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 75. Theoreme. The 99. Proposition. The square of a second mediall residuall line applied vnto a rationall line, maketh the breadth or other side a third residuall line.

SVppose that AB be a second mediall residuall line, and let CD be a rationall line. And vnto the line CD apply the parallelogramme CE equall to the square of the line AB, and making in breadth the line CF.* 1.1 Then I say, that the line CF is a third residuall line. For vnto the line AB let the line conueniently ioyned be supposed to be BG. Wherefore the lines AG & GB are mediall commensurable in power onely containing a mediall super••icies. And let the rest of the construction be as in the Proposition next going before.* 1.2 Wherefore the line CM is rationall and

[illustration]

incommensurable in length to the rationall line CD. And either of the parallelogrammes FX and NL is equall to that which is contained vnder the lines AG and GB. But that which is contay∣ned vnder the lines AG & GB is mediall. Wher∣fore that which is contained vnder the lines AG and GB twise is also mediall. Wherfore the whole parallelogramme FL is also mediall. Wherefore the line FM is rationall and incommensurable in length to the line CD. And forasmuch as the lines AG and GB are incōmensurable in length, therefore also the square of the line AG is incommensurable to the parallelogramme contay∣ned vnder the lines AG and GB. But vnto the square of the line AG are commensurable the squares of the lines AG and GB: and vnto the parallelogramme contained vnder the lines AG and GB, is commensurable that which is contained vnder the lines AG and GB twise. Wherefore the squares of the lines AG and GB are incommensurable to that which is contained vnder the lines AG and GB twise. Wherefore the parallelogrammes which are equall vnto them, namely, the parallelogrammes CL and FL are incommensurable the one to the other. Wherefore also the line CM is incommensurable in length to the line FM: and they are both rationall. Wherefore the line CF is a residuall line.* 1.3 I say moreouer, that it is a third residuall line. For forasmuch as the square of the line AG, that is, the parallelo∣gramme CH is commensurable to the square of the line BG, that is, to the parallelogramme KL, therefore the line CK is commensurable in length to the line KM. And in like sort as in the former Proposition, so also in this may we proue, that the parallelogramme contayned vnder the lines CK and KM, is equall to the square of the line NM, that is, to the fourth part of the square of the line FM. Wherefore the line CM is in power more then the line FM, by the square of a line commensurable in length to the line CM•• and neither of the lines CM nor FM is commensurable in length to the rationall line ••D. Wherefore the line CF is a third residual line. Wherfore the square of a second mediall residual line applied vnto a rationall line, maketh the breadth or other side a third residuall line: which was re∣quired to be demonstrated.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  CF concluded a residual line.