During a week in May 1977, solutions were found to the forty-year old problem of radiocarbon dating using mass spectrometry, as well as for the generalized mass spectrometry of measuring the abundances of extremely rare radioactive isotopes, such as long lived 236U. These solutions will be described together with a historical introduction, which points out the inherent advantages of some of the equipment developed for nuclear physics research that led towards the new mass spectrometry. In addition, the further developments, in what later came to be called accelerator mass spectrometry or AMS, can now be seen to have been the need to develop a better understanding of the following problems. (1) The appropriate atomic and molecular ion properties in order to expedite the separation of isobars. (2) The properties of the newer versions of the caesium sputter negative ion source that was hoped would almost eliminate the ion source memory effect. (3) The use of an improved accuracy tandem accelerator high voltage control for analysing very low final ion fluxes. Finally, (4) there was the need to assess the possible low level radiocarbon and other backgrounds of the accelerators or devices used to assist the isobar separation, after the accelerators had been used previously for many years of nuclear reaction studies. These developments were needed in order to add atomic and molecular isobar separation to the mass spectrometry for the further development of accelerator mass spectrometry (AMS) with tandem accelerators. When the basic problems had been solved, the detection and the accurate measurement of important natural very rare long-lived radio-active isotopes, such as for the sub parts per trillion (14C/C < 10−12) of natural 14C, was facilitated after further research. It was shown that there are very low levels of the atomic isobar anion 14N− from the caesium solid sample sputter ion source and that the 14C could be detected in the presence of over 1010 times as many mass 14 molecular anion hydrides 13CH−. The first and second excited states of the N anion, N(1S)− & N(1D)–, were shown later to exist but with short enough lifetimes for them not to interfere with the detection of 14C− at well below the level of 10−15 of 12C. The residual 14C background for an MP tandem accelerator, previously used for nuclear reaction studies, as well as the ion source memory effect were also found to be smaller than about 14C/C ~ 10−15 after the accelerator voltage control problem had been solved. These aims were all achieved during a week in May 1977 at the University of Rochester, leading also to the promise of the further successful development of radiocarbon dating by ion counting and the natural abundance measurements of other rare long lived radioactive isotopes by mass spectroscopy with anions or cations. In addition, the ability to date much smaller quantities of carbon, by a factor near 1000, was a huge advantage of the new method, as expected. However, it took almost a further decade of research and development to equal the accuracy of the mature beta counting radiocarbon dating method. The elimination of molecular ions from the mass spectrometry was later shown to be the key to detecting very low levels of actinides where atomic isobars are usually absent. The development of AMS for many other long lived radioactive isotopes is still continuing, as is the gradually increasing understanding of the need for an accelerator in some but not in all cases.