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# Catalogue record: 57429 — Mathematicis problema unicum
# Incipit: Invenire tres quadratos ut duorum quorumlibet intervallum
# Date: 1674-06
# Akademie-Ausgabe: AA VII,1 N. 042
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=57429&katnr=39709
# AA VII,1: its series at leibnizedition.de: https://www.leibnizedition.de/de/reihen/reihe-vii/
# Work: https://leibnizlegible.com/work/00068195
# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 2
# Folios: Bl. 212–213 (from the shelfmark LH 35, 12, 2 Bl. 212-213): 4 pages, canvases 423–426, page ids 00068195:0423 to 00068195:0426
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068195&page=423
# Lines: 358 recognised on 4 of 4 pages, mean confidence 0.78
# 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 212r — 00068195:0423
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 213v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000423.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 88 lines, status machine
Matkematicis probleum
Unicum
venire tres quadratus (entoamen
duorum quorum libet intervulla
ls
XA L qLAGL
st quadratus, et intervallum la Dzannam loge au fauxbouius S. Germam
terum ex duobus quibuscunque sit eti u derte ans cher un
quadu A.
Si nemo dabie solutione . sterru quie nict aotior
abitur Patisis solutio in suburtio spncti Germani i tu
9v b. 4
Trouver trois nombres quarrez; de oe
et 1. b v0
ruls
a. b. c. de telle vorte, que la difference l
entre deux de quelle maniere qu’on le prenne, il rara
inu
.lG  2
16.
un chacun, soubstrage d’un autre comme a = 1b = ee Rg a b = C1
a de b. et a de c. etu laisse (6uo
tousjours un nombre qui soit aussi
quarre, per consequent:  dento
tellement que  omn b minus a
4 1
et C minus a. et C minus b, soit nombres
quarrez; . Etqu’outrea, les
quarrées
11+
racines Y, dey s ces trois differences, ou
Residus estant prises comme
Rq C = ayaRg. C= b, Rg b- a.
les differences entre ces trois racines
R9. C- a, X R9 c - b. Iem
ccteut daee.
mava. Dhaquad
di
a a + 1 uingantu horum Dta 1
Zattaat 2 a +I summa quadretur
aaa t 4a+1. + 4aaax13 aaa + +aaa seu
utat taa + 1haaaut quaana
aa lib. 8.
Quadrata
V+ +d. at b+c
in
Jam saa Habemus a t ait. at a. seu 2a +
2a424  2da F2a+. Et quadratum
tertii erit aequole quadgato duorum priorum
seu  sec differentia inter quaedratur
ut ergo dra luntermin A
qada a
Lda +20
14. C b + RB=a. Item RgC. a F R,  a. Za + 1. t ajo vorum is
soient derectes nombres quarree
aa bb ee
aoa
vadratorum diffe
rentiam ese quadratum, (radiais aatad
Nunc quieriter certius C.
vonaara tnaaa iona + 4. + 1, dd qut + d
 Tria quadrata. Radtiu a a et tivtc.
adi a
aa, actitbb +. Zba 2. aatib +CharCC 2BCa
sua ptrotii a
e n. 
Herentio eltoni 
inter primum et secundum bb + Eba. Inter decundum etatentium cc. + 2 ca Y 2bC vtac no
inter primum et dertium aa bb + Zb ast cc + _ca f 2b cd Ri aua primur
hebume
autem istre differentiae sint ta necesse est sist, eas componda ex duBbus numeri si pi
planis similibus e∞ ut pt ta componitur .Ex 1b 5 + 2ba et dei duo ida
ergo termini nebent habere latera proportionalia uu  m mibna
scilicet  Ct ) BB + 2ba debet esse = fg. eto fucht =
tobi BN eBB
e
b+ 2ba = f = duout a bb+ 2dq - (d) d.  dbFEba _ nd egg dd ee
18)buie
degq nognt ceun
go g- e. Erposd = F. hrg. de de = Ex pe consequlus bb + 2b a est Gradiatum.
BiBeoe
RBCII
LiANnOrENANM
mpun
LICBE
Et B. 90d
secundo sit
totuiaqulo

## Folio 212v — 00068195:0424
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 213r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000424.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 94 lines, status machine
Nimeri Qradrati
Duo homini A. B. quorum diffra sit , data
tama taaa + F aa + 4 a + 1. da
si ex Numeri cujus dam Radiae Quadrata auferas
unitatem, residui dimidium assumas, eique
rursus unitatem addas, srducti Quadrato
Quadratum erit aliq Numerus quadcatus
cujus differentia i a Numero Quadrato
dato erit Nimerus adratus. Et contra
Ubicunque duo Numeti quadrati habent diffe-
rentiam Quadratum, necesse est, ut
maxim major sit q Numerus quadra-
1o- quadratus, vel _ Factus ex quadrato-
quadrato- quadrato, et quadrato nuneri
alterius. Resolvatur is numerus, id
est vel extrahatur ejus quadr Radirti-
biquadratica; vel extrahatur radix biqua dra-
tica factorise unjus, at guadratica factoris
alterius. Necesse est alterum rimerum
Quadratum,  cujus differentia a majore
etiam numer quadud esse debet, duorum
Enm essa cisu sciliae ipso per
vl Rogolubus differat aimajort alio
Qradrato.Quod sis is Numerus est)
evadratro- quadraticus, a radice ejus
biquadratica subtrahatur unitas, residuum
dupticetur, producto, addatur unitas (
a + a + 1. Aaa. 4a +1
 8r = 175 ara dast
4at 4a 71, dd
Nota Regulam universalem, et convertibilem.
si due sunt Numeri exodrati et a
minorem
vel brevius a. duplo ejus Radicis  +F1.
qaa. + Za +1.
2aa Laa +  +1
Uibicunqe duo Numeri quadrati habent
differentiam quadratam tunc unus
est factus ex aliquo biquadratico 8t
est duorum unum, scilicet aut
 factus ex quanato Radice quadra
ta unius factoris, nempe quadrati, et
quadrato Radicis qubiquadeaticae
duplicarae post duplicationemt ynitae
eminurae; aut est differens a
majore hoc ipso Facto. Et
contra ubicunque, id contingit duo
ind Numeri quadrati habent differen-
atiam, qu differunt Numero Quadra
to, differentia erit. is ipse factus,
aut differens a majore eo ipso
Quid vero si tres sint d Numeri quadrati
quorum quilibet a quolidem differat. Numet
quadrato id itaa assequi possibi
therit si sciticet ipsum . 
 set rumers quadratus, et A- 2B.
numerus quadratus et Quia enim A-
Est numerus Quadratus Et hujus ipsic Et
B seu B. est tem B f
Restat tantum ut se
seu A - 2B. sit qbadratus 1
Numerus Quadratus a d differentia Muntero quadratomejor sit  omnium ab omnibus differenria
Quadratus, aut scilicet ipsa differentia Numeri
dati quadrati majoris ab eralio minere. 
Quod si is Numers non est biquadratica cum
rifficiateum essetfactum ex aliquo biquadratico
12aa44 + Yaaa 4 a - 1
et aliquo quadrato, ideo eo casu, inventantur + gaaaa + 8ddaaa  49)da --
factoris, id quadratum (rit duorum ugum etc
Ut supra. et. Et quia omnis biquadratius
est factus ea biquadratico, et quadrato,
in gdda, ad ad a - 1
526 250
64 4a0
danaaa -dd = dd aaa a - 1
 Residuum post productione
Aada aaa-1. + 7ddaca p + dd ag
 seu quadratum. seu
eris quadratum,
Radices factorum, unius radix biquad raticaalitius um 137 vet 4dda
biquadratice duplum denta unitale producti
Aa Quadratum muitiplicetur radicem alterus
id est unitate, ita Regula rursus ad
generalius revocari pt retenta tantum ejus
posteriore parte hoc modo.1A AaY(HA
Ubicunque
aliquo quadrato; et alter minor
factO.
Quadratum, erit duorum unum auq scilicef b  C- B sit quadratus seu A. B.
biquadraticae subtrahatur unirat, produssi
acient
ee
(eiti
sec

## Folio 213r — 00068195:0425
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 212v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000425.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 102 lines, status machine
4aa. 4a +1
sdda aa a - 1. + dd  aca - 1. + 7dd aaad 
2...-. +.. ....... 1.. 
4a aa a -1 + aaa a - 1 7aaaa = Ce 1 4agaa + 8aaa - 4 a - 1.
e X = ABF 
raaaa + 8aa q = 4 q
ada t 294 - +
at
ee
aaanatBaaat +aa = 4a a 4a +1 +
YtB2C
Raire du sendex eti majoris d.
dd
= ee + 4
+ 4
dda
Ut d. ubique abjici possiu supponamus A. esse
Quadrato quad ratumt 
_ ZB
x-  seu , 2a +1 = _C M p. 43.
2a + Z =V
taaaa + saaa - 4a - 1 = dd
4 vaaa -4a - 1g- dd
taaaa tueaaa- + a = 2 +
gart F C = B. - 2a + sar q-  + 4a +V.
16 9
4dd
a2 F b2
 +a2 + b2 -a
 2a - b = R4 a2 +b2
4a
a + 1  aa + da Ey, 1a + 13 - a + 2a. Epo 13 = d + 2a2 ba
ga +  -  a +  FaC2 ga ( 14
iss ania - ga - a
 = + 9a sin 3 1 7dr4a a = 4d aca +
4 4
agaa  (2- 4a
1pG = a + 24  94 4
16- q ’ais qes  
= aaq pa  22- g
do
ese
dd
ce  je ce e
ec
ee
aaaa +2aaa - ra = ec
at-1 a3 ata
a+ 2 = 2C Ergo 34 + 2 = C. 3a+ 2 =2e
I +3. = 4a + 8a
13 = 4a + 8a e -99
4a
gaaua gaat agant.4aa r at 1. - ec ) 19 a 24 9 a tra - a
2a + 1
4aaag + gaa + 4a +1. Aaaa. 944
64a
a
=aa +, a2 seu
1
rarcaaa
et 8a
aa + a
aaaa in
35L11
1 16a + 32a - 134
= 1 a + 3uac - a. Eagut 1=, 10a + 5.a -a
ho  = 16a3 +32 a- 1
36aa) gaa = 4aa aa + 44442 - 3. Nu Y 43+9aa = 4a + adu
3+ 3da = 4aaaa + 9a4 a 2 - 34
ad +324 act 11.
ana aaa rqua a2
9r= aa + a2= 4aa V
2a1F + taa raa traad traatiaa aa Yaa  4
eat
 = aasatr. Ia. 1.  aa 1 4
33-9a + aa 4.1 3aa
atte (verte aa aa
F aa a
3- 2c2
da . 5+
ox
1 = 10
13+aF a +20
15+1a 164152a
aequa
e
+a = 1a 2
a - b 
20636
a a r4 - 1
a14a5 4d p2ad
13. = 44) seu
at  
a 3
q 
RE13
3.4aa
rone 13 aC, - a 2a
=16 a) 
la +raa
ta,
aat aat

## Folio 213v — 00068195:0426
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 212r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000426.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 74 lines, status machine
lino aequatio tandem hoc orta est:
2. d.
122
tan  anti a
2a a + a+215 14 +6
2aa+a
2_ + 12 a seu 12 aFI F
ranfa +E era +6
6+ 1a +12 = 2 a F at2 12a
aa
9 = 2a a + a. + 2, 2128+ 2a a +a + r6
2 a a +4
2a a + a,
tra
ILa
a
+aa + k a + a +I.
444- 2a +4 - 4
4aa +2a + a +1
tagtra haater raata
2aaa taa + 2a6.
+12a84 +6 +124 = 1.
20 asp +10f= 4aacuat raaana
zaatat
cor aa 14aa +2a
4a4 + 24
 Y zana fast, + 246
ac
a
at 1 - 4a + 1
a
aatpa +a = aatc Faa
iraa + a +2 1.2ag +4) a
aa
d
+ 2a a t a +2 6
1o= 2aa rat+ it ra saglua va ta
2aa
vaaat 104 =, 2aa.
20 a +1 =  + + + 12a + f. vel 20 4 44, 
a
p
+1 + 12a
B a + 4 a = = seu 2a +a = 7 a7 1 a  a+ 1. y6
seu ba
1 1  =d aa aaa
SBIBEIOrHEDA ino  a = 32a = 36]6
rraI
HIAHAOVEKAHA
2a s
V
sit
222
etiam affecti dummodo ab eode
Nota on aequationes in numeris folvi possunt, ut hoc loco inventa
ista aequatione p affecta 2a + a = 7 pssunt percmiri omnes
potest percurri sertes numerorum sti qui componuntur ex qudrato qu
potestate sexti et quinti gradus ejusdem ea Aaes Serescontinnuta quantum
su
satis, est dabit si modo possibilis est aequatio nucieris, eum numerum
qui faciat Z 13 Hunc in fuiem opus est mopumeris sacole trantpa
ab vabilibus in machina quadam mrobili, uut facile quae libet sers
animo offerri possit. In lineis determinatur punctum aliquod quaesit
certo motuum compositorum gentaer
N
11
a+ 1. 4 a71
9+1,
12
u
++
4a f2
+
a 250