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 67. Theoreme. The 91. Proposition. If a superficies be contayned vnder a rationall line & a first residuall line: the line which contayneth in power that superficies, is a residuall line.

SVppose that there be a rectangle superficies AB contayned vnder a rationall line AC and a first residuall line AD.* 1.1 Then I say that the line which contayneth in power the superficies AB is a residuall line. For forasmuch as AD is a first re∣siduall line,* 1.2 let the line ioyned vnto it be DG (by the line ioyned vnto it vnder∣stand

[illustration]

such a line as was spoken of in the end of the 79. pro∣position). Wherefore the lines AG and GD are rationall cōmēsurable in pow∣er only, & the whole line AG is cōmen∣surable in length to the rationall line AC, and the line AG is in power more then the line GD by the square of a line commensurable in length vnto AG, by the definition of a first residuall line. Deuide the line GD into two equall partes in the poynt E. And vpon the line AG apply a parallelogramme equall to the square of the line EG and wanting in figure by a square, and let the sayd parallelogramme be that which is cō∣tayned vnder the lines AF and FG.* 1.3 Wherefore the line AF is commensurable in length to the line FG (by the 17. of the tenth)•• And by the poyntes E, F and G, draw vnto the line AC these parallel lines EH, FI, and GK. And make perfect the parallelograme AK. And for••as much as the line AF is commensurable in length to the line FG, therefore also the whole line AG is commensurable i•• length to either of the lines AF and FG (by the 15. of the tenth). But the line AG is commensurable in length to the line AC. Wherefore either of the lines AF and FG is commensurable in length to the line AC. But the line AC is ra∣tionall, wherefore either of the lines AF and FG is also rationall. Wherefore (by the 19. of the tenth) either of the parallelogrammes AI and FK is also rationall.* 1.4 And forasmuch as the line DE is commēsurable in length to the line EG, therfore also (by the 15. of the tenth) the line DG is commensurable in length to either of the lines DE and EG. But the line DG is rationall, wherefore either of the lines DE and EG is rationall, and the selfe same line DG is incommensurable in length to the line AC (by the definition of a first residuall line, or by the 13. of the tenth••). For the line DG is incommensurable in length to the line AG, which line AG is cōmensurable in length to the line AC:* 1.5 wherfore either of the lines DE and EG is rationall and incommensurable in length to the line AC•• Wherefore (by the 21. of the tenth) either of these parallelogrammes DH and EK is mediall. Vnto the pa∣rallelogramme AI let the square LM be equall, and vnto the parallelogramme FK let the square NX be equall, being taken away from the square LM•• and ha••ing the angle LOM common to them both.* 1.6 (And to doo this, there must be founde out the meane proportionall be∣twene the lines FI and FG. For the square of the meane proportionall is equall to the pa∣rallelogramme contayned vnder the lines FI and FG. And from the line LO cut of a line equall to the meane proportionall so founde out, and descri••e the square thereof). Wherefore both the squares LM and NX are about one and the selfe same diameter (by the 20. of the

sixth) let their diameter be OR and describe the figure as it is h••••e s••t forth•• Now then for∣asmuch as the parallelogramme contayned vnder the lines AF & FG is equal to the square of the line EG,* 1.7 therefore (by the 17. of the sixth) as the line AF is to the line EG, so is the line EG to the line FG. But as the line AF is to the line EG, so is th•• parallelogramme AI to the parallelo∣gramme

[illustration]

EK. And as the line EG is to the line FG, so is the parallelogramme EK to the parallelo∣gramme FK. Wher∣fore betwene the pa∣grammes AI and FK the parallelo∣gramme EK is the meane proportionall. But (by the second part of the assumpt going before the 54. of the tenth) be••wene the squares LM and NX the parallelogramme MN is the meane proportionall. And vnto the paral∣lelogramme AI is equall the square LM, and vnto the parallelogramme FK is equall the square NX by construction. Wherefore the parallelogramme MN is equall to the parallelo∣gramme EK (by the 2. assumpt going before the 54. of the tenth). But the parallelogramme EK is (by the first of the sixth) equall to the parallelogramme DH, and the parallelogramme MN is (by the 43. of the first) equall to the parallelogramme LX. Wherefore the whole pa∣rallelogramme DK is equall to the gnomon VTZ (which gnomon consisteth of those pa∣rallelogrammes by which ye see in the figure passeth a portion of a circle greater then a se∣micircle) and moreouer to the square NX: and the parallelogramme AK is equall to the squares LM and NX by construction: and it is now proued, that the parallelogramme DK is equall to the gnomō VTZ, and moreouer to the square NX. Wherfore the residue name∣ly the parallelogramme AB is equall to the square SQ which is the square of the line LN.* 1.8 Wherefore the square of the line LN is equall to the parallelogramme AB. Wherefore the line LN contayneth in power the parallelogramme AB. I say moreouer that the line LN is a residuall line. For forasmuch as either of these parallelogrammes AI and FK is rationall•• as it is before sayd, therefore the squares LM and NX which are equall vnto them, that is, the squares of the lines LO and ON are rationall. Wherefore the lines LO and ON are also rationall. Agayne forasmuch as the parallelogramme DH that is LX is mediall, there∣fore the parallelogramme LX is incommensurable to the square NX. Wherefore (by the 1. of the sixth, and 10. of the tenth) the line LO is incōmensurable in length to the line ON•• and they are both rationall. Wherefore they are lines rationall commensurable in power onely. Wherefore LN is a residuall line by the definition, and it contayneth in power the parallelo∣parallelogramme AB. If therefore a superficies be contayned vnder a rationall line and a first residual line, the line which contayneth in power that superficies•• is a residuall line: which was required to be demonstrated.

Notes

• * 1.1

  Fourth Sena∣ry.
• * 1.2

  The ••irst par•• of the Con∣struction.
• * 1.3

  The first part of the demon∣stration.
• * 1.4

  Note. AI and FK concluded ra∣tional paralle∣logramme.
• * 1.5

  Note.

  DH and FK parallelo∣grammes me∣diall.
• * 1.6

  Second part of the construc∣tion.
• * 1.7

  Second part of the demonstra¦tion.
• * 1.8

  LN, is the onely li••e ••hat we sought & consider.