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# Title: Leibniz-Handschriften zur Mathematik LH 35, 5, 4
# Folio 2v, page id 00068022:0004
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aliquando ut
Etiam uti licet radicibus irrationalibus, si consentiat radix
numeratoris et denominatoris, c. 9.
vig.
a =X
Si - v
1 - 11 - y
et summa proveniet rationalis, duder
rationalibus irrationalibus omxta
(ainuli) +3 + tBC Ad a1y  s. -y + 1g gr
V
5,-
2
-
atque ita ratio nales atque irrationales habebuntur per intervalla
unius interstitii, quia radio quadratica, si. Eodem  modo radices
cubicae tractam possunt.
Sed si radices numeratoris et nomi-
natoris non consentiant vel si radix non sit in utroque utique patet
ubiqu percem totam feri irrationalem. Omnes autem rationales
incas sunt semper parabolverdes
Nemo usus est hactenus Hyperbo locidibus in dividendo cum tamen res
non minus succedat. sed nec adhibitae sunt divisiones; in quibus
ipsi poducti habent demmina tores vec adhisitae sunt irrationales
Imo non nisi unicum exemplum datum est, quod attulit Mercator.
Mechodus mea rvo canddi ad progressiones geometricas commodior
est altra Mercatoris per divisionem; quia, ita series qualescunque
propositae etiam Megulares satis nec ordine procedentes, ad fguram
condenientem revocantur, qualis ista est: e - V + b et
Variae aliae consunctiones institui possunt ut ista:
= - s +
3
-
+
42 +
+ , - 1 + L -
y
 = 1 - 1
10
A
s
a
11
12
is
12
-1 8c. Et ita symp Sirovae 16
108 110
erui possunt figurae. 14
sumtis seriebus fractionum quadraticarum unitate deminutarum:
a
e
∞ a
29
s
20 143 168 19
F
Omniuns terminorum punctatorum habetur summa; atem omnium Terminor
_ notatorum; lentot ac proinde est totius seriei; tem sed termini
circulo notati pendent ex quad circuli, termini notati ex quad. Hyperb.
sed quod tenmini  24
123
Unde
gnnim
omnium:
458 +
63 1o , sane sunt: 53 1638 39 1o
L
1012 ere Unde 2 -  + = -  +  -
-
b- b + b2 - b + 6
V
5
ay
12
a
10
bi2 etc. Ex summa omnium:
-y + y = y et. Jam summa
+ y7
per ergo 10 1 - y - y
- y3
erc sit ex . Et summa
-
y eu sit ex 
Jam 1- p,4y5)
1-y
7(y- yd
B X. aequatio figurae, cujus n series
quadratrie est:
Eodem modo investrgabitur figura, cujus series quadratux
3 24 63 en.
A
15 30