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

A proposition added by Campane.

If there be numbers how many soeuer in continuall proportion being the least in that proportion: a number measuring one of them, shall be a number not prime to one of the two least numbers in that proportion.

Suppose that there be numbers in continuall proportion how many soe••••r namely A, B, C, D, E which let be the least that haue the same proportion with them:* 1.1 and let the two least numbers in that proportion be F and G. And let some number as H measure some one of the numbers A, B, C, D, E, namely, C. Then I say that H is a number not prime either to F or G. Take (by the 2. of the eight) the

three least numbers in

[illustration]

the proportion of A to B: which let be P, Q, R. And afterward fower (by the same) which let be K, L, M, N: & so for∣ward till you come to the multitude of the numbers geuen A, B, C, D, E. Now it is manifest (by the demonstration of the second of the eight) that F multiplyed by P, Q, and R produceth K, L, M: and that F multiplyed by K, L, M, N produc••th A, B, C, D. And forasmuch as H measureth C: therfore H is either to F or to M not prime (by the corollary of the 32. of the seuenth added by Cāpane) If it be not prime vnto F: thē i•• that manifest which was required to be proued. But if H be not prime vnto M. Thē shal it not be prime either to F or to R (by the same corollary). If agayne it be not prime vnto F, then is that proued which was required. But if it be not prime vnto R, than (by the same corollary) shall it be a number not prime vnto G (which produceth R by the 2. of the eight)•• but G is one of the two least numbers F or G which are in the proportion of the numbers geuen at the beginning A, B, C, D, E. If therefore there be num∣bers how many soeuer. &c. which was required to be proued.

Notes

• * 1.1

  A propositio•• added by Campane.