There are certain well-known linkages for effecting the transformation of inversion, and, incidentally, by inverting a suitably situated circle, for producing straight-line motion. Reference may be made to the classic lecture by A. B. Kempe, “How to draw a straight line” (London: Macmillan, 1877). It is the object of this paper to call attention to the fact that these linkages have the corresponding property in non-euclidean geometry.