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 14, 2024.
Pages
¶ The 64. Theoreme. The 82. Proposition. Vnto a lesse line can be ioyned onely one right line incommensurable in power to the whole lyne, and making together with the whole lyne that which is made of their squares added together rationall, and that which is contayned vnder them mediall.
SVppose that AB be a lesse line, and to AB ioyne the line BC, so that let BC be such a line as is required in the Theoreme. Wherfore the lines AC and CB are incōmensurable in power, hauing that which is made of the squares of them ad∣ded together rationall, and that which is contained vnder them mediall. Then I say that vnto AB cannot be ioyned any other
[illustration]
such right line.* 1.1 For if it be possible, l••t the lyne BD be such a line. Wherfore the lines AD & DB are incommensurable in power, hauing that which is made of the squares of them added together, rationall, and that which is con∣tained vnder them mediall. And for that how much the squares of the lines AD and DB excede the squares of the lines AC and CB, so much that which is contained vnder the lines AD and DB twise, excedeth that which is contained vnder the lines AC and CB twise (by those things which were spoken in the 79. proposition) But that which is made of the squares of the lines AD and DB added together excedeth that which is made of the squares of the lines AC and CB added together by a rationall superficies, for they are either of them ratio∣nall by supposition. Wherfore that which is contained vnder the lines AD and DB twise, excedeth that which is contained vnder the lines AC and CB twise by a rationall superfi∣cies: which (by the 26. of the tenth) is impossible, for either of them is mediall by supposition. Wherfore vnto a lesse line can be ioyned onely one right line incommensurable in power to the whole line, and making together with the whole line that which is made of their squares added together rationall, and that which is contained vnder them mediall: which was re∣quired to be demonstrated.