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Page 250 - On a sheet of paper ruled in squares, and which is read as a continuous column from the bottom of one column to the top of the next...
Page 57 - ... also true of invariants (as distinguished from covariants) for quantics of the 6th order. This is how it came to pass in the infancy of the theory that the number of groundcovariants was supposed to become infinite for quantics beyond the fourth and their ground-invariants for quantics beyond the 6th order. I think it may be interesting to some of the readers of the Journal to be put in possession of the complete system of irreducible syzygies to a system of two or more quantics, and I select...
Page 99 - Alphabet. 8. Algebras may be distinguished from each other by the number of their independent fundamental conceptions, or of the letters of their alphabet. Thus an algebra which has only one letter in its alphabet is a single algebra; one which has two letters is a double algebra; one of three letters a triple algebra; one of four letters a quadruple algebra, and so on. This artificial division of the algebras is cold and uninstructive like the artificial Linnean system of botany. But it is useful...
Page 219 - 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 219 - A vid of which the square is a vid of inversion, is a vid of semi-inversion. A very general form of a vid of semi-inversion is (A : A) � (B : B) � J(C : C) � etc.
Page 97 - 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 87 - The proof in each case will consist in showing, 1st, that the proposition is (rue of the number one, and 2d, that if true of the number n it is true of the number 1 + n, next larger than n. The different transformations of each expression will be ranged under one another in one column, with the indications of the principles of transformation in another column. 1. To prove the associative principle of addition, that (x + y) + z = x + (y + z) whatever numbers x, y, and z, may be. First it is true for...
Page 97 - 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.