forms. It is extremely convenient when many similar operations are required: the whole of it was computed by the method of differences; and consequently nearly the whole of the labour may be saved by the help of the machine.
12o. Other general tables might also be here mentioned, which have been of great service in various mathematical investigations, and have been computed and printed, by different authors: such as tables of the powers of ·01, ·02, ·03, etc.: tables of the squares of the natural sines, cosines, tangents etc.: tables of figurate numbers, and of polygonal numbers: tables of the length of circular arcs: tables for determining the irreducible case of cubic equations: tables of hyperbolic functions, viz hyperbolic sines, cosines etc., and logarithmic hyperbolic sines, cosines etc. These and various other tables which it is needleſs here to mention, may be computed by the machine, with very little mental labour, and with the greatest accuracy.
Besides the general tables above alluded to, there are many others which are applicable to particular subjects only: the most important of which are those connected with Astronomy and Navigation. — When we contemplate the ease and expedition with which the seaman determines the position of his vessel, and with what confidence he directs it to the most distant quarter of the globe, we are not perhaps aware of the immense variety of tables which have been formed almost solely for his use: and without the aid of which he dare not venture on the boundleſs ocean. Not only must the general tables of the sun and moon be first computed, together with the various equations for determining their apparent places; but those places also for every day in the year are prepared solely for his use; and even for different hours in the same day. The places of certain stars must likewise be given: and, as these depend on precession, aberration and nutation, tables of this kind also must be formed for each star. Then come the lunar distances, which are computed for every third hour in the day; and which depend likewise on a variety of other complicated tables. After these come the Requisite Tables, published by order of the Board of Longitude, and the usual Logarithmic Tables for facilitating the computations, both of which are dependent on other tables from which they have been deduced or copied. Now, when it is considered that an error, in any one of these multifarious tables, will affect the last result, and thereby render the navigator liable to be led into difficulties, if not danger, it must be acknowledged that it is of very essential importance that all such tables should be computed and printed in so perfect a manner that they may in all cases be depended upon. This however, in the present mode of constructing them, is scarcely possible. I have myself discovered above five hundred errors in the work containing the Tables of the Sun et Moon, from which (still lately) the annual volumes of the Nautical Almanac have been computed: and a respectable author has asserted that, in the first edition of the Requisite Tables, published by order of the Board of Longitude, there were above a thousand errors. Many of the subsidiary tables, above alluded to, have not been computed since they were first printed: for, the mental and even manual labour of calculating them has been so great that the world has been obliged to remain contented with those original computations: and they are consequently subject to all the errors arising from subsequent editions and copies.
In the calculation of Astronomical tables, the machine will be of very material assistance: not only because an immense variety of subsidiary tables are required to determine the place of the sun, moon, and planets and even of the fixed stars but likewise on account of the frequent change which it is found necessary to introduce in the elements from which those tables are deduced: and which vary from time to time according to the improvements in Physical astronomy and the progreſs of discovery.
Within the last twenty years it has been found necessary to revise almost all the tables connected with the solar system: and already many of these have been found inefficient for the refined purposes of modern astronomy. But the great expense of time and labour and money has been the principal obstacle to the advancement of this part of the science: since each revision has been attended with the introduction of new equations, which consequently require new tables. And, to this day, we have not been furnished with any tables whatever of three (out of the four) new planets that have been discovered since the commencement of the present centring: nor can the places of many thousands of the fixed stars be readily determined for want of the subsidiary tables necessary for that purpose.
It perhaps may be proper to state that all astronomical tables (with very few trifling exceptions) are deduced by the two following methods: 1o. by the addition of certain constant quantities, whereby the mean motions of the body are determined; 2o. by certain corrections (of those motions) which depend on the sine or cosine of