Full text: Barrow, Isaac: Lectiones Opticæ & Geometricæ

_ſpiralis_, ut pro arbitrio ductâ rectâ C μ Z habeat arcus EZ ad rectam
C μ rationem aſſignatam (puta R ad S) Manifeſtum eſt lineam β YH
eſſe rectam, quoniam EZ (KO). Cμ (OY): : R. S, perpetuò. unde evoluta BMF ſit _Parabola_; quoniam axis partes AP, AD ſe
habent ut ſpatia KOY, KC β, hoc eſt ut quadrata ex ipſis OY, Cβ,
vel ex ipſis PM, DB.

67.1.

Fig. 197.
Fig. 198.

68. _Corol. Theor_. I.

Si ad figuram βCφ erigatur _cylindricus_ altitudinem habens æqua-
lem peripheriæ integræ _circuli_, cujus radius CL; erit iſte _cylindricus_
æ [?] qualis _ſolido_, quod procreatur è figurâ Cβ HK circa axem CK ro-
tatâ.

69. _Theor_. II.

Sit curva quæpiam AMB (cujus axis AD, baſis DB) & curva
AZL talis, ut liberè ductâ rectâ ZPM, ſit PZ = √ 2 APM; ſit
item alia curva OYY talis, ut ad hanc productâ rectâ ZPMY,
adſumptâque rectâ R, ſit ZP q. R q: : PM. PY; ſitque denuò DL. R: : R. LE. & per E intra angulum LDG deſcribatur _Hyper-_
_bola_ EXX; huic autem occurrat ducta recta ZHX ad AD parallela,
erit ſpatium PDOY æquale _ſpatio Hyperbolico_ LHXE.

69.1.

Fig. 195.
Fig. 199.

Hinc _ſumma_ omnium {PM/APM} = {2 LEXH/R q}.

70. _Theor_. III.

Sit curva quæpiam AMB, cujus axis AD, baſis DB; & curva
KZL talis, ut adſumptâ quâdam R, & arbitrariè ductâ rectâ ZPM
ad BD parallelâ, ſit √ APM. PM: : R. PZ; erit ſpatium ADLK
æquale _rectangulo_ ex R in 2 √ ADB; vel {ADLK/2 R} = √ ADB.

70.1.

Fig. 200.

_Exemp_. Sit ADB circuli quadrans, erit ſumma omnium {PM/APM} =
√ 2 DA x arc. AB.

71. _Theor_. IV.

Sit curva quæpiam AMB (cujus axis AD, baſis DB) ſintque
duæ lineæ EXK, GYL ità relatæ, ut in curva AMB ſumpto quopi-

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