Bremiher — Errors Affecting Logarithmic Computations. 429 



co:jtceeot:n^g the ereors by which loga- 

 rithmic COMPUTATIOISrS ARE AFFECTED. 



§ I. 



Since the exact value of any logarithm is in general an in- 

 commensnrable number, which can not be expressed except by 

 an infinite number of decimal figures, and since in the loga- 

 rithmic tables only the first decimal figures are given, which we 

 use in computation instead of the exact value, it is evident that 

 a result obtained by the use of logarithms is affected by a 

 greater or less error. But whenever we are willing to use in 

 the computation more decimal places than are called for by the 

 accuracy of our data, this error arising from the inaccuracy of 

 the logarithms may be disregarded, in comparison with that 

 which arises from the inaccuracy of the data. If, however, in 

 order to save useless labor, only as many decimal figures be 

 used in the computation as are called for by the accuracy of the 

 data,- then it will be proper before the computation is com- 

 menced to consider the theory of the errors which can arise 

 from the omitted decimals. The discussion of this theory 

 which is attempted below will show whether five, six, or seven 

 decimals ought to be used in the computation. 



§ 2. 



First, assume that the errors of all the logarithms used in 

 the computation (i. e. their true values) are known. Then by 

 the aid of the differential calculus the error of their resulting 

 sum can easily be found. For this it is sufficient to use only 

 the first differentials, since, in comparison with the true values 

 of the logarithms, the errors can be regarded as infinitesimals, 

 whose higher powers are of no weight in the computation. In 

 this computation the following equations may be used; 



