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

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III. Curva AX X talis ſit, ut PX ſecanti CS (vel CT) æquetur;
_ſpatium_ AC PX hoc eſt _Summa ſecamium ad arcum_ AM pertinen-
tium, & ad CB applicatarum) æquatur _duplo ſectori_ ACM.

Nam _ſpatium_ AF MX _segmenti_ AFM _duplum_ eſt; & 11Fig. 166.2210. Lect.
XI. _angulum_ FC PM _Trianguli_ FCM. ergo _totum ſpatium_ ACPX
totius _ſectoris_ ACM duplum eſt.

Etiam hoc è 16. hujus duodecimæ Lectionis apertè conſtat.

IV. Curva CVV talis ſit, ut PV _Tangenti_ AS æquetur; erit
_ſpatium_ CVP (hoc eſt _ſumma tangentium ad arcum_ AM _pertinen-_
33Fig. 166. _tium_, & ad rectam CB applicatarum) æquale _ſemiſſi quadrati ex_
_ſubtenſa_ AM.

Manifeſtè conſectatur ex ſeptima undecimæ Lectionis.

V. Acceptâ CQ = CP; & ductâ QO ad CE parallelâ (quæ
_byperbolæ_ LE occurrat in O) erit _ſpatium byperbolicum_ PL OQ du-
ctum in _radium_ CB (ſeu _cylindricum ad_ bafin PLOQ, altitudine
BC (duplum _ſummæ quadratorum_ ex rectis CS, ſeu PX ad _arcum_
44Fig. 166. AM pertinentibus, & ad rectam CB applicatis.

Nam quia PL. QO: : (BQ. BP. hoc eſt: :) BC + CP.
BC - CP; erit componendo PL + QO. QO: : 2 BC. BC
- CP. item eſt QO. BC: : BC. BC + CP; ergò (pares ra-
tiones adjungendo) eſt PL + QO. QO + QO. BC = 2 BC.
BC - CP + BC. BC + CP; hoc eſt PL + QO. BC: :
2 BCq. BCq - CPQ (hoc eſt: :) 2 BCq. PMq. verùm
eſt PXq. BCq: : BCq. PMq. vel(antecedentes duplando)2 PXq.
BCq: : 2BCq. PMq. ergò PL + QO. BC: : 2 PXq. BCq. vel PL x BC +
QOxBC. BCq: :2PXq. BCq. quare PL x BC + QO x BC = 2PXq.
itaque BC in omnes PL + QO ducta adæquat omnia totidem PXq.
unde conſtat Propoſitum.

VI. Hinc ſpatium αγψμ (hoc eſt _ſumma ſecantium in arcu_ AM
55Fig. 167. ad αβ applicatarum) æquatur _ſubduple ſpatio byperbolico_ PLOQ.

Nam ſumatur arcus MN indefinitê parvus, & huic æqualis recta μ ν,
ducatúrque recta NR ad AC parallela. Eſtque MN. MR: : (MC.
CF: : CS. CA: : PX. CA: :) PXq. PX x CA. adeóque
MN x PX x CA = MR x PXq. ſeu μν x μψ x CA = MR x
PXq. atqui (ex præcedente) omnium MR x PXq ſumma ſpatii
PL OQ in CA ducti ſubdupla eſt. Ergò omnia totidem μν x μ ψ
in CA ducta eidem ſubduplo æquantur. quare ſpatium αγψμ

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