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# Title: Leibniz-Handschriften zur Mathematik LH 35, 3 A 31
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Si sit series quaecunque fractionum, quadum numerator semper untas nominatores
L 4
1 e
quianque  qui seguificeentur per 1 2, 3, 4, 5, 6 et c. ut L 
2. X1 f3.2. (3. 1+2. 1) 4353) 43, +8) 4,1 -32. 1 ]8132 (8 4 43. 146)5 421. f. 3. 24 + 9321
tunc  capta e differentiarum : L) 1
Et generaliter talis repentur relatio sit  inter te rminosscaritum. sit aliquis
12. quaeritur sequens 3. 2 - (9 31 4 2.1. Hic ita it venietur
Capitum 2-1 sevibatur  terminus respondens gerier capitum sequentis, qui differt
tantum a praecedente, quod loco 1, 2,: 4 de scrbitur 2, 3, 4, C et c. et is proinde erit 3-2
praeter terminum seriei in2.
n1 delichas 3 2 habebis 3. 2 -(213. 1 +2.1 sed omissis denominatonbus
suffecerit si inveneroinus relationes numeratorum, sit ergo Numerator L nempe 2- 1
quod si jemi a
et proxime respondens seriei capitum sequentis (L) fleit, 3-2
sere proxime sequens ipsum L. in eadem sere capitum. Et quidem semper munc
consideremus autem in genere quid prodeat ex. 1. ) ∞1 ( 3. 2 - 6) 3a1 4 2. 1
si numeri 1, 2, 3, dc ponantur esse prou essionis Mithaeticae. Ante omnia autem
consideranta gendalis proprietas hujus serier () 1. 1 1. 2. 1) 1. 3, 3. 1) 1. 4 64. 1 (&c
poramus jum Nundros fictitios 1, 2, 3, 4hC esse progressionis Arichmetico aut
Que generaliter est: bXtC, tunc generalus expossio  Drminus
capitum ita poterit exprimi:  HBRFCOBKHH + sit O - bXFC
1. bx FC batC. b X. 2 +C esc - X bxtC
1. beac et Considerandum et F dictos terminis seriei capitum, +
2 1 eC
cossie ita epprimi, versi gratia alio mado, v. g. 43 + (+-  + = 6) 4 +  s .b = + (1  - 4
quemodum patet ex pagina praecedenti
Rursus considerandum ist numeros:
posse sic exprimmt generaliter (D. X
1 )1(1)) 1. 8. )) 1) t (3) () ( VV. (. (9( 2. 1)
ponamus jam Numerot fictilios 12 3 ct. esse cocgressionis qeoet Arithmelicae
L BH+C
.L + X. X - 1
C XX - x ) LX
bx+ b bet
b X + C
tunc numevos fictitios
igitur generaliter Terminus serici
+qa
1 - - X. X - 1. X - 2
bxibtc
ab) 1
bx+ ut
Hi numeri e sequentes sunt veri
Extsbrc
Et ita si semper instiluatur divi sio
in qualibet divisione prodibunt complures
XEx B
Txribe
son + 26)
intequi, respectu ipsius X, et unus fractus. quod operest
prehum foret prosequi
ri sequentu Terminus euit:
c 6x+ 26
X. X 1
1.2. bx +
seu res
iflud agitur ut dversorum terminorum relata detur inter se
co redit, ut si dectur ..  1 + 2 + 3 + 4 + 5 sC = O quorum tinguli termai sint 
se letur rela
et 1 +  + 13 + 9 4 15 &c = D
rursud detur inte  Geadem quae in ter 12 et 13 3C
Et pursus, ead Cocrpa retur relatio inter 1 et I, e quae est inter 2 et 12  3. C1)
7Xlaubus
unius secici cum relitione
habendam comminem
 uni asus istite ot sintat di senais m qubur m
et  +2 23 +24 + 25 etC 4 insielatione inser vx et v(X) jamgniter  t
inter 1 et 2 eadem quae inter 2 et 3, et quae inter 3 et q
inter 21 et 22 eadem quae inter 32 et 
EX et (y) X quo
et coparentur relatiohes
alterius sende
assumtis
 . X. 4 ,  2) XC. (posito r esisos alisue ) uc
nempe simplicis seu posito X = () vel (1) vel (t vet (3)
1, vel 2, vel 3 Xc esse
capitum erit
i.
bx+ b +C
etues
qui ita exprimuntur: F. quaum qualitet ita exprimisur X 1, X, X. L. X. a 1 X.2 HC
ubi X significant immeros naturales 0, 1, 2, 3, 4 ec. i qui sunt numeri veri
etc.
quaeritur M. numerator
puirum Nameratorem Nominatorem maximum huc pertineatem, hoc loco 3, vocando X
scubet fiet M - X L   M  M = X. L - 1. (L)
Numeratorum
1 txa
11 
bx +
t
1. 2