Abstract: Graph paper and linkages both emerged from engineering practice before being recommended as tools of mathematics instruction. Graph paper achieved an unspectacular but steady place in the curriculum, while linkages have never found a secure niche, although occasionally inspiring great enthusiasm. This talk will survey this history from the eighteenth to the twenty-first century, noting especially the roles played by research mathematicians such as P. L. Chebyshev, J. J. Sylvester, and E. H. Moore.