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 56. Theoreme. The 74. Proposition. If from a mediall line be taken away a mediall line commensurable in power onely to the whole line, and comprehending together with the whole line a rationall superficies: the residue is an irrationall line, and is called a first mediall residuall line.

SVppose that AB be a mediall line. And

[illustration]

from the line AB take away a mediall line BC commensurable in power onely to the whole line AB and comprehending toge∣ther with the line AB a rationall superficies, that is, let that which is comprehended vnder the lines AB and BC be rationall. Then I say that the line remayning, namely, AC is irratio∣nall and is called a first mediall residuall line.* 1.1 For forasmuch as the lines AB and BC are me∣diall, therefore also the squares of the lines AB and BC are mediall. But that which is con∣tayned vnder the lines AB and BC twise is rationall. Wherefore that which is composed of the squares of the lines AB and BC, that is, that which is contayned vnder AB and BC twise together with the square of the line AC is incommensurable to that which is contained vnder the lines AB and BC twise. Wherefore (by the second part the 16. of the tenth) that which is contayned vnder the lines AB and BC twise is incommensurable to the square of the line AC. But that which is contayned vnder the lines AB and BC twise it rationall, wherefore the square of the line AC is irrationall. Wherefore also the line AC is irrationall•• and is called a first mediall residuall line. This first mediall residuall line is also that part of the greater part of a first bimediall line, which remayneth after the taking away of the lesse part from the greater, wherof it hath also his name, and is called a first mediall residuall lines which was required to be proued.

Out of this proposition is taken the definition of the ninth kinde of irrationall lines, which is called a first residuall mediall line the difinition whereof is thus.

* 1.2A first residuall mediall line is an irrationall line which remayneth, when from a mediall line is taken away a mediall line commensurable to the whole in power onely, and the part taken away and the whole line contayne a mediall superficies.

An other demonstration after Campane.

Let the line DE be rationall, vpon which apply the superficies DF equall to that which is cont••i∣ned

vnder the lynes AB and BC twise, and let the superficies GE be e∣qual

[illustration]

to that which is composed of the squares of the lynes AB and BC:* 1.3 wherfore by the 7. of the second, the superficies FG is equal to the square of the lyne AC. And forasmuch as (by supposition) the superficies EG is mediall, therfore (by the 22. of the tenth) the lyne DG is rationall cōmen∣surable in power onely to the rational lyne DE. And forasmuch as by sup∣position the superficies EH is rational, therfore by the 20. of the tenth, the line DH is rational commensurable in length vnto the rationall line DE. Wherfore the lynes DG and DH are rationall commensurable in power only (by the assumpt put before the 13. of this boke). Wherfore by the 73 of this boke, the lyne GH is a residuall lyne, and is therefore irrationall. Wherfore (by the corollary of the 21. of this boke) the superficies FG is irrational. And therfore the line AC which cōtayneth it in power is irra∣tionall, and is called a first medial residuall lyne.

Notes

• * 1.1

  ••••monstra∣••ion.
• * 1.2

  Diffinition of 〈◊〉〈◊〉 ••inth irra∣tionall line.
• * 1.3

  An other de∣monstratiō af∣ter Campane.