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# Title: Leibniz-Handschriften zur Mathematik LH 35, 8, 27
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e  1676.
Ingeniosissimus Pascalius tractatum
scripsit  de Triangulo Arithmetico:
de summis, et summis summarum, et summis ex summis summarum ex Numerorum Progressionis
s,sipgular quadam ratione
Arithmeticae. Ego equad ratione incidi in Triangulum Harmonicum; de differentiis et
e
Differeatiis differentiarum, et differentiis ex differentiis differentiarum Numerorum progressionis
Harmonicae
harmonicae. Cumque Numeri progressionis Ameticae, sint Reciproci, sive fracti Numerorum
progressionis Arithmeticae; factum  est miro naturae consilio, ut Aecprori etiam reliquae
totius Trianguli Harmonici series (tantum certo modo multiplicatae) sericrum respondentium
Tr. unguli Arithmetici reciprocae essent, utque iidem sint Numeri Trianguli Arithmetici et Har-
monici, eo tantum discrimine quod Termini Tranguli Arithmetici sint  integri, id est fracti
habentes figuratum (id est Trigonalem, Pyramidalem, Trigono trigonalem, erc) pro Numeratore, et unita
tem pro Nominatore; contra Termini Trianguli Atarmonici habeantUnitatem pro Numeratore
Numerum figuratum pro Nominatore.
Reliqua ex Tabulae inspectione patent, quae dedncere demon-
s
strationibus non est operae pretrum.
Htodie enim quivis  characteristicae
et demonstrat
Analyseos cultor facile deprehendit qualia
melli multis ratiocinis Pascalius ostendit
Triang. Arithmet
Pascalii
servit ad inveniendas
summas integrorum finetrum
figuratorum, et
Quadraturas Parabolarum
Rationalium omnium
graduum, quae sunt figurae
qcareure, osymptotis
Adde quae alibi scripsi
de Triunguli Harmonici
numerorum summis
in schoda exigua, huic invluta
1d
ad
Triang. Harmon
meum
servit ad invenie
da
Summas
chorum
torum et ensircr
habentium
radraturiis
Hyperbolarum
rationalurm om
graduum sexcepta pri
nni quadrabili
quae spatia
habent finita ma
gnitudini), at ob
asumpertos infi
niter songitudins