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 19. Theoreme. The 77. Proposition. If from a right line be taken away a right line incommensurable in power to the whole line, and if that which is made of the squares of the whole line and of the line taken away added together be mediall, and the paral∣lelogramme contained vnder the same lines rationall: the line remaining is irrationall, and is called a line making with a rationall superficies the whole superficies mediall.

SVppose that AB be a right line, and from the right line AB take away a right line BC incommensurable in power to the whole line AB, and let that which is made of the squares of the lines AB and BC added together, be mediall, and the parallelogramme contained vnder the same lines rationall. Then I say, that the line remayning, namely, the line AC, is irratio∣nall,

[illustration]

and is called a line making with a rationall superficies the whole superficies mediall.* 1.1 For for∣••smuch as that which is made of the squares of the lines AB and BC added together is mediall, and that which is contained vnder the lines

AB and BC twise is rationall, therefore that which is made of the squares of the lines AB and BC added together, is incommensurable to that which is contained vnder the lines AB and BC twise. Wherefore (by the 16. of the tenth) the residue, namely, the square of the line AC is incommensurable to that which is contained vnder the lines AB and BC twise. But that which is contained vnder the lines AB and BC twise is rationall. Wherfore the square of the line AC is irrationall. Wherefore also the line AC is irrationall: and is called a line making with a rationall super••icies the whole su∣perficies

[illustration]

mediall: and is therfore so called for that that which is made of the squares of the lines AB and BC added together is mediall, & is a certaine whole superficies, part whereof is that which is contained vnder the lines AB & BC, which is a rationall superficies. For the squares of the lines AB and BC, are equall to that which is contained vnder the lines AB and BC twise, and to the square of the line AC (by the 7. of the second). Or it is therefore so called for that the square thereof added to a rationall su∣per••icies, maketh the whole superficies mediall, as shall be proued by the 109. Proposition of this booke: which was required to be proued.

In this Proposition is declared the nature of the twelueth kind of irrationall lines, which is called a line making with a rationall superficies the whole superficies mediall, whose definition is thus.

* 1.2A line making with a rationall superficies the whole superficies mediall, is an irrationall line which remaineth, whē frō a right line is taken away a right line incōmensurable in power to the whole line, and the square of the whole line & the square of the part taken away added together make a me∣diall superficies, and the parallelogramme contained of them is rationall.

This Proposition also may after Campanes way be demonstrated, obseruing the former caution.

Notes

• * 1.1

  Demonstra∣tion.
• * 1.2

  ••••••••i••ition of the twelueth ir¦ra••ionall line.