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# Title: Leibniz-Handschriften zur Mathematik LH 35, 7, 12
# Folio 2v, page id 00068083:0004
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Dispiciendum porre erit, utrum potentias reciprocorum figuratorum se pu
pronicorum Numervum summare liceat ut 1L
ad summas incundas horum duorum, 
porro exitque   -  - .
yyrdy
fiet: du2
yt y +dy
et Eyytd Equidem res reducatur
3Esgaitus num un gudu  sigatus ytedidus
3 y. yrus Nursus dy
L - 2
V1gq g.gatqus Dgidus Ergo denique
Dugedis S g. yatus 1 gtdque Ego
Cly. yrus
24+ 2
Ss r y C ady
posito s extra summam significare minimum y  inter ea quae pert ineut ad terminos quorum summa
quaeritur, at (7) esse maximum. lt seu ultimum eorum, si tamen incipiamus ab imtio
bi primum y est B. fiet: (ay
erit sis
e 3y  = 2 
Unde patet 3
Datisque sinti
Similiter d
et X.
ay - say
LTyoyndu
ll gray
dy + dy
haberi 3. u
1 est autem =
B grgatus y yt ytsqus   yrdu (33 Adus
Ergo 35 27 + sau
3 3) +dy
posse et esse  serieii summatricim 2 dy
. M dabuntur tus 1 + in dis c vel contra
= 3dy
(5)+1 vel
- 2 + 2
erminum
Ogatus nd
F2t 9. ytu Si et d  -  d  - 3d9
Bly. yotus
I y. ya dus
ac proinde si ex his duabus summa
quadratorum a reciprocis triangularibus, et summa luborum ab iisdem, alteratra
detur, etiam alteram dari seu si dentur++  +  etc. etiam
datum iri  +  +  +  dsc vel contra. Et summam cuborum a reciprocis
natuvalibus, ut io + L + erc. nihil hic prdesse a hinc accipere.
147 du
3.gr 5 yo gatu 615Y. vuts 3. Byns 9rtus
VVrdy 9d y gotus
= d-
l voget
Y Bgetus ytelgudus 1os grau
Ergo (4) du
= = + 1
Ergo dus
3 grdus y (3l yrdy
=44 d = + 4 dy
3. ydy s Byrius
()y. y+ du
Itaque recip quadrato quadrata reciprocorum triangularium, pendent a quadrato qua
dratis reciprocorum naturalium, et a quadratis rcciprocorum triangularium, quae
supra a quadratis reciprocorum naturalium pendere ostensum est. itaque
summe quadrati et cubium et quadrato quadrata ec procorum natoralium, d tricngillarium
dantur exquadratosm reciprocorum naturalium;  summae vero quadrato quadrutorum
recipiocorum trangularium dantur ex summis quadrato jam et quadrato quadra
lorum reciprocorum naturalium. Et eadem modo continuari poterit calculus
progressio constitui; vicissim quadrato sum quadrate reciprocorum naturalium
summantur sumulatis reci quadralis recipiocorum triangularium; et qua
diato- quadrasi reciprocorum naturalium summantur ex summatis quadratis
et quadrato. Quadratis reciprocorum naturalium; cubi vero reciprocorum
naturalium hic non prodeunt. Cujus crcanam aliquam causam esse suspicor
similiterus surdesolidos et alias potentias impares reciprocorum naturalium
in altioribus evitatum iri patet ex his eliudem potentiam resolvi posse
pluribus modis, easque resolutiones utiliter intera se comparari. Et ubi potentiae
Triangularium exhaustae erunt, progressum ad potentias pyramidalium fore
e series summationes
Et hac quidem quae pho modo inveniuntur osum mationes reciprocorum
faciendum.
summabiles, procedunt etiam pro seriebus infiaitis. Sed ali alia methodus investigud
est, ubimes tantum procedit in seriebus fimilis, ut si sit
fiet: 1+ 3 14 C y +  3eydy + 2edydy
e
3+tu. y+ 2t
dh ynu. geius
in aum jam  iorantosit disaartive
 nomitntar et smeriter labeunt diisrem commune
+e y + 3eau Y) + reteyhs C y + edys +. sed eta patet effici non posse ut dinta prr ertus
Fy + 2ayl
-eg - 2e dy y
 (y + 2dy
=edy y - 2C dyd
y +2dy
succedit; tentetur per y+ du
+eyt + 3edy y+ 2e dy dy) ey + 2edy +
y + u
est - eayue y
dy y = 2c dyd
ad de tatilus Terlancalis pluca aquenti soleda
3(y. y tag
PLlytiu y  qotus
duidatur per 
ero rurtus non
potest effii exuata
pute