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 18. Theoreme. The 21. Proposition. A rectangle figure comprehended vnder two rationall right lines com∣mensurable in power onely, is irrationall. And the line which in power contayneth that rectangle figure is irrationall, & is called a mediall line.

SVppose that this rectangle figure AC be comprehended vnder these rationall right lines AB and BC commensurable in power onely. Then I say, that the su∣perficies AC is irrationall: and the line which contayneth it in power is irratio∣nall, and is called a mediall line.* 1.1 Describe (by the 46. of the first) vpon the line AB a square AD.* 1.2 Wherefore the square AD is rationall. And forasmuch as the line AB is vnto the line BC incommensurable in length, for they are supposed to be commensurable in power onely, and the line AB is equall vnto the line BD, therefore also the line•• BD is

vnto the line BC incommensurable in length. And 〈◊〉〈◊〉 ••h•• lin•• 〈…〉〈…〉 is to the line •• C, so 〈◊〉〈◊〉 the square AD to the parallelogramme AC (by the first of the fiu••). Wherefore (by the 10. of the tenth) the square DA is vnto the parallelogramme AC incommensurable. But the square DA is rationall. Wherefore the pa∣rallelogramme

[illustration]

AC is irrationall. Where∣fore also the line that contayneth the super∣ficies AC in power, that is whose square is equall vnto the parallelogramme AC, is (by the Assumpt going before) irrationall. And it is called a mediall line, for that the square which is made of it, is equall to that which is contayned vnder the lines AB and BC, and therefore it is (by the second part of the 17. of the sixt) a meane proportionall line betwene the lines AB and BC. A rectangle fi∣gure therefore comprehended vnder rationall right lines which are commensurable in power onely, is irrationall. And the line which in power contayneth that rectangle figure is irra∣tionall, and is called a mediall line.

At this Proposition doth Euclide first entreate of the generation and production of irrationall lines. And here he searcheth out the first kinde of them, which he calleth a mediall line. And the definition therof is fully gathered and taken out of this 21. Pro∣position, which is this. A mediall line is an irrationall line whose square is equall to a rectangled figure contayned of two rationall lines commensurable in power onely.* 1.3 It is called a mediall line, as Theon rightly sayth, for two causes, first for that the power or square which it produceth•• is equall to a mediall superficies or parallelogramme. For as that line which produceth a rationall square, is called a rationall line, and that line which produceth an irrationall square, or a square equall to an irrationall figure gene∣rally is called an irrationall line: so i•• tha•• line which produceth a mediall square, or a square equall to a mediall superficies, called by speciall name a mediall line. Secondly it is called a mediall line, because it is a meane proportionall betwene the two lines cō∣mensurable in power onely which comprehend the mediall superficies.

¶A Corollary added by Flussates.

A rectangle parallelogramme contayned vnder a rationall line and an ••rrationall line, is irrationall.* 1.4 For if the line AB be rationall, and if the line CB be irrationall, they shall be incommensurable. But as the line BD (which is equall to the line BA) is to the line BC, so is the square AD to the parallelogrāme AC. Wherefore the parallelogramme AC shall be incommensurable to the square AD which is ra∣tionall (for that the line AB wherupon it is described is supposed to be rationall). Wherefore the pa∣rallelogramme AC which is contayned vnder the rationall line AB, and the irrationall line BC, is irrationall.

¶An Assumpt.

If there be two right lines, as the first is to the second, so is the square which is described vpon the first to the parallelograme which is contained vnder the two right lines.

Suppose that there be two right lines AB and BC.* 1.5 Then I say that as the line AB is to the line BC, so is the square of the line AB, •••• that which is contained vnder the lines AB and BC. Describe (by the 46. of the first) vpon the line AB a square AD. And make perfect the parallelograme AC. Now for that as the line AB is to the line BC (for the line AB, is equall to the line BD); so is the square AD to the parallelograme CA by the first of the six••

and AD is the square which is made of the

[illustration]

line AB, and AC is that which is contained vnder the lines BD and BC, that is, vnder the lines AB & BC: therfore as the line AB is to the line BC, so is the square described vp¦pon the the line AB to the rectangle figure contained vnder the lines AB & BC. And conuersedly as the parallelograme which is contained vnder the lines AB and BC is to the square of the line AB, so is the line CB to the line BA.

Notes

• * 1.1

  Const••uction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  Diffinition of a mediall line.
• * 1.4

  A Corollary.
• * 1.5

  This assumpt is nothing els but a part of the first pro∣position of the sixt booke.