Kitap 10
87
But E has not to CD the ratio which a square number has to a square number; therefore neither has the square on A to the square on GH the ratio which a square number has to a square number; therefore A is incommensurable in length with GH. [X. 9]
87
Therefore neither of the straight lines FG, GH is commensurable in length with the rational straight line A set out.
87
Now let the square on K be that by which the square on FG is greater than the square on GH.
87
Since then, as BC is to CD, so is the square on FG to the square on GH, therefore, convertendo, as BC is to BD, so is the square on FG to the square on K. [V. 19, Por.]
87
But BC has to BD the ratio which a square number has to a square number; therefore the square on FG also has to the square on K the ratio which a square number has to a square number.
87
Therefore FG is commensurable in length with K, [X. 9] and the square on FG is greater than the square on GH by the square on a straight line commensurable with FG.
87
And neither of the straight lines FG, GH is commensurable in length with the rational straight line A set out; therefore FH is a third apotome. [X. Deff. III. 3]
87
Therefore the third apotome FH has been found. Q. E. D.
PROPOSITION 88.
88
To find the fourth apotome.
88
Let a rational straight line A be set out, and BG commensurable in length with it; therefore BG is also rational.
88
Let two numbers DF, FE be set out such that the whole DE has not to either of the numbers DF, EF the ratio which a square number has to a square number.
