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# Catalogue record: 55735 — [Aufz. zur Zahlentheorie]
# Incipit: Qvaeritur in integris solutio huius problematis: vv=φ⁴+cψ⁴
# Date: 1714-03-28
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=55735&katnr=35779
# Work: https://leibnizlegible.com/work/00068207
# Title: Leibniz-Handschriften zur Mathematik LH 35, 15, 5
# Folios: Bl. 55–56 (from the shelfmark LH 35, 15, 5 Bl. 55-56 (=35778;35780;35781)): 4 pages, canvases 109–112, page ids 00068207:0109 to 00068207:0112
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068207&page=109
# Lines: 282 recognised on 4 of 4 pages, mean confidence 0.77
# 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.
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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 55r — 00068207:0109
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 56v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068207/jpgs/default/00000109.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 66 lines, status machine
datur
integer b
27)2 171
vaetur Numerus bC, quaeritur modus eum dividendi in duis
quadratos. Qui sint XX et yy. aX Reducantur X et y
etam ad integros, efige X = R: e sit l unitas, et fial X = (Z. t et y = lv: B
Et lb = lZZD: Et + lvv: Wn et sublatis fractionibus lottwu = UFoto orH
Pono autem Z et t, itim v et W0 esse primos inter se. Ergo 1oro dividipotest per Ht
Et rursus ff dividi potest per Wwo. Ergo + et  t= W, eritque adeo
y = lB: t et lbE L IL
Z+
Cov. seu b r–
seu b tt = lZZ + lor. itaque duo debent reperiri quadrati, quorum per
6 divilorem summa per b divisa det quadratum +E. Hinc alicujus qua
drati et 23. per b divisi residuus, una cum alterius quadrati, ut vo residuo
per b diviso, debet dare numerum divisibilem per b. imo debet horum
duorum summa aequari ipsi b. Cum enim uterque residuus sit minor b, non
AB possunt componeremubptum tin
visis sunt numero finiti et in tabula exhiberi possunt. Itaque etiam
potentiam eandem ipsir
sciri potest; an duo ex illis in unum additi faciant datum b. Ex. gro
selmaes quadrattiis sit b = 3, constet quadratis quibuscunque
per 3 divisis, refiduos non esse posse nisi C, 1, A, e, . sed bini
ex illis non possunt componere 3, Ergo numerus 3, non potest
dividi in duos quadratos. sive integros sine fractos
sine rai  quia quia au 12
10
12
19
14 15
1 a 9 16 25 36 49 64 81 100 121 194 169 196 225
∞
a 3
1 1 6 1
B
0
1232528a 12, 16 21- 27533, 40 4B
56 65 75
1
17
236 9
85
90
si duo residui componant b, seu si sit Y+ (Y) = b. quadratos ipsom
0 4+ 5
quorum sunt residui eru duo residui componentes b, sint Y et (V)
= qb +r
et vD = (9) b + (2) fiet Z2 + vD = 9b + (4) b + r + (V) Et quia
Y+ CY) = b fiet 77 + vv = q + (1) + 1, b itaque duo quadrati
 Hoc autum debet etiam sue uoru
quadrati
Z et V, quicassumantur dato numero, minores. Cum in quo
L
debeant habere residuos Y et (Y
e
ut
se quitur et MN sistra
itaque habemus theorema
habentur L
duo numeri dato numero
seldatur ale si non dantur
minores quorum quadratorum summa sit divisibilis per numerum datum
tal e.
impostibile est numerum ditum posse dividi per divelli
in duos quadratas
O

## Folio 55v — 00068207:0110
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 56r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068207/jpgs/default/00000110.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 86 lines, status machine
 + c267
Quaeritur in integiis solutio hujus problematio or . bC
1e
Oportet vV-cIp dividi posse per l:
Ergo residuns sit vo b4) +(x
Et g 64 + x. Ergo Datur alius numeus minor quam b, qui sit f,
et alius namerus minor quam b, qui sit g, ita ut f divisus per B; relinquat Y
et g divisus per b, riliaquat B +. eritque +n CF = b5. Et 
et 1= Kb + F, E Et fo- b
et = hb + Y
a = hb + Y =CKb - et seu eut
 ff - cg = hb = CKb - ) b. Ergo f Ergo dantur numen quidam 
1NC
 et 4, mirores quam b, ita ut f -cg sit divisibilis per b.
4
quid cum sit vv- bg.
dabuntur duo numeri m et R, minores quam C, ita ut sit
nm- Dn divisibilis per C. Quod si alterutiam
nor quod si tales numeri F, g. M, n, minoresuvam
laveniri non possunt problema in integras solvi non pote
4
4.
6
7 8 9 10
9 16 25 36 49 64 81
e
26.
16 81 25b 625 16
4006 656 10000
85 208 432 80 1365 2187 3333
36 65 191. 11 197 257 37
19 29 34
43 53
4 _ 10
13 0
- 3
60 67
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27
28 21)
30 31 32 33 34 35 36 37 38 39
H1 = 10+20
12= 11421
40 a1 42 43 44 43 4b 47 48 4
50 a S152 a 53. 34 55. - a
2= 20130 31 = 30+40 41 = 40 +50
1. b1220. 430
100
22 58 123 24309(565 8221146
27
 t + 2. HH3I
13 = 1. H+ 2. 2+ 1.31
1. 20+2. 30+ 1.
. 20+1.40
C (. 10r 1.20
I. 1d13. 1. 3.30F 1.40
59
= 1. 10+3. 2013.30+1.40
14 - 1. 10+ 42076. 3044. 5
15 = 1. 11+4. 21 +6. 31 +4. 91 + 1. 51
= 1. 1045, 20+ 10.30+ 10.40+5.50.
seu 1. 11 + 4. 21 +6. 31 + 4. 41 + 1. 5)
= 1. 10 + 1.20
4. 20Y4. 30
5. 30+ 6. 40
440+450
16= 1. 11. + 9. 2 + 10. 31 + n 45. 1+1. B 1. 3
16 = 1. 10+. 1. 20
5. 1045.30
10.30+1040
1920 + 1060
11
 16
0+1. 
e 
1296 800
11= 110+101110
pf 1.40
12 = 1. 10 + 2. 20+ 1. 30
(3 = 1. 1043, 20 +3.
14 = 1. 10+9. 20+ 6. 30 +4+0 + 1. 50
1G = p. 10 +5. 20 + 10. 30 + 10, 70 +550Fat 1a
18 = 1. 10 46. 20+ 15. 30+20.40 11550+6a + 1b
10+ 7. 20 +71. 30+3540 475. 50 +210+ 76+dC
B= 1. 1e+ B 20 +28. 30+56. 4047050150 a + 20+81
tia

## Folio 56r — 00068207:0111
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 55v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068207/jpgs/default/00000111.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 101 lines, status machine
Cuique p loiso plait nesidgus dsit  t   diviso per b, residuas sit F, et r- et sit divisculis per L
+G
49
441
240
384C
409b
s
2
m
81
12  64B
50d
 c
25
estienu dus
28 Martii 1714
fero Thevbald Schottne haysori
Cammathuae suter in Wien.
bitae Mam soisus ser wode bro gelgenseit dem se pater
Angustino homae a S. sosepto weisten lasen, das is zucar
es
not a nicet quit desabt, vusto problema, ada
fegosat eni Be e X +
dasbronent rmud, das)
cuna
un  weun din zase a gege bendia zorse X iu eumi
b
lqua
odi den esumnulisuit inriam.
e
alst dasi a + d. niuc quadrat fas suptimmamm
nag
pollius aubrn in achus, a Zeistu abre. genris eoe folaritit
gatont d segibem has  sne  se acia en inteqer und et sugn se
niconema, demonstricet. Onsaeet ae Bohe (b mid C e
inforet non a ino gassen, die hstmmen multi plicit ca mais
aig ius
nausus und us rram
ina
Duquum
manntum hie u von brgdu prco) st musdenscis
astupriurr ad se nali melihee
gegnit dio gegebe eszaffa. iy ende gange ziche, m un
Dmm
ex Lacporinden
inprinoe 
tnti
nu
sepFan gane gur gobes 8. aber b und c, segen dycoufalotes non a
Romman
+
t und eng gefartt ab
 un idi grau ausge
eben a, menn sit zusammen multiplicice weraden. dodgustoret unus s
din brgofito 
uno
e
Sin
gane
em rbi muso sit noe eundas
Uno 1b mas tintm normagumq
finden , 60 nusten sis zno hassen finden saeßer,
aerlite sile mre un b, und weelche rurce aeoleu nemen, f, und 9
albu
B ast balant dest
divid
f  C9 mit b gehrinet grrod
auf gege; und noch zuro andere zasen in und n, Aleiurr
ilo
ub C, ralie ast baraeau, dast mm- an udit C
ethei dividiret grrad aus gehe, soete mnc von den
id zaslus b
und
de tum exempet. N aere
rothra. senu
Amistdivisonu C ei
man ma nesme
obgo
hass erstus husenen alsonda 
ivitiones gi.
adsi
 gai ari confactores sn dE a confactores b, C, vorige
uptacores
nan molec. D est dat problem daa bigetaele sase X In
venu und wresisa
sietion est zn brabatu) E
es
nun dintes
ale
sb selbitus dat. lremi nnnae non du confactoribus ist 1, oest
 siuden
cur dem
obu moglus
su mit so est die aetee imurgue division diae mit im roogu–
nasu, ounetsi: Oduum sstu dia pragidu sat prodematis
10B 625. 654
422
F
5=
+

## Folio 56v — 00068207:0112
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 55r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068207/jpgs/default/00000112.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 29 lines, status machine
orupe
Theobald ohottelu Rayve. Cammortsuae-
sute
lianu
ritte Mam sorsus ter raeolle osubustrae sit denis
Pater Augustino a S. Thoma nnchst dieustlichus grus
Non mix brus compegausait ver tau lasdeu das
iis gwar noch nicht zit gehabt, nnsto problema
zn ende zn beingen; abee in dem ich den lerg dazu
mia boverit rinige Theoremata beobachtet, st din scien
tiam numerorum integrorum zu draemehren
Tinuen?
i habe  untro andom dianliis
gekunden in untrosuchs, in trelchen fallen ein
gegobem ganze zus in 2ergne quadratis, gange
odo banche,  geteeilet werdn
siomun. Da is dann en fordest beraeinsen, daß menn
vin baede quadrati, sollem brenche segn, und diese baenche
C
et
ad minimos terminos vebracht wroden, gt nusten
sir nothrwendius einen gemeinen nominatorem habeu
Hloces auderer habe berairsen. daßt wenn niht
muglis zus kinden Zwogahlen inde Kleiner als die
Hauin
grgebene; denen quadrati zusammen addiaeet, durs dic
gegebenesost sics o iin dividiren lason; so ist
and niht muglus die gegabene zahl in zaerene
quadratos in ganzen odee in baenehn, in wrotherlen