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 12. Theoreme. The 12. Proposition. If from vnitie be numbers in continuall proportion how many soeuer, how many prime numbers measure the least•• so many also shal measure•• the num∣ber which followeth next after vnitie.

SVppose that from vnitie be these numbers in continuall proportion A, B, C, D. Th•• I say that how many prime nūbers measure D, so many also do measure A. Suppose that some prime number namely, E, do measure D. Thē I say that E also measureth A, which is next vnto vnitie. For if E do not measure A, and E is a prime number, but eue∣ry number is to euery number which it measureth not a prime number (by the 31. of the se∣uenth). Wherefore A and E are prime numbers the one to the other. And forasmuch as E measureth D, let it measure D by the number F.* 1.1 Wherefore E multiplieng F produceth D. Againe forasmuch as A measureth D by those vnities which are in C, therefore A multipli∣eng C produceth D. But E also multiplieng F produced D, wherfore that which is produced of the numbers A, C is equall to that which is produced of the numbers E, F. Wherfore as A is to E, so is F to C. But A, E, are prime numbers, yea they are prime and the least. But the lest numbers measure the numbers that haue one and the same proportion with them equally by the 21. of the seuenth, namely, the antecedent the antecedent, and the consequent the conse∣quent. Wherfore E measureth C. Let it measure it by G. Wherefore E multiplieng G produ∣ceth C. But A also multiplieng B produceth C. Wherfore that which is produced of the num∣bers

[illustration]

A, B, is equall to that which is produced of the numbers E, G. Wherfore as A is to E, so is G to B. But A, E are prime numbers, yea they are prime and the least. But the least num∣bers (by the 21. of the seuenth) measure the numbers that haue one and the same proportion with thē equally, namely, the antece••••s the antecedē••, & the cōsequēt the conseqēt. Wherfore E measureth B. Let it measure it by H. Wherefore B multiplieng H produceth B. But A also multiplieng himselfe produceth B, wherfore that which is produced of the numbers E, H, is equall to that which is produced of the number A. Wherfore as E is to A, so is A to H. But AE are prime nūbers, yea they are prime & the least, but the least numbers (by the 21. of the se∣uenth) measure the numbers that haue one and the same proportion with thē equally, name∣ly, the antecedēt the antecedent, and the cōsequent the consequent. Wherfore E measureth A and it also doth not measure it by ••••pp••sition, which is impossible. Wherfore A and E are not prime the one to the other, wherfore they are composed. But all composed numbers are measu∣red of some prime number, wherfore A and E are measured by some prime number. And for¦asmuch as E is supposed to be a prime number. But a prime number is not (by the definition) measured by any other number but of himselfe. Wherfore E measureth A and E, wherfore B measureth A, and it also measureth D. Wherfore E measureth these numbers A and D. And in like sort may we proue that how many prime numbers measure D, so many also shall mea∣sure A: which was required to be proued.

An other more briefe demonstration after Flussates.

* 1.2Suppose that from vnitie be nūbers in cō••••nuall proportion how many so euer, namely, A, B, C, D. And let some prime nūber, namely, •• measure the last nūber which is D. Thē I say that th•• same E mea∣sureth A which is the next number vnto vnitie. For if E doo not measure A, then are they prime the one to the other by the 31. of the seuenth. And forasmuch as A, B, C, D, are proportionall from vnitie,

therefore A multiplying himselfe produceth B. Wherfore B and E ar•• prim••.

[illustration]

numbers (by the 27. of the seuenth). And forasmuch as A multiplying B produceth C, therefore C is to E also a prime number by the 26. of the se∣uenth. And likewise infinitely A multiplying C produceth D: wherefore D and E are prime numbers the one to the other (b•• 〈◊〉〈◊〉 same ••6. of th•• s••∣uenth•• Wherefore E measureth not D as it was supposed, which is absurd, wherefore the prime number E measureth A, whiche is nexte vnto vnities which was required to be proued.

Notes

• * 1.1

  Demonstra∣tion leading to an absurditie.
• * 1.2

  An other de∣monstratiō a••∣ter Flussates.