The penny magazine of the Society for the Diffusion of Useful Knowledge, issue 31, September 29, 1832

The first part of this episode takes listeners into the heart of the British Museum, where the Portland—or Barberini—vase rests on a modest table, its deep blue glass gleaming beneath delicate white bas‑relief figures. The narrator describes the vase’s elegant, flower‑shaped profile and the astonishing precision of each line, a craftsmanship that still puzzles scholars. Its journey from a 16th‑century sarcophagus near Rome, through the hands of Sir William Hamilton and the Duke of Portland, is recounted with lively historical detail, including the famous plaster moulds that once tried to capture its form.

The discussion then shifts to an age‑old mathematical curiosity: squaring the circle. The speaker explains the problem—finding a square with the same area as a given circle—and why, despite centuries of attempts, a perfect geometric solution is impossible under modern mathematics. By illustrating the gap between practical approximations and theoretical limits, the segment invites listeners to appreciate both the beauty of ancient art and the enduring allure of unsolved puzzles.