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

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194464HUGENII OBSERVATIONES eodem modo analytice componi e terminis a3 + aab, abb
+ b3 quo componitur e terminis aab + bba, 2 bba. Hæc ni-
hilominus invenitur per eandem methodum quam indi-
cat in 7. Propoſitione. Methodus autem hæc eſt. Pri-
mo quærenda eſt quantitas perquam ſi multiplices a3 + aab,
& producto addas productum abb + b3 multiplicati per da-
tam quantitatem m, ſumma æqualis ſit ſummæ duorum alio-
rum productorum, unius aab + bba multiplicati per eandem
quantitatem quæſitam, alterius 2 bba multiplicati per datam
quantitatem m. Ponamus igitur quantitatem illam æqualem
z, erit a3z + aabz + abbm + b3m = aabz + bbaz + 2 bbam;
& z = {bbm/aa + ab}: Et certum eſt, ſive multiplicetur {bbm/aa + ab} per a3 + aab
& addatur abb + b3 multiplicatum per m, ſive eadem illa
quantitas multiplicetur per aab + bba & addatur 2 bbam,
ſemper prodire eandem quantitatem 2 abbm + b3m, & conſe-
quenter ultimam hanc quantitatem componi eodem modo e
primis & ſecundis terminis progreſſionis convergentis pro-
poſitæ, quod Autor fieri poſſe negavit.
III°. Datâ autem hac quantitate 2 abbm + bbbm, ſi hac
utamur ad quærendam terminationem progreſſionis propo-
ſitæ, juxta methodum ab auctore indicatam in 7 propoſi-
tione, & in 10, reperietur = {3 aab3 + ab4 + 2 a3 bb/bb + ab + aa}; & poſito a = 1
& b = 2, illa terminatio, quæ deſignat in eo caſu ſectorem
circuli continentem {1/3} totius circuli, erit = {48/7}; & primus ter-
minus progreſſionis a3 + aab, qui deſignat {1/3} trianguli æquila-
teri inſcripti in eodem Circulo, erit æqualis 3; ita ut propor-
tio circuli ad triangulum æquilaterum inſcriptum ſit ut {48/7} ad
3@ id eſt 16 ad 7. a vero nihilominus omnes has proportiones ab-
errare facile patet.
IV°. Si examinemus, cur terminatio aliquando occurrat ve-
ra per methodum Autoris, ut in 7 Propoſitione, interdum
verò non; reperiemus id ex eo oriri quod problema 10mæ

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