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 60. Theoreme. The 78. Proposition. If from a right line be taken away a right line incommensurable in power to the whole line, and if that which is made of the squares of the whole line and of the line taken away added together be medial, and the parallelo¦gramme contayned vnder the same lines be also mediall, and incommen∣surable to that which is made of the squares of the sayd lines added toge∣ther: the line remayning is irrationall, and is called a line making with a mediall superficies the whole superficies mediall.

SVppose that AB be a right line,

[illustration]

and from AB take away a right line BC incommensurable in power to the whole line AB. And let that which is made of the squares of the lines AB and BC added together be mediall, and let the parallelograme contayned vnder the lines AB and BC be also medial, and let that which is made of the squares of the lines AB and BC added together be in∣commensurable to that which is contayned vnder the lines AB and BC. Thē I say that the line remayning, namely, the line AC i••

irrationall and is called a line making with a mediall superficies the whole superficies medial. Take a rationall line DI. And (by the 44. of the first) vnto the line DI apply the parallelo∣gramme DE equall to that which is made of the squares of the lines AB and BC added to∣gether,* 1.1 and making in breadth the line DG. And vnto the same line DI apply the paralle∣logramme DH equall to that which is contayned vnder the lines AB and BC twise and making in breadth the line DF. Wherefore the parallelogramme remayning, namely the pa∣rallelogramme FE, is equall to the square of the line AC. Wherefore the line AC contay∣neth in power the parallelograme FE.* 1.2 And forasmuch as that which is made of the squares of the lines AB and BC added together is mediall, and is equall to the parallelogramme DE, therefore also the parallelogramme DE is mediall. And the parallelogramme DE is applyed to the rationall line DI making in breadth the line DG. Wherfore (by the 22. of the tenth) the line DG is rationall and incommensurable in length to the line DI. Agayne forasmuch as that which is contayned vnder the lines AB and BC twise is mediall and is equall to the parallelogramme DH, therefore the parallelogramme DH is mediall. And the parallelograme DH is applyed vnto the rationall line DI making in breadth the line DF, wherefore the line DF is rationall and incommensurable in length to the line DI. And for∣asmuch as that which is made of the squares of the lines AB and BC added together is in∣commensurable to that which contayned vnder the lines AB & BC twise, ther••fore the pa∣rallelogramme DE is incommensurable to the parallelogramme DH. But as the parallelo∣gramme DE is to the parallelogramme DH, so (by the first of the sixt) is the line DG to the line DF: wherfore the line DG is incommensurable in length to the line DF. And they are both rationall lines. Wherefore the lines DG and DF are rationall commensurable in pow∣er onely, where••ore the line FG is a residuall line by the 73. of this booke•• But the line FH is rationall for that it is equall vnto the line DI. But a rectangle parallelogramme contayned vnder a rationall line and an irrationall line is irrationall, and the line also that contayneth in power the same parallelogramme is irrationall (by the 21. of the tenth). But the line CA contayneth in power the parallelogramme FE. Wherefore the line AC is irrationall and is called a line making with a mediall superficies the whole superficies mediall. And is therfore so called, for that that which is made of the squares of the lines AB and BC added together is mediall, & is a certayne whole superficies, part whereof is that which is cōtayned vnder the lines AB and BC, which is also mediall: you shall also by the 110. proposition of this booke vnderstand an other cause why it is so called.

This proposition may thus more briefely be demonstrated: forasmuch as that which is composed of the squares of the lines AB and BC is mediall, and that also which is contayned vnder them is me∣diall, therefore the parallelogramm••s DE and DH which are equall vnto them are mediall: but a me∣diall superficies exceedeth not a mediall superficies by a rationall superficies. Wherefore the superficies FE which is the excesse of the mediall superficies DE aboue the mediall superficies DH is irrational. And therefore the line AC which contayneth it in power is irrationall. &c.

In this proposition is shewed the conditiō and nature of the thirtenth and last kinde of irrationall lines, which is called a line making with a mediall superficies the whole superficies mediall, whose definition is thus.

A line making with a mediall superficies the whole superficies mediall is an irrationall line which remayneth,* 1.3 when from a right li••e is taken away a right line incommensurable in power to the whole line, and the squares of the whole line and of the line taken away added together make a mediall superficies, and the parallelogramme contayned of thē is also a mediall superficies, moreouer the squares of them are incommensurable to the parallelogramme contayned of them.

An assumpt of Campane.

If there be fower quātities, & if the difference of the first to the second, be as the difference of the third to the fourth, then alternately, as the difference of the first is to the third, so is the difference of the second to the fourth.

This is to be vnderstand of quātities in like sort referred the one to the other, that is if the first be greater then the second,* 1.4 the third ought to be greater then the fourth and if the first be lesse then the se∣cond, the third ought to be lesse then the fourth: and is also to be vnderstand in arithmeticiall propor∣tionality. As for example let the difference of A be vnto B as the difference of C is to D.* 1.5 Then I say that as the difference of A is to C, so is the difference of B to D. For (by this common sētence, the difference of the extreames is composed of the differences of the ex∣treames

[illustration]

to the meanes), the difference of A to C is compo∣sed of the difference of A to B and of the difference of B to C. And (by the same common sentence) the difference of B to D is composed of the difference of B to C, and of ••he difference of C to D. And forasmuch as (by supposition) the difference of A to B is as the difference of C to D, and the difference of B to C is common to them both. Where∣fore it followeth, that as the difference of A is to C, so is the difference of B to D: which was required to be proued.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  Diffinition of the thirtenth and last irra∣tionall line.
• * 1.4

  An Assumpt of Campane.
• * 1.5

  I. Dee.

  Though Cam∣panes lemma be true, ye•• the maner of de∣monstrating it, (narrowly considered) is not artificiall.