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XXXV. Observations on the Application of Machineiy to the 

 Computation of Mathematical Tables. By Charles Bab- 

 BAGE, Esq. F.R.S., 4-c. S,T* 



S^INCE I had the honour of communicating to the Astro- 

 *^ nomical Society a short account of an arithmetical engine 

 for the calculation of tables, which has been examined by se- 

 veral of the members of this society, I have not added much 

 to the practical part of the subject. I have however paid some 

 attention to the improvements of which the machinery is sus- 

 ceptible, and which will, if another engine is made, be greatly 

 improved. 



The theoretical inquiries to which it has conducted me are 

 however of a singular nature ; and I shall take this opportu- 

 nity of briefly explaining to the society some of the principles 

 on which they depend, as far as the nature of the subject will 

 permit me to do this without the introduction of too many 

 algebraic operations, which are rarely intelligible when read 

 to a large assembly. 



Of the variety of tables which are required in the present 

 state of science, by far the larger portion are intimately con- 

 nected with that department of it which it is the peculiar ob- 

 ject of this society to promote. 



The importance of astronomical science, whether viewed as 

 the proudest triumph of intellectual power, or considered as 

 the most valuable present of abstract science to the comfort 

 and happiness of mankind, equally claims for it the first as- 

 sistance from any new method for condensing the processes of 

 reasoning or abridging the labour of calculation. Astronomi- 

 cal tables were therefore the first objects on which I turned 

 my attention, when attempting to improve the power of the 

 engine, as they had formed the first motive for constructing it. 



1 have already stated to the society, in my former commu- 

 nication, that the first engine I had constructed was solely 

 destined to compute tables having constant differences. From 

 this circumstance it will be apparent that after a certain num- 

 ber of terms of a table are computed, unless, as rarely hajv 

 pens, it has a constant order of difterences, we must stop the 

 engine and place in it other numbers, in order to produce the 

 next })ortion of the table. This operation must be rejieated 

 more or less frequently according to the nature of the table. 

 Tlie more numerous the order of diflerences, the less frequent 

 will this operation become requisite. The chance of error in 

 sucli computations arises from incorrect numbers being placed 

 ill the engine: it therefore becomes desirable to limit this 

 • I'roiii tlie Mfinoir* of tlie Autiononiical Society, Part II. vol. i. p. 31 1. 



chance 



