384 : Analyses of Books. -[Nov. 
letter before us by stating that the great interest which has been 
taken by the distinguished character to whom it is addressed, in 
the success of the system of contrivances to which it relates, 
has induced him to adopt this mode of making known the prin- 
ciples and probable consequences of those contrivances. He 
observes, that the fatiguing labour and monotony of a continued 
repetition of similar arithmetical calculations, first excited the 
desire, and then suggested the idea, of a machine, which should 
become a substitute for one of the lowest operations of human 
intellect: he then proceeds as follows : . 
“The first engine of which drawings were made was one 
which is capable of computing any table by the aid of differences, 
whether they are positive or negative, or of both kinds. With 
respect to the number of the order of differences, the nature of 
the machinery did not in my own opinion, nor in that of a skilful 
mechanic whom [ consulted, appear to be restricted to any very 
limited number; and I should venture to construct one with ten 
or a dozen orders with perfect confidence. One remarkable 
property of this machine is, that the greater the number of dif- 
ferences, the more the engine will outstrip the most rapid cal- 
culator. 
“« By the application of certain parts of no great degree of 
complexity, this may be converted into a machine for extracting 
the roots of equations, and consequently the roots of numbers : 
and the extent of the approximation depends on the magnitude 
of the machine. 
“ Of a machine for multiplying any number of figures (m) by 
any other number (x), [ have several sketches; but it is not yet 
brought to that degree of perfection which | should wish to give 
it before it is to be executed. 
« T have also certain principles by which, if it should be desi- 
rable, a table of prime numbers might be made, extending from 
0 to ten millions. 
« Another machine, whose plans are much more advanced 
than several of those just named, is.one for constructing tables 
which have no order of differences constant. 
«<A vast variety of equations of finite differences may by its 
means be solved, and a variety of tables, which could be pro- 
duced in successive parts by the first machine I have mentioned, 
could be calculated by the latter one with a still less exertion of 
human thought.. Another and very remarkable point in the 
structure of this machine is, that it will calculate tables governed 
by laws which have not been hitherto shown to be explicitly 
determinable, or that it will solve equations for which analytical 
methods of solution have not yet been contrived. 
“« Supposing these engines executed, there would yet be want- 
ing other means to ensure the accuracy of the printed tables to 
be produced by them. 
“ The errors of the persons employed to copy the figures 
