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. Probleme. Th 3. Proposition. Thre numbers being geuē, not prime the one to the other: to finde out their greatest common measure.

SVppose the three numbers geuen not prime the one to the other

[illustration]

to be A, B, C. Now it is required vnto the sayd numbers A, B, C to finde out the greatest common measure. Take the greatest common measure of the two numbers A and B (by the 2 of the seuenth) which let be D: which number D either measureth the num∣ber C or not.

First let D measure C. And it also measureth the numbers A and B,* 1.1 wherfore D measureth the numbers A, B, C. Wherefore D is a common measure vnto the numbers A, B, C. Then I say also, that it is the greatest common measure vnto them. For if D be not the greatest common measure vnto the numbers A, B, C,* 1.2 let some number greater then D measure the numbers A, B, C. And let the same number be E. Now forasmuch as E measu∣reth the numbers A, B, C, it measureth also the numbers A, B. Wherefore it measureth also

the greatest common measure of the numbers A, B (by the Co∣rollary

[illustration]

of the second of the seuenth). B••t the greatest common measure of the numbers A, B, is the number D (by con∣struction). Wherefore the number E measureth the number D, namely, the greater the lesse: which is impossible. Where∣••ore no number greater thē D measureth the nūbers A, B, C. Wherefore D is the greatest common measure to the numbers A, B, C.

* 1.3But now suppose that D do not measure C. First I say that D & C are not prime num∣bers the one to the other. For forasmuch as the numbers A, B, C, are not prime the one to the other (by supposition) some one number will measure them: but that number that mea∣sureth the numbers A, B, C, shall also measure the numbers A, B, and shall likewise measure the greatest cōmon measure of AB, namely, D (by the Corollary of the second of the seuēth). And the sayd number measureth also C. Wherfore some one number measureth the num∣bers D and C. Wherefore D and C are not prime the one to the other.

Now then let there be taken (by the 2. of the seuenth) the greatest common measure vnto the numbers D and C, which let be the number E. And forasmuch as E measureth D, and D measureth the numbers A, B, therefore E also measureth the numbers A, B (by the sixt common sentēce): and it measureth also C. Wherfore E measureth the nūbers A, B, C. Wherefore E is a common measure vnto the numbers A, B, C. I say also that it is the grea∣test. For if E be not the greatest common measure vnto the numbers A, B, C, let there be some number greater then E, which measureth the nūbers A, B, C. And let the same num∣ber be ••. And forasmuch as F measureth the numbers A, B, C: it measureth also the num∣bers A, B. Wherefore also it measureth the greatest common measure of the numbers A, B (by the Corollary of the 2. of the seuenth). But the greatest common measure of the numbers A, B, is D. Wherefore E measureth D. And it measureth also the number C. Wherefore F measureth the numbers D, C. Wherefore also (by the same Corollary) it measureth the greatest common measure of the numbers D, C. But the greatest common measure of the numbers D, C, is E. Wherfore F measureth E, namely, the greater number the lesse: which is impossible. Wherefore no number greater then E shall measure the nūbers A, B, C. Wher∣fore E is the greatest common measure to the numbers A, B, C: which was required to be done.

¶Corollary.

Wherefore it is manifest, that if a number measure three numbers, it shall also measure their greatest common measure. And in like sort more num∣bers being geuē not prime the one to the other may be found out their grea∣test common measure, and the Corollary will followe.

Notes

• * 1.1

  Two cases in this Proposi∣tion.
• * 1.2

  The first case.
• * 1.3

  The second case.