Full text: Barrow, Isaac: Lectiones Opticæ & Geometricæ

XXVIII. Sit _Circulus_ AMB, cujus _Radiui_ CA, & ad hunc per-
pendicularis recta DBE; ſit item curva ANE talis, ut ductâ utcun-
que rectà PMN ad DE parallelâ (quæ circulum ſecet in M, dictam
curvam in N) ſit recta PN æqualis _Arcui_ AM; ſit demum _axe_
AD _baſe_ DE deſcripta _Parabola_ AOE, hæc extra curvam AN E
tota cadet.

42.1.

Fig. 151

Nam ſecet recta PN parabolam in O; & connectantur ſubtenſæ
AB, AM; eſtque DE. PN: : arc AB. arc. AM & gt; AB. AM
: : DE. PO. quare PN& lt; PO; unde liquet Propoſitum.

XXIX. Exhinc (& è vulgò notis _ſpatiorune_ ADB, ADE _dimen-_
_ſionibus_) facilè colligitur hæc regula: {3 CAx DB/2 CA+CD} & lt; arc. AB.

Fig. 152.

Porrò ſi ponatur arc. AB = 30 grad. ſitque 2 CA = 113; juxta
regulam iſtam computando, proveniet _tota circumferentia_ major quàm
355, minus fractione unitatis.

XXX. Hinc etiam _dato arcu_ AB, nominatiſque AB = p; CA = r; & DB = _e_, ad inveniendum _ſinum rectum_ DB adhibebitur hæc æqua-
tio; {3 _rrpp_/_9rr_ + _pp_} = {12 _rrp_/9 _rr_ + _pp_} _e_-_ee._ vel ponendo _k_ = {3 _rrp_/9 _rr_ + _pp_}; erit
_kp_ = 4_ke_ - _ee._ vel 2 _k_ - √ 4 _kk_ - _kp_ = _e._

XXXI. Sit AMB _Circulus_, cujus Radius CA, & huic perpendi-
cularis recta DBE; ſit item curva ANE pars _Cycloidis_ ad _Circulum_
AMB pertinentis; demum ad axem AD, baſin DE ſtatuatur _Para-_
_bola_ AOE; hæc intra _Cycloidem_ tota cadet.

42.1.

Fig. 153.

Etenim utcunque ducatur recta PM ON ad DE parallela, lineas
expoſitas ſecans, ut cernis; connectantúrque _ſabtenſæ_ AB, AM; eſtque DE. PO: : AB. AM : : curv. AE. AN & gt; DE. PN; adeoque PO & lt; PN. unde conſtat Propoſitum.

XXXII. Exhinc, & è _notis ſegmentorum circular is atque Cycloida-_
_lis dimenſionibus_, hæc elicitur _Regula_ {2CA x DB + CD x DB/CA + 2CD}
& gt; arc. AB.

Porrò ſi fuerit arc. AB = 30 grad. & ponatur 2 CA = 113; è
regula hac conſectatur fore _totam circumferentiam_ minorem quam
355, plus fractione.

Vides igitur ut è propoſitis duabus regulis ſtatim emergit _Diametri_
ad _Circumferentiam Proportio Metiana_.

XXXIII. Quoniam exorbitanti ſe obviam dedit _Cyclois_ hoc adno-
tabo _@ beorema_, neſcio an uſpiam ab illis, qui de _Cycloide_ tam fusè
ſcripſerunt, animadverſum; Completo _Rectangulo_ ADEG, _ſpatium_

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