360 Mr. F. Baily on Mr. Babbage's new Machine Jor calculating 



may be required by the public. It is probable that the pre- 

 sent tables, if extended from 108,000 to 1,000,000 would be of 

 greater utility than an extension of the present tables to a 

 larger number of figures. The quantity of mental labour 

 saved by the machine may be estimated in the following man- 

 ner. Suppose a machine constructed, capable of computing 

 with five orders of differences; it would be necessary to cal- 

 culate those differences for every thousandth logarithm only: 

 consequently, if the table extended from 10,000 to 10,0000, 

 there would be but ninety sets of differences to compute. 

 Any one of these sets being placed in the machine, with its 

 first five differences, it will deliver the 500 preceding loga- 

 rithms and also the 500 succeeding ones ; thus producing a 

 thousand logarithms : at the end of which term, another set 

 of differences must be substituted. With five orders of diffe- 

 rences, a table of logarithms may be computed to eight places 

 of figures, which shall be true to the last figures, and it would 

 not require more than half an hour to compute each set of 

 differences ; particularly as the higher numbers require very 

 little labour, two or three terms of the series being quite suffi- 

 cient. 



9°. Tables of Logarithmic Sines, Cosines, Tangents and 

 Cotangents. The remarks which have been made in the pre- 

 ceding article, will apply with nearly equal propriety to the 

 tables here alluded to. The mental labour required for their 

 construction by the machine is reduced to a very insignificant 

 quantity, when compared with the prodigious labour employed 

 in the usual way. 



10°. Tables of Hyperbolic Logarithms. Some small tables 

 of this kind have been printed in several works, and are useful 

 in various integrations ; but the most comprehensive set was 

 computed by Mr. Barlow, which contains the hyperbolic lo- 

 garithms of all numbers from 1 to 10,000. The labour of 

 computing them is very great, which is the cause of their not 

 being more extended. From a slight examination of the sub- 

 ject, it would appear that the mental labour may, in this case, 

 be reduced by the machine to about a two hundredth part of 

 what was formerly necessary. 



1 1°. Tables for finding the Logarithms of the sum or dif- 

 ference of two quantities, each of whose logarithms is given. 

 This table, which was first suggested by Mr. Gauss, has been 

 printed in at least three different forms. It is extremely con- 

 venient when many similar operations are required : the whole 

 of it was computed by the method of differences ; and con- 

 sequently nearly the whole of the labour may be saved by 

 the help of the machine. 



12°. Other 



