27986
II.
Iiſdem poſitis, atque paratis;
_ſummarectangulorum_ AZ x AE
+ BZ x BF + CZ x CG, & e. æquatur _trienti cubi_ ex baſe
11Fig. 122.DH.
+ BZ x BF + CZ x CG, & e. æquatur _trienti cubi_ ex baſe
11Fig. 122.DH.
Nam ob HL.
LG:
: PD.
DH:
: PD x DH.
DHq;
erit HL x
DHq = LG x PD x DH. hoc eſt HL x HOq = DC x Dψ x
DH. Similíque diſcurſu, LK x LYq = CB x CZ x CG. & KI
x KYq = BA x BZ x BF, & c. Verùm HL x HOq + LK x
LYq + KI x KYq, & c. adæquant trientem cubi ex DH; itaque
liquet Propoſitum.
DHq = LG x PD x DH. hoc eſt HL x HOq = DC x Dψ x
DH. Similíque diſcurſu, LK x LYq = CB x CZ x CG. & KI
x KYq = BA x BZ x BF, & c. Verùm HL x HOq + LK x
LYq + KI x KYq, & c. adæquant trientem cubi ex DH; itaque
liquet Propoſitum.
III.
Simili ratione conſtabit ſummam AZ x AEq + BZ x BFq
+ CZ x CGq, & c. æquari τῶ{DH_qq_; /4} & eſſe ſummam AZ x
AE cub. + BZ x BE cub. + CZ x CG cub & c. = {DH {5/ }/5}; ac
eodem in continuum tenore.
+ CZ x CGq, & c. æquari τῶ{DH_qq_; /4} & eſſe ſummam AZ x
AE cub. + BZ x BE cub. + CZ x CG cub & c. = {DH {5/ }/5}; ac
eodem in continuum tenore.
IV.
Exhinc conſectantur haud aſpernanda _Theoremata_:
Sit
VDψφ ſpatium quodlibet, cujus axis VD, ut dictum, æquiſectus;
22Fig. 122. ſi concipiantur ſingula ſpatia VAZφ, VBZφ, VCZφ, & c. in
ſuas ordinatas AZ, BZ, CZ, & c. reſpectivè ſingulas duci, quæ pro-
veniet ſumma adæquabitur ipſius ſpatii VDψφ ſemiquadrato.
VDψφ ſpatium quodlibet, cujus axis VD, ut dictum, æquiſectus;
22Fig. 122. ſi concipiantur ſingula ſpatia VAZφ, VBZφ, VCZφ, & c. in
ſuas ordinatas AZ, BZ, CZ, & c. reſpectivè ſingulas duci, quæ pro-
veniet ſumma adæquabitur ipſius ſpatii VDψφ ſemiquadrato.
Nam (utì priùs oſtenſum) figuræ VDψ φ adaptari poteſt ſpatium
VDH; tale nimirum ut ductà quâvis ad curvam VH perpendiculari,
ceu EP, ſit AP ſibireſpondenti applicatæ AZ æqualis; unde 33_Præced_. Lect. X.441 _hujus_.55Fig. 122. ſpatium VAZ φ = {AE_q_/2}; & VBZ φ = {BF_q_; /2} & VCZ φ = {CG_q_/2}
& c. quapropter omnia VAZφ x AZ + VBZφ x BZ + VCZφ
x CZ, & c. æquabuntur omnibus {AE_q_ x AZ + BF_q_ x BZ + CG_q_ x CZ/2}
hoc eſt τῶ {DH_qq_/4 x 2}; hoc eſt τῶ {VDψφ x VDψφ/2. }
663 _hujus_.VDH; tale nimirum ut ductà quâvis ad curvam VH perpendiculari,
ceu EP, ſit AP ſibireſpondenti applicatæ AZ æqualis; unde 33_Præced_. Lect. X.441 _hujus_.55Fig. 122. ſpatium VAZ φ = {AE_q_/2}; & VBZ φ = {BF_q_; /2} & VCZ φ = {CG_q_/2}
& c. quapropter omnia VAZφ x AZ + VBZφ x BZ + VCZφ
x CZ, & c. æquabuntur omnibus {AE_q_ x AZ + BF_q_ x BZ + CG_q_ x CZ/2}
hoc eſt τῶ {DH_qq_/4 x 2}; hoc eſt τῶ {VDψφ x VDψφ/2. }
V.
Quòd ſi ducantur omnia √ VAZφ, √ VBZφ, √ VCZφ,
& c. in ſuas applicatas AZ, BZ, CZ, & c. reſpectivè proveniet ag-
gregatum æquale duabus tertiis radicis quadratæ facti ex ipſo ſpatio
VDψφ cubato (τῶ {2/3} √ VDψφ {3/ })
& c. in ſuas applicatas AZ, BZ, CZ, & c. reſpectivè proveniet ag-
gregatum æquale duabus tertiis radicis quadratæ facti ex ipſo ſpatio
VDψφ cubato (τῶ {2/3} √ VDψφ {3/ })
Nam adaptatâ curvâ VH, eſt √ VAZ φ = AE√ {1/2};
&
√ VBZφ
= BF √ {1/2}, & VCZφ = √ CG √ {1/2}, & c. Cùm itaque
= BF √ {1/2}, & VCZφ = √ CG √ {1/2}, & c. Cùm itaque
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