construct a square
==================

.. index:: construction, square

.. image:: elem.1.prop.46.png
   :align: right
   :width: 300px

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`. [^I.46:1]

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] <pb n="348"/>

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

[I.11]: /elem.1.11 "Book 1 - Proposition 11"
[I.31]: /elem.1.31 "Book 1 - Proposition 31"
[I.34]: /elem.1.34 "Book 1 - Proposition 34"
[I.29]: /elem.1.29 "Book 1 - Proposition 29"

## Footnotes

[^I.46:1]: construct
    Proclus (<xref n="Proc. p. 423, 18" from="ROOT" to="DITTO">p. 423, 18 sqq.</xref>) note the difference between the word <em>construct</em> (<foreign lang="greek">συστἡσασθαι</foreign>) applied by Euclid to the construction of a <em>triangle</em> (and, he might have added, of an <em>angle</em>) and the words <em>describe on</em> (<foreign lang="greek">ἀναγράφειν ἀπό</foreign>) used of drawing a square on a given straight line as one side. The <em>triangle</em> (or <em>angle</em>) is, so to say, pieced together, while the describing of a square on a given straight line is the making of a figure <quote>from</quote> <em>one</em> side, and corresponds to the multiplication of the number representing the side by itself.