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 3. Probleme. The 11. Proposition. Vnto two right lines geuen, to finde a third in proportion with them.

SVppose that there be two right lines geuen BA and AC, and let them be so put that they comprehend an angle howsoeuer it be. It is required to finde vnto BA and vnto AC a third line in proportion.* 1.1 Produce y^e lynes AB and AC vnto the pointes D and E. And vnto the

[illustration]

line AC (by the 2. of the first) put an equall line BD, and draw a lyne from B to C. And by the pointe D (by the 31. of the first) draw vnto the lyne BC a parallel lyne DE, which let concurre with the line AC in the point E.* 1.2 Now forasmuch as vnto one of the sides of the triangle ADE, namely, to DE is drawne a parallel line BC: therfore as AB is in proportion vnto BD, so (by the 2. of the sixt) is AC vnto CE. But the lyne BD is equall vnto the line AC. VVherfore as the lyne AB is to the line AC, so is the line AC to the line CE. VVherfore vnto the two right lines geuen AB and AC is found a third line CE in proportiō with them: which was required to be done.

¶ An other way after Pelitarius.

Let the lines AB and BC be set directly in such sort that they both make one right line.* 1.3 Then frō the point A erect the lyne AD makyng with the lyne AB an angle at all aduentures. And put the lyne AD equall to the lyne BC. And draw a right line from D to B which produce beyond the poynt B vnto the point E. And by the point C draw vnto the lyne DA a parallel lyne CE concurring with the lyne DE in the point E. Then I say that the line CE is the third line proportionall with the lines AB and BC. For for••asmuch

as by the 15. of the first, the angle B of the

[illustration]

triangle ABD is equall to the angle B of the tri∣angle CBE, and by the 29. of the same, the angle A is equall to the angle C, and the angle D to the angle E: therefore by the 4. of this booke AB is to DA, as BC is to CE. Wherfore (by the 11. of the fifth) AB is to BC as BC is to CE: which was required to be done.

¶ An other way also after Pelitarius.

Let the lines AB and BC be so ioyned toge∣ther,* 1.4

[illustration]

that they may make a right angle, namely, ABC. And drawe a line from A to C, and from the point C drawe vnto the line AC a perpendicular CD (by the 11. of the first) And produce the lyne CD till it concurre with the line AB produced vn∣to the pointe D. Then I say that the line BD is a third lyne proportionall with the lines AB and BC: which thing is manifest by the corollary of the 8. of this booke.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  An other way after Pelita¦rius.
• * 1.4

  An ot••e•• way after Pelitarius.