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 21. Theoreme. The 23. Proposition. Vpon one and the selfe same right line can not be described two like and vnequall segmentes of circles, falling both on one and the selfe same side of the line.

* 1.1FOr if it be possible, let there be described vpon the right line AB two like & vnequall sections of circles, namely, ACB & ADB, falling both on one and the selfe same side of the line AB. And (by the first petition) drawe the right line ACD, and (by the third petition) drawe right lines from C to B, and from D to B. And for asmuch as the segment ACB is like to the segment ADB: and like

[illustration]

segmētes of circles are they which haue equall angles (by the 10. definition of the third). Where••ore the angle ACB is equall to the angle ADB, namely, the outward angle of y^e triangle CDB to the inward angle: which (by the 16. of the first) is impossible. Wherfore vp∣on one and the self same right line can not be described two like & vnequall seg∣mentes of circles, falling both on one & the selfe same side of the line: which was required to be demonstrated.

* 1.2Here Campane addeth that vpon one and the selfe same right lyne cannot be described two like and vnequall sections neither on one and the selfe same side of the lyne, nor on the opposite side. That they can not be described on one and the selfe same side, hath bene before demonstrated, and that neither also on the oppo∣site side, Pelitarius thus demonstrateth.

Let the section ABC be set vppon the lyne AC, and vpon the other side let be set the section ADC vppon the selfe same lyne AC,

[illustration]

and let the section ADC be lyke vnto the secti∣on ABC. Then I say that the sections ABC and ADC being thus set are not vnequal. For if it be pos∣sible let the section ADC be the greater. And de∣uide the line AC into two equal partes in the point E. And draw the right lyne BED deuiding the lyne AC right angled wise. And draw these right lynes AB, CB, AD and CD.* 1.3 And forasmuch as the section ADC is greater then the section ABC, the perpen¦dicular lyne also ED shall be greater then the per∣pendicular lyne EB: as is before declared in the ende of the definitions of this third booke. Wher∣fore

from the lyne ED ••ut of a ly••e equall to the lyne EB: ••hich 〈◊〉〈◊〉 be EF. And draw these right lynes AF and CF. Now then (by the 4. of the first) the triangle AEB shall be equall to the triangle AEF•• and the angle EBA shall be equall to the angle EFA. And by the same reason the angle EBC shall be equall to the angle EFC. Wherefore the whole angle ABC is equall to the whole angle AFC. But by the 21. of the first, the an∣gle AFC is greater then the angle ADC. Wherfore also the angle ABC is greater then the angle ADC. Wherefore by the definition the sections ABC and ADC are not lyke, which is contrary to the supposition. Wherefore they are not lyke and vnequall: which was required to be proued.

Notes

• * 1.1

  Demons••ra∣tion leading to an impos∣sibilitie.
• * 1.2

  An ad••ition of Campane d••mo••strated by Pelitari••s.
• * 1.3

  Demonstra¦tion leading to an impossi∣bilitie.