“ The exact limitations of algebra are not generally agreed upon by mathematicians, and hence various definitions have been proposed for the science. It has been proposed to limit it to the theory of equations, as the etymology of the word would suggest; but this has become a separate branch of mathematics. Perhaps the most satisfactory definition, especially as it brings out the distinction between algebra and arithmetic, is that of Comte: "Algebra is the calculus of functions, and arithmetic is the calculus of values." ”

“ It was only after the opening of the nineteenth century that Abel, by the use of the theory of gi-oups discovered by Galois, gave the first satisfactory proof of the fact, anticipated by Gauss and announced by Rulfini, that it is im- possible to express the roots of a general equation as algebraic functions of the coefficients when the degree exceeds the fourth. ”