Full text: Barrow, Isaac: Lectiones Opticæ & Geometricæ

omnia AZ x AE + BZ x BF + CZ x CG, & c. = {DHcub. /3}
crunt omnia AZ x √ VAZ φ + BZ x √ VBZφ + CZ x
√ VCZφ, & c. = {DHcub/3} √{1/2} = √ {DH 6 /18. } Eſt autem DH_q_ =
2 VD ψ φ, vel DH 6 = 8VD ψ φ {3/ }; quapropter omnia AZ x
√ VAZ φ + BZ x √ VBZφ + CZ x √ VCZφ, & c. = √
{8/18} VD ψ φ {3/ } = {2/3} √ VDψφ {3/ }.

VI. _Exempla._ Sit VDψ circuli quadrans (cujus radius dicatur
R, & Peripheria P) ſegmenta VAZ, VBZ, VCZ, & c. in ſi-
nus rectos AZ, BZ, CZ, & c. ducta conficient {R_q_P_q_/8. }

Fig. 123.

Item Summa AZ √ VAZ + BZ √ VBZ + CZ √ VCZ,
& c. = {2/3} √{R 3 P 3 /8. } = √ {R 3 P 3 /18. }

Si VD ψ ſit parabolæ ſegmentum, factum è ſegmentis in applicatas
erit {2/9} VD_q_ x Aψ_q_; ac è radicibus ſegmentorum in applicatas factum
erit {2/3} √ {8/2} { /7} VD 3 x Dψ 3 √ {3/2} {2/4} { /3} VD 3 x Dψ 3 .

Similia plura de factis è _Segmentorum poteſtatibus, autradicibus_
_aliis in applicatas, aut ſinus ductis_, hinc extundi poſſent.

VII. E dictis porrò ſequitur, ſi omnes ( vertici, & perpendicula-
ribus interjectæ) VP per reſpectiva puncta A, B, C, & c. Concipian-
tur applicatæ, puta ut AY, BY, CY, & c. reſpectivis VP æquentur; erit è ſic applicatis _conſtitutum ſpatium_ ADξθ _æquale ſemiſſe quadrati_
_ex ſubtenſa_ VH.

Nam, ob omnes VA + VB + VC, & c. = {VD_q_; /2} & omnes
AP + BP + CP& c. = {DH_q_,/2} liquet fore omnes VP =
{VH_q_. /2}

VIII. Porrò, ſi (poſitis iiſdem) ſit curva RXXS talis, ut ſit IX
= AP, & KX = BP; & LX = CP, & c. erit _ſolidum factum ex_

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