This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1880 Excerpt: ...with the elements of the Differential Calculus will see that all the results of the present Chapter may be obtained from Taylor's Theorem; and thus these results may be easily retained in the memory, or at least readilyrecovered when required. For example, consider the natural sine; we have by Taylor's Theorem sin(0 + ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1880 Excerpt: ...with the elements of the Differential Calculus will see that all the results of the present Chapter may be obtained from Taylor's Theorem; and thus these results may be easily retained in the memory, or at least readilyrecovered when required. For example, consider the natural sine; we have by Taylor's Theorem sin(0 + A)=sin0 + Acos0-jj-sin + XA), where A is some proper fraction. This formula shews that if we put sin (6 + h) = sin 6 + h cos 6 h' the error is less than. Moreover we see that when 6 is small the principle of proportional parts is especially applicable, for then the term sin (6 + Xh) is extremely small in comparison with h cos 6; and on the other hand, when 6 is nearly the principle h' is not so appropriate, because then-sin (6 + Xh) may be sensible u in comparison with h cos 6. Again, by Taylor's Theorem, we have log sin (6 + h) log sin 6 + fth cot 6----cosec (6 + XA), where /t is the modulus and X some proper fraction. This equation shews that the principle of proportional parts is in general applicable for the logarithmic sine, but that the differences of consecutive logarithmic sines are irregular when the angles are small, and insensible and irregular when tho angles are nearly right angles. 210. The following application of Taylor's Theorem will give a good mode of estimating the amount of error involved in tho principle of proportional parts. Take the logarithmic sine for example; we have log sin (6 + h) = log sin 6 + ph cot (6 + Xh), where is some proper fraction. Thus the approximation uses cot 6 instead of cot (6 + h). The true value in fact of log sin(0 + h)--log sin 6 must lie between fih cot 6 and fih cot (6 +h), so that the error is less than fih cot 8--cot (6 + h). MISCELLANEOUS EXAMPLES. 1. From one of the angles of a re...
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