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 2. Theoreme. The 3. Proposition. If there be numbers in continuall proportion how many soeuer, and if they be the lest of all numbers that haue one and the same proportion with thē: their extremes shall be prime the one to the other.

SVppose that the numbers in continuall proportion being the least of all numbers that haue the same proportion with them be A, B, C, D.* 1.1 Then I say that their extremes A and D are prime the one to the other. Take (by the 2. of the eight, or by the 35. of the seuen••h) the two least numbers that are in the same proportion that A, B, C, D, are, and let the same be the numbers E, F. And after that take thre numbers

[illustration]

G, H, K, and so alwayes forward on more (by the former proposition) vntill the multitude ••aken be equall to the multitude of the numbers geuen A, B, C, D. And let those numbers be L, M, N, O. Wherfore (by the 29. of the seuenth) their extremes L, O,* 1.2 are prime the one to the other. For forasmuch as E and F are prime the one to the other, and eche of them mul∣tiplieng

himselfe produced G and K, & likewise ech of these, G & K multiplieng himself pro∣duced L & O, therfore (by the 29 of the seuenth) G & K are prime the 〈◊〉〈◊〉 to th•• other, & so likewise are L and O prime the one to the other. And forasmuch as A, B, C, D, are the least of

[illustration]

all numbers that haue the same proportion with them, and likewise L, M, N, O, are the least of all numbers that are in the same proportion that A, B, C, D, are, and the multitude of these numbers A, B, C, D, is equall to the multitude of these L, M, N, O: therfore euery one of these A, B, C, D, is equall vnto euery one of these L, M, N, O. Wherefore A is equall vnto L, and D is equall vnto O. And forasmuch as L and O are prime the one to the other, and L is e∣quall vnto A, and O is equall vnto D: therfore A and D are prime the one to the other: which was required to be proued.

Notes

• * 1.1

  This proposi∣tion is the ••••∣uerse of the first.
• * 1.2

  Demonstra∣tion••