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 29. Theoreme. The 41. Proposition. If two right lines incommensurable in power be added together, hauyng that which is composed of the squares of them added together mediall, and the parallelogramme contayned vnder them mediall, and also incommen∣surable to that which is composed of the squares of them added together•• the whole right line is irrationall, and is called a line contayning in power two medials.

LEt these two right lines AB and BC being incommensurable in power, and hauing that which is composed of the squares of the lines AB and BC mediall, and the pa∣rallelogramme which is contayned vnder the lines AB and BC mediall, and also incommensurable to that which is composed of the squares of the lines AB and BC added together,* 1.1 be added together (the 35. of the tenth teacheth to finde out two such lines) Then I say that the whole line AC is irrationall. Take a rationall line DE, and (by ••he 44. of the first) vpon the line DE apply the parallelogramme DF equall to the squares of the lines AB and BC. And vpon the line GF, which is equall to the line DE, apply the parallelogrāme GH equall to that which is contained vnder the lines AB and BC tw••se. Wherefore the

whole parallelogramme DH is equall to the square of the line AC (by the 4. of the second.* 1.2) And forasmuch as that which is composed of the squares of the lines AB and BC is mediall and is equall to the parrllelogramme DF: therfore DF also is mediall (by that which was said in the 38. proposition of this booke). And it is

[illustration]

applied vpon the rationall line DE. Wherefore the line DG is rationall and incommensurable in lēgth vnto y^e line DE (by the 22. of the tēth). And by the same reason the lyne GK is rational & incōmesura∣ble in length vnto the line GF, that is, vnto the line DE. And forasmuch as that which is composed of y^e squares of the lines AB & BC added together, is by supposition incōmensurable to that which is cōtained vnder the lines AB and BC twise, therfore also the parallelogramme DF is incommensurable vnto the parallelogramme GH. Wherfore also the line DG is incommensurable vnto the line GK (by the first of the sixt) and by the tenth of the tenth. But it is now proued that they are rationall. Wherfore the lines DG and GK are rationall commensurable in power onely. Wherfore (by the 36. of the tenth) the whole line DK is rationall, and is called a binomiall line, but the line DE is irrationall. Wherfore the parallelogramme DH is irrationall (by the corollary added after the 22. propositiō of the tenth). Wherfore also the line which containeth it in power is irrationall: but the line AC containeth it in power. Wherfore the line AC is irrationall, and is called a line contayning in power two medials. It is called a line containing in power two medials, for that it contay∣neth in power two mediall supersicieces; one of which is composed of the squares of the lynes AB and BC added together; and the other is that which is contained vnder the lines AB & BC twise: which was required to be demonstrated.

In this proposition is taught the nature of the 7. kinde of irrationall lines which is called a line whose power is two medials. The definition whereof is taken of this pro∣position after this maner.

A line whose power is two medials, is an irrationall line which is composed of two right lines in∣commensurable in power,* 1.3 the squares of which added together, make a mediall superficies, and that which is contained vnder them is also mediall, and moreouer it is incommensurable to that which is composed of the two squares added together.

The reason why this line is called a line whose power is two medials, was before in the ende of the demonstration declared.

And that the said irrationall lines are deuided one way onely, that is, in one point onely, into the right lines of which they are composed, and which make euery one of the kindes of those irrationall lines, shall straight way be demonstrated: but first will we demonstrate two assumptes here following.

¶An Assumpt.

Take a right line and let the same be AB, and deuide it into two vnequall partes in the point C,* 1.4 and againe deuide the same line AB into two other vnequal partes, in an other point namely, in D, and let the line AC (by supposition) be greater then the line DB. Then I say

that the squares of the lines AC and BC added together, are greater then the squares of the lines AD and DB added together. Deuide the line AB (by the 10. of the first) into two e∣quall partes in the point E. And forasmuch as the line AC is greater then the line DB, take away the line DC which is common to them both: wherfore the residue AD is greater then the residue CB, but the line AE is equall to the line EB. Wherfore the line DE is lesse then the line EC. Wherfore the pointes C and D are not

[illustration]

equally distant from the point E, which is the point of the section into two equall partes. And foras∣much as (by the 5. of the second) that which is con∣tayned vnder the lines AC and CB together with the square of the line EC is equall to the square of the line EB. And by the same reason that which is contayned vnder the lynes AD and DB together with the square of the line DE, is also equall to the self same square of the line EB: wherfore that which is contained vnder the lines AC and CB together with the square of the line EC is equall to that which is contained vnder the lines AD and DB to∣gether with the square of the line DE: of which the square of the line DE is lesse then the square of the line EC (for it was proued that the line DE is lesse then the line EC). Wher∣fore the parallelogramme remayning, contayned vnder the lines AC and CB is lesse thē the parallelogramme remayning contayned vnder the lines AD and DB. Wherfore also that which is contayned vnder the lines AC and CB twise is lesse then that which is contayned vnder the lines AD and DB twise. But (by the fourth of the second) the square of the whole line AB is equall to that which is composed of the squares of the lines AC and CB toge∣ther with that which is contained vnder the lynes AC and CB twise, and by the same reason the square of the whole line AB is equall to that which is composed of the squares of the lines AD and DB together with that which is contayned vnder the lynes AD and DB twise: wherfore that which is composed of the squares of the lynes AC and CB together with that which is contayned vnder the lynes AC and CB twise, is equall to that which is composed of the squares of the lynes AD and DB, together with that which is con∣tayned vnder the lynes AD and DB twise. But it is already proued that that which is con∣tayned vnder the lynes AC and CB twise, is lesse then that which is contayned vnder the lines AD & DB twise. Wherfore the residue, namely, that which is composed of the squares of the lines AC and CB is greater then the residue, namely, then that which is composed of the squares of the lines AD and DB: which was required to be demonstrated.

¶An Assumpt.

A rationall superficies exceedeth a rationall superficies, by a rationall su∣perficies.

Let AD be a rationall superficies, and let it exceede AF

[illustration]

being also a rationall superficies by the superficies ED. Then I say that the superficies ED is also rationall. For the paral∣lelogramme AD is commensurable to the parallelogramme AF, for that either of them is rationall. Wherefore (by the second part of the 15. of the tenth) the parallelogramme AF is commensurable to the parallelogramme ED. But the the parallelogramme AF is rationall. Wherfore also the pa∣rallelogramme ED is rationall.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  Diffinition of a li••e contai∣ning in power two medials.
• * 1.4

  An Assumpt.