It is written in two distinct parts.
Part I: Search By the Mule is about The Mule's search for the elusive Second Foundation, with the intent of destroying it. In the end the First Speaker of the Second Foundation telepathically modifies the Mule to make him not care about finding the Second Foundation. Search by the Mule was originally published in the January 1948 issue of Astounding Science Fiction under the title Now You See It—.
Part II: Search By the Foundation takes place sixty years after the first part, fifty-five years after the Mule's death (by natural causes). The members of the (First) Foundation are now fully aware that the Second Foundation is out there (they had known of its existence all along, but had not known its purpose or nature until the Mule). Also concurrent with this plot thread is the Foundation's ongoing conflict with the Mule's former imperial capital at Kalgan. The ensuing war is won by the Foundation, and is listed in the Encyclopedia Galactica as the last major conflict before the rise of the Second Empire.
After inventing a device that can jam telepathic abilities and can even be used to cause telepaths great pain, the Foundation finds and locates telepaths on Terminus, "at the other end of the galaxy" (from the First Foundation, also at Terminus). Since, as Arkady Darell puts it, "a circle has no end", then by tracing the disc of the galaxy around its edge, one would come back to Terminus. Thus, they declare the Second Foundation destroyed after finding the roughly 50 mentalist agents on Terminus, and are content to forget the matter. Finally, in response to the question "Where is the Second Foundation?", the First Foundation had found an answer that fit. However, although this was "the answer that satisfied", this was not "the answer that was true".
The Second Foundation was actually located on Trantor, at the centre of the galaxy. It was called "Star's End," due to the ancient saying that "All roads lead to Trantor, and that is where all stars end". The location was also said to fit the "other end of the galaxy" location since the galaxy is not in fact a disc, but a spiral — and from the edge, the other end of a spiral lies at the centre. The book also noted that Hari Seldon was a social scientist, not a physical one. When the two Foundations were founded, they could be described as being at opposite social ends of the Galaxy — with Trantor at the very center of galactic power and prestige, and Terminus at the other extreme.
