Notices respecting New Books. 



313 



rithras, and are equal for a considerable series of numbers. Hence 

 it is plain that the logarithms of the differences of the successive reci- 

 procals can be obtained by addition, and the calculation conducted 

 in a tabular form. Suppose n to equal 62500, then will K equal 

 408 2330, and the calculation will stand thus : — 



Numbers. 



Logs, of diff. of 

 reciprocals. 



Diff. of 

 reciprocals. 



Reciprocals. 





K= 



408 2330 







62500 



Diff. of logs. 







•00001600 0000000 



62499 



139 



408 2469 



256004 



0256004 



98 



139 



2608 



12 



0512016 



97 



139 



2747 



20 



0768036 



96 



139 



2886 



29 



1024065 



95 



139 



3025 



37 



1280102 



The reciprocals entered in the Table are, of course, 



1600 000, 1600 026, 1600 051, 1600 077, 

 1600 102, 1600 128, &c. 



In the same tabular calculation the reciprocals of the half and 

 quarter numbers are found by simple multiplication. 



The chief merit of a work of this kind is, of course, accuracy. To 

 secure this, every precaution seems to have been taken. " To pre- 

 vent error," says Colonel Oakes, " the co. logarithms were checked 

 independently at each50th term. In taking out the numbers, the pro- 

 gression of their differences was kept in view, so that no material 

 error could occur. The summation of the differences was checked 

 at every 10th term by a subordinate summation, and by comparison 

 with Barlow's Tables ; and wherever the seventh figure could be un- 

 certain, it was determined by actual division. Finally, every hun- 

 dredth term was computed by division. The whole of the calcula- 

 tions were performed in duplicate, and when the proofs were set up 

 from one manuscript they were read with the other ; and second 

 and third proofs were also each examined by addition of the printed 

 differences, and by comparison with Barlow's Table at each 10th 

 term. Lastly, the proofs were again examined, and the whole Table 

 virtually recomputed by summation on the Arithmometre of M. 

 Thomas de Colmar." 



It is proper to add that the work was undertaken at the suggestion 

 of Professor De Morgan, who says that it is, as far as he knows, 

 " the largest which has ever been attempted," and that " it is a very 

 useful Table, and that its applications are far too little known and 

 thought of." 



