Harriot's calculation of the meridional parts as logarithmic tangents

Abstract

Many approximate solutions of the problem of constructing a true sea-chart or Mercator's projection were worked out in the sixteenth and seventeenth centuries. It was not until after the middle of the latter century that it became widely known that the solution was very closely related to logarithmic tangents. However, unpublished manuscripts in private possession in England show that the problem had been completely dealt with much earlier by Thomas Harriot (1560–1621), the contemporary of Napier and Briggs, who is best known in the history of mathematics for his work on algebra. Harriot was successively scientific and navigational adviser to Sir Walter Ralegh and under the patronage of the earl of Northumberland. As a young man he produced an approximate solution of the Mercator problem, which is now lost. He then developed theories of the conformality of stereographic projection, the rectification and quadrature of the equiangular (logarithmic) spiral, used the exponential series, devised interpolation formulae, and applied these results to the calculation of the so-called meridional parts (latitudes croissantes) used in the construction of a Mercator chart, which was most probably completed in 1614. Transcriptions of many of the relevant manuscripts are included in the article.