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 15. Proposition. If two magnitudes commensurable be composed, the whole magnitude com∣posed also shall be commensurable to either of the two partes. And if the whole magnitude composed be commensurable to any one of the two partes, those two partes shall also be commensurable.

LEt these two commensurable magnitudes AB and BC, be composed or added toge∣ther. Then I say, that the whole magnitude AC is cōmensurable to either of these partes AB and BC: For forasmuch as AB and BC are commensurable, therfore (by the first definition of the tenth) some one magnitude measureth them both.* 1.1 Let there be a magnitude that measureth them, and

[illustration]

let the same be D. Now forasmuch as D measureth AB and BC, it shall also measure the whole magni∣tude composed AC, by this common sentence, what soeuer magnitude measureth two other magnitudes, shall also measure the magnitude composed of them. But the same D measureth AB and BC (by supposition). Wherefore D measureth AB, BC, and AC. Wherefore AC is commensu∣rable to either of these magnitudes AB and BC.

But now suppose that the whole composed magnitude AC be commensurable to any one of these two magnitudes AB or BC, let it be commensurable I say vnto AB.* 1.2 Then I say, that the two magnitudes AB and BC are commensurable. For forasmuch as AB and AC are commonsurable, some one magnitude measureth them (by the first definition of the tenth). Let some magnitude measure them, and let the same be D. Now forasmuch as D measureth AB and AC, it also measureth the residue BC, by this common sentence, what soeuer mea∣sureth the whole and the part taken away, shall also measure the residu••. But the same D mea∣sureth the magnitude AB (by supposition). Wherefore D measureth either of these magni∣tudes AB and BC. Wherefore the magnitudes AB and BC are commensurable. If ther∣fore two magnitudes commensurable be composed, the whole magnitude composed also shall be commensurable to either of the two partes. And if the whole magnitude composed be com∣mensurable

to any one of the two partes, those two partes shall also be commensurable: which was required to be demonstrated.

¶ A Corollary added by Montaureus.

* 1.3If an whole magnitude be commensurable to one of the two magnitudes which make the whole magnitude, it shall also be commensurable to the other of the two magnitudes. For if the whole mag∣nitude AC be commensurable vnto the magnitude BC, then by the 2. part of thys 15 Proposition: the magnitudes AB and BC are commensurable. Wherefore (by the first part of the same) the magnitude AC shall be commensurable to either of these magnitudes AB and BC. This Corollary Theon vseth in the demonstration of the 17. Proposition and also of other Propositions. Howbeit Euclide left it out, for that it seemed easie as in a maner do all other Corollaryes.

Notes

• * 1.1

  Demonstra∣tion of the first part.
• * 1.2

  Demonstrati∣on of the se∣cond pa••t which is the conuerse of the first.
• * 1.3

  A Corollary.