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 15, 2024.
Pages
¶The 46. Theoreme. The 64. Proposition. The square of a line contayning in power a rationall and a mediall super∣ficies applied to a rationall line, maketh the breadth or other side a fift bi∣nomiall line.
SVppose that the line AB be a line contayning in power a rationall and a mediall superficies, and let it be supposed to be deuided into his partes in the point C, so that let AC be the greater part, and take a rationall line DE. And (by the 44. of the first) vnto the line DE apply the parallelogramme DF equall to the square of the line AB, and making in breadth
[illustration]
the line DG. Then I say, that the line DG is a fift binomiall line.* 1.1 Let the selfe same cōstructi∣on be in this, that was in the former. And for∣asmuch as AB is a line contayning in power a rationall and a mediall superficies,* 1.2 and is deui∣ded into his partes in the poynt C, therefore the lines AC & CB are incōmensurable in power, hauing that which is made of the squares of thē added together mediall, and that which is con∣tayned vnder then rationall. Now forasmuch as that which is made of the squares of the lines AC and CB added together is mediall, there∣fore also the parallelogramme DL is mediall. Wherefore (by the 22. of the tenth) the line DM is rationall and incommensurable in length to the line DE. Againe forasmuch as [ 1] that which is contayned vnder the lines AC and CB twise, that is, the parallelogramme MF, is rationall, therefore by the 20. the line MG is rationall & cōmensurable in length [ 2]
descriptionPage [unnumbered]
to the line DE. Wherefore (by the 13. of the tenth) the line DM is incommensurable in [ 3] length to the line MG. Wherefore the lines DM and MG are rationall commensurable in power onely. Wherefore the whole line DG is a binomiall line. I say moreouer, that it is a fift binomiall. For, as in the former, so al∣so
[illustration]
in this may it be proued, that that which is contayned vnder the lines DK and KM is e∣quall to the square of MN the halfe of the lesse: and that the line DK is incommensurable in length to the line KM. Wherefore (by the 18. of the tenth) the line DM is in power more thē the line MG by the square of a line incommen∣surable in length to the line DM. And the lines DM and MG are rationall commensurable in power onely, and the lesse line, namely, MG is commensurable in length to the rationall line geuen DE. Wherefore the line DG is a fift binomiall line: which was required to be demonstrated.