In July 1925 Heisenberg published a paper [Z. Phys. 33, 879-893 (1925)] 
which ended the period of `the Old Quantum Theory' and ushered in the 
new era of Quantum Mechanics. This epoch-making paper is generally 
regarded as being difficult to follow, perhaps partly because Heisenberg 
provided few clues as to how he arrived at the results which he reported. 
With David MacManus and Thomas Snyder we have reconstructed  details of calculations 
of the type which, we suggest, Heisenberg may have performed. Our main 
aim is to make his remarkable paper more accessible. We emphasize that 
the results are precisely those obtained in ordinary standard mechanics, 
and we suggest that some discussion of the approach - based on direct 
computation of transition amplitudes - could usefully be included in 
undergraduate courses in quantum mechanics. Our paper is published 
in  the American Journal of Physics: volume 72 (2004), 
pages 1370-1379.