Descartes's Logarithm Graph

Rene Descartes, in his Geometry (1637), invented a machine for solving the “mean proportionals” problem. In modern terms, that means finding a geometric sequence with n terms that starts at a number a and ends at b. See the GeoGebra file Descartes's Mean Proportionals Machine That machine can be repurposed to construct the graph of a logarithm function. (See David Dennis’s article  A logarithm table is constructed as a geometric sequence (x) paired with an arithmetic sequence (log(x)), then interpolated using square roots, repeatedly, to fill in the gaps. This spreadsheet file gives more information. This demonstration constructs a geometric sequence along the x axis using Descartes’s machine, constructs an arithmetic sequence along the y axis, then matches the values to give points on a logarithm graph.