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 4. Proposition. Euery lesse number is of euery greater number, either a part, or partes.

* 1.1SVppose there be two numbers A and BC. Of which let BC be the lesse. Then I say, that BC is either a part or partes of A. For the numbers A and BC are ei∣ther prime the one to the other, or not. First let A and BC be prime the one to the other.* 1.2 And deuide the number BC into those vnities which are in it. Now euery one of

the vnities which 〈◊〉〈◊〉 in BC i•• s••me certaine part of A•• Wherefore BC are partes of A.

But now suppose that the number•• A and BC 〈◊〉〈◊〉 not prime the one to the other. Then BC either measureth A or not measureth it.* 1.3 If BC measure:

[illustration]

A, then is BC a part of A. But if not•• take (by the 2. of the seuenth) the greatest common ••eas••r•• of A and BC, and let the same be D. And let BC be deuided into as man•• pa••te•• as it hath equall vnto D, that is into BE, EF•• and FC. And forasmuch as D measureth A, therefore D i•• a part of A. But D is equall vnto euery one of these partes BE, EF, and FC. Wherfore also euery one of these partes BE, EF, and FC, is a part of A. Wher∣fore the number BC is partes of A. Wherefore euery lesse number is of euery greater nūber, either a part or partes: which was required to 〈◊〉〈◊〉 proued.

Notes

• * 1.1

  Two cases in this Proposi∣tion.
• * 1.2

  The first case.
• * 1.3

  The second case.