The progress of algebra in recent times may be considered under three principal heads: the study of fundamental laws and the birth of new algebras, the growth of the theory of equations, and the development of what is called modern higher algebra.
We have already spoken of George Peacock and D. F.
Gregory in connection with the fundamental laws of algebra.
Much was done in this line by De Morgan.
Augustus De Morgan (1806–1871) was born at Madura (Madras),
and educated at Trinity College, Cambridge. His scruples about the doctrines of the established church prevented him from proceeding to the M.A. degree, and from sitting for a fellowship. In 1828 he became professor at the newly established University of London, and taught there until 1867, except for five years, from 1831–1835. De Morgan was a unique, manly character, and pre-eminent as a teacher. The value of his original work lies not so much in increasing our stock of mathematical knowledge as in putting it all upon a thoroughly logical basis. He felt keenly the lack of close reasoning in mathematics as he received it. He said once: "We know that mathematicians care no more for logic than
logicians for mathematics. The two eyes of exact science are mathematics and logic: the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye; each believing that it can see better with one eye than with two." De Morgan saw with both eyes. He analysed logic mathematically, and studied the logical analysis of the laws, symbols, and operations of mathematics; he wrote a Formal Logic as well as a Double Algebra, and corresponded both with Sir William Hamilton, the metaphysician, and Sir William
Rowan Hamilton, the mathematician. Few contemporaries were as profoundly read in the history of mathematics as was De Morgan. No subject was too insignificant to receive his attention. The authorship of "Cocker's Arithmetic" and the work of circle-squarers was investigated as minutely as was
the history of the invention of the calculus. Numerous articles of his lie scattered in the volumes of the Penny and English Cyclopædias. His Differential Calculus, 1842, is still a standard work, and contains much that is original with the author. For the Encyclopædia Metropolitana he wrote on the calculus of functions (giving principles of symbolic reasoning)
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,