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

* 1.11 Magnitudes commensurable are suchwhich one and the selfe same mea∣sure doth measure.

First he sheweth what magnitudes are commensurable one to an other. To the better and more cleare vnderstanding of this definition, note that that measure whereby any magnitude is measured, is lesse then the magnitude which it measureth, or at least e∣quall vnto it. For the greater can by no meanes measure the lesse. Farther it behoueth, that that measure if it be equall to that which is measured, taken once make the mag∣nitude which is measured: if it be lesse, then oftentimes taken and repeted, it must pre∣cisely render and make the magnitude which it measureth. Which thing in numbers is easely sene, for that (as was before said) all numbers are commensurable one to an o∣ther. And although Euclide in this definition comprehendeth purposedly, onely mag∣nitudes which are continuall quantities, as are lines, superficieces, and bodies, yet vn∣doubtedly the explication of this and such like places, is aptly to be sought of numbers as well rationall as irrationall. For that all quantities commensurable haue that pro∣portion the one to the other, which number hath to numbers. In numbers therfore, 9 and 12 are commensurable, because there is one common measure which measureth them both, namely, the number 3. First it measureth 12, for it is lesse then 12. and be∣ing taken certaine times, namely, 4 times, it maketh exactly 12: 3 times 4 is 12, it also measureth 9, for it is lesse then 9. and also taken certaine times, namely, 3 times, it ma∣keth precisely 9: 3 times 3 is 9. Likewise is it in magnitudes, if one magnitude measure two other magnitudes, those two magnitudes so measured, are said to be commensura∣ble. As for example, if the line C being dou∣bled,

[illustration]

do make the line B, and the same lyne C tripled, do make the line A, then are the two lines A and B, lines or magnitudes com¦mensurable. For that one measure, namely, the line C measureth thē both. First, the line C is lesse thē the line A, and alsolesse thē the line B, also the line C taken or repeted certaine times, namely, 3 times, maketh precisely the line A, and the same line C taken also certain times, namely, two times, maketh pre∣cisely the line B. So that the line C is a common measure to them both, and doth mea∣sure them both. And therfore are the two lines A and B lines commensurable. And so i∣magine ye of magnitudes of other kyndes, as of superficiall figures, and also of bodies.

Notes

• * 1.1

  The f••rst definition.