Full text: Volumen secundum. Opera geometrica. Opera astronomica. Varia de optica. (2)

ET HYPERBOLÆ QUADRATURA. omnes ſeriei convergentis terminationem eodem modo eſſe
compoſitam ex terminis convergentibus primis quo ex termi-
nis convergentibus ſecundis, tertiis, vel quartis, & c.

78. PROP. XI. THEOREMA.

Dico ſectorem circuli, ellipſeos vel hyperbolæ A B I P
non eſſe compoſitum analyticè à triangulo
A B P & trapezio A B F P.

78.1.

TAB. XLIII.
Fig. 1. 2. 3.

Ponatur triangulum A B P a & trapezium A B F P b : ma-
nifeſtum eſt ex prædictis trapezium A B I P eſſe Vqab & polygonum A B D L P {2 ab/a + Vqab}, item ſectorem A B I P eſſe
hujus ſeriei convergentis terminationem. ut ex ſeriei termi-
nis auferantur ſigna radicis & fractionis, pro a & b primis
ſeriei terminis convergentibus, hoc eſt pro triangulo A B P
& trapezio A B F P ponantur a 3 + a 2 b & ab 2 + b 3 ; erunt-
que ſecundi ſeriei termini convergentes, hoc eſt trapezium
A B I P & polygonum A B D L P, ba 2 + b 2 a & 2 b 2 a, di-
co ſeriei convergentis (cujus primi termini convergentes ſunt
a 3 + a 2 b, ab 2 + b 3 & ſecundi ſunt ba 2 + b 2 a, 2 b 2 a) termina-
tionem non eſſe compoſitam analyticè a terminis a 3 + a 2 b,
ab 2 + b 3 : ſi enim componatur prædicta terminatio analyticè a
terminis convergentibus a 3 + a 2 b, ab 2 + b 3 ; componetur etiam
eadem terminatio analyticè & eodem omnino modo à termi-
nis convergentibus ba 2 + b 2 a, 2b 2 a; & proinde eadem quan-
titas, nempe prædicta terminatio, eodem modo componitur
analyticè ex terminis a 3 + a 2 b, ab 2 + b 3 , quo componitur ex
terminis ba 2 + b 2 a, 2b 2 a,
ſed nulla quantitas poteſt
eodem modo analyticè com-
poni ex terminis a 3 + a 2 b,
ab 2 + b 3 , quo componitur
ex terminis ba 2 + b 2 a, 2b 2 a, quod ſic demonſtro. ſi analy-
ticè componeretur quantitas ex terminis a 3 + a 2 b, ab 2 + b 3 ,

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