Page:Theory of the motion of the heavenly bodies moving about the sun in conic sections- a translation of Gauss's "Theoria motus." With an appendix (IA theoryofmotionof00gaus).pdf/58

 we shall pay no attention to this augmentation of the error, since there is no objection to our affixing one more than another decimal figure to the proportional part, and it is very evident that, if the proportional part is exact, the interpolated logarithm is not liable to a greater error than the logarithms given directly in the tables, so far indeed as we are authorized to consider the changes in the latter as uniform. Thence arises another increase of the error, that this last assumption is not rigorously true; but this also we pretermit, because the effect of the second and higher differences (especially where the superior tables computed by TaYtor are used for trigonometrical functions) is evidently of no importance, and may readily be taken into account, if it should happen to turn out a little ton great. In all cases, therefore, we will put the maximum unavoidable error of the tables $=\omega,$ assuming that the argument (that is, the number the logarithm of which, or the angle the sine etc. of which, is sought) is given with strict accuracy. But if the argument itself is only approximately known, and the variation $\omega^{\prime}$ of the logarithm, etc. (which may be defined by the method of differentials) is supposed to correspond to the greatest error to which it is liable, then the maximum error of the logarithm, computed by means of the tables, can amount to $\omega+\omega^{\prime}.$

Inversely, if the argument corresponding to a given logarithm is computed by the help of the tables, the greatest error is equal to that change in the argument which corresponds to the variation $\omega$ in the logarithm, if the latter is correctly given, or to that which corresponds to the variation $\omega+\omega^{\prime}$  in the logarithm, if the logarithm can be erroneous to the extent of $\omega^{\prime}.$  It will hardly be necessary to remark that $\omega$  and $\omega^{\prime}$  must be affected by the same sign.

If several quantities, correct within certain limits only, are added together, the greatest error of the sum will be equal to the sum of tlie greatest individual errors affected by the same sign; wherefore, in the subtraction also of quantities approximately correct, the greatest error of the difference will be equal to the sum of the greatest individual errors. In the multiplication or division of a quantity not strictly correct, the maximum error is increased or diminished in the same ratio as the quantity itself.