# How to Describe a Square on a Given Line AB

Euclid, in his treatise of geometry, "Elements", gave in Book I Proposition XLVI, the method "On a given line (AB) to describe a square." You will learn in this article to make a Euclidean square.

### Part 1 The Tutorial

1. 1
Erect AD at right angles to AB [XL.], and make it equal to AB [III.].
2. 2
Draw DC parallel to AB through D d[XXXI.], and draw BC through B parallel to AD, then AC is the square required.
3. 3
Demonstrate that because AC is a parallelogram, AB is equal to CD [XXXIV.]; and AB is equal to CD [XXXIV.]; and AB is equal to to AD (by construction); therefore AD is equal to CD, and AD is equal to BC [XXXIV.]. Hence the four sides are equal; therefore AC is a lozenge, and the angle A is a right angle. Therefore AC is a square (Def. XXX.).

### Part 2 Supporting Propositions

1.            1
The supporting Propositions are provided above, in reverse Proposition order (not by order of mention):

1. 1
Make use of helper articles when proceeding through this tutorial:

## Things You'll Need

• Sharp pencil and paper
• Rule
• Compass
• Microsoft Excel, or an application like it.
• Handy reference to Euclid's "Elements" would be nice.

## Article Info

Categories: Geometry | Mathematics