# Page:On Mr. Babbage's new machine for calculating and printing mathematical and astronomical tables.pdf/6

subsidiary tables in order to produce the necessary corrections; a labour so gigantic as to preclude all hope of seeing it accomplished by the pen. By the help of the machine, however, the manual labour vanishes, and the mental labour is reduced to a very insignificant quantity. For, as I have already stated, astronomical tables of every kind are reducible to the same general mode of computation: viz. by the continual addition of certain constant quantities, whereby the mean motions of the body may be determined ad infinitum; and by the numerical computation of certain circular functions for the correction of the same. The quantities depending on these circular functions, let them arise from whatever source they may, or let them be dependent on any given law whatever, are deducible with equal ease, expedition and accuracy by the help of the machine. So that in fact there is no limit to the application of it, in the computation of astronomical tables of every kind.

I might now draw your attention to those other subjects of a particular nature, to which the machine is applicable; such as the tables of Interest, Annuities etc. etc.: all of which are reducible to the same general principles, and will be found to be capable of being computed by the machine with equal facility and safety. But, I trust that enough has been said to show the utility and importance of the invention; and invitation, inferior to none, of the present day: and which, when followed up by the construction of a machine of larger dimensions now in progreſs (by which alone its powers and merit can be duly appreciated) will tend considerably to the advancement of science, and add to the reputation of its distinguish inventor.

I have omitted to state that this machine computes, in all cases, to the nearest figure, whatever it may be. That is, after the required number of figures are computed, if the next following figure should be a 5 or upwards, the last figure is increased by unity, without any attention on the part of the operator.

But, it is not in these mechanical contrivances alone, that the beauty and utility of the machine consist. Mr. Babbage, who stands deservedly high in the mathematical world, considers these but of a secondary kind, and has met with many curious and interesting results, which may ultimately lead to the advancement of science. The machine which he is at present constructing will tabulate the equation ${\displaystyle \bigtriangleup ^{4}u_{3}=c}$: consequently there must be a means of representing the given constant c, and also the four arbitrary ones introduced in the integration. There are five axes in the machine, in each of which one of these may be placed. It is evident that the arbitrary constant must be given numerically, although the members may be any whatever. The multiplication is not like that of any other machines with which I am aequainted, viz. a repeated addition — but is an actual multiplication: and the multiplier as well as the multiplicand may be decimal. A machine possessing five axes (similar to the one now constructing) would tabulate, according to the peculiar arrangement, any of the following equations

 ${\displaystyle \bigtriangleup ^{5}u_{3}=au_{3+1}}$ ${\displaystyle \bigtriangleup ^{5}u_{3}=au_{3+2}}$ ${\displaystyle \bigtriangleup ^{5}u_{3+1}=au_{3}+\bigtriangleup ^{4}u_{3}}$ ${\displaystyle \bigtriangleup ^{5}u_{3+1}=a\bigtriangleup ^{2}u_{3+1}+\bigtriangleup ^{4}u_{3}}$

If the machine possessed only three axes, the following series, amongst others, might be tabulated

 ${\displaystyle \bigtriangleup ^{2}u_{3+1}=a\bigtriangleup u_{3}+\bigtriangleup ^{2}u_{3}}$ ${\displaystyle \bigtriangleup ^{3}u_{3}=au_{3}}$

If there were but two axes, we might tabulate

${\displaystyle \bigtriangleup ^{2}u_{3}=au_{3+1}}$

These equations appear to be restricted; and so they certainly 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 twin the handle of the instrument, it becomes 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 ${\displaystyle \bigtriangleup ^{4}u_{3}=c}$: 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 equations ${\displaystyle \bigtriangleup ^{2}u_{3}=c}$: but it may be represented in its whole extent by the equation ${\displaystyle \bigtriangleup ^{2}u_{3}=au_{3+1}}$. 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 θ 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 complicated to place three such machines by the side of each other,