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 1. Probleme. The 2. Proposition. Two numbers being geuen not prime the one to the other, to finde out their greatest common measure.

SVppose the two numbers geuen not prime the one to the other to be AB and CD. It is required to finde out the greatest common measure of the said numbers AB and CD.* 1.1 Now the numbe•• CD either measureth

[illustration]

the number AB or not. If CD measure AB it also mea∣sureth it selfe.* 1.2 Wherefore CD is a common measure to the numbers CD and AB. And it is manifest also that it is the greatest common measure•• for there is no number greater then CD that will measure CD.

* 1.3But if CD do not measure AB, then if of the numbers AB

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and CD, the lesse be continually taken away from the greater, there will before you come to vnitie, be left a number, which will measure the number going before (by the 1. of the seuenth). For if there should not, then should the numbers AB and CD be prime the one to the other, which is contrary to the supposition. Let the sayd number left by the continuall substraction of the lesse number out of the greater be FC. So that le•• the number CD measuring AB, and subtrahed out of it as often as you can le••e a lesse number then it selfe, namely AE. And let AE measuring CD, and subtrahed out of it

as often as you can leaue a lesse-then it selfe namely, CF. And suppos•• th••t CF do so mea∣sure AE that there remayne nothing. Then I say that CF is a common measure to the numbers AB and CD. For forasmuch as CF measureth AE, and AE measureth DF,* 1.4 therefore CF also measureth DF (by the fifth common sentence of the seuenth) and it likewise measureth it selfe, wherfore it also measureth the whole CD (by the sixth common sentence of the seuenth••)•• but CD measureth BE, wherefore CF also measureth BE (by the ••ifte common sentence of the seuenth). And it measureth also EA:* 1.5 wherefore it also measureth the whole BA (by the sixth common sentence of the seuenth): and it also measu∣reth CD as we haue before proued: wherefor•• the number CF measureth the numbers AB & CD wherfore the number CF is a commō measure to the numbers AB & CD.

I say also that it is the greatest common measure. For if CF be not the greatest commō measure to AB and CD,* 1.6 let there be a number greater then

[illustration]

CF, which measureth AB and CD: which let be G. And forasmuch as G measureth CD, and CD measureth BE, therefore G also measureth BE (by the ••••ft common sentence of the seuenth). And it measureth the whole AB, where∣fore also it measureth the residue, namely, AE (by the 4. common sentence of the se∣••enth). But AE measureth DF, wherefore G also measureth DF (by the foresayd 5. com∣mon sentence of the seuenth). And it measureth the whole CD. Wherefore it also measu∣reth the residue FC: namely, the greater number the lesse: which is impossible. No number therefore greater then CF shall measure those numbers AB and CD: wherefore CF is the greatest common measure to AB and CD: which was required to be done.

Corrolary.

Hereby it is manifest, that if a number measure two numbers it shall also measure their greatest common measure. For if it measure the whole & the part taken away, it shall alwayes measure the residue also, which residue is at the length, the greatest common measure of the two numbers geuen.

Notes

• * 1.1

  Two cases in this probleme.
• * 1.2

  The first case.
• * 1.3

  The second case.
• * 1.4

  Demonstratiō of the second case.
• * 1.5

  That CF is a common mea∣sure to the numbers AB and CD.
• * 1.6

  That CF is the greatest common mea∣sure to AB and CD.