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 11. Theoreme. The 11. Proposition. If in a circle hauing a rationall line to his diameter be inscribed an equila∣ter pentagon: the side of the pentagon is an irrationall line, and is of that kinde which is called a lesse line.

SVppose that in the circle ABCDE hauing a rationall line to his diameter be inscribed a pentagon figure ABCDE. Then I say that the side of the pentagon figure ABCDE, namely, the side AB, is an irrational line of that kinde which is called a lesse line. Take (by the 1. of the third) the centre of the circle, and let the same be the point F, and draw a right line from the point A to the point F,* 1.1 and an other from the point F to the point B, and extend those lines to the pointes G and H. And draw a right line from the point A to the point C. And from the semidiameter FH take the fourth part (by the 9. of the sixt) and let the same be FK: But the line FH is rationall (for that it is the halfe of the diameter which is supposed to be rationall), wherefore also the

line FK is rationall. And the line or semidiameter BF is rationall. Wherefore the whole line BK is rationall.* 1.2 And forasmuch as the circumference ACG is equall to the circumference ADG, of which the circumference ABC is equall to the circumference AED, wherefore the residue CG is equall to the residue GD. Now if we drawe a right line from the point A to the point D, it is manifest that the angles ALC and ALD are right angles. For foras∣much as the circumference CG is equall to the circumference GD, therefore (by the last of the sixth) the angle CAG is equall to the angle DAG. And the line AC is equall to the line AD, for that the circumferences which they subtend are equall, and the line AL is c••mmon to them both, therefore there are two lines AC and AL equall to two lines AD and AL, and the angle CAL is equall to the angle DAL. Wherefore (by the 4. of the first) the base CL is equall to the base LD, and the rest of the angles to the rest of the angles, and the line CD is double to the line CL. And by the same reason may it be proued, that the an∣gles at the point M are right angles, and that the line AC is double to the line CM. Now for∣asmuch as the angle ALC is equall to the angle AMF, for that they are both right angles, and the angle LAC is common to both the triangles ALC and AMF: wherefore the an∣gle remayning, namely, ACL, is equal to the angle remayning AFM, by the corollary of the 32. of the first. Wherefore the triangle ACL is equiangle to the triangle AMF. Wherefore proportionally, by the 4. of the sixth, as the line LC is to the line CA, so is the line MF to the line FA. And in the same proportion also are the doubles of the antecedents LC and MF (by the 15. of the fifth). Wherefore as the double of the line LC is to the line CA, so is the double of the line MF to the line FA. But as the double of the line MF is to the line FA, so is the line MF to the halfe of the line FA, by the 15. of the fifth, wherefore as the double of the line LC is to the line CA, so is the line MF to the halfe of the line FA, by the 11. of the fifth. And in the same proportion, by the 15. of the fifth, are the halues of the consequents, namely, of CA and of the halue of the line AF. Wherefore as the double of the line LC is to the halfe of the line AC, so is the line MF to the fourth part of the line FA. But the double of the line LC is the line DC, and the halfe of the line CA is the line CM, as hath before bene proued, and the fourth part of the line FA is the line FK (for the line FK is the fourth part of the line FH by construction). Wherfore as the line DC is to the line CM, so is the line MF to the line FK. Wherfore by composition (by the 18. of the fifth) as both the lines DC and CM are to the line CM, so is the whole line MK to the line FK.

[illustration]

Wherefore also (by the 22. of the sixt) as the squares of the lines DC and CM are to the square of the line CM, so is the square of the line MK to the square of the line FK. And forasmuch as (by the 8. of the thirtenth) a line which is subten∣ded vnder two sides of a pentagon figure, as is the line AC, being diuided by an extreame & meane propor••ion, the greater segment is equall to the side of the pentagon figure, that is, vnto the line DC: and (by the 1. of the thirtenth) the greater segment hauing added vnto it the halfe of the whole, is in power quintuple to the square made of the halfe of the whole: and the halfe of the whole line AC is the line CM. Wherefore the square that is made of the lines DC and CM, that is, of the greater segment and of the halfe of the whole, as of one line, is quintuple to the square of the line CM, that is, of the halfe of the whole. But as the square made of the lines DC and CM, as of one line, is to the square of the line CM, so is it proued, that the square of the line MK is to the square of the line FK. Wherefore the square of the line M∣K is quintuple to the square of the line FK. But the square of the line KF is rationall, as

hath before bene proued, wherefore also the square of the line MK is rationall, by the 9. diffi∣nition of the tenth, for the square of the line MK hath to the square of the line KF that proportion that number, hath to number, namely, that 5. hath to 1. and therefore the sayd squares are commensurable, by the 6. of the tenth. Wherefore also the line MK is rationall. And forasmuch as the line BF is quadruple to the line FK (for the semidiameter BF is equal to the semidiameter FH), therfore the line BK is quintuple to the line FK. Wherefore the square of the line BK is 25. times so much as the square of the line KF, by the corollary of the 20 of the sixt. But the square of the line MK, is quintuple to the square of the FK, as is proued. Wherfore the square of the line BK is quintuple to the square of the line KM. Wher∣fore y^e square of the line BK, hath not to y^e square of the line KM, that proportiō that a square number hath to a square number, by the corollary of the 25. of the eight. Wherefore (by the 9. of the tenth) the line BK is incommensurable in length to the line KM, and either of the lines is rationall. Wherefore the lines BK and KM are rationall commensurable in power onely. But if frō a rationall line be taken away a rationall line being commensurable in power onely to the whole, that which remayneth is irrationall, and is (by the 73. of the tenth) cal∣led a residuall line. Wherefore the line MB is a residuall line. And the line conueniently ioy∣ned vnto it, is the line MK. Now I say that the line BM is a fourth residuall line. Vnto the excesse of the square of the line BK aboue the square of the line KM, let the square of the line N be equall (which excesse how to finde out, is taught in the assumpt put after the 13. pro∣position of the tenth). Wherefore the line BK is in power more then the line KM by the square of the line N. And forasmuch as the line KF is commensurable in length to the line FB, for it is the fourth part thereof, therefore (by the 16. of the tenth) the whole line KB is commensurable in length to the line FB. But the line FB is commensurable in length to the line BH, namely, the semidiameter•• to the diameter: wherefore the line BK is commensura∣ble in length to the line BH, by the 12. of the tenth. And forasmuch as the square of the line BK is quintuple to the square of the line KM, therefore the square of the line BK hath to the square of the line KM that proportion that fiue hath to one. Wherefore by conuersion of proportion (by the corollary of the 19. of the fifth) the square of BK hath to the square of the line N, that proportion that fiue hath to fower: & therfore it hath not that proportiō that a square number hath to a square number, by the corollary of the 25. of the eight. Wherfore, the line BK is incommensurable in length to the line N (by the 9. of the tenth). Wherfore the line BK is in power more then the line KM, by the square of a line incommensurable in length to the line BK. Now then forasmuch as the whole line BK is in power more then the line conueniently ioyned, namely, then KM, by the square of a line incōmensurable in length to the line BK, and the whole line BK is commensurable in length to the rationall line geuen BH: therefore the line MB is a fourth residuall line, by the diffinition of a fourth residuall line. But a rectangle parallelogramme contayned vnder a rationall line and a fourth residual line, is irrationall, and the line which contayneth in power the same parallelogramme is also irrationall, and is called a lesse line (by the 94. of the tenth). But the line AB contayneth in power the parallelogramme contayned vnder the lines HB and BM (for if we drawe a right line from the point A to the point H, the triangle ABH shall be like to the triangle ABM, by the 8. of the sixth. For from the right angle BAH is drawen to the base BH a perpendicu∣lar line. And therefore as the line BH is to the line BA, so is the line AB to the line BM. this followeth also of the co••ollary of the sayd 8. of the sixth. Wherefore the line AB which is the side of the pentagon figure, is an irrationall line of that kinde which is called a lesse line. If therefore in a circle hauing a rationall line to his diameter be inscribed an equilater pen∣tagon, the side of the pentagon is an irratio••all line, and is of that kinde which is called a lesse line: which was required to be demonstrated.

Notes

• * 1.1

  Construstion.
• * 1.2

  Demonstra∣tion.