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.]]
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Subject terms
Geometry -- Early works to 1800.
Link to this Item
http://name.umdl.umich.edu/A00429.0001.001
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"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 18. Theoreme. The 18. Proposition. Two numbers being geuen, to searche out if it be possible a third number in proportion with them.
SVppose that the two numbers geuen be A and B. It is required to searche out if it be possible a third number proportionall with them. Now A, B are either prime the one to the other or not prime.* 1.1 If they be prime, then (by the 16. of the ninth) it is mani∣nifest that it is impossible to finde out a third number proportional with them. But now sup∣pose that AB be not prime the one to the other. And let B multiplieng himselfe produce C. Now A either measureth C, or measureth it not. First,* 1.2 let it measure it and that by D. Wher∣fore A multiplieng D produceth
[illustration]
C.* 1.3 But B also multiplieng himself produced C. Wherfore that which is produced of A into D, is equall to that which is produced of B in∣to himselfe. Wherefore (by the se∣cond part of the 19. of the seuēth) as A is to B, so is B to D. Wherfore vnto these numbers A, B is found out a third number in proportion, namely, D.
But now suppose that A do not measure C••* 1.4 Then I say that it is impossible to ••inde out a third nū∣ber in proportion with these num∣bers A, B. For if it be possible, let there be found out such a number, and let the same be D.
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Wherfore that which is produced of A into D, is equall to that which is produced of B into himselfe, but that which is produced of B into himselfe is C. Wherfore that which is produced of A into D is equall vnto C. Wherfore A multiplieng D produced C. Wherefore A measu∣reth G by D. But it is supposed also not to measure it, which is impossible. Wherefore it is not possible to finde out a third number in proportion with A & B, whensoeuer A measureth not C: which was required to be proued.