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 32. Theoreme. The 44. Proposition. A second bimediall line is in one poynt onely deuided into his names.

SVppose that the line AB being a second bimediall line, be deuided into hys names in the poynt C: so that let the lines AC and CB be mediall lines com∣mensurable in power onely, comprehending a mediall superficies. It is manifest that the poynt C deuideth not the whole line AB into two equall partes.* 1.1 For the lines AC and CB are not commensurable in length the one to the other. Now I say that the line AB cannot be deuided into his names in any other poynt but onely in C. For if it be possible, let it be deuided into his names in the poynt D, so that let not the line AC be one and the same, that is, let it not be equall, with the line DB. But let it be greater then it. Now it is manifest (by the first assumpt going before the 42. proposition of this booke) that the squares of the lines AC and CB are greater then the squares of the lines AD and DB. And also that the lines AD and DB are mediall lines commensurable in power onely, comprehending a mediall supersicies. Take a rationall line EF. And (by the 44. of the first) vpon the line EF apply a rectangle parallelograme EK equall to the square of the line AB. From which pa∣rallelograme take away the parallelograme EG equall to the squares of the lines AC and CB Wherefore the residue, namely, the parallelograme HK is equall to that which is contai∣ned

vnder the lines AC and CB twise. Agayne from the the parallelograme EK take away the parallelograme EL equall to the squares of the lines AD and DB which are lesse then the squares of the lines AC and CB. Wherefore the re∣sidue,

[illustration]

namely, the parallelograme MK is equall to that which is contayned vnder the lines AD and DB twise. And forasmuch as the squares of the lines AC and CB are mediall, therefore the parallelograme EG also is me∣diall.* 1.2 And it is applyed vpon the rationall line EF•• where∣fore the line EH is rationall and incommensurable in length to the line EF. And by the same reason, the parallelograme HK is mediall (for that which is e∣quall vnto it, namely, that which is contayned vnder the lines AC and CB twise is mediall) therefore the line HN is also rationall and incommensurable in length vn∣to the line EF. And forasmuch as the lines AC and CB are mediall lines commensurable in power onely, therefore the line AC is incommensurable in length vnto the line CB. But as the line AC is to the line CB, so is the square of the line AC to that which is contayned vnder the lines AC and CB (by the 1. of the sixt). Wherefore the square of the line AC is incommensurable to that which is contayned vnder the lines AC and CB. But (by the 16. of the tenth) vnto the square of the line AC are commensurable the squares of the lines AC and CB added to∣gether, for the lines AC and CB are commensurable in power onely. And vnto that which is contayned vnder the lines AC and CB is commensurable that which is contayned vn∣der the lines AC and CB twise. Wherefore that which is composed of the squares of the lines AC and CB, is incommensurable to that which is contained vnder the lines AC and CB twise. But to the squares of the lines AC and CB is equall the parallelograme EG, and to that which is contained vnder the lines AC and CB twise is equall the para∣lelograme HK. Wherfore the parallelograme EG is incommensurable to the parallelograme HK. Wherefore also the line EH is incommensurable in length to the line HN. And the lines EH and HN are rationall. Wherefore they are rationall commensurable in power one∣ly: but if two rationall lines commensurable in power onely be added together, the whole line is irrationall, and is called a binomiall line (by the 36. of the tenth). Wherefore the binomiall line EN is in the poynt H deuided into his names. And by the same reason also may it be proued that the lines EM and MN are rationall lines commensurable in power onely. Wher¦fore EN being a binomiall line is deuided into his names in sundry poyntes, namely, in H and M, neither is the line EH one and the same, that is, equal with MN. For the squares of the lines AC and CB are greater then the squares of the lines BD and AD (by the 1. as∣sumpt put after the 41. of the tenth). But the squares of the lines AD and DB are greater then that which is contayned vnder the lines AD and DB twise (by the assumpt put after the 39. of the tenth). Wherefore the squares of the lines AC and CB, that is, the parallelo∣grame EG is much greater then that which is contained vnder the lines AD and DB twise, that is then the parallelograme MK. Wherfore (by the first of the sixt) the line EH is greater then the line MN. Wherefore EH is not one and the same with MN. Wherefore a binomiall line is in two sundry poyntes deuided into his names. Which is impossible. The selfe same absurditie also will follow if the line AC be supposed to be lesse then the line DB. A second binomiall line therefore is not deuided into his names in sundry poyntes. Wherefore it is deuided in one onely: which was required to be demonstrated.

Notes

• * 1.1

  Demonstratiō leading to an impossibili••ie.
• * 1.2

  Demonstrati∣on leading to an imp••ssibi∣litie.