Proposition I.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. 1

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 AB, DE,

  • 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.

References

Footnotes

  1. construct
    Proclus (p. 423, 18 sqq.) note 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. ↩