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# Title: Leibniz-Handschriften zur Mathematik LH 35, 14, 1
# Folio 275r, page id 00068201:0549
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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