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.]]
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 6, 2024.

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The 18. Theoreme. The 18. Proposition. If magnitudes deuided be proportionall: then also composed they shall be proportionall.

SVppose that the magnitudes deuided being proportionall, be AE, EB, CF, & FD, so that as AE is to EB, so let CF be to FD.* 1.1 Then I say, that composed also they shall be proportionall, that is, as AB is to BE, so is CD to DF. For if AB be not vnto BE, as CD is to FD,

[illustration]

then shall AB be vnto BE,* 1.2 as CD is either vnto a magnitude lesse then FD, on vnto a magnitude greater. Let it first be vnto a lesse, namely, to DG.* 1.3 And forasmuch as, as AB is to BE, so is CD to DG: the composed magnitudes therefore are proportio∣nall, wherefore deuided also they shall be proportionall (by y^e 17. of the first). Wherefore as AE is to EB, so is CG to GD. But by supposition as AE is to EB, so is CF to FD. Wherefore (by the 11. of the fift) as CG is to GD, so is CF to FD. Now then there are foure magnitudes, CG, GD, CF, and FD: of which the first CG is greater then the third CF. Wherefore (by the 14. of the fift) the second GD is greater then the fourth FD. But it is also put to be lesse then it: which is impossible. Wherfore it can not be that as AB is to BE, so is CD to a magnitude lesse then FD. In like sort may we proue, that it can not be so to a magni∣tude greater then FD. For by the same order of demonstration, it would follow that FD is greater then the sayd greater magnitude: which is impossible. Wher∣fore it must be to the selfe same. If therefore magnitudes deuided be proportio∣nall, then also composed they shall be proportionall: which was required to be proued.

Notes

* 1.1

Demonstrati∣on of propor∣tion by com∣position.

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* 1.2

This proposi∣tion is the con¦uerse of the former.

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* 1.3

Demonstratiō ••e••aing to an ••mpossibility.

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