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# Title: [Traite Du Triangle Arithmetique, Avec Quelques Autres Petits Traitez Sur La mesme Matiere] Traité Dv Triangle Arithmetiqve, Avec Qvelqves Avtres Petits Traitez Svr La mesme Matiere (shelfmark Nm-A 605)
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C OM BTHA TTOHES.
Primo, quod ex le parce, in primo trianculo eam proxortionem con-
rinoere Summa enim cellularum rnicxe sux Perieinem pe numerus pri-
mae cellulae C idestrnicas, aequatur mulcieudini combinationum ex-
ponenris seriei, in exponenre crianguli, hi enim exponences lune vni-
taces. Vnicas vero in rnicare rnico modo ex lemm. 2. huins combinacur.
Secundo, Si ea proportio in aliquo trianculo contincat; Id est si sum-
ma cellularum vuiuseujuscunque seriei trianculi cujusdam, aequetur
mulcirudini combinationum exponentis seriei in exponente trianeuli
Dico ex eandem proportionem in trianculo proxime sequenti contin-
gere.
Flis astumpris, Facile ostenderur in lingulis eriangulis eam proporrio-
nem concingere, conrinsir enim in primo, ex primo assumpro immo
de manifecta quoque ipta elt in secundo criangulo, ergo ex secundo
assumpro e in sequenri rriangulo contingie, quare de in lequenci dein
inAnitum.
Tocum ergonesorium in secundi assumpri demonstratione consistie,
quod ita expedieeur.
Sic criangulus quilibet v. 8. Terrius in quo supponitur HXC PFOPOE-
eio, id est, Cummam cellularum seriei prima C F0 f x aequari mul-
ticudini combinarionum numeri, I exponentis seriei in nUmeTO, 3.
exponente triancali. Summam vero cellularum secundae seriei e +
aequari mulcicudini combinarionum numeri 2 exponentis seriei in nu-
mero z exponente rrianculi, summam vero cellularum terriae seriei,
nempe cellulam, A, aequari combinationibus numeri z exponentis
Seriei in 3 exponente trianculi Dico et eandem proportionem con-
cingere ex in lequenci criangulo quario, id est, Cummam cellularum v.
8. Secundae Periei e Y i F B, aquari mulcieudini combinarionum nu-
meri 2 exponentis seriei in numero 4 exponente trianeuli.
Frenim o Y L. aequacur multieudini combinarionum numeri 2 in 3
ex hyposh. Cellula vero B Xquaeur ex reneratione trianeuli arith. cel
lulis C + ∞ F ∞ hae vero cellulae aequantur ex Byp. mulricudini com-
binationum numeri F in 3. Frgo cellulae o f. L F O aequancur mulcieu.
dini combinationum numeri 2 in 3 plus multicudine combinationum
numeri r in 3, hae aurem mulcieudines eequaneur ex quario lemmate
hacius mulcicudini combinarionum numeri 2 in 4. Eroo lumma cellu-
lacum e Y il F O aequatur mulciendini combinationum numeri 2 in
4. Q. F. D.
Idem Lemma 5. Problematice enuntiatum.
Daris duobus numeris inxqualibus invenire in triansul.
arith. Quoc modis minor in majore combinetur.
Propolici line duo numeri r. 8. 4 ex 6, oporret reperire in criangulo
arich. quoc modis 4 combinetur in 6.