¶The 31. Probleme. The 43. Proposition. A first bimediall line is in one poynt onely deuided into his names.
SVppose that AB be a first bimediall line, and let it be deuided into his partes in the point C, so that let the lines AC and CB be mediall cōmensurable in power onely, and containing a rationall superficies. Then I say that the line AB can not be deuided into his names in any other poynt then in C. For if it be possible let it be deuided into his names in the poynt D, so that let
AD &
DB be mediall lines commensurable in power onely, comprehending a rationall superficies. Now for∣asmuch as how much that which is contayned vnder the lines
AD and
DB twise di
••ferreth from that which is contayned vnder the lines
AC and
CB twise, so much differreth that which is composed of the squares of the lines
AD and
DB from that which is composed of the squares of the lines
AC and
CB: but that which is contayned vnder the lines
AD and
DB twise differreth from that which is contay¦ned vnder the lines
AC and
CB twise, by a rationall superficies (by the second assumpt go∣ing before the 41. of the tenth). For either of those superficieces is rationall. Wherefore that which is composed of the squares of the lines
AC and
CB differeth from that which is com¦posed of the squares of the lines
AD and
DB by a rationall superficies, when yet they are both mediall superficieces: which is impossible. Wherefore a first bimediall line is in one poynt onely deuided into his names: which was required to be proued.