4
nempe. porro 41 est residnum de quadrato 109 cujqus latus 13
proximus impar est 2. 13 +1 seu 27, hinc numerus 26 etia est duplu
latus ipsius 169.
625 habet latus 25, et erit n = 25. 13-, 
et fit a+ R X _1 + (25- 13) 2. 13 + sq = 113 + 4. 128
A
Ros  Retid (15) + (25- 13) 2. 13. +(25 -13) 2) = Refid. ex 252
41
312
144
4 31 100
d
1
to
1
127
ae
128
EiS
04
r
16
+ 0 101018d50+
2t
Aao
DEB + Resid. 13) + (25 - 13)2. B + (25- 13): C+
497
f. 128 = 25= C. 128
Ret.io (15) ( 2. 13, 25) (9)131 4 25 (113) - 2. 2413 = 25 + f- e) 128
128 (P.5
qum
498 13 - Resid. 13 = ((e s) 1 hhoc per se patet

## Folio 23v — 00068255:0046
# Source image: https://digitale-sammlungen.gwlb.de/content/00068255/jpgs/default/00000046.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 6 lines, status machine
im = g2 - v t v Ergo s B I Refid. , si n par est resid. nn
residucrum
Hinc in paria n paribus medions reciproci medietas est radem, in
e
sin n sit impar est resid- 
imporibus simul conficiunt 2. seu unus residuus est complementi alterius