# Understanding a medieval approximation

A medieval text (Maimonides's commentary to chapter 2 of Eruvin in my retranslation from the Hebrew) discusses a rectangle whose area is $5000$ square cubits. It reads in relevant part:

… that the length should not exceed the width except to such an extent that the diagonal is double the width. Then the length of the region will be $93\frac1{27}$…. All these reckonings are approximate….

"The diagonal is double the width" amounts to $\sqrt{L^2+W^2}=2W$ of course. The length $L$ is then $\sqrt{5000\sqrt3}\approx93.0605$, between $93\frac1{17}$ and $93\frac1{16}$.

Where might he have gotten $1/27$ from?

Note that I'm not asking "How could he be so far off?". I'm asking specifically about the number $93\frac1{27}$ and why he may have that as a solution as opposed to, say, $93\frac1{24}$ or $93\frac1{30}$.

• It could be that he was implicitly using successive divisions into thirds, and this was the appropriate approximation. Put another way, working in base $3$, $10110.001_3 = 93 + \tfrac{1}{27}$ is $\sqrt{5000\sqrt{3}}$ rounded down to the nearest $0.001_3$. – Travis Willse May 5 '16 at 13:52
• I'm not sure, answering that at best probably requires knowing something specific about the mathematical practices of his historical context. – Travis Willse May 5 '16 at 14:36
• Or he could have made an arithmetical error. What numerical system was he using? – DanielWainfleet May 6 '16 at 1:21
• AS I recall the gemara there uses a pretty good approximation of $\sqrt{2}$ but still off somewhat. Perhaps Rambam was using that value and therefore arrived at the figure you mention. I would suggest using the gemara's approximation of root 2 to see what you get. – Mikhail Katz Jun 30 '17 at 7:06
• Are you sure of this? The very little Arabic I know could be just enough to tell whether he wrote $1/3\times 1/9$ or $1/27$, but according to the Wikipedia entry for Mishneh Torah it's written "in Hebrew in the style of the Mishnah", even though "His previous works had been written in Arabic." – Noam D. Elkies Jul 4 '17 at 14:35