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 (แอฟริกา) 1:

Stelling 6
As 'n rasionele Reguit-kaplyn is in Extreme en Gemiddelde
verhouding dan elk van die stukke is dat irrasionele (reguitlynmetode) genoem 'n Apotome.
Laat AB 'n Rasionele Reguitlyn Sny in Extreme en Gemiddelde verhouding by C en. laat AC wees die groter stuk. Ek sê dat AC en CB is dat elke irrasionele (Reguitlyn)
genoem 'n Apotome.
Want BA gewees, en laat AD gemaak (gelyk) aan helfte van BA. Daarom, omdat die reguit-lyn AB is Sny in Extreme en Gemiddelde verhouding by C,
en AD, wat is die helfte van die AB, is by die Groter stuk AC, die (vierkante) op CD is dus Vyf keer die (vierkant. ) op DA [Prop. 13.1]. Dus, die (vierkante) op
CD het om die (vierkante) op DA die verhouding wat 'n aantal (het) om 'n nommer. Die (vierkante) op CD (is), dus ge- meetbaar met die (vierkante) op DA [Prop. 10.6]. En
die (vierkante) op DA (is) Rational. Vir DA [is] rasionele, wat die helfte van AB, wat rasionele. Dus, die (vierkante) op CD (is) ook rasionele [Def. 10.4]. So, CD is ook rasionele. En aangesien die (vierkante) op CD nie hoef te die (vierkante) op DA die verhouding wat 'n vierkantige num-ber (het) om 'n vierkant getal, CD (is), dus incommensu- rable in lengte met DA [Prop. 10.9]. So, CD en DA is rasionele (reguit-lyne wat) meetbaar in net vierkant. So, AC is 'n apotome [Prop. 10,73].
Weereens, aangesien AB is in Sny Extreme en Gemiddelde verhouding, en AC is die Groter stukke, die (Reghoek Vervat) deur AB en BC is dus gelyk aan die (vierkante) op AC [Def. 6.3 Prop. 06:17]. Dus, die (vierkante) op die apotome AC, toegepas op die rasionele (reguitlyn) AB, BC maak as breedte. En die (vierkante) oor 'n Apotome, toegepas op 'n
Rasionele (Reguitlyn), Maak 'n Eerste Apotome as breedte [Prop. 10,97]. So, CB is 'n eerste apotome. En CA Wasalso getoon (om te wees) 'n Apotome
So, as 'n rasionele Reguitlyn is in Extreme Sny Mean verhouding en dan elk van die stukke is dat irrasionele (Reguitlyn) genoem 'n Apotome.