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 24. Probleme. The 90. Proposition. To finde out a sixth residuall line.

TAke a rational line and let the same be A,* 1.1 And

[illustration]

take three numbers E, BC, and CD, not ha∣uing the one to the other that proportion that a square number hath to a square number. And let not the number BC haue to the number BD that proporti∣on that a square number hath to a square number. And let the number BC be greater then the number CD, & as the number E is to the number BC, so let the square of the line A be to the square of the lyne FG. And as the number BC is to the number CD, so let the square

of the line FG be to the square of the line GH. Now therfore for that as the number E is to the nūber BC,* 1.2 so is the square of the line A to the square of the line FG, therfore the square of the line A is commensurable to the square of the line F G. Wherfore the square of the line FG is rationall, and the line FG is also rationall. And for that the number E hath not to the number BC that proportion that a square number hath to a square number, therfore the line A is incommensurable in length to the line FG. Againe for that as the number BC is to the number CD, so is the square of the line FG to the square of the line GH, therefore the square of the line FG is commensurable to the square of the line GH. But the square of the line FG is rationall, wherfore the square also of the line GH is rationall, wherfore the line GH is also rationall. And for that the number B•• hath not to the number CD that propor∣tion that a square number hath to a square number, therfore the line FG is incommensura∣ble in length to the line GH, and they are both rationall. Wherefore the lines FG and GH are rationall commensurable in power onely. Wherfore

[illustration]

the lyne FH is a residuall line. I say moreouer that it is a sixt residuall line. For for that as the number E is to the number BC, so is the square of the line A to the square of the line FG, and as the number BC is to the nūber CD, so is the square of the line FG to the square of the line GH, therefore by equalitie of proportion as the number E is to the number CD, so is the square of the line A to the square of the line GH. But the num∣ber E hath not to the number CD that proportion that a square number hath to a square number. Wherefore the line A is in••••mmensurable in length to the line GH, and neither of these lines FG nor G•• is commensurable in length to the rationall line A. And forasmuch as the square of the line FG is greater then the square of the line GH, vnto the square of the line FG let the the squares of the lines GH and K be equall. Now therfore for that as the number B•• is to the number CD, so is the square of the line FG to the square of the line GH, therefore by conuersion of proportion as the number BC is to the number BD, so is the square of the line FG to the square of the line K. But the number BC hath not to the number BD that pro∣portion that a square number hath to a square number, therfore the line FG is incommen∣surable in length to the line K. Wherfore the line FG is in power more then the lyne GH by the square of a line incommensurable in length to the line FG, and neither of the lines FG nor GH is commensurable in length to the rationall line A. Wherfore the line FH is a sixt residual line. Wherfore there is found out a sixt residuall line: which was required to be done.

* 1.3There is also a certayne other redier way to finde out euery one of the forsayd sixe residu∣all lines which is after this maner. Suppose that it were required to finde out a first residuall line: Take a first binomiall line AC, & let the greater name

[illustration]

thereof be AB. And vnto the line BC let the line BD be equall. Wherefore the lines AB and BC, that is the lines AB and BD are rationall commensurable in power onely, and the line AB is in power more then the line BC, that is, then the line BD by the square of a line commensurable in length to the line AB. And the line AB is commensurable in length to the rationall line geuen. For the line AC is put to be a first binomiall line. Where∣fore the line AD is a first residual line. And in like maner may ye finde out a second, a third, a fourth, a fift, and a sixt residuall line, if ye take for eche a binomiall line of the same order.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tio••.
• * 1.3

  An other more redie way to finde out the sixe re∣siduall lines.