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

* 1.16 Lines which are commensurable to this line, whether in length and power, or in power onely, are also called rationall.

This definition needeth no declaration at all, but is easily perceiued, if the first de∣finition be remembred, which ••heweth what magnitudes are commensurable, and the third, which ••heweth what lines are commensurable in power. Here not••, how aptly & naturally, Euclide in this place vseth these wordes commensurable either in length and power, or in power onely. Because that all lines which are commensurable in length, are also commensurable in power•• when he speaketh of lines commensurable in lēgth, he euer addeth and in power, but when he speaketh of lines commensurable in power, he addeth this worde Onely, and addeth not this worde in length, as he in the other added this worde in power. For not all lines which are commensurable in power, are straight way commensurable also in length. Of this definition, take this example. Let the first line rationall of purpose,

[illustration]

which is supposed and laide forth, whose partes are certaine & known, and may be expressed, named, and nūbred be AB, the quadrate wher∣of let be ABCD: then suppose a∣gaine an other lyne, namely, the line EF, which let be commensurable both in length and in power to the first rationall line, that is, (as before was taught) let one line measure the length of eche line, and also l••t one super••icies measure the two squares of the said two lines, as here in the example is supposed and also ap∣peareth to the eie, then is the line E F also a rationall line.

Moreouer if the lyne EF be commensurable in power onely to the rationall line AB first set and supposed, so that no one line do measure the two lines AB and EF: As in example y•• see to be (for that the line EF, is made equall to the line AD, which is the diameter of the square ABCD, of

[illustration]

which square the line AB is a side, it is certayne that the ••ide of a square is incōmēsurable in lēgth to the diameter of the same square: if there be yet founde any one su∣perficies, which measureth the two squares ABCD, and EFGH (as here doth the triangle ABD, or the triangle ACD noted in the square ABCD, or any of the foure triangles noted in the square EFGH, as appeareth somwhat more manifestly in the second example, in the declaration of the last definition going before) the line EF is also a rational line. Note that these lines which here are called rationall lines, are not rational lines of pur∣pose, or by supposition, as was the first rationall line, but are rationall onely by reason of relation and comparison which they haue vnto it, because they are commensurable vn∣to it either in length and power, or in power onely. Farther here is to be noted, that these wordes length, and power, and power onely, are ioyned onely with these worde•• commensurable or incommensurable, and are neuer ioyned with these woordes ratio∣nall or irrationall. So that no lines can be called rational in length, or in power, nor like wise can they be called irrationall in length, or in power. Wherin vndoubtedly Campa∣nus was deceiued,* 1.2 who vsing those wordes & speaches indifferently, caused & brought in great obscuritie to the propositions and demonstrations of this boke, which he shall

easily see which marketh with diligence the demonstrations of Campanus in this booke.

Notes

• * 1.1

  The sixth de••inition.
• * 1.2

  Camp••nus ••ath caused much o••scuri∣tie in this tenth booke.