How to Describe a Square on a Given Line AB

Three Parts:The TutorialSupporting PropositionsHelpful Guidance

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

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    Erect AD at right angles to AB [XL.], and make it equal to AB [III.].
  2. Image titled Describe a Square on a Given Line AB Step 2
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    Draw DC parallel to AB through D d[XXXI.], and draw BC through B parallel to AD, then AC is the square required.
  3. Image titled Describe a Square on a Given Line AB Step 3
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    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

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

Part 3
Helpful Guidance

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    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.

Sources and Citations

Article Info

Categories: Geometry | Mathematics