Trigonometry and Double Algebra (Google eBook)

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Taylor, Walton and Maberly, 1849 - Algebra - 167 pages
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Page vi - Syllabus of a Course of Lectures upon Trigonometry and the Application of Algebra to Geometry. 8vo. 7*. 6d. MECHANICS AND HYDROSTATICS. Elementary Hydrostatics. By WH BESANT, MA, Late Fellow of St John's College. [Preparing. Elementary Hydrostatics for Junior Students. By R. POTTER, MA late Fellow of Queens...
Page vi - O'Brien (Rev. M.) Treatise on Plane Coordinate Geometry ; or the Application of the Method of Coordinates to the Solution of Problems in Plane Geometry.
Page 101 - C, etc., might stand for virtues and vices, the reader might believe him, or contradict him, as he pleases, but not out of this chapter. The one exception above noted, which has some share of meaning, is the sign placed between two symbols, as in A B. It indicates that the two symbols have the same resulting meaning, by whatever steps attained. That A and B, if quantities, are the same amount of quantity ; that if operations, they are of the same effect, etc.
Page v - J.) on the objections against the geometrical representation of the square roots of negative quantities ii. 371 ; on the geometrical representation of the powers of quantities whose indices involve the square roots of negative quantities, ii.
Page 104 - De Morgan professes to give a complete inventory of the laws which the symbols of algebra must obey, for he says ' ' Any system of symbols which obeys these rules and no others, except they be formed by combination of these rules, and which uses the preceding symbols and no others, except they be new symbols invented in abbreviation of combinations of these symbols, is symbolic algebra.
Page 41 - Frend's papers, and dated November 16, 1801, distinctly lays it down that, in these matters, it is not the principles which prove the conclusions, but the truth of the conclusions which proves that there must, somewhere or other, be principles. " Whether or not," says he, " I have found a logic, by the rules of which operations with imaginary quantities are conducted, is not now the question ; but surely this is evident, that, since they lead to right conclusions, they must have a logic.
Page 92 - As soon as the idea of acquiring symbols and laws of combination, without given meaning, has become familiar, the student has the notion of what I will call a symbolic calculus; which, with certain symbols and certain laws of combination, is symbolic algebra: an art, not a science; and an apparently useless art, except as it may afterwards furnish the grammar of a science.
Page 101 - It is a mode of combination represented by -) ; when + receives its meaning, so also will the word addition. It is most important that the student should bear in mind that, with one exception , no word nor sign of arithmetic or algebra has one atom of meaning throughout this chapter, the object of which is symbols, and their laws of combination, giving a symbolic algebra which may hereafter become the grammar of a hundred distinct significant algebras. If any one were to assert that + and ...
Page 99 - The next and second step, .... consisted in treating the results of algebra as necessarily true, and as representing some relation or other, however inconsistent they might be with the suppositions from which they were deduced. So soon as it was shewn that a particular result had no existence as a quantity, it was permitted, by definition, to have an existence of another kind, into which no particular inquiry was made, because the rules under which it was found that the new symbols would give true...
Page 98 - Ignorance, the necessary predecessor of knowledge, was called nature ; and all conceptions which were declared unintelligible by the former, were supposed to have been made impossible by the latter. The first who used algebraical symbols in a general sense, Vieta, concluded that subtraction was a defect, and that expressions containing it should be in every possible manner avoided. Vitium negationis, was his phrase.

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