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THE work before the reader is entirely new, not being in any sense a second edition of that which I published on the same subject in 1837.It consists of two books. In the first, I have endeavoured to give the student who has a competent knowledge of arithmetic and algebra -- as much for instance as is contained in my works on those subjects, to which reference is made in various places -- a view of trigonometry, as a branch of algebra and a constituent part of the foundation of the higher mathematics. In the second, I have given an elementary view of algebra in its purely symbolic character, with the application of that geometrical basis of significance which affords explanation of every symbol.The term double algebra has not yet obtained currency, though that of triple algebra has, of late years, been much employed. It means algebra in which each symbol stands for an object of thought having two distinct and independent qualities: just as the symbol of a straight line, to be perfect, must designate both the length and direction of the line. I have not, after much thought, and some discussion, been able to fix on a better name of sufficient brevity. If, by the application of a somewhat startling adjective to the word algebra, any of those who are still bewildered by an art in which impossible quantities, or quantities which are not quantities, are made objects of reasoning, should become aware that by slow degrees, and the union of many heads, the art has become a science, and the impossibilities possible, they, at least, will have no objection to the phrase.A. DE MORGAN

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