after heisenberg wrote his breakthrough paper, he gave it to one of his teachers to fix up and went on a vacation. max born was puzzled by the equations and the non-commuting equations that even heisenberg thought were a problem. after several days born realized that these equations were directions for writing out matrices. matrices were new and strange, even for mathematicians of that time, but how to do math with them was already clearly known. he and a few others worked everything out in matrix form before heisenberg came back from his time off, and within a few months the new quantum mechanics in matrix form gave them the basis for another paper.