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 7. Theoreme. The 7. Proposition. If an equilater Pētagon haue three of his angles, whether they follow in or∣der,

or not in order, equall the one to the other: that Pentagon shalbe equi∣angle.

SVppose that ABCDE, be an equilater pentagon. And let the angles of the sayd Pentagon,* 1.1 namely, first, three angles folowing in order, which are at the points A, B, C, be equal the one to the other. Then I say that the Pentagon ABC∣DE is equiangle.* 1.2 Draw these right lines AC, BE, and FD. Now forasmuch as these two lines CB,* 1.3 and BA, are equall to these two lines BA, and AE, the one to the other, and the angle CBA is equall to the angle BAE:* 1.4 therefore (by the 4. of the first) the base A∣C is equall to the base BE, and the triangle ABC is equall to the triangle ABE, and the rest of the angles are equal to the rest of the angles, vnder

[illustration]

which are subtended equall sides. Wherefore the an∣gle BCA is equall to the angle BEA, and the angle ABE to the angle CAB. Wherefore also the side A∣F, is equall to the side BF (by the 6. of the first). And it was proued that the whole line AC is equal to the whole line BE. Wherefore the residue CF is equall to the residue F••. And the line CD, is equall to the line DE. Wherefore these two lines FC, and CD are e∣quall to these two lines FE, & ED, and the base F∣D, is common to them both. Wherefore the angle F∣CD, is equall to the angle FED (by the 8. of the first). And it is proued that the angle BCA, is equal to the angle AEB. Wherefore the whole angle BCD is equall to the whole angle AED. But the angle BCD, is supposed to be equall to the angles A, and B. Wherefore the angle AE∣D, is equall to the angles A and B. In like sort also may we proue that the angle CDE, is equall to the angles A, and B. Wherefore the Pentagon ABCDE is equiangle.

* 1.5But now supp••se that three angles, which folow not in order, be equall the one to the o∣ther, namely, let the angles A, C, D, be equall. Then I say that in this case also the Pentagon ABCDE is equiangle. Draw a right line from the point B, to the point D. Now forasmuch as these two lines BA, and A••, are equall to these two lines BC, and CD, and they compre∣hende equall angles, therefore (by the 4. of the first) the base BE, is equall to the base BD. And the triangle ABE, is equall to the triangle BDC, and the rest of the angles are equall to the rest of the angles, vnder which are subtended equall sides: wherefore the angle AEB, is equall to the angle CDB. And the angle BED, is equall to the angle BDE, (by the 5. of the first) for the side BE, is equall to the side BD. Wherefore the whole angle AED, is equal to the whole angle CDE. But the angle CDE, is supposed to be equall to the angles A, and C. Wherefore the AED, is equall to the angles A, and C. And by the same reason also the an∣gle ABC, is equall to the angles A; C, and D. Wherefore the Pentagon ABCDE is equian∣gle. If therefore an equilater Pentagon haue three of his angles, whither they follow in order, or not in order, equall the one to the other: that Pentagon shalbe equiangle: which was re∣quired to be proued.

Notes

• * 1.1

  Two cases in this proposi∣tion.
• * 1.2

  Construction.
• * 1.3

  Th•• first case.
• * 1.4

  Demonstra∣tion.
• * 1.5

  The second case.