The primitive level of geodesy at the time the young Caspar Wessel became assistant surveyor is described. Fundamental geometric concepts were not yet developed and the shape of the Earth still debated. Many computations tacitly assumed a flat Earth. When Caspar Wessel took over the triangulation of Jutland he proposed a division of the country. His plan was approved by the Royal Danish Academy of Sciences and Letters at a meeting on February 5 1779. Although this plan is often mentioned, the content of it is not known. Inspection of the published maps, however, shows that it was a rational division, still in use, which allowed the sheets to be joined. This system is continued to the present day. The kind of applied mathematics which filled Wessel's professional life is indicated by Thomas Bugge's writings. A new geometric theorem shows simply, in terms of the Gauss curvature, the size of the error in regarding the Earth as flat. This "gap theorem" is invented for the present paper.
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