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 beginning of the Senaries by Composition. ¶ The 2••. Theoreme. The 36. Proposition. If two rationall lines commensurable in power onely be added together:* 1.1 the whole line is irrationall, and is called a binomium, or a binomiall line.

〈…〉〈…〉

B and BC is incommensurable to the square of the

[illustration]

line BC. But vnto the parallelograme contained vnder the lines AB and BC is commensurable the parallelograme contained vnder AB and BC twise (by the 6. of the tenth): wherefore that which is contained vnder AB and BC twise is in∣commensurable to the square of the line BC (by the 13 of the tenth). But vnto the square of the line BC is commensurable that which is composed of the squares of the lines AB and BC (by the 15. of the tenth), for by supposition the lines AB and BC are commensurable in power onely. Wherefore (by the 13. of the tenth) that which is composed 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) that which is contained vn∣der AB and BC twise together with the squares of the lines AB and BC, which (by the 4. of the second) is equall to the square of the whole line AC, is incommensurable to that which is composed of the squares of AB and BC added together. But that which is composed of the squares of AB and BC added together is rationall, for it is commensurable to either of the squares of the lines AB and BG of which either of them is rationall by supposition: wherfore the square of the line AC is (by the 10. definition of the tenth) irrationall. Wherefore the line AC also is irrationall, and is called a binomiall line.

* 1.2This proposition sheweth the generation and production of the second kinde of ir∣rationall lines which is called a binomium, or a binomial line. The definition whereof is fully gathered out of this proposition, and that thus.

A binomium or a binomiall line, is an irrationall line composed of two rationall lines commensu∣rable the one to the other in power onely. And it is called a binomium, that is, hauing two names, because it is made of two such lines as of his partes which are onely commensu∣rable in power and not in length: and therefore ech part or line, or at the least the one of them, as touching length, is vncertaine and vnknowne. Wherefore being ioyned to∣gether their quantitie cannot be expressed by any one number or name, but ech part remayneth to be seuerally named in such sort as it may. And of these binomiall lines there are sixe seuerall kindes,* 1.3 the first binomiall, the second, the third, the fourth, the fifth, and the sixt, of what nature and condition ech of these is shalbe knowne by their definitious which are afterward set in their due place.

Notes

• * 1.1

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

  Diffinition of a binomiall line.
• * 1.3

  Sixe kindes of binomiall lines.