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.
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) onCD 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]. Andthe (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].อีก เนื่อง จากมีการตัด AB ในอัตราส่วนที่มาก และหมายถึง AC เป็นชิ้นใหญ่ (เหลี่ยมอยู่) โดย AB และ BC ได้ดังเท่ากับ (ตาราง) บน AC [เริ่มต้น 6.3, Prop. 6.17] ดังนั้น (สี่เหลี่ยม) ใน apotome AC กับเชือด AB (เส้นตรง) ทำให้ BC เป็นความกว้าง และ (สี่เหลี่ยม) ในการ apotome กับการทำให้ (เส้นตรง), เชือด apotome แรกเป็นความกว้าง [Prop. 10.97] ดังนั้น CB เป็น apotome แรก CA wasalso แสดงการ) และการ apotome ดังนั้น ถ้าเป็นเชือดแบบเส้นตรงจะตัดในอัตราส่วนที่มาก และเฉลี่ย แล้วแต่ละชิ้นเป็นว่า จำนวนอตรรกยะ (เส้นตรง) เรียกว่า apotome เป็นการ
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