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PROPOSITION 46.

On a given straight line to describe[1] a square.

Let AB be the given straight line;

thus it is required to describe a square on the straight line AB.

Let AC be drawn at right angles to the straight line AB from the point A on it [I. 11], and let AD be made equal to AB;

through the point D let DE be drawn parallel to AB, and through the point B let BE be drawn parallel to AD. [I. 31] Therefore ADEB is a parallelogram;

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[I. 34]therefore AB is equal to DE, and AD to BE.

But AB is equal to AD;

therefore the four straight lines BA, AD, DE, EB are equal to one another;

therefore the parallelogram ADEB is equilateral.

I say next that it is also right-angled. For, since the straight line AD falls upon the parallels AB, DE,

[I. 29]the angles BAD, ADE are equal to two right angles.

But the angle BAD is right;

therefore the angle ADE is also right.

And in parallelogrammic areas the opposite sides and angles are equal to one another; [I. 34]

therefore each of the opposite angles ABE, BED is also right.

Therefore ADEB is right-angled.

And it was also proved equilateral.

Therefore it is a square; and it is described on the straight line AB.

[1]

Proclus (p. 423, 18 sqq.) notes the difference between the word construct (συστήσασθαι) applied by Euclid to the construction of a triangle (and, he might have added, of an angle) and the words describe on (ἀναγράφειν ἀπό) used of drawing a square on a given straight line as one side. The triangle (or angle) is, so to say, pieced together, while the describing of a square on a given straight line is the making of a figure “from” one side, and corresponds to the multiplication of the number representing the side by itself.

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See Commentary on Proposition I.46.