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not going especially well', Heisenberg wrote to his father at the end of June. A little more than a week later, he had finished the paper that ushered in a new era in quantum physics. Still uncertain about what he had done and its true significance, Heisenberg sent a copy to Pauli. Apologising, he asked him to read and return the paper within two or three days. The reason for the haste was that Heisenberg was due to give a lecture at Cambridge University on 28 July. With other commitments he was unlikely to return to Göttingen until late September and wanted 'either to complete it in the last days of my presence here or to burn it'.43 Pauli greeted the paper 'with jubilation'.44 It offered, he wrote to a colleague, 'a new hope, and a renewed enjoyment of life'.45 'Although it is not the solution to the riddle,' Pauli added, 'I believe that it is now once again possible to move forward.' The man who took those steps in the right direction was Max Born.

He had little inkling of what Heisenberg had been doing since returning from the little island in the North Sea. Born was therefore surprised when Heisenberg gave him the paper and requested that he decide whether it was worth publishing or not. Tired by his own exertions, Born put the paper to one side. When a couple of days later he sat down to read it and pass judgement on what Heisenberg had described as a 'crazy paper', Born was immediately captivated. He realised that Heisenberg was being uncharacteristically hesitant in what he was putting forward. Was it a consequence of having to employ a strange multiplication rule? Heisenberg was still groping even at the conclusion of the paper: 'Whether a method to determine quantum-mechanical data using relations between observable quantities, such as that proposed here, can be regarded as satisfactory in principle, or whether this method after all represents far too rough an approach to the physical problem of constructing a theoretical quantum mechanics, an obviously very involved problem at the moment, can be decided only by a more intensive mathematical investigation of the method which has been very superficially employed here.'46

What was the meaning of the mysterious multiplication law? It was a question that so obsessed Born, he could think of little else during the days and nights that followed. He was troubled by the fact that there was something vaguely familiar about it, but he could not pinpoint exactly what. 'Heisenberg's latest paper, soon to be published, appears rather mystifying, but is certainly true and profound', Born wrote to Einstein, even though he was still unable to explain the origin of the strange multiplication.47 Praising the young physicists at his institute, especially Heisenberg, Born admitted 'that merely to keep up with their thoughts demands at times considerable effort on my part'.48 After days of considering nothing else, the effort on this occasion was rewarded. One morning, Born suddenly recalled a long-forgotten lecture he had attended as a student and realised that Heisenberg had accidentally stumbled across matrix multiplication in which X times Y does not always equal Y times X.

On being told that the mystery of his strange multiplication rule had been solved, Heisenberg complained that 'I do not even know what a matrix is'.49 A matrix is nothing more than an array of numbers placed in a series of rows and columns, just like the arrays that Heisenberg constructed in Helgoland. In the mid-nineteenth century the British mathematician Arthur Cayley had worked out how to add, subtract, and multiply matrices. If A and B are both matrices, then A×B can yield a different answer from B×A. Just like Heisenberg's array of numbers, matrices do not necessarily commute. Although they were established features of the mathematical landscape, matrices were unfamiliar territory for the theoretical physicists of Heisenberg's generation.

Once Born had correctly identified the roots of the strange multiplication, he knew that he needed help to turn Heisenberg's original scheme into a coherent theoretical