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Proposition 46.

On a given straight line to describe 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;

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

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

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

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.

Q. E. F.


1 Proclus (p. 423, 18 sqq.) notes the difference between the word construct (συστἡσασθαι) 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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