SECT. 1.] TO POSITION IN THE ORBIT. 31 



30. 



Since none of the numbers which we take out from logarithmic and trigo 

 nometrical 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 to, 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 inter 

 polation, this error may be slightly increased from two causes. 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 



