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 15, 2024.

Pages

¶The 11. Theoreme. The 13. Proposition. If there be foure numbers proportionall: then alternately also they shall be proportionall.

SVppose that there be foure numbers proportional,* 1.1 A, B, C, D, so that as A is to B, so let C be to D. Then I say that alternately also they shalbe proportional, that is. as A is to C, so is B to D. For forasmuch as (by supposition) as A is to B, so is C to D, therfore (by the 21. definition of this booke) what part or partes

[illustration]

A is of B the selfe same part or partes is C of D. Therfore alternate∣ly what part or partes A is of C the selfe same part or partes is B of D (by the 9. of the seuenth) & also (by the 10. of the same): wherfore as A is to C, so is B to D (by the 21. definition of this booke): which was required to be proued.

Here is to be noted, that although in the foresayd example and demonstration the number A be supposed to be lesse then the number B, and so the number C is lesse then the number D:* 1.2 yet will the same serue also though A be supposed to be greater then B, wherby also C shall be greater then D, as in th••s example here put. For for that (by supposition) as A is to B, so is C to D, and A is supposed to be greater then B, and C greater then D: therefore (by the 21. definition of this

[illustration]

Booke) how multiplex A is to B, so multiplex is C to D, and therefore what part or partes B is of A, the selfe same part or partes is D of C. Wherefore alternately what part or partes B is of D, the selfe same part or partes is A of C, and therefore by the same definition, B is to D, as A is to C. And so must you vnderstand of the former Proposition next going before.

Notes

* 1.1

This in dis∣crete quanti••y answereth to the sixtenth proposition of the fifth booke in continuall quantitie.

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* 1.2

Note.

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