Continuous and discontinuous proportion; dividing a line

[I]

Continuous.

2 4 8 16

[II]

Discontinuous.

2 4 3 6

Four line segments, arranged in parallel, and labeled c, d,1 a, and b. The last two segments are also indicated with 6 and 4 (both executed in the non-specular sense). Beside the latter two segments, the caption Lines. Two more line segments, arranged in parallel, and labeled e and f. The segments are indicated with 3 and 2 (both executed in the non-specular sense); and a caption to the left reads Numbers.2 [III]

…can be3 drawn from the given point to any of the said partitions, this line can likewise be divided exactly by each of these partitions.

[IV]

Let us say that the straight line that is to be proved is bf, that the given point is a, that the distance from the point to the end of the line is ab, and that the partitions b, c, d, e, and f, are each, of themselves, equal to ab. I say that line ac is twice the distance ab, that line ad is three times, that ae is four times, and that af is five times.

[I]

[II]

[IV]

[I]

[II]

Parallel segments captioned "Lines" (c, d, a, b) and "Numbers" (e, f) [III]

[IV] The proof that ac, ad, ae and af are 2, 3, 4 and 5 times a–b

Continuous and discontinuous proportion; dividing a line

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Continuous versus discontinuous proportion

Dividing line b–f from the point a