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.
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http://name.umdl.umich.edu/A00429.0001.001
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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
¶The 27. Theoreme. The 32. Proposition. Parallelipipedons being vnder one and the selfe same altitude, are in that proportion the one to the other that their bases are.
SVppose that these parallelipipedons AB and CD be vnder one & the selfe same altitude. Then I say that those parallelipipedons AB and CD are in that pro∣portion the one to the other, that their bases are, that is, that as the base AE is to the base CF, so is the parallelipipedon AB to the parallelipipedon CD.* 1.1 Vpon the line FG describe (by the 45.
[illustration]
of the first) the parallelo∣gramme FH equall to the parallelogramme AE and e∣quiangle with the parallelo∣gramme CF. And vpon the base FH describe a paralleli¦pipedō of the selfe same alti∣tude that the parallelipipedō CD is, & let ye same be GK.* 1.2 Now (by the 31. of the ele∣uenth) the parallelipipedon AB is equall to the parallelipipedon GK, for they consist vpon e∣quall
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bases, namely, AE and FH, and are vnder one and the selfe same altitude. And foras∣much as the parallelipipedon CK is cut by a plaine superficies DG, being parallel to either of the opposite plaine super••icieces, therfore (by the 25. of the eleuenth) as the base HF is to the base FC, so is the parallelipipedon GK to parallelipipedon CD: but the base HF is equal to the base AE, and the parallelipipedon GK is proued equall to the parallelipipedon AB. Wherfore as the base A••E is to the base CF, so is the parallelipedon AB, to the parallelipipe∣don CD. Wherfore parallelipipedons being vnder one and the selfe same altitude, are in that proportion the one to the other that their bases are: which was required to be demonstrated.
I neede not to put any other figure for the declaration of this demonstration, for it is easie to see by the figure there described. Howbeit ye may for the more full sight therof, describe solides of pasted paper according to the construction there set forth, which will not be hard for you to do, if ye remem∣ber the descriptions of such bodies before taught.
A Corollary added by Flussas.
Equall parallelipipedons cōtained vnder one and the selfe same altitude, haue also their bases equal. For if the bases should be vnequall, the parallelipipedons also should be vnequal by this 32 propositiō. And equall parallelipipedons hauing equall bases, haue also one and the selfe same altitude. For if they should haue a greater altitude, they should exceede the equall parallelipipedons which haue the selfe same altitude: But if they should haue a lesse they should want so much of those selfe same equal parallelipipedons.