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The theoretical part of this little book is an elementary exposition of the nature ofthe number concept, of the positive integer, and of the four artificial forms of numberwhich, with the positive integer, constitute the \number-system of algebra, viz. thenegative, the fraction, the irrational, and the imaginary. The discussion of the artificialnumbers follows, in general, the same lines as my pamphlet: On the Forms of Numberarising in Common Algebra, but it is much more exhaustive and thorough-going. Thepoint of view is the one first suggested by Peacock and Gregory, and accepted bymathematicians generally since the discovery of quaternions and the Ausdehnungslehreof Grassmann, that algebra is completely defined formally by the laws of combinationto which its fundamental operations are subject; that, speaking generally, these lawsalone define the operations, and the operations the various artificial numbers, as theirformal or symbolic results. This doctrine was fully developed for the negative, thefraction, and the imaginary by Hankel, in his Complexe Zahlensystemen, in 1867, andmade complete by Cantor's beautiful theory of the irrational in 1871, but it has notas yet received adequate treatment in English.

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