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 2. Theoreme. The ••. Proposition. If a right line, be in power quintuple to a segment of the same line: the dou∣ble of the sayd segment is deuided by an extreame and meane proportion, and the greater segment thereof is the other part of the line geuen at the be¦ginning.

* 1.1SVppose that the right line

[illustration]

DC be in power quin∣tuple to a segment of the same line, namely, to A∣D, and let the double of the line AD be the line AB. Then I say that the line AB is deuided by an extreme and meane proportion, and the greater segment thereof is the lyne AC.* 1.2 Describe on either of the lines AB and CD squares, namely AE and D∣F. And in the square DF make per∣fect the figure, and extend the line FC to the point G.* 1.3 And forasmuch as the square DF is quintuple to the square DH, by suppossition, therfore the gno∣mon MNX is quadruple to the square DH. And forasmuch as the line AB is double to the line AD, therefore the square of the line AB is quadruple to the square of the line AD (by the 20.

of the sixt), that is, the square AE to the square DH. And it is proued that the gnomon M∣NX is quadruple to the square DH. Wherefore the gnomon MNX is equall to the square AE. And forasmuch as the line AB is double to the line AD, but the line AB is equall to the line AK, and the line AD to the line AH: therefore the line AK is double to the line A∣H: wherefore also (by the first of the sixth) the parallelogramme AG is double to the paral∣lelogramme CH. But the parallelogrammes LH and CH are double to the parallelogramme CH (by the 43. of the first): wherefore the parallelogramme AG is equall to the parallelo∣grammes LH and CH. And it is proued that the whole gnomon MNX is equall to the whole square AE. Wherefore the residue HF is equall to the parallelogramme CE. And CE is that which is contained vnder the lines AB and CB, for the line AB is equall to the line BE, and HF is the square made of the line AC. Wherefore that which is contayned vnder the lines AB and BC, is equall to the square of the line AC. Wherfore as the line AB is to the line AC, so is the line AC to the line CB. * 1.4 But the line A•• i•• greater then the line AC, wherefore the line AC is greater then the line CB. Wherefore the line AB is deuided by an extreme and meane proportion, and the greater segment thereof is the line AC. If therfore a right line be in power quintuple to a segment of the same line, the double of the sayd segment is deuided by an extreame & meane proportion, and the greater segment thereof is the other part of the line geuen at the beginning: Which was required to be proued.

* 1.5 Now, that the double of the line AD (that is AB) is greater then the line AC may thus be proued. For if not, then if if it be possible let the line AC be double to the line AD, wherefore the square of the line AC is quadruple to the square of the line AD. Wherefore the squares of the lines AC and AD are quintuple to the squares of the line AD. And it is supposed that the square of the line DC is quintuple to the square of the line AD, wherefore the square of the line DC is equall to the square of the lines AC and AD: which is impos∣sible (by the 4. of the second). Wherefore the line AC is not double to the line AD. In like sorte also may we proue that the double of the line AD is not lesse then the line AC, for this is much more absurd: wherefore the double of the line AD is greater thē the line AC•• which was required to be proued.

This proposition also is an other way demonstrated after the fiueth proposition of this booke.

Two Theoremes, (in Euclides Method necessary) added by M. Dee.

A Theoreme. 1.

A right line can be deuided by an extreame and meane proportion, but in one onely poynt.

Suppose a line diuided by extreame and meane proportion, to be AB. And let the greater segment be AC. I say, that AB can not be deuided by the sayd proportion, in any other point then in the point C. If an aduersary woulde contend that it may, in like sort, be deuided in an other point: let his other point, be supposed to be D: making AD; the greater segment of his imagined diuision. Which AD, also, let be lesse then our AC: for the first discourse. Now, forasmuch as by our aduersaries opinion, AD, is the greater segment, of his diuided line•• the parallelogramme conteyned vnder AB, and DB, is equall to the square of AD, by the third definition and 17. proposition of the sixth Booke. And by the same definition and proposition, the parallelogramme vnder AB, and CB, conteyned, is equall to the square of our greater segment AC. Wherefore, as the parallelogramme, vnder AB, and D••, is to the square of AD: so i•• 〈◊〉〈◊〉 parallelogramme, vnder AB, and CB, to the square of AC. For proportion of equality, is con∣cluded

[illustration]

in them both. But, foras∣much as D•••• i•• (by•• * 1.6 supposition) greater thē CB, the parallelogrāme vnder AB, and DB, is greater then the parallelogramme vnder AC,

and CB: by the first of the sixth (for

[illustration]

AB is their equall heith.) Where∣fore, the square of AD, shalbe grea∣ter then the square of AC: by the 14. of the fifth. But the line AD, is lesse then the line AC, by supposi∣tion: wherefore the square of AD is lesse then the square of AC. And it is concluded also to be greater then the square of AC: Where∣fore the square of AD, is both greater, then the square of AC•• and also lesse. Which is a thing impos∣sible. The square therefore of AD, is not equall to the parallelogramme vnder AB, and DB. And there∣fore by the third definition of the sixth, AB is not deuided by an extreame and meane proportion, in the point D: as our aduersary imagined. And (Secondly) in like sort will the inconueniency fall out: if we assigne AD, our aduersaries greater segment, to be greater then our AC. Therefore seing neither on the one side of our point C: neither on the other side of the same point C, any point can be had, at which the line AB can be deuided by an extreame and meane proportion, it followeth of nec••ssitie, that AB can be deuided by an extreame and meane proportion in the point C, onely. Therefore, a right line can be deuided by an extreame and meane proportion, but in one, onely point: which was requi∣site to be demonstrated.

A Theoreme. 2.

What right line so euer, being deuided into two partes, hath those his two partes, proportionall, to the two segmentes of a line deuided by extreame and meane proportion: is also it selfe deuided by an ex∣treame and meane proportion: and those his two partes, are his two segments, of the sayd proportion.

Suppose, AB, to be a line deuided by an extreame and meane proportion in the point C, and AC to be the greater segment. Suppose also the right line DE, to be deuided into two partes, in the point F: and that the part DF, is to FE, as the segment AC, is to CB: or DF, to be, to AC, as FE is to CB. For so these pa••tes are proportionall, to the sayd segmentes. I say now, that DE is also deuided by an extreame and meane proportion in the point F. And that DF, FE, are his segmentes of the sayd pro∣portion. For, seing, as AC, is to CB: so is DF, to FE: (by supposition). Therfore, as AC, and CB (which is AB) are to CB: so is DF, and FE, (which is DE) to FE: by the 18. of the fifth. Wherefore (alternate∣ly) as AB is to DE: so is CB, to FE. And

[illustration]

therefore, the residue AC, is to the residue DF, as AB is to DE, by the fifth of the fift. And then alternately, AC is to AB, as DE, is to DF. Now therefore backward, AB is to AC, as DE is to DF. But as AB is to A∣C, so is AC to CB: by the third definition of the sixth booke. Wherefore DE is to DF, as AC is to CB: by the 11. of the fifth. And by suppositi∣on, as AC is to CB, so is DF to FE: wherefore by the 11. of the fifth, as DE is to DF: so is DF to FE. Wherefore by the 3. definition of the sixth, DE is deuided by an extreame and meane proportion, in the point F. Wherefore DF, and FE are the segmentes of the sayd proportion. Therefore, what right line so euer, being deuided into two partes, hath those his two partes, proportionall to the two seg∣mentes of a line deuided by extreame and meane proportion•• is also it selfe deuided by an extreme and meane proportion, and those his two partes are his two segmentes, of the sayd proportion•• which was requisite to be demonstrated.

Note.

Many wayes, these two Theoremes, may be demonstrated: which I leaue to the exercise of young studentes. But vtterly to want these two Theoremes, and their demonstrations: in so principall a line, or rather the chiefe piller of Euclides Geometricall pallace,* 1.7 was hetherto, (and so would remayne) a great disgrace. Also I thinke it good to note vnto you, what we meant, by one onely poynt. We m••••••••, that the quantities of the two segmentes, can not be altered, the whole line being once geuen. And though, from either end of the whole line, the greater segment may begin:* 1.8 And so as it were the point of secti∣on may seeme to be altered: yet with vs, that is no alteration: forasmuch as the quantities of the seg∣mentes, remayne all one. I meane, the quantitie of the greater segment, is all one: at which end so euer it be taken: And therefore, likewise the quantitie of the lesse segment is all one, &c. The like confidera∣tion may be had in Euclides tenth booke, in the Binomiall lines. &c.

Io••n Dee. 1569. Decemb. 18.

Notes

• * 1.1

  This proposi∣tion is the con∣uerse of the former.
• * 1.2

  Construction.
• * 1.3

  Demonstra∣tion.
• * 1.4

  An Assūpt.
• * 1.5

  The Assūpt proued.
• * 1.6

  Because AC is supposed grea∣ter then AD: therefore his residue is lesse, then the residue of AD, by the common sen∣tence. Where∣fore, by the sup∣position: DB is greater then ••C.
• * 1.7

  The chie••e line in all Euclides Geometrie.
• * 1.8

  What is ment here by, A secti∣on in one onely poi••t.