"Bürgi’s algorithm... reverses the process of forming second differences, i.e., performs up to sign some form of two-fold discrete integration—with the right normalization at the start and end of the sequence. Of course, our Perron-Frobenius eigenvector v of M is also an of M−1, but now for the smallest eigenvalue. Bürgi’s insight must have been that the study of iterations of M is much more useful than those of M−1 in order to approximate the entries of the critical eigenvector. ...[T]his process has unexpected stability properties leading to a quickly convergent sequence of vectors that approximate this eigenvector and hence the sine-values. The reason for its convergence is more subtle than just some geometric principle such as exhaustion, monotonicity, or , it rather relies on the equidistribution of a diffusion process over time—an idea which was later formalized as the and studied... in the theory of s. ...Bürgi’s insight anticipates some aspects of ideas and developments that came to full light only at the beginning of the 20th century."