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# Catalogue record: 57974 — [De irrationalibus aequationum radicibus]
# Incipit: Duas oportet comminisci radices quadraticas tales, ut quadrata singula
# Date: 1675-08
# Akademie-Ausgabe: AA VII,2 N. 036
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=57974&katnr=Cc2+1029
# AA VII,2: its series at leibnizedition.de: https://www.leibnizedition.de/de/reihen/reihe-vii/
# Work: https://leibnizlegible.com/work/00068201
# Title: Leibniz-Handschriften zur Mathematik LH 35, 14, 1
# Folios: Bl. 275–276 (from the shelfmark LH 35, 14, 1 Bl. 275-276): 4 pages, canvases 549–552, page ids 00068201:0549 to 00068201:0552
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068201&page=549
# Lines: 351 recognised on 4 of 4 pages, mean confidence 0.73
# 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 275r — 00068201:0549
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 276v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000549.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 76 lines, status machine
Duns oportet comminisci radices quadraticas tales, ut quadrata singularum, et cubi, et facta sub
ipsis. sint rationalia, si certis modis mnttiphiata invicem jungantur:
1 +m + s0, 1 + n +hum m. seu n+ n +3nn +2na aequandus quadrato:
(5) bc + s0bc. Ejus cubus: bc + bc + 3bc
F19bc + 0sObc + l.. unte quad. . 86 sebc + fb s0b  + 1
e sit alia L s, bc + + (b + m
erit cubus inde.
ponendo 9pf. itemque teph. et
componendo fient aequationes:
f + 3h m 1, 0. sive n A
1. 1
ritit tc
etem
+3n h
+.14
3t be
+3h bra
2m t
1259bc
R3 B
 bc
1bac nbe
demque 11f  2t 3hibc (+3 mi L  35  0 er ponendo: x  eel: fr v h. Rej: 2Iv + 3 hEbc   0
9h4
quod si adtum ponasBp h hd fiet: t lino si jam hponas  v fiei 2l + 31 bc +V3  4
H sive Et + hY bc + h2V. et faciendo Y ∞Y. fiet 2 + hbc + h A, et aursus faciendo
 +) _h. fiei. 2co + hp n ed: fier: 2 +  bc + ∞2. Sed ut haec muttamus
sufficit fieri ber, . 54 - 18IH. sive faciendo 1g Af. et hp e f fiet: - 3 - 18s n. sed satius ista ditferre
19h0
ad finem. Erisque pono b+C⊓) = 12 ( 2f bc) + f a + st f bc
fiei bter 15 3+19d e 
27h
27 ∞ - 45 - 61  fp +, Nf n 41
303
95 23 +54 A 
Hinc s ut obiter dicam videndum an effici possit in numeris ut habectur numers  rationalis cutus
qui non sit quadratus, et cujus cubus sit quadratus B. sed hoc  est impossibile. Imo eodem argumento
video id in irationalibus quoque fore imposribile.
Videndum an tales fieri posiut irrationales quadraticae, ut quadratum eatum sit rationale et factus
quoque sub ipsis rationalis, et tamen cantera
a qVb + 56- c + 4S +b- B E + qu 1IX.
pX e1 pb nac 20S6 C pb (, Y6- c) +2 Cp +pu + inj sb+S. + npsb - s.
X  +2b+2c 1b + 5... + n
Are Y
sb - F..
+ b n + 3 n Sb + 5..
Sb - 5..
fier: 2b + 3n + 2np + 9 B  et rursus n + 6 ba + 2p b + pn + gn + r
b- bac, H + 3n + 2np + 4 + n + pn + 9n +, sive 9n +6n + 2gn
X, 3n +p
+2p2.
Hn +ptgnur
5 125 n + b4
soc pro 2 N scribendo _, N
fier N + 2p N + 9N + p9 B ponamus et X.
fiet: 4b2 + ac +8bc + nt + + 4b n + 4t 45 2 b ne sib. + F. + ab n p XE
b- si
sed hini semper ad aliam uscendetur cubica,  apus est ergo irutionalibus quadcaticis cubi as neludentibus
Nihil. hac.
notabile
Exemplum hic rraninalis in speciem, nempe Sb +J6- C + Sb-sB 
17 2b 42c. Itaque etsi  sit majoi quam b. non tamen rdeo quantitas tota erit
imaginatia. SXx nsy e b. Lxy n c quo ( x + (3y 1 Z.
BxV ei 1DC quaeritur Z.
10b. 5b = c + sab - s ex
mb +21 + 2b - 2c + 4 S C Y 2b + 2c
 ReculX + cy 3
X2 + y +
omnis si irrationalis ducatur inseipsam, multiplicatur fractio quae ejus est exponens. si in aliam adduntur expo–
nentes: Exponens ab (3) est Z. exponens à SD est = 4 L +: 3 b 1 2c. si 3 postet componi ex
dinita quantitate numerorum progressionis Geometricae duplae, possemus explicare sic Cubamam ipsius
umagmariae apparentis radicem. Sed infinita ad eam rem serie opus. Nam:
3B + N 4 ist 116 450
dC 5 + 1rg + f- 143 Y 1 2+ + 1415
crisg. s
+hi =( 2e m3 = 5  5

## Folio 275v — 00068201:0550
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 276r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000550.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 90 lines, status machine
y2
FC
+ & (0 bc + l
2 19bC + + ( bc +m  cd Bet. L + plest 4 q0 q trto Z+ sc 
N   (4 bc + 29l se bc
Eodem modo habetur 1S, pro f. g. E
ponendo tantum
2
c
1fgbc.
+31+
ela
h. t. m
16..
359 bc..
+ 3Vg
+ fcb
Arb 19 b2
a
e
c
+ 2s9 bc
+ 12
+ 13
13 f.bl 10 b2
+597d.
a
+ 6f9l bc
lc.
(BoI bc
4
3c
1 bc
ba
at
+4529bc 14...
+ 4  3b..
+212b...
pc...
+14
+4f3 b2
+ 4fg ( b)
 a
4 c 2t b 19bc
35 9bc
ns
5 batc.
2t8bc
312
X
XpI
sal
4
tet guc)
+41s
pro
.
1
ponendo h. t.. m
 eodem modo explicabitur,
. 1.
(a5lbc
+af q bc)
Falu
F 6. m..
+ 15 96c. . . .
13129.
2
6 bet
13f14 68
EGlalbC
73
0Z4 m
aq sr aeguio. X ap t qt ureuso
3fg bc.
alf.
r 134
6f gl bck
39lt c2
3t lF bc
+ f2 + f1 - f1
13 + f2l -f12
32ct
3fl9bc
312f
bc
snsh bc
1359bc
3119
YcoZ . eodem modo Exprinietur, matatis muirundis

## Folio 276r — 00068201:0551
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 275v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000551.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 66 lines, status machine
pone fB9. et h⊓ t. et si 910. tunc componendo columnas 1. et 2. fiet
per ststractionem fier. 4f + sex 45 m + pf em +  P +r hl + 3pf Ih +
+ 3rh mf. B
tortemf bc
Et residua aequario erit:
12f3Ibc 412shem be tspf mbc + 3rh3 L bc, +9ps Ihb
 + 4(2f + asnh + p3 3pl fm t pBh
hrueht rris 1I
quae posterior aequatio destructis destmendis eleganti quadam ratione respondet priori
eritque: If⊓ - + stm - pns f - rhihl - 3rh mf- 4su h - p V Ih - rnemf- 3rm hl
4 f + 3pfh
H - 4sh. pf Brhf
45 + rhi + 3pfh
I 4(2 + 3pIm
componantur duae aequationes C et D. subtrahendo; et faciamus, m h.
habebi mu:
+fl  +ph +3psti IVH quae aequatio dividi potest
+fl +1 ph  plifl eio
sive ph fen1B + aflF
+2phfl - 2 rti 3
e
( =  h3
+ fl
2h0 S
per fel et fiet:
et fiet sabt a est sit As C
t
45. 1 pur bi tuc p a setrp rtes
h
Si jam duas aequationes componas addendo, fiet:
4f1), + 85ht + et ) S 4 paf 15
sive  frt BI + ast sut (2  + alid sel + 8sh
85. I. 3H
adeoque phef et 1 + + fl fE  - 4 3 set - 35h + 3f 2 - 2ph fl + 2r h3
C + 1
et fiet f+ l p 8fI - 8sht
Crh. 2phifl Et rusus f. d. F), + ast fil + as 1
247 1
ap t est + arti
tite
xin qua posteriore aequatione, ut utrobique possit extrahi radix, excitabimus radicem, ab:
a
a
sive Or B p quod efficere non est in nostra potestate.
faciemus: (4 f 1)  - 2fl + 2rh
An ergo aliteo rectius
+4 fle + et, eritq
Sl  1nh - e, 1 .+ . avnge e f+t (+ 2 f1) p C + (2A
Examinandum ram superest, an in valore ipsius F+l. paulo ante incento: SP. sht
depresio quaedam inveniri possit. Sed video ipsam h. non posse servire grli - sl. 
2t2
+f 1 + peh 2fl, et p e p 3rh C + + sht + P
vid endum ergo an explicando ipsam h effici possit, ut pet + asti + 3rhe, sit quadratur;
x F
seo hic rursus vider h fieri inutilem, nam  242 2 15 rebet esse p 2r Quod luisus in
iste in reno intime penetravent
cuum calculus
F a  se +a: jam signa ipsium f. C. variant: S _ sit . et e + fer: + se r a
si jam- fe + a
+fe -a, erit
X tium valorum posriatium
19.
f sit = et e = fiet: + fe + a
f sit  et C = fiet : - fe + a
f sit = et e+ fiet- fe + a
+ e

## Folio 276v — 00068201:0552
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 275r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000552.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 119 lines, status machine
Videndum an possit novi quiddam ciria qcubicas aequationes fieri facta
positione ejus modi.
ap
qX 1 dqu F (0bc  5, +bc + la + 04(bc + d (Obc + m qu
erit PX 2f 196C + F 19bC + la + 2dg s0bc + Z Bbe tm 4
+2dupt . . . . + 2 n q (Obc + q 1Dbc + m + n p
a s.
df (3)bc + f19bc + la SF 12) BC + F 15) bc + la
a3
+B3) 4 13 bc + 2 (3)bc + in
d f136C + f 13bc + la 15 bC f13b0 + la
+ 3d q 1 bc + 8 (0bc + m
(BBC Q 13) bc + m
Sb 10bc +28 bc + 2gm (8bc
a 510BC + 5 s0betia
+q2
+
ag 5 10bc + f 1.bc + la
+34 n. 5 s8bc +f50bC rta +6dn  3 (bc + 2 19b2+ m
+3n d (5 1Bbc + f (3b0 + la + 3n
AN H
+n
+gn,
+   bC +29m 1bC
ut q
Ponendo jam 3n p - p. et efficiendo: np + a +YF0 sive A P - P +Y 10 sive 2P +Y10
tunc ante omnia simul e fractiones et criationales quadraticae tollentur ex aequatione: Videndum
autem an hoc sit in potestate:
X y+ ∞ ah y + a l. 10 VY X + Co. X + 3C0X + 3co d + co
et debet esse.  1 = r. sive C)  Co ( 1) ratico - at
Unde nr -co 27
et fiet co  ab co a at A0 Quae aequatio potest I na ctabilis esse etiam num data non
est tractabilis et fiet : 2, 27C3 - 3c0 + ahco  - c + an Cd = a l. Unde esse deberet 170
Quid si sit S +y + ab y + a 70 fiet et y, X + co. fier
X3 + 3CO X + 3c0 X + c. et fiet: E(∞, 27 C (3, 9Co  3, 313 3 n 2P
+: 2dc0. +E C I
tah _ rahico
a
C
M
Cec-co (34)- 
C2o)
dud
+at
MH
( ( c0) 4 atica ract
e qn
et tentandum aufieri possit esse : 2c0 3  - 9ah + 17 ast pjo sed haec conditio priori
cincidit. Quid si faciamus: in aeq. y x - ah y + al: H. VH 
3
PH   + ScX YS  inde aeqpuio  X 3
Unum adhuc tentandum superest, si
3 
ah Ex at c .. - a β3C0 X
o
aequationem cubicam per qubotraria mulliplicater X3 +Y3c. 2r3c. +Yco
ex. V7
X
ex quadentica rursus ad cubicam reduramus. sit. A yZ -ah y + a 10. in Y  12. dabic.
rnV X = mati. + mas frso Unde per resulam Huddcuii, cubica:
A y X - ah y + ady
fiets
fi et culica haec: X3 + ph X A Smfr X = mA
+27 X +D et +D 27lmA
et fisi aequatio: m +  +2 h  + hm - 2ti tapst +A
et sublato termino secundo
M X + h M -oto
Noti penultunum coati
M affinnation redoere velle
= 4h
 2742
redit ad determinationem Cardarii
Terminus illimus est quardratus a 1 4h - 27A
Termipenult. est: h2, w7N- at3 23a sa
27 A2
(2FY 13
(16 H
8, 17 N s C
+27, 173
27, 27 43
14
7 h3 2b3
cpnregut
 m A
nA
pa Fsbc a flObe ala
+29 bc + 2yn 36
+C (3) bc regn. (b2
+ m
at f16C + 10bc rla
n
bs01
=ah. =ahico
rebi mirabili
calculi exitu
destrnitur cd
adeoque nihil hinc
po est duu
Nisi nos putemus
duci ad calulli praepa–
rationem
N r
4h3- e  vocando
+4+ 22F2
8172 3.
= ha2kr
2h
ea tinro
 sed sed hic rursus co eoamesct
et β Credo dacet
adem quod ante
(xtimA
rah
 neto
-h-ni
+3mA h
+m 62
sive detet esse  + hd - 25 de p 4h
qod est oospiee