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 28. Theoreme. The 30. Proposition. If two numbers be prime the one to the other: then both of them added to∣gether, shall be prime to either of them. And if both of them added toge∣ther be prime to any one of them, then also those numbers geuen at the be∣ginning, are prime the one to the other.

SVppose that these two numbers AB and BC being prime numbers be added to∣gether. Then I say, that both these added together, namely, the number ABC, is prime to either of these AB, and BC. For if CA and AB be not prime the one to the other, some number then shall measure them.* 1.1 Let some number mea∣sure them, and let the same be D. Now then forasmuch as D

[illustration]

measureth the whole CA and the part taken away AB, it mea∣sureth also the residue CB (by the 4. common sentence). And it measureth BA. Wherfore D measureth these numbers AB and BC, being prime the one to the other: which is impossible (by the 13. definition of the se∣uenth). Wherefore no number measureth these numbers CA and AB. Wherefore CA and AB are prime the one to the other. And by the same reason also may it be proued, that CA and BC are prime the one to the other. Wherefore the number AC is to either of these num∣bers AB and BC, prime.

But now suppose that the numbers CA and AB be prime the one to the other. Then I say, that the numbers AB and BC are also prime the one to the other, For if, AB & BC be not prime the one to the other: some one number measureth these numbers AB and BC••

Let some one number measure them, and let the same be D. And forasmuch as D measureth either of these numbers AB and BC, it shall also measure the whole CA (by the 6. common sentence).* 1.2 And it also measureth AB. Wherefore D measureth these numbers CA and AB being prime the one to the other: which is impossible (by the 13. definition of the seuenth). Wherefore no number measureth these numbers AB and BC. Wherefore AB and BC are prime the one to the other, which was required to be proued.

Notes

• * 1.1

  Demonstra∣tion of the first part lea∣ding to an absurditie.
• * 1.2

  Demonstra∣tion of the second part which is the con••c••se of the first, lean••ng also to an ab∣surditi••.