The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed
About this Item
- Title
- The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed
- Author
- Euclid.
- Publication
- Imprinted at London :: By Iohn Daye,
- [1570 (3 Feb.]]
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- Subject terms
- Geometry -- Early works to 1800.
- Link to this Item
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http://name.umdl.umich.edu/A00429.0001.001
- Cite this Item
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"The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed." In the digital collection Early English Books Online. https://name.umdl.umich.edu/A00429.0001.001. University of Michigan Library Digital Collections. Accessed June 15, 2024.
Pages
Page 425
FOrasmuch as in the 4. of this booke, it hath bene proued, that one and the self same cir∣cle containeth both the triangle of an Icosahedron, and the pentagon of a Dodecahe∣dron described in one and the selfe same Sphere: Wherefore the circles, which cōtaine those bases, being equall, the perpendiculars also which are drawen from the centre of the Sphere to those circles, shall be equall (by the Corollary of the Assumpt of the 16 of the twelfth). And therefore the Pyramids set vpon the bases of those solides haue one and the selfe same altitude: For the altitudes of those Pyramids concurr•• in the centre. Wherefore they are in proportion as their bases are, by the 5. and 6. of the twelfth. And therefore the pyramids which compose the Dodecahedron, ar•• to the pyramids which compose the Icosahedron, as the bases are, which bases are the superficieces of those solides. Wherefore their solides are the one to the other, as their superficieces are. But the superficies of the Dodecahedron is to the superficies of the Icosahedron, as the side of the cube is to the side of the Icosahedron, by the 6. of this booke. Wher∣fore by the 11. of the fifth, as the solide of the Dodecahedron is to the solide of the Icosahedron, so is the side of the cube to the side of the Icosahedron, all the said solides being inscribed in one and the selfe same Sphere. Wherefore the solide of a Dodecah••dron is to the solide of an Icosahedron: as the side of a cube is to the side of an Icosahedron, all those solides being described in one and the self same Sphere: which was required to be proued.
A Corollary.
The solide of a Dodecahedron is to the solide of an Icosahedron,* 1.2 as the su∣perficieces of the one are to the superficieces of the other, being described in one and the selfe same Sphere: Namely, as the side of the cube is to the side of the Icosahedron, as was before manifest: for they are resolued into pyramids of one and the selfe same altitude.
Notes
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* 1.1
This Campane putteth a•• a Co∣rollary in the 9. proposition af∣ter his order.
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* 1.2
This Corolla∣ry is the 9. proposition after Cam∣pane.