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# Title: Leibniz-Handschriften zur Mathematik LH 35, 14, 1
# Folio 239v, page id 00068201:0478
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Honamus jam  quantitelem binomiam unde radix extrahendnest, compositam esse ex duobus-
quadratis (17) b + d P Eut B+
(14) d + b2
b+ Na 1 d + s
Praterea nemsse est quemndundum est ba  b +  ita potest feri, 4 + B
Ergo by + 4oque
Id AC efei
hint ergo
bira
dyze + b y. Eigo d 4q. beys
dyz byZ. Quod si sim ponamus
Ponamus gain esse y in certa ratione ad Z seu faciumus Z
(3
byrdy +4
.Ego Nb2+ 
Nitil ergo exhis magnopere duci potest
contumemus divisionem, fiei
ac ( ac rV +ac
b + ....
by ady + a
y fier:
sive Vb d p, d + by + dy rd
Semta jam radice: b + ac
b+ A ac
1a)- ac S. . st ac b ra) rac i a (  bi  S.
b + V. 
1b + Sin
2b ac, 1tac -  b2  2a +  - aC Cbac su (+ ac B bt abacfa ) adC At biet
(- ac 2b2+ 26 ac + a S.)
+ S
b + S. v
 fiet ZH ac - ac
(21)
b32
Itaque si sit. eae y fen  V +3 saey, -
ac Z f 4c 74 ac 45  ac 76 etc.
b5r 662 b72.
y, +a y s et. Videanus jam
an quadratura figurarum haberi possit, quarum aequationes termi ordinatae alicui terminis hujus progressionis
aequales, sane  stdat parabolam _ N Sau  X, fiet X a  X at, sive:
Pone X  +Y, unde fier:  +- r +r
qat
VA a y , sive Xyt
l’ary’ay
a+ y
Sane omnium Terminitum rationalium hod loco exposit figurae quadiari possunt; sed
imationales quadrentur, snihit alud quaeritur, quam sa solidi facti ex circulo in igura
rationalem terminum irrationalem, cujus figurae quadratura quaeritur praecedentem.
quaeramus haum figurrum fangentes fiet: eg. N  4y a -6 ,7  l, et pro
2Xa
X ponendo eu valorem, fier: 4yt a2-076
2aV Sr 1- sed nihil exhis.
seneri teminorum saltem ex parte rationalim temino der calionali aequalium exhibuisse,
Redemido ad Aequationem 20, simuma ineatur separatim harum quantitatum:
23) ac + ac q + ac g4 + ac y6 en Hec summa autem cum sit progressionis sgometucae
b52
112 F
b7L, sub ratione Z facile mniri potest. erit enim:
ac proinde summa quae sita erit: 
sive ab
sux
abio
acb. sive acb
V2Z
24) b. Aqu hoc puinde non procedit, insi acb a2 cum quanlifasiun ad. tpre
serie 4 ubi - y Haciio
B- Bac sive
negativn est ut in sumima
quia series ista non23 no est, decrescens sed crescers, posito jam hulis serici summam esse b
nisi ac sit quantitas negativa, sed tunc, sume quoque ista non possumus ab ea nicise, munifestum
est seprem: Y + Y 2e st +   erc. esse decrescentem. He proinde erit
summa ad , ut Y ad _ 2a est Ergo sumu erit: a  a a y sive a s
Ego foret sat  K Saon  quodia pesesm serie termini rationales
quorum summan invenum ducti ino rationalem; sed ista sunt absuda, ita enim
valor rationatis haberetur ipsius irrationalis Error ergo incalculo. I vero in eo
imo male
consistit, quod tunt loco b  ac d scibendum b.  s.
hnterum Mettodus osta cum.  ac est quantitas affirmativa, servit, ad approx. Interim in sene
erescente hoc serviuit ad approqinquationes nam inde a terimtio dato retio summa haberi
potest, cui uddatae apsamet duora inhadurem; habetur aequalio cum ipsa radrie unde quantitas rationalis
ipsi aequalis prope vera
eodem plane res redit, si dividendo incipias a Radice
nisi quod tunc radicis multipli sunt in mnentoo nominatore trminorm, cia b vel a in nomil
nators quod antea erat contra
sun tureniemo ne tunc quidem, et ratio est, cur ista non sucedam
a tyrer a
b + b +a
arabierae AS (t a a b arb ae + ac B )   a bt biacr 
b+ sud
Eit en et ita in infinitum eritque sars et compendi carsa, radicem  inrationalem appellando
 
in signis invensus
n
25
(267
2ap =X BY