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

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.

About the Authors

Matthew Haines is Professor of Mathematics at Augsburg University in Minneapolis, Minnesota. He was a member of the 1998-99 Institute for the History of Mathematics and Its Use in Teaching led by Victor Katz and Fred Rickey. If readers find themselves unexpectedly or otherwise in Minneapolis or neighboring areas, they are invited to stop by Matt's office for HoM conversation, chocolate, and an espresso. 

Jody Sorensen is Associate Professor of Mathematics at Augsburg University in Minneapolis, Minnesota. She's interested in dynamical systems, history of mathematics, and expository writing. This article brings Jody one step closer to her goal of completing projects with each of her departmental colleagues!