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
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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
http://name.umdl.umich.edu/A00429.0001.001
Cite this Item
"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 7, 2024.
Pages
¶ The 14. Theoreme. The 14. Proposition. Cones and Cylinders consisting vpon equall bases, are in proportion the one to the other as their altitudes.
SVppose that the cylinders FD, and EB, and the cones AGB, and CKD, do consiste vpon equall bases, namely, vpon the circles AB, and CD.* 1.1 Then I say that as the cylinder EB is to the cylinder FD, so is the axe GH to the axe KL. Extende the axe KL directly to the poynte N, and vnto the axe GH, put the axe LN equall: and about the axe LN imagine a cylinder CM.* 1.2 Now forasmuch as the cilinders EB, and CM, are vnder equall altitudes, therefore (by the 11. of the twelueth) they are in the proportion the one to the other as their bases are. But the bases are equall the one to the other. Wherefore also the cylinders EB, and CM are equall the one to the other. And forasmuch as the whole cylinder FM, is diuided by a playne superficies CD being
descriptionPage 374
a parallell to either of the opposite plaine su∣perficieces:
[illustration]
therefore (by the 13. of the twel∣ueth) as the cylinder CM is to the cylinder FD, so is the axe LN to the axe LK. But the cylinder CM is equal to the cylinder EB, and the axe LN to the axe GH. Wherefore as the cylinder EB is to the cylinder FD, so is the axe GH to the axe KL.
But as the cylinder EB is to the cylinder FD, so (by the 15. of the fift) is the cone ABG to the cone CDK,* 1.3 for the cylinders are in treble proportion to the cones (by the 10. of the twelueth). Wherefore (by the 11. of the fift) as the axe GH is to the axe KL, so is the cone ABG to the cone CDK, & the cylinder EB to the cylinder FD. Wherfore cones & cy¦linders consisting vpon equal bases are in pro∣portion the one to the other as their altitudes: which was required to be demonstrated.