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 1, 2024.

Pages

¶The 21. Theoreme. The 23. Proposition. Numbers prime the one to the other: are y^e least of any numbers, that haue one and the same proportion with them.

* 1.1SVppose that A and B be numbers prime the one to the other. Then I say that A and B are the least of any numbers that haue one and the same proportion with them. For if A and B be not the least of any numbers that haue one and the same propor∣tion with them, then are there some numbers lesse then A and

[illustration]

B, being in the selfe same proportion that A and B are. Let the same be C and D. Now forasmuch as the least numbers in any proportion measure any other numbers hauing the same pro∣portion equally, the greater the greater, and the lesse the lesse (by the 21. of the seuenth) that is, the antecedent the antece∣dent, and the consequent the consequent: therefore C so many times measureth A, as D measureth B. How many times C measureth A, so many vnities let there be in E.* 1.2 Wherefore D measureth B by those vnities which are in E. And forasmuch as C measureth A by those vnities which are in E, therefore E also measureth A by those vnities which are in C. And by the same reason E measureth B, by those vnities which are in D. Wherefore E measureth A and B being prime numbers the one to the other which (by the 13. definition of the seuenth) is impossible. Wherefore there are no other numbers lesse then A and B, which are in the selfe same proportion that A and B are. Wherefore A and B are the least numbers that haue one and the same proportion with them: which was required to be demonstrated.

Notes

* 1.1

This and the eleuen propo∣sitions follow∣ing, declare the p••ssions and properties of•• prime nūbers.

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* 1.2

Demonstra∣tion leading to an impossibi∣lity.

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Pages

¶The 21. Theoreme. The 23. Proposition. Numbers prime the one to the other: are ye least of any numbers, that haue one and the same proportion with them.

Notes

¶The 21. Theoreme. The 23. Proposition. Numbers prime the one to the other: are y^e least of any numbers, that haue one and the same proportion with them.