# Using Leonardo of Pisa's table of Chords

I saw a problem asking to use Leonardo's table of chords to find the arc length cut off by a chord with length 8 rods, 3 ft, and 16 2/7 unciae in a circle of diameter 10.

I do not understand at all how to read Leonardo's table. Can anyone help explain?

This is a picture of Leonardo's table of chords:

• Tip: post an image (or a link to it), it would make your question far more interesting. Jan 5 at 1:37

Here is my guess. Consider a circle of the diameter 42. It's circumference would be 132. (Note that $$\pi \approx 132/42$$, also note that 132 feet is 8 rods, if we use Wikipedia's value for rod as 8.5 feet). Then the table shows how long is a chord of an arc with a given length. That is, what is $$f(x)=42\sin(\pi\frac{x}{132})$$?
The first column is $$x$$. The second column is just a supplementary angle of x (i.e. 132-x). The rest columns are the value of $$f(x)$$ in some units. It looks like the units that were used are feet (3rd column), feet/6 (3th column), feet/6/18 (4th column). I'm not sure about the last column, maybe it feet/6/18/18 or feet/6/18/20. While third and fourth column is calculated correctly (except obvious typographical mistakes in 3rd column in 55th and 56th rows, where it should be 40 instead of 41 - otherwise the values do not even monotonically increase), the last two columns contain errors. Here is my recalculation of this table in google docs.