366 Mr. F. Baily on Mr. Babbage's new Machine. 



If the machine possessed only three axes, the following series, 

 amongst others, might be tabulated, 



A 2 «*+i = aAu z + A 7 u x A ? « s = au z . 

 If there were but two axes, we might tabulate 



These equations appear to be restricted ; and so they cer- 

 tainly are. But, since they can be computed and printed by 

 machinery, of no very great complication, and since it is not 

 necessary (after setting the machine at the beginning) to do 

 any thing more than turn the handle of the instrument, it be- 

 comes a matter of some consequence to reduce the mode of 

 calculating our tables to such forms as those above alluded to. 



A table of logarithms may be computed by the equation 

 A*w z = c : but in this case the intervals must not be greater 

 than a few hundred terms. Now, it may be possible to find 

 some equation, similar to those above mentioned, which shall 

 represent a much more extensive portion of such tables, — 

 possibly many thousand terms : and the importance that would 

 result from such an equation renders it worthy the attention 

 of mathematicians in general. 



A table of sines may, for a small portion of its course, be 

 represented by the equation A 2 "* = c- but it may be repre- 

 sented in its whole extent by the equation A J "s; = au x + i. 

 Now, this is precisely one of the equations above quoted : and 

 if a proper machine were made (and it need not be a large 

 one) it would tabulate the expression A sin from one end of 

 the quadrant to the other, at any interval (whether minutes or 

 seconds) by only once setting it. It would not be very com- 

 plicated to place three such machines by the side of each other, 

 and cause them to transfer their results to a common axis, 

 with which the printing apparatus might be connected. Such 

 a machine would, amongst other tables, compute one from the 

 expression 



A sin + B sin 2d + C sin 30 

 the utility of which, in astronomy, is well known. In fact, 

 Mr. Babbage is of opinion that it would not be impossible to 

 form a machine which should tabulate almost any individual 

 equation of differences. 



Amongst the singular and curious powers produced by 

 small additions to the machinery, may be reckoned the possi- 

 bility of tabulating series expressed by the following equations : 



A*« s = the ui.its figure of u z , 



A'« s = 2 x the figures found in the tens place of u z + l , 



A 5 "* = 4 x the figures found in the units and tens place of 



and many others similar thereto. Again : 



