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
An other demonstration.
Suppose that A be a lesse line, and vnto A let the line B be commensurable whether in length and power, or in power onely. Then I say that B is a lesse line. Take a rationall line CD.* 1.1 And vnto the line CD apply (by the 44 of the first) the parallelogramme CE equall to the square of the line A, and making in bredth the line CF. Wherefore (by the 100. propo∣sition)
descriptionPage 302
the line CF is a fourth residuall line. Vnto the line FE apply
[illustration]
(by the same) the parallelogramme EH equall to the square of the line B, and making in breadth the line FH. Now forasmuch as the line A is commensurable to the line B,* 1.2 therefore also the square of the line A is cōmensurable to the square of the line B. But vnto the square of the line A is equall the parallelogramme CE, & vnto the square of the line B is equal the parallelogramme EH. Wherfore the parallelogramme CE is commensurable to the parallelogramme EH. But as the parallelogramme CE is to the parallelogramme EH, so is the line CF to the line FH. Wherfore the line CF is commen∣surable in length to the line FH. But the line CF is a fourth resi∣duall line. Wherfore the line FH is also a fourth residuall line (by the 103. of the tenth): and the line FE is rationall. But if a superfi∣cies be contained vnder a rationall line, and a fourth residuall lyne, the line that containeth in power that superficies is (by the 94. of the tenth) a lesse lyne. But the line B containeth in power the superficies EH. Wherfore the line B is a lesse line: which was required to be pro∣ued.