The extension which is required to be given to the meaning of algebraic addition and subtraction, for the purpose of representing a vector by a polynomial expression (by the algebraic sum, namely, of components differing in direction from one another), renders it necessary to investigate into the question of what the geometrical signification of the result of the other elementary operations of algebra will become when these operations are performed on those polynomial expressions?

In February 1831 the late Mr Witham read to the Royal Society of Edin burgh, a paper of much interest on two fossil trees of great size which had been brought to light, the one in 1826, and the other in 1830, during the excavations carried on in Craigleith sandstone quarry, in the immediate neighbourhood of our city. In March of the same year this paper, with the addition of several chemical analyses of the fossils, was read also before the Natural History Society of Northumberland. In 1833 he included his observations on these fossils in a separate and more comprehensive treatise on “The Internal Structure of Fossil Vegetables found in the Carboniferous and Oolitic Deposits of Great Britain.” The main purpose of Mr Witham's researches—in addition to an accurate description and delineation of natural objects previously little investigated—was to show that fossil vegetables in the oolitic and carboniferous formations were not, as had been generally supposed prior to his researches, always ferns, tree-ferns, lycopodiums, and other acrogenous plants, but on the contrary many of them woody exogenous trees, showing longitudinal ligneous bundles, transverse medullary rays, concentric annual layers, and other characters belonging to the intimate organisation of our existing forest trees. Mr Witham even went so far as to identify the structure of the Craigleith fossils with that of our modern pines, and to assign them to two separate species.