Barrow, Isaac - Lectiones opticae & geometricae : in quibus phaenomenon opticorum genuinae rationes investigantur, ac exponuntur: et generalia curvarum linearum symptomata declarantur

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Addenda Lectionibus Geometricis.

Inſervit hoc ſuperficiebus deſignandis, quarum in promptu ſit di-
menſio, etenim (ductâ ME ad AD parallelâ) Superficies Solidi
ex plani BM E circa axem DB rotatu progeniti adæquat {Periph/Rad}
x GD X; ut habetur in 11a Lectionis XII.

In Lect. XI. appendice, numero XXXIII. de Cycloide profer-
tur Tbeorema quoddam, id quod ex bujuſmodi generaliori
Tbeoremate deduci potuiſſet.

SI t AM B curva quælibet, cujus Axis AD , baſis DB , ſit item
11Fig. 223. curva AN E talis, ut ſi arbitrariè ducatur PM N ad DB E pa-
rallela, poſitoque rectam TN curvam AN E tangere, ſit TN parallela
ſubtenſæ AM ; completo Rectangulo AD EG erit Spatium trili-
neum AE G æquale Segmento AD B.

Huic ſuppar Theorema tale eſt: liſdem poſitis, ſi tam Segmentum
AD B, quam Spatium AE G circa Axem AG convertantur; erit
productum è Segmento AD BS olidum producti ex AE G duplum.

E tangentium porrò contemplatione ſuborta eſt methodus, per
quam expediſſimè plurima circa maximas quantitates Theoremata
deducuntur; quæ certè ſi tempeſtivè ſe objeciſſent, digna cenſuiſſem
quæ Lectionibus inſererentur, ex iis indigitabo nonnulla.

Sit curva quæpiam AL B, cujus Axis AD , baſis DB ; & huic
22Fig. 224. parallelæ LG , λ γ; item LT curvam tangat.

_Theor_. I.

Sit _m_ numerus quicunque, poteſtates exponens; ſi ponatur
DG {_m_ - 1/} x TG = GL {_m_/}, erit DG {_m_/} + GL {_m_/} maximum, ſeu
majus quam D γ {_m_/} + γ λ {_m_/}.

_Theor_. II.

Itidem ſumpto numero _m_, ſi ponatur BL {_m_ - 1/} x TL = GL {_m_/};
erit GL {_m_/} + BL {_m_/} maximum ſeu majus quam γ λ {_m_/} + B λ {_m_/}.

_Theor_. III.

Sint numeri quilibet _m_, _n_; ſi ponatur _m_ x TG = _n_ x DG , erit
DG {_m_/} x GL {_n_/} maximum, ſeu majus quam D γ {_m_/} x γ λ {_n_/}.

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