The 15. Theoreme. The 15. Proposition. Like partes of multiplices, and also their multiplices compa∣red together, haue one and the same proportion.
SVppose that AB be equemultiplex to C,
[illustration]
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
- 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 6, 2024.
Pages
as DE is to F. Then I say, that as C is to F, so is AB to DE. For forasmuch as how multiplex AB is to C,* 1.1 so multiplex is DE to F: therfore how many magnitudes there are in AB equall vnto C, so many are there in DE equall vnto F. Deuide AB into the magnitudes equall vnto C, that is, into AG, GH, and HB: and likewise DE into the magnitudes equall vnto E, that is, into DK KL, and LE. Now then the multitude of these AG,* 1.2 GH, and HB, is equall to ye multitude of these DK KL, and LE. And forasmuch as AG, GH, and HB, are equall the one to ye other and likewise DK, KL, and LE, are also equall the one to ye other: ther∣fore as AG is to DK, so is GH to KL, and HB
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to LE. Wherefore (by the 12. of the fift) as one of ye antecedentes is to one of the consequentes, so are all the antecedentes to all the consequentes. Wherfore as AG is to DK, so is AB to DE. But AG is equall vnto C, and likewise DK to F. Wherefore as C is to F, so is AB to DE. Like partes therefore of multiplices and also their multiplices compared together, haue one and the same proportion that their equemultiplices haue: which was required to be demonstrated.
Notes
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* 1.1
Construction.
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* 1.2
Demonstra∣tion.