Linear Associative Algebra

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Van Nostrand, 1882 - Algebra, Universal - 133 pages

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Page 2 - Mathematics," according to Prof. Peirce, "belongs to every inquiry, moral as well as physical. Of some sciences, it is so large a portion that they have been quite abandoned to the mathematician. Such is the case with geometry and analytic mechanics. But in many other sciences, as in all those of mental philosophy and most of the branches of natural history, it is of no practical value [at present] to separate the mathematical portion and subject it to isolated discussion.
Page 2 - Mathematics, under this definition, belongs to every enquiry, moral as well as physical. Even the rules of logic, by which it is rigidly bound, could not be deduced without its aid. The laws of argument admit of simple statement, but they must be curiously transposed before they can be applied to the living speech and verified by observation. In its pure and simple form the syllogism cannot be directly compared with all experience, or it would not have required an Aristotle to discover it. It must...
Page 121 - A very general form of a vid of inversion is (A:A) (B:B) (C:C) &c., in which each doubtful sign corresponds to two cases, except that at least one of the signs must be negative. The negative of unity might also be regarded as a symbol of inversion, but cannot take the place of an independent vid. Besides the above vids of inversion, others may be formed by adding to either of them a vid consisting of two different letters, which correspond to two of the one-lettered vids of different signs...
Page 121 - Thus (A:A) + (B.:B)-(C:C)+x(A:C)+y(B:C) is a vid of inversion. The new vid which Professor Clifford has introduced into his biquaternions is a vid of inversion. SEMI-INVERSION. A vid of which the square is a vid of inversion, is a vid of semiinversion. A very general form of a vid of semi-inversion is (A: A) (B:B) ^/-l(C:C) &c.
Page 1 - Mathematics is the science which draws necessary conclusions. This definition of mathematics is wider than that which is ordinarily given, and by which its range is limited to quantitative research. The ordinary definition, like those of other sciences, is objective; whereas this is subjective. Recent investigations, of which quaternions is the most noteworthy instance, make it manifest that the old definition is too restricted. The sphere of mathematics is here extended, in accordance with the derivation...
Page 1 - Mathematics is not the discoverer of laws, for it is not induction; neither is it the framer of theories, for it is not hypothesis; but it is the judge over both, and it is the arbiter to which each must refer its claims; and neither law can rule nor theory explain without the sanction of mathematics.
Page 3 - BOOK I* THE LANGUAGE OF ALGEBRA 5. The language of algebra has its alphabet, vocabulary, and grammar. 6. The symbols of algebra are of two kinds: one class represent its fundamental conceptions and may be called its letters, and the other represent the relations or modes of combination of the letters and are called the signs. 7. The alphabet of an algebra consists of its letters; the vocabulary defines its signs and the elementary combinations of its letters; and the grammar gives the rules of composition...
Page 119 - But there is no difficulty in reducing them to a linear form, and, indeed, my algebra (e3) is the simplest case of Hankel's alternate numbers ; and in any other case, in which n is the number of the Hankel elements employed, the complete number of vids of the corresponding linear algebra is 2" — 1 . The limited character of the algebras which I have investigated may be regarded as an accident of the mode of discussion. There is, however, a large number of unlimited algebras suggested by the investigations,...
Page 6 - В . 20. The sign + is called plus in common algebra and denotes addition. It may be retained with the same name, and the process which it indicates may be called addition. In the simplest cases it expresses a mere mixture, in which * This, of course, supposes that С does not vanish.

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