(A) ALGEBRA -- INTRODUCTION

ALGEBRA is that branch of the mathematical sciences which has for its object the carrying on of operations either in an order different from that which exists in arithmetic, or of a nature not contemplated in fixing the boundaries of that science. The circumstance that algebra has its origin in arithmetic, however widely it may in the end differ from that science, led Sir Isaac Newton to designate it "Univresal Aritmetic," a designation which, vague as it is, indicated its character better than any other by which it has been attempted to express its functions-better certainly, to ordinary minds, than the designation which has been applied to it by Sir William Rowan Hamilton, one of the greatest mathematicians the world has seen since the days of Newton - "the Science of Pure Time;" or even than the title by which De Morgan would paraphrase Hamilton's words - "the calculus of Succession."

To express in few words what it is which effects the transition from the science of arithmetic into a new field is not easy. It will serve, probably, to convey some notion of the position of the boundary line, when it is stated that the operation of arithmetic are all capable of direct interpretable only by comparison with the assumptions on which they are based. For example, multiplication of fractions - which the older writers on arithmetic, Lucas de Burgo in Italy, and Robert Recorde in England, clearly perceived to be a new application of the term multiplication, scarcely at first sight reconcilable with its original definition as the exponent of equal additions, - multiplication of fractions becomes interpretable by the introduction of the idea of multiplication into the definition of the fraction itself. On the other hand, the independent use of the sign minus, on which Diophantus, in the 4th century, laid the foundation of the science of algebra in the West, by placing in the forefront of his treatise, as one of his earliest definitions, the rule of the sign minus, "that minus multiplied by minus produced plus" - this independent use of the sign has no originating operation of the same character as itself, and might, if assumed in all its generality as existing side by side with the laws of arithmetic, more especially with the commutative law, have led to erroneous conclusions. As it as, the unlimited applicability of this definition, in connection with all the laws of arithmetic standing in their integrity, pushes the dominion of algebra into a field on which the oldest of the Greek arithmeticians, Euclid, in his unbending march, could never have advanced a step without doing violence to his convictions.

In asserting that the independent existence of the sign minus, side by side with the laws of arithmetic, might have led to anomalous results, had not the operations been subject to some limitation, we are introducing no imaginary hypothesis, but are referring to a fact actually existing. The most recent advance beyond the boundaries of algebra, as it existed fifty years ago, is that beautiful extension to which Sir W.R.Hamilton has given the designation of Quaternion, the very foundation of which requires the removal of one of the ancient axioms of arithmetic, "that operations may be performed in any order."





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