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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 3 (shelfmark LH 13, 3)
# Folio 204r, page id 00068200:0409
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Atrithmeticorum o+I=
34b 43C+3 1+
Si primus terminus sit 0, secundus 1, ultimus l, erit summa L+1, l
 vel posito numerorum
terminorum esse n = 1+ 1, fiet ex theoremate sx: nl = 1: 2 seu (x = nl: 2 = l+ 1, l: 2 (3+ 3
Quadraticorum 2+1 = 3 +  0+ 1r 4 =  - 5 + oFirara - 
SXX: nH = 13 + A a seu SXX = IH, H,  + d st
Cubitorum e+ 18
OH + 4 + 9 + 16 - 30
=3 + 8 1onorib netib 4u a
Ot 1+8 = 9 = L -t  ot 1+8+27 = 36 = = =  + 1. oideoque sx. nl- 4t4
8+8 +8 e Autululus Kapinorum Z7, 12 +7+ 27
slous Quod si L sit infinihus evanescent t,
C, et. Itaque sisit series infinita quam
utatum a puncto seu o incipientium et continue prescentium,  in ratione primanorum secundanorum, terrianorum, quarta
norum etc, erit caseries ad sertes aequalium ad eamseriem, ut t ad  reciproca ad 2, L, F, E cH. (seu sa six, sX SX etC)
seu meo stylost.s Xax SXXdx Sxdx Sx dx EtC = Z, ZXX,  X, x x B et generaliter (Xdx =  Xet 1
Hinc autem et datur summa subsecundanorum, subtertianorum, etc dux Sdx x, sax Yx (Ax Ex.
Nam sit X = y fiet X = y Datur _. Sydx = L XC+1
Sed sydx + sxdy = xy seu sxdy = xy - sydx
peri - = XeY = Xet, est 1 - D. Ergo sy e du =  , X
quae regula comprehendi petest sub priore SXex =
Tabula sesti sar, 
Hin stritX. sis
f
six ix Exe ix Ex. EN
Y 2x 3X 2
EX Z  u. 36
Y 4X + X a G X 1 414
YEx EX  EX 35 B
quae regula Adeoque et
y. yte = e
pe dx = +
Ye+ LB x i Aoet ns petu anti primanes, autisecudan antinteriomos
tantum pro e augando
Tabula mea Recipiocorum. Hi teres Edrisetur minino equalium
A series aequalium mulcato serie pitmranorum, secundanorum, 3tianorum exc. quae ranturque summae, fiet
S1 -X, aX S IXX dx  f. ( - X, dx Si X, dx
seriei aepo potenti a serie apstomurum, seu residuorum a pimanis 1.X, quaeeatur summa
X1 -x 1 - 2X LXC 1 - 3X + 3XX + ∞ 1 - ax + 6XX + 4 X + X
Siax, 1 - XX IAXXFX
Sdx, 1- e
Sdx, 1- X 1- 2X  X
Su x, 1- X
Et generaliter, 6. 1
L- 3 43 -
1-3X + 3x = X
53+2-2
1.3iaKS. 1X)
1- 2 +6 - 4 +
(- 4XX+6X- 4 X6 + X
- 3 +6-2 +
gxi aperix
 4 +6, - 4 +.
G. IX, (1 - XC) = en, en , en -2e, en- 3e,  ete
Hinc quantitas qua sinita CA, 1, Cn. e+ B en 2C + 1, en. 3e+ etC
Hinc quantitas ista finita , et haberi potest, modo n sit integeri ita
enim mavitesum est exhuuriri tandem multiclicatores. Evenit antem
pulchre, ut quoties l est, ,,  etc. evanescant cumeratores expli-
catione facta, it nominatores sint sigurati, seu Arithmetice replicati
10 13∞, as
Seidx, (X - Xn = en, en e, en 2C, en - 3C, et
entMI entne entnHe, eninHL. extnt1- A
uodrepere donec scilicet in numeratore en -. e est 1, et donec in
numeratore en -. e est o. quia tunc superest n+I
5, dX, (XX- xer2)n =en en- e, en 2e, en-3C, et
 SX x =
Et generaliteo extrati, ontrume entenr Ze, entent-ge, entin +1 - 2e, t
Indx (X ern = en en e, en 2, en - 3C, etc
entratius entrnti-e, entrnti- e, entruti 3e, entrntiae,  ecorit metur
donec scilicet in numerature, minimus multiplicator fiat 1, et in menominatore. rn +
Patet etiam numeratorem semper esse eundem, utcunue variet r. Et ut
sastu ntiteee s nnor muctitioinem qumiener dentmnts
0+ 142. - 2
3+3t243=  et ila porro
oncralivores
foii minde
Semper reduca
potest ad
SX dx (.xe)n
duii quadrati, cubi; Biquadrati
Sndx (1fXe)n
15 Et1, ZH, 3+
ds pratin, at
1t1, 2+ 12 4, 21
bene compinuum
ae intra et quae suera primanos
am unam Tabulam, nam indices supra sunt progrissiones
Hart Arithmeticae, intra prodressionis harmonice
 subtecandani, primani, sesquip
   in cntisiundum
_3, uet L vel Z vor le HC selundrase hon abie
imltatos sero pruimanoum
per autem denominatores ultiplica ooces multi
ne appeat emem utrbra
arpicatorum motitudtntm.