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 12. Theoreme. The 17. Proposition. If there be three right lines in proportion, the rectangle figure comprehended vnder the extremes, is equall vnto the square that is made of the meane. And if the rectangle figure which is made of the extremes, be equal vnto the square made of the meane, then are those three right lines proportional.

SVppose that there be three lines in proportion A, B, C, so that as A is to B, so let B be to C. Then I say that the rectangle figure comprehended vnder y^e lines A and C is equall vnto y^e square made of the line B.* 1.1 Vnto the line B (by the 2. of the

[illustration]

first) put an equall line D. And because by supposition) as A is to B, so is B to C, but B is equall vnto D, where∣fore (by the 7. of the fifth) as A is to B, so is D to C, but if there be foure right lines pro¦portionall, the rectangle fi∣gure comprehended vnder

the extremes is equall vnto the rectangle figure comprehended vnder the meanes (by the 16. of the sixt). Wherfore that which is contained vnder the lines A and C is equall vnto that which is comprehended vnder the lines B and D. But that which is contained vnder the lines B and D, is the square of the line B, for the line B is equall vnto the line D. Wherfore the rectangle figure comprehēded vn∣der the lines A and C is equall vnto the square made of the lyne B.

* 1.2But now suppose that that

[illustration]

which is comprehended vn∣der the lines A & C be equal vnto the square made of the line B. Then also I say, that as the line A is to the line B, so is the line B to the lyne C. The same order of constru∣ction that was before, beyng kept, forasmuch as y^t which is contained vnder the lynes A and C is equall vnto the square which is made of the line B. But the square which is made of the line B is that which is contained vnder y^e lines B & D, for the line B is put equall vnto the line D. Wherefore that which is contayned vn∣der the lines A and C is equall vnto that which is contayned vnder the lines B and D. But if the rectangle figure comprehended vnder the extremes, be equall vnto the rectangle figure comprehended vnder the meane lynes, the foure right lines shall be proportionall (by the 16. of the sixth) Wherfore as the line A is to the line B, so is the line D to the line C. But the line B is equall vnto the lyne D. Wherfore as the line A is to the lyne B, so is B to the line C. If therefore there be three right lynes in proportion, the rectangle figure comprehended vnder the ex∣tremes, is equall vnto the square that is made of the meane. And if the rectangle figure which is contayned vnder the extremes, be equall vnto the square made of the meane, then are those three right lines proportional: which was required to be demonstrated.

* 1.3¶ Corollary added by Flussates.

Hereby we gather that euery right lyne is a meane proportionall betwene euery two right lines which make a rectangle figure equall to the square of the same right lyne.

Notes

• * 1.1

  The first part of th•••• Theoreme.
• * 1.2

  The second part which is the conuerse of the first.
• * 1.3

  A Co••ollary.