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# Title: Leibniz-Handschriften zur Mathematik LH 35, 15, 5
# Folio 71r, page id 00068207:0141
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Vin Amelissime Consultissime
Domine Gratiosissime
Per varias ambages quibus me in Arithmetica Dgadica frustratus fui tanden
in rem ipsam penetrasse mihi videor et filum Ariadmeum invenisse quod non
solum in proposito Problemate, sed e in multis similibus rem expediat
o oe 
(07) (80) (87) (010D)
Edo 20B0) Eab4s oEpoBSYapodore
(20) (83) (407) (807) (980) (483) (8807
Totus scopus harum signaturarum est, ut diseamus (00d
= pr 1- 2C, pN8q
Datis certis signaturis quot dari possunt in denionibus
invenire omnes signaturas que dari possunt in Septer
Scio a simplicioribus incipiendum. Sed quia non omnes Casus explicandi,
occurrunt in simplicioribus hine ut me ubique explicem Septeniones
sumsi e senionibus inveniendes, in ipsa praxi utique a simplicioribus incipias
Ponamus ergo jam inventes esse omnes Seniones; hoc modo perge
(∞) = O + p -2pC ex dignaturis
p = 9) est noru additu quod ans
Omittitur, quia manet senio, quaerutur vero seeteniones si
plomillitur, quia solum est expletivum mutas a in
axo est dsenio phrabiref pt Exmultielicatus dat +64 abidesus Les 
=D + pO-2pOD ex Signaturis. Sed D. omillitur per s.
pO est OSenio = - 32 abidef. in (p. =9) duitus fact = 32 abcdetq
-2eOD est 0D Senic = -20 abedes in (2P) duitus Facit 140 abedetG
Horum Summa est
+Babid etus =() Septenioni,
(0D) = 0D + C D- 26OD ex Signaturis Sed ORmittitur persi
PD. est D Senio =  Habidef in(p = 9) duitus Tacit abidefue 
-200D. est ODLenio= 2oabidef in 7 (20) ductus Facet quabedetus
(8) = 02+pOD. 2PODS ex Signaturis sed st omittitur perX
=+36abidetus -(CD
p0D est ODsenio --20 abedef insp -9) ductqs Facit-20abidefy() Sepfenioni
Nam alterum mebrum 2pODd in Senionibus adhuc deest seu nihilvales Invent.4
(0) = 50 + P0- 2000 ex signaturis omisso S0per X.
po est S Sento = Hoabedet in (p- g) ductus Farit +ivabidefgi
-2005 est 00 Genio = -6 abides in (2) ductus
Fact +12 abiderq
(SD) = p 8D. +p O0 2p0DS, omissis 6D et -2∞0D8 per X. et 7
Ergo tatum Sumitur p0o seu 0 Senio = - babedet, in(e) Facil -babidefg - (85)
(80D) = 50D + p0- 200DS omissis 80D et 2p0d per
est nne
pOD est SD Senio =abidet per (p =9) ducbus Tac. abidetge -(0D) septen
Sic iterum ordine collocari possunt septeniones jam inventi et
ex iis eluui octeniones Methodo, Simillima.
(() sumit  -2p0 = - 12B abedefg= (0) Octenien.
(D) sumit po: Habedetgh
OD = - 72 abidefgt
I: Babidetgn=DVctenion.
(0D) sumit exsignat unis (PD = 8abe defghi–
inventu 
S... (e0D = +36 abedesgh7 L2pCD- - iE abidesat) = -bgabidetah = (∞ D) Octenion
b 200D6 -- 2 abidetgt) = +34 abcd efgh.
(80) sumit.-
Secr --20 abcdefgh
(5) Octenioni
(5D. (e0o rrabidetu L. 2PDS =  4a abid esut) (5C) Octenioni = -64 abid
2pC Dot = - 2 abidefgh) = 20 abcdefgh (orDjoctonigni
(0DS) (10D =-Babedeta
-200D8-- rabioe fgh
= -habedetgh = (038) Octenioni
(Numit p0Do= abidefgh. omisso altero- 2p0D0 per 
Et quia Xudi coefficientes quaeruntur  acera in Calculo omint
possunt, hac ratione in Novenionithine Literae in Calculo oilit
-20 = 4756 = ((
20 5) =0 () -1030 -  ) -120- (O∞Y BD- -62
O018 - - 6
-2707515 = + 16)- 48-0
2095 = + 16) 28 = D62 Dst  20
2006= + 16), 36 - (0D 0080 = - 8-
Ex ita inventis Novenionibus via datur ad Deniones et ponro
constituendos
Quod ut expeditius fiat pauca notentum
de Signaturis lnitialibus
Simplices 
e ve
2001 20 (
Intemedia OD
etc:. ex LuBaISi NIS Fundamentis
Coudum ante innotuerunt
fit si totam Columnam Olis per totam Columnam
Dnae muttiplices, ubi sex membra Producti se mutu
tollunt sola membra per nudum Dmultiplicata manent
Et sic in omnibus aliis Intermediis Compositis Columnis tria membra per
tria membra multiplicando tantum retineo tria membra Producti
Intermediarum signaturarum Coefficientes quaerere, propter se
non opnes est. Sed quia Simplices Columnae, C D. S2 etc: a multis membris
Compositarum dependent ideo e Compositas in promtu esse opus est,
quod quamdin negrexeram e. B ficultatibus elictar non potui
50 
Eo o 286.) 1. 29 00otiedo (p0vo
Geniones dati.
O- - 32 abidef
D= - 4 abcdes
OD= -20 abidet
02 = + 10 abedes
80--6 abedet
0D- abedes
flenionibus est
Septeniones datu
0- +6q abidefuinverut.
D = +B abidefq. Inv. 2
OD = + 36 abedetque Inv. 3
et = - 20 abedetque Uno. 4
50= +22 abi desge lno 5
5D = - babidesget Inv. B
508 abedetg. Inb. 7
Octeniones datu
Ov -64
o5 138
ND= +20
600 -8
septenioni