# 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].

We first describe the Bürgi transform $\mathcal{B}(\overrightarrow{v})$ of an $n$-dimensional real vector $\overrightarrow{v}$: in the first step, halve the last coordinate and compute 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

1. 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. 