Huygens, Christiaan - Christiani Hugenii opera varia; Bd. 2: Opera geometrica. Opera astronomica. Varia de optica

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150423ET HYPERBOLÆ QUADRATURA. quos terminos æquales appellamus ſeriei terminatio-
nem.

PROP. VII. PROBLEMA.

Oportet prædictæ ſeriei terminationem
invenire.

Ut huic problemati ſatisfiat, oportet primò invenire
quantitatem quæ eodem modo componitur ex termi-
nis convergentibus a, b, quo ex terminis convergentibus
{ca + bd - ad/c}, {bc - bc + ae/c}, hoc autem facile fit hoc modo: inveniatur
quantitas quæ multiplicata in a & addita b multiplicata in
quantitatem data m, eandem quantitatem facit ac ſi multi-
plicaretur in {ca + bd - ad/c} & adderetur {bc - be + ae/c} multiplicata etiam
in eandem quantitatem data\m m. ſit quantitas illa z, & pro-
inde za + bm æquatur {zca + zbd - zad + mbe - mbe + mae/c}, & æquatione
reducta invenitur {z = mac - mbe/ad - bd}; quæ quantitas ſive multiplica-
ta in a & addita m, ſive multiplicata in {ca + bd - ad/c} & addita
{mbe - mbe + mae/c} efficit eandem in utroque caſu quantitatem nempe
{maae - mbae + mbad - mbbd/cd - bd}: & proinde prædicta quantitas eodem mo-
do componitur ex terminis convergentibus a, b, quo compo-
nitur ex terminis convergentibus {ca + bd - ad/c}, {bc - be + ae/c}. atque a
& b quoniam ſunt quantitates indefinitæ poſſunt eſſe quili-
bet totius ſeriei termini convergentes, modò termini con-
vergentes immediatè ſequentes ſint {ca + bd - ad/c} & {bc - be + ae/c}, &
proinde quantitas {maae - mbae + mbad - mbbd/cd - bd} eodem modo componi-
tur ex quibuslibet totius ſeriei terminis convergentibus quo
componitur ex terminis convergentibus a, b; & igitur

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