Suggestions for Further Exploration

Greek ladders seem to beg for further exploration and generalization. For example, can a  Greek Ladder be defined that matches a sequence of Newton's  Method estimates exactly? Can a Greek Ladder be defined and associated with Newton's Method for cube roots, fourth roots, or fifth roots? (A Greek Ladder  that estimates fourth roots is defined in the next section.) Perhaps the reader can find some hidden treasures! Greek ladders are fun and accessible to students at many different levels.

Conclusion: A Little Story

It seems that Isaac Newton and his assistant Roger Cotes, who was also Newton's student (and, according to the Mathematics Genealogy Project, Newton only had two graduate students), were seeking in one of their research projects a simple rational approximation for \(\sqrt[4]{2}\). Newton came up with a couple of estimates that were not very good. But very quickly, Cotes offered the simple estimate of \(\frac{44}{37}\), which is surprisingly good for a two-digit denominator, in that \[ \left(  \frac{44}{37}\right)^{4}\approx1.999879 \] Cotes did not reveal how he got this, and many have tried to explain it, but no one seems to have thought of Greek Ladders. Here are just the first three steps of the classical  \(\sqrt[4]{2}\) Greek ladder, whose initial rung is \(\langle1\quad1\quad1\quad1\rangle\) and with each rung \(\langle a\quad b\quad c\quad d\rangle\) being followed by \(\langle a+b+c+d\quad2a+b+c+d\quad 2a+2b+c+d\quad2a+2b+2c+d\rangle\), with the approximations being \(\frac{d}{c}\). The first three rungs can be quickly written and are \[ \begin{array}{cccc} 1 & 1 & 1 & 1\\ 4 & 5 & 6 & 7\\ 22 & 26 & 31 & 37 \end{array} \] So \(\frac{7}{6}\) and \(\frac{37}{31}\) are two estimates, and their Farey mean is \[ \frac{7+37}{6+31}=\frac{44}{37} \] With such a simple computation, invoking the principle of Occam's Razor makes for a good guess that Cotes used this procedure. Cotes died young, and of him, Newton is reported to have said, "If Cotes had lived, we might have known something."

Addendum

Since this paper appeared, I have discovered how Cotes apparently arrived at his approximation of \(\frac{44}{37}\) for \(\sqrt[4]{2}.\) Cotes was seemingly quite attracted to continued fractions as a method for approximating algebraic irrationals. Indeed, according to David Fowler [8, p. 312], Cotes obtained his estimate for the fourth root of 2 by means of continued fractions, thus making my conjecture -- that he may have used a Greek ladder -- utterly false. My Greek ladder conjecture seemed plausible enough, but according to Fowler, it is simply wrong, and the error is regretful.

As a humorous finish, though, we point out that Cotes could have arrived at the famous fraction \(\frac{44}{37}\) even more simply by noting that Newton had offered as approximations to \(\sqrt[4]{2}\) the fractions \(\frac{6}{5}\), \(\frac{13}{11}\), and \(\frac{25}{21}\) [9], then computing their Farey mean; to wit, \(\frac{6+13+25%}{5+11+21}=\frac{44}{37}\).