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 and on the theory of probability. Celebrated is his Budget of Paradoxes, 1872. He published memoirs "On the Foundation of Algebra" (Trans. of Cam. Phil. Soc., 1841, 1842, 1844, and 1847).

In Germany symbolical algebra was studied by Martin Ohm, who wrote a System der Mathematik in 1822. The ideas of Peacock and De Morgan recognise the possibility of algebras which differ from ordinary algebra. Such algebras were indeed not slow in forthcoming, but, like non-Euclidean geometry, some of them were slow in finding recognition. This is true of Grassmann's, Bellavitis's, and Peirce's discoveries, but Hamilton's quaternions met with immediate appreciation in England. These algebras offer a geometrical interpretation of imaginaries. During the times of Descartes, Newton, and Euler, we have seen the negative and the imaginary, $\scriptstyle{\sqrt{-1}}$,|undefined accepted as numbers, but the latter was still regarded as an algebraic fiction. The first to give it a geometric picture, analogous to the geometric interpretation of the negative, was H. Kühn, a teacher in Danzig, in a publication of 1750–1751. He represented $$\scriptstyle{a\sqrt{-1}}$$ by a line perpendicular to the line $\scriptstyle{a}$, and equal to $$\scriptstyle{a}$$ in length, and construed $$\scriptstyle{\sqrt{-1}}$$ as the mean proportional between $$\scriptstyle{+1}$$ and $\scriptstyle{-1}$. This same idea was developed further, so as to give a geometric interpretation of $\scriptstyle{a+\sqrt{-b}}$,|undefined by Jean-Robert Argand (1768–?) of Geneva, in a remarkable Essai (1806).[70] The writings of Kühn and Argand were little noticed, and it remained for Gauss to break down the last opposition to the imaginary. He introduced $$\scriptstyle{i}$$ as an independent unit co-ordinate to 1, and $$\scriptstyle{a+ib}$$ as a "complex number." The connection between complex numbers and points on a plane, though artificial, constituted a powerful aid in the further study of symbolic algebra. The mind required a visual representation to aid it. The notion of what we now call vectors was growing upon mathematicians,