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

¶The 4. Theoreme. The 6. Proposition. If there be numbers in continuall proportion how many soeuer, and if the first measure not the second, neither shall any one of the other measure any one of the other.

SVppose that there be numbers how many soeuer in continuall proportion, name∣ly, f••ue, A, B, C, D, E. And suppose that A measure not B. Then I say, that nei∣ther shall any other of the numbers A, B, C, D, E, measure any one of the other. That A, B, C, D, E, do not in continuall order measure one the other, it is mani∣fest: for A measureth not B. Now I say, that neither shall any other of them measure any o∣ther of them. I say that A shall not measure C. For how many in multitude A, B, C, are, take so many of the lest numbers that haue one and the same proportion with A, B, C, (by the 35. of the seuenth) and let the same be F, G, H.* 1.1 And forasmuch as F, G, H are in the selfe same

[illustration]

proportion that A, B, C, are: and the multitude of these numbers A, B, C, is equall to the mul∣titude of those numbers F, G, H, therefore of equalitie (by the 14. of the seuenth) as A is to C, so is F to H. And for that as A is to B, so is F to G, but A measureth not B, therefore nei∣ther doth F measure G. Wherefore F is not vnitie. For if F were vnitie, it should measure any number. But F and H are prime the one to the other (by the 3. of the eight). Wherefore F measureth not H: & as F is to H, so is A to C, wherefore neither doth A measure C. In like sort may we proue that neither shall any other of the numbers A, B, C, D, E, measure any other of the numbers A, B, C, D, E: which was required to be demonstrated.

Notes

• * 1.1

  Demonstration.