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 14. Theoreme. The 15. Proposition. In a circle, the greatest line is the diameter, and of all other lines that line which is nigher to the centre is alwayes greater then that line which is more distant.

SVppose that there be a circle

[illustration]

ABCD, and let the diameter thereof be the line AD, and let the centre thereof be the poynt E. And vnto the diameter. AD let the line BC be nigher then the line FG. Then I say that the line AD is the greatest, and the line BC is greater then y^e line FG. Draw (by the 12. of the first) from the centre E to the lines BC and FG per∣pendicular lines EH and EK.* 1.1 And for asmuch as the line BC is nigher vn∣to the centre then the line FG, therfore

(by the 4. definition of the third) the line EK is greater then the line EH. And (by the third of the first) put vnto the line EH an equall line EL. And (by the 11. of the first) from the point L raise vp vnto the line EK a perpen∣dicular line LM: and extend the line LM to the poynt N. And (by the first petition) draw these right lines, EM, EN, EF, and EG. And for asmuch as the line EH is equall to the line EL, therefore (by the 14. of the third,* 1.2 and by the 4. definition of the same) the line BC is equall to the line MN. Againe for asmuch as the line AE is equall to

[illustration]

the line EM, and the line ED to the line EN, therefore the line AD is e∣quall to the lines ME and EN. But the lines ME and EN are (by the 20. of the first) greater then the line MN. Wherefore the line AD is greater then the line MN. And for asmuch as these two lines ME and EN are equall to these two lines FE and EG (by the 15. definition of the first) for they are drawen from the centre to the circumfe∣rence, and the angle MEN is greater then the angle FEG, therefore (by the 24. of the first) the base MN is greater then the base FG. But it is proued that the line MN is equall to the line BC: Wherefore the line BC also is grea∣ter then the line FG. Wherefore the diameter AD is the greatest, and the line BC is greater then the line FG. Wherefore in a circle, the greatest line is the diameter, and of all the other lines, that line which is nigher to y^e centre is alwaies greater then that line which is more distant: which was required to be proued.

¶An other demonstration after Campane.

In the circle ABCD, whose centre let be the poynt E, draw these lines, AB, AC, AD, FG, and HK, of which let the line AD be the diameter of the circle.* 1.3 Then I say that the line AD is the greatest of all the lines.

[illustration]

And the other lines eche of the one is so much greater then ech of the other, how much nigher it is vnto the centre. Ioyne together the endes of all these lines with the centre, by drawing these right lines EB, EC, EG, EK, EH, and EF. And (by the 20. of the first) the two sides EF and EG of the triangle EFG, shall be greater then the third side FG. And for asmuch as the sayd sides EF & EG are equall to the line AD (by the definition of a circle) therefore the line AD is greater then the line FG. And by the same reason it is greater then euery one of the rest of the lines, if they be put to be bases of tri∣angles: for that euery two sides drawen frō the

centre are equall to the line AD. Which is

[illustration]

the first part of the Proposition. Agayne, for as∣much as the two sides EF and EG of the tri∣angle EFG, are equall to the two sides EH and EK of the triangle EHK, and the angle FEG is greater then the angle HEK, therfore (by the 24. of the first) the base FG is greater then the base HK. And by the same reason may it be proued, that the line AC is greater then the line AB. And so is manifest the whole Pro∣position.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  An other de∣monstration after Cam∣pane.