Elements of trigonometry, and trigonometrical analysis, preliminary to the differential calculus (Google eBook)

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1837
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Page 73 - The first was tacitly to contend for the principle that human faculties, at the outset of any science, are judges both of the extent to which its results can be carried, and of the form in which they are to be expressed. 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...
Page 74 - 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 97 - To multiply by 10, move the decimal point one place to the right, annexing a zero if necessary.
Page 74 - This ought to have been the most startling part of the whole process. That contradictions might occur, was no wonder ; but that contradictions should uniformly, and without exception, lead to truth in algebra, and in no other species of mental occupation whatsoever, was a circumstance worthy the name of a mystery. Nothing could prevail against the practical result that theorems so produced were true ; and at last, when the interpretation of the abstract negative quantity shewed that a part at least...
Page 74 - The next and second step, though not without considerable fault, yet avoided the error of supposing that the learner was a competent critic. It 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...
Page 64 - That is, any two sides are proportional to the sines of the opposite angles. This is the formula upon which all others relative to triangles will be made to depend.
Page 74 - Thus, 1 2, and a (a + b), appeared under the name of negative quantities, or quantities less than nothing. These phrases, incongruous as they always were, maintained their ground, because they always produced a true result, whenever they produced any result at all which was intelligible: that is, the quantity less than nothing, in defiance of the common notion that all conceivable quantities are greater than nothing, and the square root of the negative quantity, an absurdity constructed upon...
Page 63 - that is, tan. A + tan. B .f- tan. C = tan. A tan. B tan. C ; a remarkable property of the angles of a plane triangle.

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