32 RELATIONS PERTAINING SIMPLY [BOOK I. 



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 propor 

 tional part, and it is very evident that, if the proportional part is exact, the inter 

 polated 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 TAYLOR 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 too great. In all cases, therefore, we will put the maximum unavoid 

 able error of the tables =co, 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 a/ 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 m -\- a/. 



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 argu 

 ment which corresponds to the variation to in the logarithm, if the latter is cor 

 rectly given, or to that which corresponds to the variation w -j- w in the loga 

 rithm, if the logarithm can be erroneous to the extent of w . It will hardly be 

 necessary to remark that w and a/ 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 the 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. 



