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/57

 30.

Since none of the numbers which we take out from logarithmic and trigonometrical tables admit of absolute precision, but are all to a certain extent approximate only, the results of all calculations performed by the aid of these numbers can only be approximately true. In most cases, indeed, the common tables, which are exact to the seventh place of decimals, that is, never deviate from the truth either in excess or defect beyond half of an unit in the seventh figure, furnish more than the requisite accuracy, so that the unavoidable errors are evidently of no consequence: nevertheless it may happen, that in special cases the effect of the errors of the tables is so augmented that we may be obliged to reject a method, otherwise the best, and substitute another in its place. Cases of this kind can occur in those computations which we have just explained; on which account, it will not be foreign to our purpose to introduce here some inquiries concerning the degree of precision allowed in these computations by the common tables. Although this is not the place for a thorough examination of this subject, which is of the greatest importance to the practical computer, yet we will conduct the investigation sufficiently far for our own object, from which point it may be further perfected and extended to other operations by any one requiring it.

31.
Any logarithm, sine, tangent, etc. whatever, (or, in general, any irrational quantity whatever taken from the tables,) is liable to an error which may amount to a half unit in the last figure: we will designate this limit of error by $\omega,$ which therefore is in the common tables $=0.00000005.$  If now, the logarithm, etc., cannot be taken directly from the tables, but must be obtained by means of interpolation, this error may be slightly increased fron two eauses. In the first place, it is usual to take for the proportional part, when (regarding the last figure as unity) it is not an integer, the next greatest or least integer; and in this way, it is easily perceived, this error may be increased to just within twice its actual amount. But