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
About this Item
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.]]
Rights/Permissions
To the extent possible under law, the Text Creation Partnership has waived all copyright and related or neighboring rights to this keyboarded and encoded edition of the work described above, according to the terms of the CC0 1.0 Public Domain Dedication (http://creativecommons.org/publicdomain/zero/1.0/). This waiver does not extend to any page images or other supplementary files associated with this work, which may be protected by copyright or other license restrictions. Please go to http://www.textcreationpartnership.org/ for more information.
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 23. Probleme. The 89. Proposition. To finde out a fift residuall lyne.
TAke a rational line and let the same be A,* 1.1 and vnto it let the line CG be commensurable in length. Wherefore the line CG is rationall. And take two numbers DF and FE, which let be such, that the number DE haue to neither of these numbers DF nor FE that proportion that a square number hath to a square number. And as the number FE is to the number DE, so let the square of the line CG be to the
[illustration]
square of the line BG.* 1.2 Wherefore the square of the line CG is commensurable to the square of the line BG•• Wherefore the square of the line BG is rationall, and the line BG is also rational. But the numbers DE and EF haue not that proportion the one to the other that a square number hath to a square nūber. Wherfore the lines BG and GC are ra∣tionall commensurable in power onely. Wher••fore the line BC is a residuall line. I say moreouer that it is a fift resi∣duall line. For forasmuch as the square of the line BG is greater then the square of the line GC, vnto the square of the line BG let the squares of the lines GC and H be equal. Now ther∣fore for that as the number DE is to the number EF, so is the square of the line BG to the square of the line GC, therfore by conuersion of proportion, at the number DE is to the nū∣ber DF, so is the square of the line BG to the square of the line H. But the numbers DE & DF haue not that proportion the one to the other that a square number hath to a square number. Wherefore the line BG is incommensurable in length to the line H. Wherefore the line BG is in power more then the line CG by the square of a line incommensurable in lēgth to the line BG, and the line CG which is ioyned to the residual line is commēsurable in lēgth to the rationall line A. Wherefore the line BC is a fift residuall line. Wherefore there is found out a fift residuall line: which was required to be done.