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 14, 2024.

Pages

¶ The 9. Theoreme. The 11. Proposition. Betwene two square numbers there is one meane proportional number. And

Page 207

a square number to a square, is in double proportion of that which the side of the one is to the side of the other.

SVppose that there be two square numbers A and B, and let the side of A be C, & let the side of B be D. Then I say that betwene these square numbers A and B, there is one meane proportionall number, and also that A is vnto B in double proportion of that which C is to D. Let C multiplieng D produce E.* 1.1 And foras∣much as A is a square nūber, & the side thereof is C,

[illustration]

therfore C multiplieng himselfe produced A. And by the same reason D multiplieng himselfe produced B. Now forasmuch as C multiplieng C produced A, and multiplieng D produced E, therfore (by the 17. of the seuenth) as C is to D, so is A to E. Againe forasmuch as C multiplieng D produced E, and D multiplieng himselfe produced B, therefore these two numbers C and D multiplieng one number, namely, D, produce E and B. Wherfore (by the 18. of the seuenth) as C is to D, so is E to B. But as C is to D, so is A to E. Wherefore as A is to E, so is E to B. Wherefore betwene these square numbers A and B, there is one meane proportionall number, namely, E.* 1.2 Now also I say that A is vnto B in double proportion of that which C is to D. For forasmuch as there are three numbers in continuall proportion, A, E, B, therfore (by the 10. definition of the fift) A is vnto B in double proportiō of that which A is to E. But as A is to E, so is C to D. Wherefore A is vnto B in double proportion of that which the side C is vnto the side D: which was required to be proued.

Notes

* 1.1

The first part of this propo∣sition demon∣strated.

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* 1.2

The second part demon∣strated.

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