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# Title: Leibniz-Handschriften zur Mathematik LH 35, 2, 1
# Folio 214r, page id 00067993:0429
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TXXXTDON ONNV
exoacxo
OLLINALS LON LALLIAS OAEDAOHIOS
Sngeuones suse
N2 LAIADSMASSIS NA EVSAVAO
cendens, inter alia habetur per agrationes gradus indetiniti tractenns
a nemine con(ILCTATAS, ut R HE XX F. X aqVaL. 30, _ qVATaTIE Z, rC-
Periecur este 3, quia 53, (3 eAt 27,f3 sirC30: quales aegrationes
Pro circulo dabimus suo loco. Algebraica expressio sic per numeros, li-
cecircarionales, rulgares, seu per radices aequacionum communium
Qrae quidem pro gradratura Sheta  ecors suipossibilis
valorem circuli exactum proxressione cermitoru, inprimis racio
natium, qualem hoc locoproxonam.
Inveniigitur: vid. fg. 2.
Oracrato Diamerri existente 
circuli aream Lore:  3 ∞3 = E8 retis
nempe gradeacum Diametri inceorum demes (ne nunius
Hac valor) ejus cerria parce, addit a ruruS C quia niIII
demsimus) quinca, demeagre iterum (quia nimium re. ad.
secimus) septima; de ita POrrO.
Ericesi valor jasto mojor 2 excore camen existence intca=
minor Z
or =
ninort 5A+ 
XC
Foca ergo leries continet omnes approxingrationes limutilire valores
justo majores dejusto minores: prour enim lonse conneala uelligitur,
eric errorminor Fractione daca,ac proinded minor daca quans quallii.
eace. Grare cora series exactum exprimie ralorem. Ex licec uno numeto
Cimma ejus Periei exprimi non possie, de Peries in infnitum producatur:
qroniam eumen un a lese progrectionis constac, cora Patis mente perci.
vitur. Nam, siquidem Circulus non est gracraco commenturabilis,
non porest uno numero exprimi, sed in rationalibus necestario per
seriemr exthiberi deber; quemadmodum ec Diagonalis quacdrati, et sc.
Atio extrema de media catione facta, qram aliqui divinam rocant,
aliae qre mulcx quancitates, quae sune irrarionales. Ex quidem ALu
dolphus porisset regulam dare, qra in infinirum concinuarentur nu-
meri 31 4159 dc, dedisset nebis guacraturam Arichmericam exa-
ctam in inceeris, quam nos extribenus in tractis.
Ne quis aurem in his parum versatus pucet, seriem ex inllucis
terminis conttancem non posse agrari ciculo, qui est quamcivoe ti
Superest Quadratura Prichmetica, quae Talcem per series sc, exhriben
bec
nica: Sciendum ek, mulcas series numero terminorum insinitas, etseir
summâ qracitates sinicas. Exempli Facillimi loco sit series ab unicare de
crelcens, procressionis geometrica deplae, F f. Z HF  ra L5 ~ Fo+ Xc
in intinicum; quae tamen non facit plus quam L. Nam in adjecti
linea recta AB (Fig. 4.) quae sICL CrIC A C Z, a residuum (CB) bile
candoin D, habebis CD Z; dc reliduum (BB) bisecando in E, tabetis
DE, E; de residuum (LB) bilecanoo in F, habebis FF, 2G; dC it
a ine alibi octentum est.
persequi non vacAC: hoc tamen praeterire non opoctet, celthosse-
riei noctra, 2, 5, F, n 3dc, elle progrenton Montcusr in con-
tinua proporcione harmonica, ut ex petsenes palebul: Orin de per
_ 2, i 22.  29_c est itidem tertes Matonue propoccnalium.
Tcagi cum Circulus sic F L 5 F5F 5 F  dc, 3 7. XX. 15 d
posteriorem seriem parcialem, a petost Ludalle HaO, erie maxnicudo
Circuli, differencia duacum serierum proeressionis harmonica. El
voniam quoicunque terminorum numero tntorum progrectionis
hiarmonicae summa, compendio aliquo iniri pouull, hinc appropin-
quationes compendiolae, (A POIL L Put AlliS etlet OpuS) duci
octene
F r8, de ita porro, in unut ducenae, subeb leriem novam
pro magucudine Circuli, nempe = (la Cll 2 5) f. 5 (id est 7 3)
1459 (i est 3 FF.) eec. itaque:
Quadrato inscripio existente. 2
Erit Area Circuli 5L5, 59L35FZES _c.
Sune aurem numeri 3. 57. 99. 195. 313 _C, excerpei per Lau X
Eerie numerorum gradratorum (4. 9. 16. 25 8cC.) unitace minutoeu
unde Sc Series: 3. 8. 15. 24. 35. 48 63. 80. 99. 120. 143. 168. 195. 224. 273.
288. 323, 360. 399 deci ex cujus seriei numeris, quarcus quisque pou
primum noster est. Inveni aucem (quod memorabile ett)
nica 3t s t es tas tis tao todo bs tot de  qun
concinuando sine sine, nunqram cgredieris cerminum B. Idem in
Fractionibus numerorum siguracorum seu eriangulo Flarmonico fieri,
Mulca nocari possent circa hanc Oradracuram, sed qua nunc
Patrum, F 2, Z. 2, Z, dc Mela tettes prosienanis harmonica;
Si quis in serie nosra cerminos signo n aftectos collere volet,
is duos proximosif. Z Z, icem L 5. - Z, iem HL. 8 - . d r 15 -  ee
ei in-