28188
_ſpatio_ VDψ φ _circa axem_ VD _rotato ſubduplnm ſolidi ex ſpatio_
DRSH, _itidem circa axem_ VD _rotato, confecti._
DRSH, _itidem circa axem_ VD _rotato, confecti._
Nam ob HL.
LG:
: PD.
DH:
: Dψ.
DH:
: Dψ_q_.
Dψ x
DH: : Dψ_q_. HS x DH; erit HL x HS x DH = LG x Dψ_q_.
= DC x Dψ_q_. Simili planè diſcurſu erit LK x LX x DL =
CB x CZ_q_; & KI x KX x DK = BA x BZ_q_, & c. atqui ſoli-
dum ptius eſt {ῶ/δ}: AZ_q_ + BZ_q_ + CZ_q_, & c. & ſolidum poſte-
rius eſt {2 ῶ/δ}: DI x IX + DK x KX + DL x LX, & c. itaque
conſtat Propoſitum.
DH: : Dψ_q_. HS x DH; erit HL x HS x DH = LG x Dψ_q_.
= DC x Dψ_q_. Simili planè diſcurſu erit LK x LX x DL =
CB x CZ_q_; & KI x KX x DK = BA x BZ_q_, & c. atqui ſoli-
dum ptius eſt {ῶ/δ}: AZ_q_ + BZ_q_ + CZ_q_, & c. & ſolidum poſte-
rius eſt {2 ῶ/δ}: DI x IX + DK x KX + DL x LX, & c. itaque
conſtat Propoſitum.
IX.
Hæc itidem omnia ſimili ratione vera ſunt, etiam ſi curva VEH
11Fig. 124. rectæ VD convexas ſuas partes obvertat; nempe quovis in curva ac-
cepto puncto E; & per hoc ductâ EP ad curvam VEH perpendicu-
lari, & EAY ad rectam VD normali, factáque AZ = AP; erit
ſpatium VDψ = {DH_q_; /2} Sin quoque fiat AY = VP; erit ſpati-
um VD ψ = {VH_q_; /2} Et pariter quoad cætera.
11Fig. 124. rectæ VD convexas ſuas partes obvertat; nempe quovis in curva ac-
cepto puncto E; & per hoc ductâ EP ad curvam VEH perpendicu-
lari, & EAY ad rectam VD normali, factáque AZ = AP; erit
ſpatium VDψ = {DH_q_; /2} Sin quoque fiat AY = VP; erit ſpati-
um VD ψ = {VH_q_; /2} Et pariter quoad cætera.
Ex his verò _Theorematis quam innumerarum magnitudinum_ (ex
ipſarum immediatè conſtructione) _dimenſiones innoteſcant_, ab expe-
rientia facilè comperietur.
ipſarum immediatè conſtructione) _dimenſiones innoteſcant_, ab expe-
rientia facilè comperietur.
X.
Sit rurſus curva quæpiam VH (cujus axis VD, baſis DH)
22Fig. 125.& linea DZZO talis, ut a curvæ puncto quopiam, cen E, ductâ
rectâ ET, quæ curvam tangat, & recta EIZ ad baſin parallelâ, ſit
qerpetuò IZ æqualis ipſi AT; dico _ſpatium_ DHO _ſpatio_ VDH
_æquari_.
22Fig. 125.& linea DZZO talis, ut a curvæ puncto quopiam, cen E, ductâ
rectâ ET, quæ curvam tangat, & recta EIZ ad baſin parallelâ, ſit
qerpetuò IZ æqualis ipſi AT; dico _ſpatium_ DHO _ſpatio_ VDH
_æquari_.
Æquiſecetur enim recta DH indefinitè, punctis I, K, L, per quæ
ducantur rectæ EIZ, FKZ, GLZ ad VD parallelæ, curvæque oc-
currentes ad E, F, G, unde ducantur rectæ EA, FB, GC ad HD
parallelæ, rectæque ET, FT, GT (ut & HT) _curvam tangentes;_
lineæ verò ſe, ut Schema monſtrat, interſecent. Eſtque jam triangu-
lum GLH ſimile triangulo TDH (nam ob diviſionem iſtam indefi-
nitam arculus GH rectæ inſtar cenſeri poteſt, eatenus tangenti HT
coincidens) quare LG. LH: : TD. DH; & LG x DH = LH
x TD; ſeu CD x DH = LH x HO. ſimili ratiocinio eſt BC
ducantur rectæ EIZ, FKZ, GLZ ad VD parallelæ, curvæque oc-
currentes ad E, F, G, unde ducantur rectæ EA, FB, GC ad HD
parallelæ, rectæque ET, FT, GT (ut & HT) _curvam tangentes;_
lineæ verò ſe, ut Schema monſtrat, interſecent. Eſtque jam triangu-
lum GLH ſimile triangulo TDH (nam ob diviſionem iſtam indefi-
nitam arculus GH rectæ inſtar cenſeri poteſt, eatenus tangenti HT
coincidens) quare LG. LH: : TD. DH; & LG x DH = LH
x TD; ſeu CD x DH = LH x HO. ſimili ratiocinio eſt BC
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