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# Title: Leibniz-Handschriften zur Mathematik LH 35, 4, 9
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Demonstrato, quod columnae serierum
exhibentium potestates ab Arithmefficis
aut numetos ex his conflatos, sint periodicae.
Omnis series numerorum rationalium, qui sint arithme
tioorum potestates ejusdem gradus (exempli gratia
series omnium X, posito X esse progressionis Arithme-
tica numeros) continuatis quantum opus differentia
tionibus dat differentiam constantem. Fultima diffe-
rentia nabur alium est 1, quadratorum ab ipsis, Z, cuborum
6, quadrato quadratorum 24, surde solidorum 1o eeo
id est l; 1. 2; 1. Z. 3; 1.2. 3. 43 1. 2. 3. 4 5. ( Simile aliquidsit
Si arithmetici non sint Naturales, sed alic quicunque,
verbi gratia impares ordine positi, 13,517, cec 7
Idem est, si series exprimi possit per formulam
analyticam integram ex potestatibus conflatam, verbi gr.
X 3 6X = 7X + 1.
dem est, etsi formula habeat numeros in denomina-
na multiplicata
per denominutorem communem dabit seriem arithmetico-
rum. Ergo et non multiplicata, id enim aequalitatem
differentiae non mutat. Exempli gratia Numeri XX+X
sunt semper integri itaque horum series aliquum
differentiam (hoc loco secundam) habet constantem,
primae vero differentia sunt progressionis Arithmelicae.
Hinc etiam omnes series Liguratorum continuatis
differentiationibus dabunt seriem Arithmelicorum
et generaliter omnis series Neumerorum integrorum
qua Analytice expressa per formulam, incognita non cudit
vel in denominatorem (ut fit in progressione tarmonica) vel
in Exponentem (ut in Geometrica) continuatis quantum
roduc 
vel quod idem est, vice versâ omnis sujs modi series fit per
Summaetionem arithmeticorum replicatam. Omnis enim