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 7, 2024.

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¶ The 20. Theoreme. The 20. Proposition. Prime numbers being geuen how many soeuer, there may be geuen more prime numbers.

SVppose that the prime numbers geuen be A, B, C.* 1.1 Then I say, that there are yet more prime numbers besides A, B, C. Take (by the 38. of the seuenth) the lest number whom these numbers A, B, C do measure, and let the same be DE. And vnto DE adde vnitie DF. Now EF is either a prime number or not. First let it be a prime number,* 1.2 then are there found

[illustration]

these prime numbers A, B, C, and EF more in multi∣tude then the prime numbers ••irst geuen A, B, C.

But now suppose that EF be not prime.* 1.3 Wherefore some prime number measureth it (by the 24. of the se∣uenth). Let a prime number measure it, namely, G. Then I say, that G is none of these numbers A, B, C. For if G be one and the same with any of these A, B, C. But A, B, C, measure the nūber DE: wher∣fore G also measureth DE: and it also measureth the whole EF. Wherefore G being a num∣ber shall measure the residue DF being vnitie•• which is impossible. Wherefore G is not one and the same with any of these prime numbers A, B, C: and it is also supposed to be a prime number. Wherefore there are ••ound these prime numbers A, B, C, G, being more in multitude then the prime numbers geuen A, B, C: which was required to be demonstrated.

A Corollary.

By thys Proposition it is manifest, that the multitude of prime numbers is infinite.

Notes

* 1.1

Two cases in this Proposi∣tion.

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* 1.2

The first case.

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* 1.3

The second case.

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