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# Piece: https://leibnizlegible.com/api/records/36811/text
# Catalogue record: 36811 — Tentamen ad problema sex quadratorum
# Incipit: Aeqvationem hanc habemus:
# Date: 1672-09
# Akademie-Ausgabe: AA VII,8 N. 052
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=36811&katnr=39702
# AA VII,8: the volume in the Repositorium des Leibniz-Archivs (CC BY-NC 4.0): https://www.gwlb.de/leibniz/digitale-ressourcen/repositorium-des-leibniz-archivs/laa-bd-vii-8
# In the Akademie-Ausgabe: AA VII,8 (PDF, Repositorium des Leibniz-Archivs, CC BY-NC 4.0): https://www.gwlb.de/fileadmin/Leibniz/repositorium-des-leibniz-archivs/LAA-BdVII8.pdf#52
# Work: https://leibnizlegible.com/work/00068195
# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 2
# Folios: Bl. 197–198 (from the shelfmark LH 35, 12, 2 Bl. 197-198): 4 pages, canvases 393–396, page ids 00068195:0393 to 00068195:0396
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068195&page=393
# Lines: 374 recognised on 4 of 4 pages, mean confidence 0.75
# 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 197r — 00068195:0393
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 198v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000393.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 99 lines, status machine
Aequationem hanc habemus:
4aarr. 1 90at1 - ari
a an - a = 1 =
d’g alis diga
2d cun +i
20 1 3c0a
da a 3. 1, - d = 1 18 10 i1a 18 184
a + 2aY-dd -d= 4X + 49 +
 a 19d 9d9d a
Reassum am.
srpositae duae aequationes findamentales hae sunt:
264a + 64 a (7 325) vel  1.t
136
18 1B13
32 a + 69 vel  vel
18c,
ee = 16a5 + 32 a + 26 Ergo
+a + 8a  - 44 +1 ) = 16a + 32a + 16
F+F FP Cdd9 = 16a 32 a F 16
Fi
9d2
ut 2 F 1 = 4a +E
a + 1.
A3
9d2 at1
ut 2 = 4a 2 + 4a  - 
odrnat. a +1.
vel 3a +1 = 4a +4a 2 + +a +.
a+d+ +2a3d2 = 4a7 +a + 2X +a) f + 4)7 + 2a3 
4a3d2 - a4d2. 2a2d2, d2
8
16d
4a + 8a3 = 4a +1 = e
ee =
11
aa + ar art
vel a3+1 = 4a +
d’9 d9 d9. = et vel a +1. + a
3a3 H2 - 1 - a = H 4 + 
a = 43-1, - 1 =
e
tiet d’a2 4 d2-
a +2a74 -7 = 4a + 49 + 4 + 1 Y2a + a5
da a +1.
1a2
a A + + a d - ad- d = 4a + 4a + + - a a
d2a
4294
a
942
2a +1. - 4a - 4a + = 4a
+a
4a5 - 44+ 382
n
50
4 +
1esal6
1d5a
1b b
16]4 c6
4
60115, 60. 18
1
5d(20. 170 15
44+8 + 
150
Ergo ddg 16a + 32a + 16 4a + 8a + 4+1.
152 4 a3 8a2 + 4a+ 1  a4 + 2a +44+1 : 9 (a +2074 + 49
gd= i Notundum hic est rem
esse per utilm ab nun parte termiinum,
ab altra raonem habere. Quia ratio
gam per multiplicationes et divisiones a
radicum extractiones tractari poliri re-
duci _ Salva raves identitate, termino
alto intacto Est ergo rad materia aequa-
tronum in aequave. Idque hic egregio
exemplo patet cum enim 4a +8a + 4 suit sequalia od
a2+ 243 +4a + 1.
quadrato Necesser esttrus ravis terminu esse
aequalem cuidam quadrato, neque videndum est
quae quibus divita producant quadratum, et
ante omnia cectum est quadratum divisum per
quadratum producerequadratum. Quare quanqua
vel 2a +
3
3a  ar 1.
ia
tisic
e id
Aa +4a +5X2
A + aet 2a Ht
1 45
15
6-35
1
4 + b. eH.
na aa + bh +rab
ab ta bb:
ate

## Folio 197v — 00068195:0394
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 198r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000394.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 91 lines, status machine
talis solutio non fulura sit recpoca
ita ut extra eam alia no esse non possit
erit tamen vera, si 4a +8a + 4 censentur aequale = ff. Cto.
et  a2 + 243+ 4a + 1 o censeatur  qub = gg B
aequatio ergo est: 4a2 + 8 a + 4) = ff
a +2a +2 = H - 2
a at2 = q s4
Nota si a sumatur 1. tuperior quidem
terminus est quadratum sed non inferior
si tamen ei adjiciatur 1. est ipse
tum 9. Iovertatur rasio, et loco
dd + +a +8a +4
ar+ 2a3F ∞a+1
duatur, a+ 2a+ a +I
ded qapea +t
addatur supeion 1. erit ex ipse utum sumo
sito quod a sit 1. et acedet toti
= a 12a 740 +2 sed ita
qa e8a +4
Omna perturbabuntu
Alia ergo alia ratio investrganda est
Qua ex duo divisione numeri dati per
datum fit Quadratum. A= cc. Ergo non posserior.
L =c. Similiter ergo ut rod utistium
Casum redeamus, si 4a + 8a +4 = f dd.
Sd tra +4a f2.
Ego 4a +9a +4 it cut = d.
a2 +2a3+4a f2
Ergo
4 a +8a +4 -  d
ard + 2a d + 4 2d
4a+8a
= d+ 4
a3 + 2ad + adF 2d aa + 2a + 4ad f 2d
4u
a trrad aadted
Si Numerus qui per alium divisus producit qua
dratum, est numerus 8 f4
purso
est numerus qui quadralum multiplicanis, producit
dividendum ergo
4 a +2a + 4a +23d- 4a +6a +4.
16 110re 1 
4a + ba
13d ga
4∞ + 8a + 41 + 1
2a
a evel
24 Aat utu
14
144)
Exhhac duitone spatet, ut eatsas 
it quadratum 2a
4a +
Aa+ _ + 4
xa harca
+c
 a
seu 30 7(=4a.
3a
sErgo aC).
3a+4+1 = 4a +3
3a +2 = fa) 3a +a
Ergo 3a + 4 = 4a +3.
Ergo 3a + 3a + 10
si jam a. sipponatur esse 2. tunc 4
4a + 8a + 4 + 1 erit tunc prorz quide
terminus 4a+8a
.C. verit quadralis sed
64
+ 8a + 5
1 1
4a + aHt
A∞ + X + +
Ex hoc
Calculo se qui
4a +8a +5 sit
Numers quadratus qualiscunque suppo–
natur a, quia scilicet, orretur
aequatio inter 4 et = 2+1. seu .
quod est impossibile. Notamdus est hit
modus satis elesans probandi an aliqud
possit exprim calculo 
sed vereor ut sit planie sufficiens
luo est, cum subtenis a quo subtra
hendum subtrahendo major
elud dum minor, tunc enim mutam
sipotus radto
6d1 16+16+5
a A 
e p

## Folio 198r — 00068195:0395
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 197v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000395.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 91 lines, status machine
B
Nunc ex ota fractione em quadtatam
extrahamus, si modo id possibile est: t id
ita teutabimus Nota f represeutat nomimatore nosra fatar ore tobies sentibpus sit
8a
r
videat an ex tato possit ladix quadhat a
ditisore
34
ga +. 8d + 3ba + 9
317
9
su +
16427
Ami equitur, si  tottum sit q Nueius
(Duadratus, necesse esse, ut 2 3aF9
6a +6a semeti metiatr 33a +9.
sen uit 64 a+ 1. + et pr 1F 11. cex. = 33 at9
ddg = 1ba 2 +3ra
d3= Rg
+a +8a +5
4a +2a+ + a +1
4a +2a+ 44 +1, 3f = 4a 18 a f5
4a 2f, 2a f tga ff + ff = 4a +8a +5
Ego 4a25 + 2af 14a+ . = 447 18af. 15]
40++2a
= 4a +84,5 4f
4a2de3 + 2a23 d1 1a3d +Bd = 4a + 8a +
12 ad +6 a d + 12 ad + 3d - 4 a7 3 + L
seum
di
a
3
ex
G_- 4 3d = 12ad + bad + 12ad - 4a, 8a
2
Dividantur omnia per d. fiet:
3d
= 3 = 1242+ 6a3 + 12a +4a - 84
2
3dd
3dd
442
sdd aud
hact et Ses majius quam y dd.
3
3dd 3d4
ergo d’est fractio
3d
hoc tas majii illa
d
t
et
ata2
Vi ab aequalibus ansenas tirqualia
retidue sunt inaequalik odem
sed indagandum est tu, habeantue
gequam raveno intir se. Hoc
difficile erit determinatu, quia nec
12 a42 5 a2+ 12 a 3dd 5
+a + 6a
3de
A
12 a 16424 12 59d2
9dd 20a, + 40 a
14
+3
seu 20a5 a +1
12 at +6a  164 + 12a4+ 124 Gd4
20a 20 4
12a++ 6a + 124 +) (12+6+12) 309d4
5484. 760c
((20 bd9- 361 720d
24+ a + 2a + 5 sic d
1360d.
19 1360
seu 6a a + 1, + 1. etc. = 33 a + 9
 Na 2F1215 = 33a4+9
6a s+6ar = 33a+9
33a - ba r + 6a s - 9. Paese (6)2
11 a = 2 a7 + 2as - 3 (1s 15  3
18
4+6 = 15+ 15 4
(2
si
966+ 6. = 12 + 12
43
s.
ex Ravibus partim datis inveniri pte
Orad dotorum. si parressummari
in cinum non possint

## Folio 198v — 00068195:0396
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 197r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000396.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 93 lines, status machine
e
s 2
60 d
636
ni
16B.+ . FhC F b0r 8tF ag  e
243 8,
t
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lt.
5
u
213
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DuO 5
totum 2 2 2 etc. ralet 1. Totum:
tolum 
n y
1, 17
27 27
3+ 3. 2 1t 2, + (5, etc 57
eitac
3 125. = 5.
e
L’et
Ergpo. inter 1. + 3  22 differentiae est 1
1- 
Ergo abjecto o 1. a majie fiet aequale
2r es
tium quae addita prinis dant summam
Hinc inveniri t quantitas differentiam, servren-
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