# 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.

## Steps

### Part 1 The Tutorial

- 1
**Erect AD at right angles to AB [XL.**], and make it equal to AB [III.]. - 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
**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

### Part 3 Helpful Guidance

- 1
**Make use of helper articles when proceeding through this tutorial:**- See the article How to Create a Spirallic Spin Particle Path or Necklace Form or Spherical Border for a list of articles related to Excel, Geometric and/or Trigonometric Art, Charting/Diagramming and Algebraic Formulation.
- For more art charts and graphs, you might also want to click on Category:Microsoft Excel Imagery, Category:Mathematics, Category:Spreadsheets or Category:Graphics to view many Excel worksheets and charts where Trigonometry, Geometry and Calculus have been turned into Art, or simply click on the category as appears in the upper right white portion of this page, or at the bottom left of the page.

## 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

- The First Six Books of the Elements of Euclid by John Casey; public domain resource - this eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at Gutenberg.
- http://www.britannica.com/EBchecked/topic/194880/Euclid

## Article Info

Categories: Geometry | Mathematics