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 numberCD (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
ThusIf a rational straight-line is cut in extreme and mean ratio then each of the pieces is that irrational (straight-line). Called an apotome.
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