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 third Theoreme. The sixt Proposition. If a triangle haue two angles equall the one to the other: the sides also of the same, which subtend the equall angles, shalbe equall the one to the other.

SVppose that ABC be a triangle, hauing the angle ABC equall to the angle ACB. Then I say that the side AB is equall to the side AC. For if the side AB be not equal to the side AC, then one of them is greater. Let AB be the greater. And by the third proposition,* 1.1 from AB be∣ing the greater cut of a line equal to the lesse line, which is AC•• And let the same be D B. And drawe a line from the poynt D to the poynt C.* 1.2 Now forasmuch as the

[illustration]

side DB is equall to the syde AC, and the line BC is common to thē both: therefore these two sydes DB and BC are equall to these two sydes AC & CB the one to the other. And the angle DBC is by supposytion equall to the angle ACB. VVher∣fore (by the 4 proposytion) the base DC is equall to the base AB: & (by the same) the triangle DBC is equall to the triangle ACB: namely, the lesse triangle vnto the greater triāgle, which is impossi¦ble. VVhe••fore the syde AB is not vnequal to the side AC. VVherfore it is e∣qual. If therfore a triangle haue two angles equall the one to the other: the sydes also of the same, which subtende the equall angles, shall be equall the one to the other•• which was required to be demonstrated.

* 1.3In Geometrie is o••tentimes vsed conuersion of propositions. As this propo∣sition is the conuerse of the proposition next before. The chiefest and most pro∣per kind of conuersion is, when that which was the thing supposed in the former proposition, is the conclusion of the conuerse and second proposition: and con∣trary wise that which was concluded in the first, is the thing supposed in the se∣cond As in the fifth proposition it was supposed the two sides of a triangle to be equal, the thing concluded is, that the two angles at the base are equall & in this proposition, which is the conuerse therof is supposed that the angles at the base be equall. VVhich in the former proposition was the conclusion. And the con∣clusion is, that the two sydes subtending the two angles are equall, which in the former proposition was the supposition. This is the chiefest kind of conuersion vniforme and certayne.

There is an other kind of conuersion,* 1.4 but not so full a conuersion nor so per∣fect as the first is. VVhich happeneth in composed propositions, that is, in such, which haue mo suppositions then one, and passe from these suppositions to one conclusion. In the cōnuerses of such propositiōs, you passe from the conclusion of the first proposition, with one or mo of the suppositions of the same: & con∣clude some other supposition of the selfe first proposition: of this kinde there are many in Euclide. Therof you may haue an example in the 8. proposition be∣ing the conuerse of the four••h. This conuersion is not so vniforme as the other; but more diuers and vncertaine according to the multitude of the things geuen, or suppositions in the proposition.

But because in the fifth proposition there are two conclusions,* 1.5 the first, that the two angles at the ba••e be equall: the second, that the angles vnder the ba••e are equall: this is to be noted, that this sixt proposition is the conuerse of the ••ame fifth as touching the first conclusion onely.* 1.6 You may in like maner make a con∣uerse of the same proposition touching the second conclusion therof. And that after this maner.

THe two sides of a triangle beyng produced, if the angles vnder the base be equall, the said triangle shall be an Isosceles triangle. In which propositiō the supposition is, that the angles vnder the base are equall: which in the fifth proposition was the conclusion•• & the conclusion in this proposition is, that the two sides of the triangle are equal, which in the fift proposition was the supposition. But now for proofe of the said proposition:

Suppose that there be a triangle ABC, & let the

[illustration]

sides AB•• and AC be produced to the poyntes D and G, and let the angles vnder the base be equall, namely, the angles DBC, and GCB. Then I say that the triangle ABC is an Isosceles triangle.* 1.7 For take in the line AD a point which let be E. And vnto the line BE put the line CF equall (by the 3. propositio••).* 1.8 And draw these lines EC, BF, and EF. Now forasmuch as BE is equall to CF, and BC is common to thē both•• therfore these two lines BE & BC, are equall to these two lines CF and CB the one to the other, & the an∣gle EBC is equall to the angle FCB by supposition. Wherfore (by the 4. proposition) the base of the one is equall to the base of the other, and the one triangle is equall to the other triangle, & the other angles re∣mayning are equal vnto the other angles remayning, the one to the other, vnder which are subtended equall sides•• Wherfore the base EC is equall to the base FB, and the angle BEC to the angle CFB, and the angle CBF to the angle BCE. But the whole angle EBC is equall to the whole angle FCB, of which the angle FBC is equa••l to the angle ECB•• wherefore the angle remayning EBF is e∣quall to the angle remayning FCE. But the line BE is ••quall to the line CF, & the line BF to the line CE, and they contayne equall angles: wherfore by the same fourth pro∣position the angle BEF is equall to the angle CFE. Wherfore by this sixt proposition the side AE is equall to the side AF•• o•• whiche B•• is equall to CF, by construction: wherfore (by the third common sentence) the residue AB is equall to the residue AC Wherfore the triangle ABC is an Isosceles triangle. If therfore the two sides of a trian∣gle being produced, the angles vnder the base be equall, the sayd triangle shall be an I∣sosceles triangle: which was required to be proued.

This moreouer is to be noted, that in this proposition there may be an other case•• for in taking an equall line to AC from AB,* 1.9 you may take it from the poynte A and

not from the poynt B. And yet though this supposition

[illustration]

also be put the selfe same absurdity will follow.

For suppose that AC be equall to AD: and produce the line CA to the poynt E: and put the line AE equall to the line DB (by the third proposition) wherefore the whole line CE is equall to the whole line AB (by the se∣cond common sentēce) Draw a line from the poynt E to the point B. And forasmuch as the line AB is equall to the line EC, and the line BC is common to them both, and the angle ACB is supposed to be equall to the an∣gle ABC: Wherfore (by the fourth proposition) the tri∣angle EBC is equall to the triangle ABC, namelye, the whole to the part: which is impossible.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstration leading to an impossib••l••ty.
• * 1.3

  The chiefest and 〈◊〉〈◊〉 proper kind of conuersion.
• * 1.4

  An other kind of conuersion not so perfect as the first.
• * 1.5

  Two conclusions in the fifth pro∣position.
• * 1.6

  The sixt propo∣sition i•• the con∣uerse as tou∣ching the first conclusion onely. The conuerse as touching the se∣cond conclusion.
• * 1.7

  Construction.
• * 1.8

  Demonstration••
• * 1.9

  An other case in this sixt proposi∣tion.