Proposition 6 If a rational straigh

Proposition 6
If a rational straight-line is cut in extreme and mean
ratio then each of the pieces is that irrational (straight- line) called an apotome.
Let AB be a rational straight-line cut in extreme and mean ratio at C, and let AC be the greater piece. I say that AC and CB is each that irrational (straight-line)
called an apotome.
For let BA have been produced, and let AD be made (equal) to half of BA. Therefore, since the straight-line AB has been cut in extreme and mean ratio at C,
and AD, which is half of AB, has been added to the greater piece AC, the (square) on CD is thus five times the (square) on DA [Prop. 13.1]. Thus, the (square) on
CD has to the (square) on DA the ratio which a number (has) to a number. The (square) on CD (is) thus com- mensurable with the (square) on DA [Prop. 10.6]. And
the (square) on DA (is) rational. For DA [is] rational,being half of AB, which is rational. Thus, the (square) on CD (is) also rational [Def. 10.4]. Thus, CD is also rational. And since the (square) on CD does not have to the (square) on DA the ratio which a square num-ber (has) to a square number, CD (is) thus incommensu- rable in length with DA [Prop. 10.9]. Thus, CD and DA are rational (straight-lines which are) commensurable in square only. Thus, AC is an apotome [Prop. 10.73].
Again, since AB has been cut in extreme and mean ratio, and AC is the greater piece, the (rectangle contained) by AB and BC is thus equal to the (square) on AC [Def. 6.3,Prop. 6.17]. Thus, the (square) on the apotome AC, applied to the rational (straight-line) AB, makes BC as width. And the (square) on an apotome, applied to a
rational (straight-line), makes a first apotome as width [Prop. 10.97]. Thus, CB is a first apotome. And CA wasalso shown (to be) an apotome
Thus, if a rational straight-line is cut in extreme and mean ratio then each of the pieces is that irrational (straight-line) called an apotome.

ผลลัพธ์ (อังกฤษ) 2:

Proposition 6
If a Rational Straight-Cut line is in Extreme and Mean
ratio then each of the pieces is that irrational (Straight- line) Called an Apotome.
Let AB be a Rational Straight-line Cut in Extreme and Mean ratio at C, and. let AC be the greater piece. I Say that AC and CB is that each irrational (Straight-line)
Called an Apotome.
For Let BA have been produced, and Let AD be Made (Equal) to Half of BA. Therefore, since the Straight-line AB has been Cut in Extreme and Mean ratio at C,
and AD, which is Half of AB, has been added to the Greater piece AC, the (square) on CD is thus Five times the (square. ) on DA [Prop. 13.1]. Thus, the (square) on
CD has to the (square) on DA the ratio which a Number (has) to a Number. The (square) on CD (is) thus com- mensurable with the (square) on DA [Prop. 10.6]. And
the (square) on DA (is) Rational. For DA [is] rational, being half of AB, which is rational. Thus, the (square) on CD (is) also rational [Def. 10.4]. Thus, CD is also rational. And since the (square) on CD does not have to the (square) on DA the ratio which a square num-ber (has) to a square number, CD (is) thus incommensu- rable in length with DA [Prop. 10.9]. Thus, CD and DA are rational (straight-lines which are) commensurable in square only. Thus, AC is an apotome [Prop. 10.73].
Again, since AB has been in Cut Extreme and Mean ratio, and AC is the Greater piece, the (Rectangle Contained) by AB and BC is thus Equal to the (square) on AC [Def. 6.3, Prop. 6:17]. Thus, the (square) on the apotome AC, applied to the rational (straight-line) AB, makes BC as width. And the (square) on an Apotome, Applied to a
Rational (Straight-line), Makes a First Apotome as width [Prop. 10.97]. Thus, CB is a first apotome. And CA Wasalso shown (to be) an Apotome
Thus, if a Rational Straight-line is in Extreme Cut Mean ratio and then each of the pieces is that irrational (Straight-line) Called an Apotome.