How to Create Higher Exponential Powers Geometrically

Two Parts:The tutorialHelpful Guidance

This method of multiplication was used by Descartes and is from Euclid's "Elements", Book VI, Proposition 12. It is based on similar triangles. It may very well be the way Mother Nature accomplishes Multiplication! One imagines that Nature might be able to create straight lines through the emission of rapid vibrations through tightly packed particles or molecules. See the article Center a Circle and think how it might work in reverse in order to accomplish just this requirement.

However, this is only a theory, a possibility; Science knows that Nature accomplishes mathematical wonders, such as phyllotaxis, and growth patterns very much like fractal iterative patterns, but is still debating just how She accomplishes that! It's well worth thinking about and devising experiments and empirical evidence for proof.


  • Become familiar with the basic image and concept:
    Similar Triangles

Part 1
The tutorial

  1. 1
    Look at these similar triangles, and therefore the proportion {DG}/{DH}={DE}/{DF}. You can use it to perform multiplication and division. Open a new workbook in Excel and copy the drawing to experiment with it.
  2. 2
    Make horizontal line DH of length 1, extend DF by length x from DH and raise DG of length y at an angle above horizontal DF. Draw HG and construct a line through F parallel to HG. Let it intersect DG at E.
  3. 3
    Create exponential powers by merely making DG of length y from horizontal DF equal to DF of length x from DH. Then DE will have length x^2.
  4. 4
    Repeat the process for x^3, x^4, x^5, ..., using the new y length each time. For example, to obtain x^3, use the relation DE/DH = DC/DF which becomes DE/1 = DC/DF, which becomes DE*DF = DC, and since DE = x^2 and DF = x, then DC = x^3. See Diagram below. This works because the lines EH and CF are parallel and so the triangles are similar and Euclid's theorem holds.
  5. Image titled EUCLID EXPONENTIATION.png
  6. 6
    Obtain x^4 by connecting HC and drawing the parallel to HC up from F to point B extended on ray DC to create line FB. Use relation DC/DH = DB/DF; DC/1 = DB/DF; DC*DF = DB; thus x^3 * x = x^4.

Part 2
Helpful Guidance

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

Article Info

Categories: Mathematics | Geometry