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 1, 2024.
Pages
¶ The 8. Theoreme. The 10. Proposition. If foure magnitudes be proportionall, and if the first be commensurable vnto the second, the third also shal be commensurable vnto the fourth. And if the first be incommensurable vnto the second, the third shall also be in∣commensurable vnto the fourth.
SVppose that these foure magnitudes A, B, C, D, be proportionall. As A is to B, so let C be to D, and let A be commensurable vnto B. Then I say that C is also commensurable vnto D.* 1.1 For forasmuch as A is commensurable vnto B, it hath (by the fift of the tenth) that proportion that number hath to number. But as A is to B, so is C to D. Wherfore C also hath vnto D that pro¦portion
[illustration]
that number hath to number. Wherfore C is commen∣surable vnto D (by the 6. of the tenth). But now suppose that the magnitude A be incommensurable vnto the magnitude B.* 1.2 Then I say that the magnitude C also is incommensurable vnto the magnitude D. For forasmuch as A is incommensura∣ble vnto B, therfore (by the 7. of this booke) A hath not vnto B such proportion as number hath to number. But as A is to B, so is C to D. Wherefore C hath not vnto D such proportion as number hath to number. Wherfore (by the 8. of the tenth) C is incommensurable vnto D. If therefore there be foure magni∣tudes proportionall, and if the first be commensurable vnto the
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second, the third also shall be commensurable vnto the fourth. And if the first be incommen∣surable vnto the second, the third shall also be incommensurable vnto the fourth: which was required to be proued.
¶ A Corollary added by Montaureus.
If there be foure lines proportionall, and if the two first, or the two last be commensurable in power onely, the other two also shall be commensurable in power onely.* 1.3 This is proued by the 22. of the sixt, and by this tenth proposition. And this Corollary Euclide vseth in the 27. and 28. propositions of this booke, and in other propositions also.