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 13. Probleme. The 48. proposition. To finde out a first binomiall line.

* 1.1TAke two numbers AC and CB, & let them be such, that the number which is made of them both added together, namely, AB, haue vnto one of them, 〈◊〉〈◊〉, vnto BC that proportion that a square number hath to a square numb••r. ••ut vnto the other, namely, vnto CA let it not haue that proportion that a square number hath to a square number (such as is euery square num∣ber which may be deuided into a square number and into a number not square).* 1.2 Take also a certayne rationall line, and let the

[illustration]

same be D. And vnto the line D let the line EF be commensurable in length. Wherefore the line EF is ra¦tionall. And as the number AB is to the nūber AC, so let the square of the line EF be to the square of an other ••i••e, namely, of FG (by the co∣rollary of the sixt of y^e tēth). Wher∣fore the square of the line EF hath to the square of the line FG that proportion that number hath to number. Wherefore the square of the line EF is commensurable to the square of the line FG (by the 6. of this booke) And the line EF is rationall.* 1.3 Wherefore the line FG also is rationall. And forasmuch as the number AB hath not to the number AC, that proportion that a square number hath to a square number, neither shal the square of the line EF haue to the square of the line FG that proportion that a square number hath to a square number. Wherefore the line EF is incom∣mensurable in length to the line FG (by the 9. of this booke). Wherefore the lines EF and FG are rationall commensurable in power onely. Wherefore the whole line EG is a binomi∣all line (by the 36. of the tenth). I say also that it is a ••irst binomiall line. For for that as the 〈◊〉〈◊〉 BA is to the number AC, so is the square of the line EF to the square of the line ••G: but the number BA is greater then the number AC: wherefore the square of the line ••F is also greater then the square o•• the line FG. Vnto the square of the line EF let the squares of the lines FG and H be equall (which how to finde out is taught in the assumpt put

••ft••r the 13. of the t••nth). And f••r that as th•• number BA is to the number AC, so is the square of the line EF to the square of the line FG: therefore (by co••uersion or e••ersion of proportion (by the corollary of the 19. of the fift) as the number AB is to the number BC, so is the square of the line EF to the square of the line H. But the number AB hath to the num¦ber BC that proportion that a square number hath to a square number. Wherefore also the square of the line EF hath to the square of the line H that proportion that a square number hath to a square number. Wherefore the line EF is commensurable in length to the line H (by the 9. of this booke. Wherefore the line EF is in power more then the line FG by the square of a line commensurable in length to the line EF. And the lines EF and FG are ra∣tionall commensurable in power onely. And the line EF is commensurable in length to the rationall line D. Wherefore the line EG is a first binomiall line: which was required to be doone.

Notes

• * 1.1

  The third Se∣nary by com∣position.
• * 1.2

  Construction.
• * 1.3

  Demonstra∣ti••n.