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 57. Theoreme. The 75. Proposition. If from a mediall lyne be taken away a mediall lyne commensurable in power only to the whole lyne, and comprehending together with the whole lyne a mediall superficies, the residue is an irrationall lyne, and is called a second mediall residuall lyne.

SVppose that AB be a mediall line, and from AB take away a mediall line CB commensurable in power onely to the whole line AB, and comprehending toge∣ther with the whole line AB a mediall superficies, namely, the parallelogramme contained vnder the lines AB and BC.* 1.1 Then I

[illustration]

say that the residue, namely; the line AC is irrationall, and is called a second mediall residuall line. Take a rationall line DI, and (by the 44. of the first) vnto the line DI apply the pa∣rallelogramme DE equall to the squares of the lines AB & BC, and making in bredth the line DG.* 1.2 And vnto the same line DI apply the parallelogramme DH equall to that which is cōtained vnder the lines AB & BC twise, and makyng in breadth the line DF. Now the parallelogramme DH is lesse then the parallelogramme DE, for that also the square of the lines AB and BC are greater then that which is contained vnder the lines AB and BC twise, by the square of the line AC by the 7. of the second. Wherfore the parallelogramme remayning, namely, FE, is equal to the square of the line AC. And forasmuch as the squares of the lines AB and BC are medi∣all, therfore also the parallelogramme DE is mediall, and is applied to the rationall line DI, making in bredth the line DG. Wherfore (by the 22. of the tenth) the line DG is rational and incommensurable in length to the line DI. Againe forasmuch as that which is contai∣ned vnder the lines AB and BC is mediall, therfore also that which is contained vnder the lines AB and BC twise is mediall, but that which is contained vnder the lines AB and BC twise is equall to the parallelogramme DH. Wherfore the parallelogramme DH is mediall and is applied to the rationall line DI making in breadth the line DF. Wherfore the line DF is rationall and incommensurable in length to the line DI. And forasmuch as the lines AB and BC are cōmensurable in power onely, therfore the line AB is incommēsurable in lēgth to the line BC. Wherfore (by the assumpt going before the 22. of the tenth, and by the 10. of the tenth) the square of the line AB is incommensurable to that which is contained vnder the lines AB and BC. But vnto the square of the line AB are commensurable the squares of AB and BC (by the 15. of the tenth) And vnto that which is contained vnder the lines

AB and BC is commēsurable to that which is contained vnder the lines AB and BC twise. Wherfore the squares of the lines AB and BC are incommensurable to that which is contai∣ned vnder the lines AB and BC twise. But vnto the squares of the lines AB and BC is equal the parallelogrāme DE, and to that which is cōtained vnder the lines AB and BC twise, is equall the parallelogramme DH. Wherefore the parallelogrāme DE is incōmēsurable to the parallelogramme DH. But as the parallelogramme DE is to the parallelogramme DH, so i•• the line GD to the line DF. Wherfore the line GD is incom∣mensurable

[illustration]

in lēgth to the line DF. And either of thē is ra∣tionall. Wherfore the lines GD and DF are rationall com∣mensurable in power onely. Wherfore the line FG is a residu∣all line (by the 73. proposition of the tenth) And the line DE is a rationall line. but a supe••ficies comprehended vnder a rationall line, and an irrationall line is irrationall (by the 21 of the te•••••••• and the line which containeth in power the same super••icies is irrationall (by the assumpt going before the same) Wherfore the parallelograme FE is irrationall. But the line AC containeth in power the parallelogramme FE. Wherfore the line AC is an irrationall line and is called a second mediall residuall line. And this second mediall residuall line is that part of the greater part of a bimediall line which remayneth after the taking away of the lesse part from the greater: which was required to be proued.

An other demonstrtion more briefe after Campane.

* 1.3Suppose that AB be a mediall line, from which take away the me∣diall

[illustration]

line GB commensurable vnto the whole line AB in power onely and contayning with it •• mediall superficies, namely, that which is con∣tayned vnder the lines AB and BG. Then I say that the residue AG is an i••rationall line, and is called a second mediall residua••l line. Take a ratio∣nall line DC, vpon which apply a parallelogramme equall to that which is composed of the squares of the lines AB and BG, which by the 45. of the first let be DCEI. Agayne let the parallelogramme ZFEI be equall to that which is contayned vnder the line•• AB and BG twise. Wherfore the superficies remaining DF is equal to the square of the line AG by the 7. of the second. (For that which is contayned vnder the lines AB and BG twise together with the square of the line A•• is equall to that which is composed of the squares of the lines AB and BG). And forasmuch as the squares of the lines AB and BG are mediall, for that they are descri∣bed of medial lines: the parllelogrāme DE which is equal vnto thē, shall also be mediall. And foras••uch as that which is cōtained vnder the lines AB and BG is by supposition mediall, therfore the superficie•• Z•• which is double vnto it is also mediall. But the mediall superficies DE excedeth not the mediall supe••ficies ZE by a rational superficies (by the 26. of this booke). Wherfore the excesse, namely, the super••icies D•• is irrationall, vnto which the square of the line A•• is equall: wherefore the square of the line A•• is ir••ationall, and therefore the line AG which contayneth it in power is irratio∣nall by the assumpt put before the 21. of this booke, and is called a second mediall residuall line.

This proposition setteth forth the nature of the tenth kinde of irrational lines, which is called a second residuall mediall line, which is thus defined.

* 1.4A second residual•• mediall lyne is an irrationall lyne which remayneth, when from a medial line is taken away a mediall lyne commensurable to the whole in power onely, and the part taken away & the whole lyne contayne a mediall superficies.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  An other de∣mon••••ra••ion af••er Cam∣pane.
• * 1.4

  Diffinition of the tenth ir••a¦tionall line.