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. Theoreme. The 1. Proposition. Two vnequall magnitudes being geuen, if from the greater be taken away more then the halfe, and from the residue be againe taken away more then the halfe, and so be done still continually, there shall at length be left a cer∣taine magnitude lesser then the lesse of the magnitudes first geuen.

SVppose that there be two vnequall magnitudes AB, and C, of which let AB be the greater. Then I say, that if from AB, be taken away more then the halfe,* 1.1 and from the residue be taken a∣gaine more then the halfe, and so still continually, there shall at the length be left a certaine magnitude, lesser then the lesse mag∣nitude geuē, namely, then C. For forasmuch as C is the lesse mag∣nitude, therefore C may be so multiplyed, that at the length it will be greater then the magnitude AB (by the 5. definition of the fift booke).* 1.2 Let it be so multiplyed, and let the multiplex of C grea∣ter then AB, be DE. And deuide DE into the partes equall

[illustration]

vnto C, which let be DF, FG, and GE. And from the mag∣nitudes AB take away more then the halfe, which let be BH: and againe from AH, take away more then the halfe, which let be HK. And so do continually vntill the diuisions which are in the magnitude AB, be equall in multitude vnto the diui∣sions which are in the magnitude DE. So that let the diuisions AK, KH, and HB, be equall in multitude vnto the diuisions DF, FG, and GE. And forasmuch as the magnitude DE is greater then the magnitude AB, and from DE is taken away lesse then the halfe, that is, EG (which detraction or taking a∣way is vnderstand to be done by the former diuision of the mag∣nitude DE into the partes equall vnto C: for as a magnitude is by multiplication increased, so is it by diuision diminished) and from AB is taken away more then the halfe, that is, BH: therefore the residue GD is greater then the residue HA (which thing is most true and most easie to conceaue, if we remēber this principle, that the residue of a greater magnitude, after the ta∣king away of the halfe or lesse then the halfe, is euer greater then the residue of a lesse magni∣tude, after the taking away of more then the halfe). And forasmuch as the magnitude GD is greater then the magnitude HA, and from GD is taken away the halfe, that is, GF: and from AH is taken away more then the halfe, that is, HK: therefore the residue DF is greater then the residue AK (by the foresayd principle). But the magnitude DF is equall vnto the magnitude C (by supposition). Wherefore also the magnitude C is greater then the magnitude AK. Wherefore the magnitude AK is lesse then the magnitude C. Wherefore of the magnitude AB is left a magnitude AK lesse then the lesse magnitude geuen, namely, then C which was required to be proued. In like sort also may it be proued if the halfes be taken away.

A Corollary.

* 1.3Of this Proposition it followeth, that any magnitude being geuen how litle soeuer it be, there may be geuen a magnitude lesse then it: so that it is impossible that any mag∣nitude

should be geuen then which can be geuen no losse.

¶An other demonstration of the same.

Suppose that the two vnequall magnitudes geuen be AB and C. And let C be the lesse. And forasmuch as C is the lesse, ther••fore C may so be multiplyed, that it shall at the length be greater then A••.* 1.4 Let it be so multiplyed, and let the multiplex of C exceding AB be the magnitude FM. And deuide FM into his partes equall vnto C, that is, into the magni∣tudes MH, HG, and GF. And from AB take away more then the halfe, which let be the magnitude BE: and likewise from EA take away againe more then the halfe, name∣ly, the magnitude ED. And thus do continually vntill the diuisions which are in the magnitude FM, be equall in multitude to the diuisions which are in the magnitude AB: and let those diuisions be the magnitudes BE, ED, and DA. And how multiplex the magnitude FM is to the magnitude C, so multiplex let the magnitude KX be to the magnitude DA.* 1.5 And deuide the magnitude KX into the magnitudes equall to the magnitude DA: which let be KL, LN, and NX. Now then the diuisions which are in the magnitude KX, are equall vnto the diuisi∣ons

[illustration]

which are in the magnitude MF. And forasmuch as BE is greater then the halfe of AB, therefore BE is greater then the residue EA Wherefore BE is much more greater then DA. But DA is equall vnto XN. Wherefore BE is greater then XN. Againe forasmuch as DE is greater then the halfe of EA, therefore DE is greater then the residue DA: but DA is equall vnto LN: wherefore DE is greater then LN. Wher∣fore the whole magnitude DB is greater thē the whole magnitude XL. But DA is equall vnto LK. Where∣fore the whole magnitude AB is greater then the whole magnitude KX. And the magnitude MF is greater then the magnitude BA: wherefore MF is much greater thē KX. And forasmuch as those mag∣nitudes XN, NL, & LK, are equall the one to the o∣ther, & likewise these magnitudes MH, HG, and GF, are equal the one to the other, & the multitude of those magnitudes which are in MF, is equall to the multi∣tude of those magnitudes which are in KX: therfore as KL is to FG, so is LN to GH, and NX to HM. Wherefore (by the 12. of the fift) as one of the antecedentes, namely, KL, is to one of the consequences, namely, to FG, so are all the antecedentes, namely, the whole KX to all the consequentes, namely, to the whole FM. But FM is greater then KX. Wherefore FG is greater then LK. But FG is equall vnto C: and KL vnto DA (by supposition). Wherefore the magnitude C is greater then the magnitude AD: which was required to be proued.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  A Corollary.
• * 1.4

  Construction.
• * 1.5

  Demonstra∣tion.