If as the whole (of a number) is to

If as the whole (of a number) is to the whole (of another), so a (part) taken away (is) to a (part) taken away,then the remainder will also be to the remainder as the whole (is) to the whole.
Let the whole AB be to the whole CD as the (part) taken away AE (is) to the (part) taken away CF. I say that the remainder EB is to the remainder FD as the whole AB (is) to the whole CD.
If as the whole (of a number) is to the whole (of another), so a (part) taken away (is) to a (part) taken away,then the remainder will also be to the remainder as the whole (is) to the whole.
Let the whole AB be to the whole CD as the (part) taken away AE (is) to the (part) taken away CF. I say that the remainder EB is to the remainder FD as the whole AB (is) to the whole CD.(For) since as AB is to CD, so AE (is) to CF, thus which(ever) part, or parts, AB is of CD, AE is also the same part, or the same parts, of CF [Def. 7.20]. Thus, the remainder EB is also the same part, or parts, of the remainder FD that AB (is) of CD [Props. 7.7, 7.8].Thus, as EB is to FD, so AB (is) to CD [Def. 7.20]. (Which is) the very thing it was required to show.

ผลลัพธ์ (อังกฤษ) 2:

If as the whole (of a number) is to the whole (of another), so a (part) taken away (is) to a (part) taken away, then the remainder will also be to the remainder as the whole (is. ) to the whole.
Let AB be the whole to the whole as the CD (Part) taken Away AE (is) to the (Part) taken CF. Away I Say that the remainder EB is to the remainder FD as the whole AB (is) to the whole CD.
If as the whole (of a Number) is to the whole (of another), so a (Part) taken Away (is. ) to a (Part) taken Away, then the remainder Will also be to the remainder as the whole (is) to the whole.
Let the whole AB be to the whole CD as the (Part) taken Away AE (is) to the. (part) taken away CF. I say that the remainder EB is to the remainder FD as the whole AB (is) to the whole CD. (For) since as AB is to CD, so AE (is) to CF, thus which (ever) part, or parts. , AB is of CD, AE is also the same part, or the same parts, of CF [Def. 7:20]. Thus, the remainder EB is also the same part, or parts, of the remainder FD that AB (is) of CD [Props. 7.7, 7.8] .Thus, as EB is to FD, so AB (is) to CD [Def. 7:20]. (Which is) the very thing it was required to show.

ผลลัพธ์ (อังกฤษ) 3:

If as the whole (of a number) is to the whole (of another), so a (part) taken away (is) to a (part), taken away then the. Remainder will also be to the remainder as the whole (is) to the whole.
Let the whole AB be to the whole CD as the (part). Taken away AE (is) to the (part) taken away CF. I say that the remainder EB is to the remainder FD as the whole AB (is). To the whole CD.
.If as the whole (of a number) is to the whole (of another), so a (part) taken away (is) to a (part), taken away then the. Remainder will also be to the remainder as the whole (is) to the whole.
Let the whole AB be to the whole CD as the (part). Taken away AE (is) to the (part) taken away CF. I say that the remainder EB is to the remainder FD as the whole AB (is). To the whole CD.(For) since as AB is, to CD so AE (is), to CF thus which (ever), parts part or, is of, AB CD AE is also the, same part or. The, same parts of CF [Def. 7.20]. Thus the remainder, EB is also the part same, parts or, the of remainder FD that AB (is). Of CD [Props. 7.7 7.8]. Thus, as EB, is, to FD so AB (is) to CD [Def. 7.20]. (Which is) the very thing it was required to. Show.
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