Dear Professor Greitzer - Introduction

Editor’s note: This article was published in 2006.

Authors' Note: Professor Sam Greitzer was a giant among the U.S. math community, as an author, as a teacher, and as a coach for the US Math Olympiad team for much of the second half of the 20th century. We knew him through the Arbelos, a mathematics journal “produced for precollege philomaths” which he published and for which he wrote most of the articles. Although the journal was filled with an amazingly wide spectrum of mathematics jewels, his favorite field of play was Geometry, and every issue had something interesting in that valley. His death was a great loss for the philomath community. This article is in tribute to those contributions, the best way we know how to show him our respect. (For more about Professor Greitzer, click here.)

Dear Professor Greitzer,

In your articles you lament the state of Geometry education nowadays; and perhaps you’re right, Professor. Our classes may not know the difference between Ceva’s Theorem and Ptolemy’s Theorem. But we’re explorers, and that’s central to mathematics. We are writing to tell you a tale that will warm your heart, and a tale that will perhaps give us all some insight into the thought processes of some creative minds of the mathematical past. We do so at your request, quoting you on page 15 of the January 1984 edition of the Arbelos in an article demonstrating methods of capturing the value of \(\pi\):

Other mathematicians have used other formulas [to approximate \(\pi\)]. For example, Dase based his evaluation on the formula

$${{\pi}\over{4}}=\arctan{\frac{1}{2}}+\arctan{\frac{1}{5}} +\arctan{\frac{1}{8}}\quad {\rm !}$$

... and Machin used the formula

$${\frac{\pi}{4}}=4\arctan{\frac{1}{5}}-\arctan{\frac{1}{239}} \quad {\rm !!!}$$

One wonders how these mathematicians derived their initial formulas (hence the exclamation points).

One can find biographies of Zacharias Dase and of John Machin in the MacTutor History of Mathematics Archive.

Dase was not a mathematician, but rather a savant, hired to utilize formulae like that above to generate useful tables of values. Machin was a renowned mathematician of the early 1700s, and very likely discovered the series formula ascribed to him above, although proofs of its correctness are attributed instead to Abraham De Moivre, Johann Bernoulli, and Jakob Hermann.

We’d like to show you a path our Geometry classes walked that effectively discovers these formulae, and uses only elementary geometric tools, which we know will appeal to you, since you frequently eschewed the Calculus when simpler tools would get the job done.