Estimates of historical human population size
Although I cannot find the Scientific American article to which the poster refers, I assume that the “historic human population size” is the size of the human population in Africa before it underwent the expansion that accompanied the emergence from the African continent. Some important papers on this subject were published in the late 1990s (somewhat later than 30 years ago) and there is a suggestion in one of earlier estimates that I have not yet found a reference for.

There is an implication in the Harpending et al. paper (below) of earlier estimates of the order of 1,000. I will try to find the source (help on this welcomed).

Harpending et al. published a paper in PNAS of February 1998 in which they conclude an effective population size of the order of 10,000 individuals for most of the Pleistocene.

Reich and Goldstein published a paper in PNAS of July 1998 in which they conclude a maximum pre-expansion population size of 5,900.

A more recent study by Huff et al. published in PNAS in February 2010in which they estimated that the effective population size of human ancestors living before 1.2 million years ago was 18,500, and in which they rejected all models where the ancient effective population size was larger than 26,000


Why should the later estimate be more likely than the earlier ones?
Although the details of the calculations are complex (or at least too complex for me), it appears that the genetic markers that are used are crucial, and that it is important to have markers that represent rare mutational events. The earliest studies apparently used mitochondrial genes and the non-combining parts of the Y chromosome as markers. (These are used and are valid for studies of more recent time.) The limitations in these resulted in a shift to Alu repeats and microsatellites in the two 1998 papers, respectively. The 2010 paper used the insertion of the longer LINE repeats, which occur about one tenth as often as Alu insertions.
There is a Wikipedia article on this topic that may be also be of interest.