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 87. Theoreme. The 111. Proposition. A residuall line, is ••ot one and the same with a binomiall lyne.

SVppose that AB be a residuall line. Then I say that AB is not one and the same with a binomiall line. For if it be possible let it be a binomiall line. And take a ra∣tionall line DC.* 1.1 And (by the 44. of the first) vnto the line CD apply a rectangle parallelogramme CE equall to the square of the line AB, and making in bredth the line DE.* 1.2 And forasmuch as AB is a residuall line, therfore (by the 97. of the tenth) the line DE is a first residuall line. Let the line co••••••niently ioyned vnto it be E••. Wherfore the lines D

F and FE are rationall commensurable in power ••n∣ely,

[illustration]

and the line DF is in power more then the line FE by the square of a line commensurable in length to the line DE & the line DF, is cōmensurable in lēgth to the rational line put DC. Again forasmuch as AB is by position a binomiall line, therefore (by the 60. of the tenth) the line DE is a first binomiall line. De∣uide it into his names in the point G. And let DG be the greater name. Wherfore the lines DG and ••E are rationall commensurable in power onely. And the line DG is in power more then the line GE by the square of a line commensurable in length to the lyne DG, and the line DG is commensurable in length to the rationall line put DC. Wherefore the line DF is commensurable in length to the line DG. Wherfore (by the 13. of the tenth) the whole line DF is commensurable in lēgth to the line remaining, namely, ••o the line GF. And forasmuch as the line DF is cōmēsurable to the line FG, but the line FD is rationall. Wherfore the line FG is also rationall. And forasmuch as the line FD is commensurable in length to the line FG, but the line DF is incommensurable in length to the line FE. Wher∣fore the line FG is incommensurable in length to the line FE (by the 13. of the tenth) and they are both rationall lines. Wherfore the lines GF and FE are rationall commensurable in power onely. Wherfore (by the 73. of the tenth) the line EG is a residuall line, but it is also rationall (as before hath bene proued): which is impossible, namely, that one & the same line should be both rationall and irrationall. Wherfore a residuall line is not one and the same with a binomiall line, that is, is not a binomiall line: which was required to be demonstrated.

* 1.3¶ A Corollary.

A residuall lyne and the other fiue irrationall lynes following it, are neither mediall lines, nor one and the same betwene themselues•• that is, one is vtterly of a diuers kinde frō an other. For the square of a mediall line applied to a rationall line, maketh the breadth rationall and incommensurable in length to the rationall lyne, whereun∣to it is applied (by the 22. of the tenth) The square of a residuall line applied to a rationall line, maketh the breadth a first residuall line (by the 97. of the tenth). The square of a first mediall residuall line applied to a rationall line, maketh the breadth a second residuall lyne (by the 98. of the tenth) The square of a second mediall residuall line applied vnto a ratio∣nall line, maketh the breadth a third residuall line (by the 99. of the tenth) The square of a lesse line applied to a rationall line, maketh the breadth a fourth residuall line (by the 100. of the tenth) The square of a line making with a rationall superficies the whole superficies mediall applied to a rationall line, maketh the breadth a ••ift residuall line (by the 101. of the tenth) And the square of a line making with a mediall super••icies the whole superficies medi∣all applied to a rationall line, maketh the breadth a sixt residuall line (by the 102. of the tēth) Now forasmuch as these foresaid sides which are the breadthes differ both from the first breadth, sor that it is rational, and differ also the one frō the other, for that they are residuals of diuers orders and kindes, it is manifest that those irrationall lines differ also the one from the other. And forasmuch as it hath bene proued in the 111. proposition, that 〈◊〉〈◊〉 residual 〈◊〉〈◊〉 is not one and the same with a binomiall line, and it hath also bene proued that the 〈…〉〈…〉 of a residuall line and of the fiue irrationall lines that follow it being applied to a rational line do make their breadthes one of the residuals of that order of which they were, whose square•• were applied to the rationall line, likewise also the squares of a binomiall line, and of the fiue irrationall lines which follow it, being applied to a rationall line, do make the breadthes one

of the binomials of that order of which they were, whose squares were applied to the rationall line. Wherfore the irrationall lines which follow the binomiall line, and the irrationall lines which follow the residuall line, differ the one from the other, so that all the irrationall lynes are 13. in number, namely, these.

• 1 A mediall line.
• 2 A binomiall line.
• 3 A first bimediall line.
• 4 A second bimediall line.
• 5 A greater line.
• 6 A line containing in power a rationall superficies and a mediall superficies.
• 7 A line contayning in power two mediall superficieces.
• 8 A residuall line.
• 9 A first mediall residuall line.
• 10 A second mediall residuall line.
• 11 A lesse line.
• 12 A line making with a rationall superficies the whole superficies mediall.
• 13 A line making with a mediall superficies the whole superficies mediall.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstrati∣on on leading to an impossibi∣litie.
• * 1.3

  A Corollary.