# The Root of the Matter: Approximating Roots with the Greeks - Further Exploration and References

Author(s):
Matthew Haines and Jody Sorensen (Augsburg University)

### Further Exploration

It's always interesting to analyze historical ideas using modern lenses, and this can lead to unexpected connections and insights. There are more areas to read about and explore regarding Theon's method. In 1951, Vedova [5] explored the method algebraically and made a connection to continued fractions. In their 1967 article, Waugh and Maxfield [6] provided a stimulating historical perspective of side-and-diagonal numbers including alternate ways to view the matrices and speculative evidence that Archimedes used side-and-diagonal numbers in his approximations of radicals. We plan to investigate the efficiency of Theon's method, and how the $2x^2=y^2 \pm 1$ equation varies when approximating other roots. Theon's method provides fertile ground to connect many seemingly unrelated areas of mathematics at an undergraduate level.

We have created and tested a class activity for Linear Algebra which explores Theon's method and convergence. (Download our Approximating Square Roots with Linear Algebra classroom activity here. The applet students are asked to use to assist them with step 8 of this activity is here on the Linear Algebra page of this article, just under Figure 5.)

### References

[1] David Flannery, The Square Root of 2: A Dialogue Concerning a Number and a Sequence, Copernicus (2006).

[2] Kurt Herzinger and Robert Wisner. Connecting Greek Ladders and Continued Fractions. MAA Convergence, 2014. DOI:10.4169/convergence20140102.

[3] Carl Huffman. Pythagoreanism. In Edward N. Zalta, editor, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, Winter 2016 edition. https://plato.stanford.edu/archives/win2016/entries/pythagoreanism/.

[4] Theon of Smyrna. Mathematics Useful for Understanding Plato (Robert and Deborah Lawlor, translators). Wizards Bookshelf, 1979.

[5] G. C. Vedova. Notes on Theon of Smyrna. The American Mathematical Monthly, 58, no. 10:675–683, 1951.

[6] Frederick V. Waugh and Margaret W. Maxfield. Side-and-diagonal Numbers. Mathematics Magazine, 40, no. 2:74–83, 1967.