# How Jost Bürgi Computed the Sines of All Integer Angles in 1586

### Grégoire Nicollier

University of Applied Sciences of Western Switzerland

Switzerland

January 31, 2016

Jost Bürgi (1552-1632) was a Swiss clockmaker and autodidact mathematician who computed the very first table of logarithms in 1588. The present sine algorithm was lost until 2013 when Meno Folkerts discovered Bürgi's work *Fundamentum Astronomiae* in the university library of Wroclaw (former Breslau) in Poland [Folkerts et al].

*backwards*the vector of the partial sums of the modified vector; compute then

*forwards*the vector of the partial sums of the first step. Consider for example $\overrightarrow{v}=(1,\,2,\,3,\,4)$:

$ \begin{array}{rrrrrl} \overrightarrow{v}= &1&2&3&4\;\phantom{\leftarrow}&\\ &8&7&5&2\leftarrow&\\ &\rightarrow8&15&20&22\;\phantom{\leftarrow}&=\mathcal{B}(\overrightarrow{v}) \end{array}. $

If $\overrightarrow{v}$ is *any* nonzero $n$-dimensional vector with nonnegative coordinates, the iterated transform $\mathcal{B}^{(k)}(\overrightarrow{v})$ divided by its last coordinate converges for $k\to\infty$ to the vector $\left(\sin\ell\frac\pi{2n}\right)_{\ell=1}^{n}.\;$ (The proof, which was of course unknown to Bürgi, is shortly described below.)

For the initial vector

$\overrightarrow{v}=(2,\,4,\,6,\dots,\,60,\,61,\,62,\dots,\,119,\,120),$

which approximates $\left(120\sin\ell^\circ\right)_{\ell=1}^{90}$ with correct values at $30^\circ$ and $90^\circ$, $\mathcal{B}^{(4)}(\overrightarrow{v})$ divided by its last coordinate gives for example rational approximations of $\sin\ell^\circ$, $1\le\ell\le90$, that are all within $2\times10^{-7}$ of the exact values.

Here is the proof of the algorithm's convergence [Folkerts et al]. The Bürgi transform is the linear transformation $\mathcal{B}(\overrightarrow{v})=\overrightarrow{v} B$ given by the matrix

$\displaystyle B=\left(b_{ij}\right)_{1\le i,\,j\le n}=\left(\frac{\min(i,\,j)}{1+\left\lfloor \frac in\right\rfloor}\right)_{1\le i,\,j\le n}.$

Using the formula

$2\sin\alpha\sin\beta=\sin\left(90^\circ-\alpha+\beta\right)-\sin\left(90^\circ-\alpha-\beta\right)$

-- which was used before the logarithms to compute a product with a sum! -- one sees easily that $\left(\sin\ell\frac\pi{2n}\right)_{\ell=1}^n$ is a left eigenvector of $B.\;$ Now everything follows immediately from the Perron-Frobenius theorem about matrices with positive elements.

### References

- M. Folkerts, D. Launert, and A. Thom: Jost Bürgi Method for Calculating Sines, preprint, 17 pages, October 2015, https://arxiv.org/abs/1510.03180, accessed on January 31, 2016.

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