Bürgi's Kunstweg to Calculate Sines

[Thomas Klemm]

Double Difference

We can use the addition formualas to simplify the double difference:

\(
\sin(\alpha \pm \beta) = \sin(\alpha) \cos(\beta) \pm \cos(\alpha) \sin(\beta)
\)

This leads to:

\(
\begin{align}
\Delta^2 \sin(\alpha)
&:= \left[\sin(\alpha + \beta) - \sin(\alpha)\right] - \left[\sin(\alpha) - \sin(\alpha - \beta)\right] \\
&= \sin(\alpha + \beta) - 2 \sin(\alpha) + \sin(\alpha - \beta) \\
&= \sin(\alpha) \cos(\beta) + \cos(\alpha) \sin(\beta) - 2 \sin(\alpha) + \sin(\alpha) \cos(\beta) - \cos(\alpha) \sin(\beta) \\
&= 2 \sin(\alpha) \left(\cos(\beta) - 1\right) \\
\end{align}
\)

We notice that it is proportional to \( \sin(\alpha) \) independent of \( \alpha \).

In the limit of \( \beta \to 0 \) we get:
\(
\lim_{\beta \to 0} \frac{\cos(\beta) - 1}{\beta^2} = - \frac{1}{2}
\)

This leads to the 2nd derivative of the sine function:
\(
\sin(\alpha){''} = \lim_{\beta \to 0} \frac{\Delta^2 \sin(\alpha)}{\beta^2} = - \sin(\alpha)
\)

Conclusion

If we apply the double difference on a sequence \( \{a_j\} \) where \( j \in \{1, \cdots, n\} \) and the values are sines of angles in arithmetic progression, we get a sequence that is proportional to the original.

The proportional factor is:

\(
2 \left(\cos(\beta) - 1\right)
\)

Here \( \beta \) is the difference between consecutive angles.

However we have to consider two cases at the boundary.

Lower bound: j = 1

In this case we have:

\(
\alpha = \beta
\)

This leads to:

\(
\begin{align}
\sin(\alpha + \beta) - 2 \sin(\alpha) + \sin(\alpha - \beta)
&= \sin(2 \alpha) - 2 \sin(\alpha) + \sin(0) \\
&= 2 \sin(\alpha) \cos(\alpha) - 2 \sin(\alpha) \\
&= 2 \sin(\alpha) \left(\cos(\alpha) - 1 \right) \\
&= 2 \sin(\alpha) \left(\cos(\beta) - 1 \right) \\
\end{align}
\)

We end up with the same result as before.

Upper bound: j = n

In this case we have:

\(
\alpha = 90^\circ
\)

Here we calculate only the single difference:

\(
\sin(\alpha) - \sin(\alpha - \beta) = 1 - \cos(\beta)
\)

We notice that apart from the factor \( -2 \) we get the same result since \( \sin(\alpha) = 1 \).

Matrix Notation

We can therefore describe the double difference operation with the matrix \( \Delta \):

\(
\Delta = \left[\begin{matrix}
2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
-1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 2
\end{matrix}\right]
\)

Bürgi seems to have noticed that the result was less precise than before.
So he reversed the process.

\(
\Sigma = \Delta^{-1} = \left[\begin{matrix}
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \frac{1}{2}\\
1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 1\\
1 & 2 & 3 & 3 & 3 & 3 & 3 & 3 & \frac{3}{2}\\
1 & 2 & 3 & 4 & 4 & 4 & 4 & 4 & 2\\
1 & 2 & 3 & 4 & 5 & 5 & 5 & 5 & \frac{5}{2}\\
1 & 2 & 3 & 4 & 5 & 6 & 6 & 6 & 3\\
1 & 2 & 3 & 4 & 5 & 6 & 7 & 7 & \frac{7}{2}\\
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 4\\
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & \frac{9}{2}
\end{matrix}\right]
\)

Calculating the Sines

We can simply copy and paste the following matrices into Free42:

1 1 1 1 1 1 1 1 0.5
1 2 2 2 2 2 2 2 1
1 2 3 3 3 3 3 3 1.5
1 2 3 4 4 4 4 4 2
1 2 3 4 5 5 5 5 2.5
1 2 3 4 5 6 6 6 3
1 2 3 4 5 6 7 7 3.5
1 2 3 4 5 6 7 8 4
1 2 3 4 5 6 7 8 4.5

ENTER
ENTER
ENTER

2
4
6
7
8
9
10
11
12

×

63
124
181
232
276
312
339
356
362

×

2064
4065
5942
7638
9102
10290
11166
11703
11884

×

67912
133760
195543
251384
299587
338688
367499
385144
391086

×

2235060
4402208
6435596
8273441
9859902
11146776
12094962
12675649
12871192


Note: What appeared to be simple turned out to be a problem.
The separator in the matrix has to be a tabulator.
But I couldn't figure out how to do that when you copy it from this page.
It is always replaced by a blank.
Thus my recommendation for now is to copy the matrix into a text-editor, replace the blanks by tabs and copy the result into Free42.

References

• The mathematics behind Jost Bürgi’s method for calculating sine tables
• Jost Bürgi’s skillful computation of sines