Abstract
About the middle of last century, there were invented several forms of multiple algebra, some of which are extensively used today in mathematical physics. The two types discussed here are quaternions, a quadruple algebra, and vector analysis, a triple algebra. The notations and forms of analysis used in the two systems are contrasted, using for each their representation of scalar and vector products and of Gauss's, Stokes's and Green's theorems. Maxwell made use of the vector part of quaternions in his Treatise on Electricity, and it was largely due to this that Gibbs came to invent his form of vector analysis. By 1890, about fifty years after the invention of quaternions, it became apparent that Gibbs's vector analysis was being used in place of quaternions, and this gave rise to a heated controversy in the pages of Nature. In the seventy years since this controversy, it is apparent that Gibbs's work and point of view have been completely vindicated.
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