Full text: Barrow, Isaac: Lectiones Opticæ & Geometricæ

CG = KL x LZ; & AB x BF = IK x KZ, & VA x AE =
DI x IZ. Verùm ſumma CD x DH + BC x CG + AB x
BF + VA x AE à ſpatio VDH minimè differt; & ſumma LH x
DO + KL x LZ + IK x KZ + DI x IZ à ſpatio DHO mi-
nimè differt. itaque ſpatio VDH, DHO æquantur.

Hoc _perutile Theorema_ doctiſſimo Viro D. _Gregorio Aberdonenſi_
debetur; cui ſequentia ſubnectimus.

XI. Iiſdem poſitis; ſolidum ex ſpatio DHO circa axem VDR
rotato factum duplum erit ſolidi facti ex ſpatio VDH itidem circa ax-
em VD rotato.

41.1.

Fig. 125.

Nam eſt HL. LG: : (DH. DT: : DH. HO: :) DHq. DH x HO. unde HL x DH x HO = LG x DHq = CD x
DHq. Similíque diſcurſu ſunt LK x DL x LZ = BC x CGq. & KI x DK x KZ = AB x BFq. & demum ID x DI x IZ =
VA x AEq. Eſt autem (ut vulgò notatum habetur) ſumma CD
x DHq + BCB x CGq + AB x BFq + VA x AEq dupla
ſummæ DI x IE + DK x KF + DL x LG, & c. Quare ſolidum
ex ſpatio HDO circa axem DR converſo factum duplum eſt ſolidi,
quod è ſpatio VDH circa VD converſo producitur.

XII. Hinc, ſumma DI x IZ + DK x KZ + DL x LZ, & c. æquatur ſummæ quadratorum ex applicatis ad VD; ſcilicet ipſis AEq
+ BFq + CGq, & c.

XIII. Simili ratiocinio conſtabit ſummam DIq x IZ + DKq x
KZ + DLq x LZ, & c. triplam eſſe ſummæ DIq x IE + DKq
x KF + DLq x LG, & c. hòc eſt æqualem ſummæ cuborum ab
omnibus AE, BF, CG, & c. ad VD applicatis. Idem quoad _re-_
_liquas poteſtates_ obſervabilis eſt Concluſionum tenor.

XIV. Iiſdem poſitis; ſi DXH ſit linea talis, ut quævis ad DH
o [?] rdinata, ceu IX, ſit media proportionalis inter ſibi congruas ordi-
natas IE, IZ; erìt ſolidum ex ſpatio VDH circa axem DH rotato
duplum ſolidi ex ſpatio DXH circa eundem axem DH converſo pro-
creati.

Nam ob VA x AE = DI x IZ, erit VA x AE x EI = DI x IZ x IE = ID x
IXq. Similíque de cauſa AB x BF x FK = IK x KXq; & BC
x CG x GL = KL x LXq, & c. Eſt autem ſumma VA x AE
x EI + AB x BF x FK + BC x CG x GL, & c. Subdupla ſum-

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