208 Mr. Babbage on the Applicatioyi of Machinery 



chance as much as possible. In examining the analytical 

 theory of the various differences of the sine of an arc, I no- 

 ticed the property which it possesses of having any of its even 

 orders of differences equal to the sine of the same arc increased 

 by some multiple of its increment multiplied by a constant 

 quantity. With the aid of this principle an engine might be 

 formed which would require but little attendance, and I be- 

 lieve that it might in some cases compute a table of the form 

 A sin d from the 1st value of 6 = up to 6 = 90° with only one 

 set of figures being placed in it. 



It is scarcely necessary to observe what an immense number 

 of astronomical tables are comprised under this form, nor the 

 great accuracy which must result from having reduced to so 

 few a number the preliminary computations which are requisite. 

 In pursumg hito its detail the principle to which I have al- 

 luded, which lends itself so happily to numerical application, 

 I have traced its application to other species of tables, and am 

 enabled to point out a course of analytical investigation which 

 will in all probability afford ready methods for constructing 

 tables, even of the most complicated transcendent, in a man- 

 ner equally easy. 



■ I will now advert to another circumstance, which, although 

 not immediately connected with asti'onomical tables, resulted 

 from an examination of the engine by which they can be 

 formed. 



On considering the arrangement of its parts, I observed 

 that a different mode of connecting them would produce ta- 

 bles of a new species altogether different from any with which 

 I was acquainted. I therefore computed with my pen a small 

 table such as would have been formed by the engine bad it 

 existed in this new shape, and I was much surprised at dis- 

 covering that no analytical method was yet known for deter- 

 minins its ?ith tei-m. The following is the first series I wrote 

 down : 



Series. Diff. Series. Diff. 



10 ... 222 42 20 . . . 924 8f5 



264 46 1010 86 



.310 46 1096 92 



356 52 25 . , . 1188 100 



15 . . . 408 60 1288 108 



468 68 1396 114 



536 74 1510 114 



610 74 1624 118 



684 78 30 . . . 1742 120 



20 . . . 762 80 1862 122 



u . . io» iu 842 82 1984 



The equation of finite differences from which it is produced 

 A*M = units fig. of « , , 



which 



