and print trig Mathematical and Astronomical Tables. ?59 



5°. Tables of the Square Roots and Cube Roots of Num- 

 bers. A table of the first kind has been given by Mr. Lam- 

 bert: and a more extended one by Mr. Barlow, in his Col- 

 lection. The latter writer constructed his table by means of 

 differences; an advantage which may be applied with greater 

 effect to the table of Cube Roots, on account of the greater 

 convergency of this order of differences. 



6°. Tables of the Reciprocals of Numbers. These are 

 amongst the most simple but most useful of arithmetical tables ; 

 and are peculiarly valuable in converting various series into 

 numbers, — thus facilitating the calculation of differences for the 

 more ready construction of other tables. In order, however, 

 to be employed in such operations, it is absolutely necessary 

 that they should be infallible. Several tables of this kind have 

 been printed : the most recent and extensive of which are those 

 of Mr. Barlow and Mr. Goodwin. 



7°. Tables of Natural Sines, Cosines, Tangents, &c. The 

 utility of tables of this kind is evident from the variety of 

 forms in which they have been, from time to time, printed : 

 and it is needless to insist on their importance at the present 

 day, since no seaman dare venture out of sight of land with- 

 out a knowledge of their use. In order to be of any real ser- 

 vice, however, they should be accurate ; and diligently revised 

 from time to time : otherwise they may be worse than useless. 

 The labour of computing tables of this kind will vary accord- 

 ing to the number of figures contained in the result. It ap- 

 pears desirable that the larger tables of this sort should be 

 printed with their several orders of differences to a much 

 greater extent than formerly, for the purpose of making other 

 tables, and for executing several mathematical operations be- 

 neficial to science. It would be difficult to state precisely the 

 quantity of mental labour saved by the machine, in construct- 

 ing tables of the kind ; but, I believe, it may be fairly reduced 

 to the two thousandth part of the whole. 



8°. Tables of the Logarithms of Numbers. Tables of 

 this kind are in the hands of every person engaged in nume- 

 rical investigations: and it is needless to dwell on their 

 utility and importance. The logarithms of number from 1 to 

 108,000 have been already computed, with a greater or less 

 number of figures; but this has been the work of various 

 authors, and of several successive years: the labour is so im- 

 mense that no human being has ventured to undertake the 

 whole. The tables which now exist are chiefly copies from 

 those original and partial computations. By the help of the 

 machine, however, this immense labour vanishes, and new 

 tables may be readily computed and re-computed as often as 



may 



