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 77. Theoreme. The 101. Proposition. The square of a lyne making with a rationall superficies the whole super∣ficies mediall applied vnto a rational line, maketh the breadth or other side a fift residuall lyne.
SVppose that AB be a line making with a rationall superficies the whole superficies mediall, and let CD be a rationall line. And vnto the line CD apply the parallelo∣gramme CE equall to the square of the line AB, and making in breadth the line CF. Then I say that the line CF is a fift residuall line. For vnto the line AB let the line con∣ueniently ioyned be supposed to ••e BG. Wherfore the lines AG and GB are incommensura∣ble in power, hauing that which is made of their squares added together mediall, and that which is contained vnder them rationall. Let the rest of the construction be in this as it was in the former propositions. Wherfore the whole parallelogramme CL is mediall. Wherefore the line CM is rationall and incommensurable in length to the line CD. And either of the
descriptionPage [unnumbered]
parallelogramme FX & NL is rationall•• Wher∣for••
[illustration]
the whole parallelogramme FL is also ratio∣nall. Wherfore also the line FM is rationall and commensurable in lēgth to the line CD. And for∣asmuch as the parallelogramme CL is mediall, and the parallelogramme FL is rationall, there∣fore CL and FL are incommensurable the one to the other, and the line CM is incommensurable in length to the line FM, and they are both ratio∣nall. Wherfore the lines CM and FM are ratio∣nall commensurable in power onely. Wherfore the lyne GF is a residuall line. I say moreouer that it is a ••i••t residual line.* 1.1 For we may in like sort proue, that the parallelograme contained vnder the lines CK and KM, is equall to the square of the line NM, that is, to the fourth part of the square of the lyne FM. And forasmuch as the square of the line AG, that is, the paral∣lelogramme CH is incommensurable to the square of the line BG, that is to the parallelo∣gramme KL, therfore the line CK is incommensurable in length to the line KM. Wherfore (by the 18. of the tenth) the line CM is in power more then the line FM, by the square of a line incommensurable in length to the line CM. And the line conueniently ioyned, namely, the line FM is commensurable in length to the rationall line CD. Wherfore the line CF is a ••i••t residuall line. Wherfore the line CF is a fift residuall line. Wherfore the square of a line making with a rationall superficies the whole superficies medial, applied vnto a rational line maketh the breadth or other side a fift residuall lyne: which was required to be demonstrated.