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

11 And these lines whose poweres they are, are irrationall. If they be squares, then are their sides irrationall. If they be not squares,* 1.1 but some other rectiline figures, then shall the lines, whose squares are equall to these rectiline figures, be irrationall.

Suppose that the rationall

[illustration]

square be ABCD. Suppose also an other square, name¦ly the square E, which let be incōmēsurable to the ra¦tionall square, & therefore is it irrationall: and let the side or line which produ∣ceth this square be the line FG: then shall the line FG by this diffinition be an ir∣rationall line: because it is the side of an irrationall square. Let also the figure H being a figure on the one side longer (which may be any other rectiline figure rectangled or not rectangled, triangle, pentagone, trapezite, or what so euer ells) be incommensurable to the rationall square ABCD, then because the figure H is not a square, it hath no side or roote to produce it yet may there be a square made equall vnto it: for that all such figures may be reduced into triangles, and so into squares, by the 14. of the second. Suppose that the square Q be equall to the irrationall figure H. The side of which figure Q let be the line KL: then shall the line KL be also an irrational line, because the power or square thereof, is equal to the irrationall figure H: and thus conceiue of others the like.

These irrationall lines and figures are the chiefest matter and subiect, which is en∣treated of in all this tenth booke: the knowledge, of which is deepe, and secret, and pertaineth to the highest and most worthy part of Geometrie, wherein standeth the pith and mary of the hole science: the knowlede hereof bringeth light to all the bookes following, with out which they are hard and cannot be at all vnderstoode. And for the more plainenes, ye shall note, that of irrationall lines there be di••ers sortes and kindes. But they, whose names are set in a table here following, and are in number 13. are the [ 1] chiefe, and in this tēth boke sufficiently for Euclides principall purpose, discoursed on. [ 2]

• A mediall line. [ 3]
• A binomiall line. [ 4]
• A first bimediall line. [ 5]
• A second bimediall line. [ 6]
• A greater line. [ 7]
• A line containing in power a rationall superficies and a mediall superficies. [ 8]
• A line containing in power two mediall superficieces. [ 9]
• A residuall line. [ 10]
• A first mediall residuall line. [ 11]
• A second mediall residuall line. [ 12]
• A lesse line. [ 13]
• A line making with a rationall superficies the whole superficies mediall.
• A line making with a mediall superficies the whole superficies mediall.
• Of all which kindes the diffinitions together with there declarations shalbe set here after in their due places.

Notes

• * 1.1

  The eleuenth de••inition.