a given arc: and which are called equations of the mean motions. The mean motions of any of the celestial bodies may be computed by means of the machine, without any previous calculation: and those quantities depending of the sine and cosine may in all cases be computed by the machine with the help of two previous calculations of no great length or labour, and in most cases with the help of one only.
In the year 1804 Baron de Zach published his Tables of the Sun: and within two years of that date, Mr. Delambre published similar tables. In 1810 Mr. Carlini published his Tables of the Sun, on a new construction: so that within the space of six years it was considered necessary by the distinguished astronomers to publish these three interesting and highly useful works.
In the year 1806 Mr. Bürg published his very valuable Tables of the moon; a work which superseded the use of Mason's tables, and was rewarded with a double prize by the French government: It was received with gratitude by the scientific in every nation, and opened a new æra in the history of astronomy and navigation. These were followed by the Tables of Burckhardt in 1812; which are still more accurate than those of Burg: and, at te present moment, the elements of some new tables have been deduced by Mr. Damoiseau. But it is the elements only which have yet been deduced: since it is these alone which can be expected to engage the attention of the profound mathematician. Nevertheleſs laborious, yet useful, operation of computation cannot safely be left to inferior hands. The merit of each is however very unequally estimated by the world. Euler had three hundred pounds granted him by the English government for furnishing the elements, and Mayer three thousand for the actual computation of the tables of the Moon, which were published by the Board of Longitude in the year 1770.
The elements of Mr. Damoiseau have been already two years before the public: but the time and labour necessary to compute the tables therefrom are so great that they have not yet appeared. In order to deduce the place of the moon from these elements, no less than 116 different equations are requisite, all depending on the sine or cosine of different arcs. The labour of computing each equation, with the pen, would be immense; and liable to innumerable errors: but, with the assistance of the machine, they are deduced with equal facility and safety, and without much previous computation.
In the year 1808 Mr. Bouvard published his tables of Jupiter and Saturn: but in 1821, owing to the progreſs of discovery and the advancement of physical astronomy, it was found necessary to revise the elements; and an entire new set of tables was then published by this distinguished astronomer. In order to deduce the geocentric places of these planets, it is requisite to compute no leſs than 116 tables depending on the sine or cosine of certain arcs.
I shall not intende further on the time of your readers by alluding to the tables of the other planets, which are all liable to similar observations: but I shall take the liberty of calling their attention to those very useful tables which have, from time to time appeared for determining the apparent places of the fixed stars; and which generally assume the title of „Tables of Aberration et Nutation.“ Tables of this kind are of vast importance to the practical astronomer, since they save a great deal of time and labour in the reduction of observations: and it is believed that many valuable observations remain unreduced, for want of convenient tables of this sort.
The first general tables of this kind were published by Mr. Mezger at Mannheim in 1778, and contained the corrections of 352 stars. In 1807 Mr. Cagnoli extended these tables to the corrections of 501 stars: and in the same year Baron de Zach published at Gotha his „Tabulae speciales Aberrationis et Nutationis“ which contain the corrections for 494 zodiacal stars. But, already these tables have nearly outlived their utility. Independent of their very limited extent, the elements from which they were deduced, have been superseded by others more agreable to actual observation; which, together with their exclusion of the solar nutation, and other minute quantities which cannot safely be neglected in the present state of astronomy, renders these tables of doubtful utility to the practical astronomer.
The number of zodical stars (without including the very minute ones) is considerably above a thousand: each of which may, in the course of a revolution of the nodes, suffer an occultation by the moon. These occultations are ascertained to be visible at sea, even from the unsteady deck of a vessel under sail: and afford the surest means of determining the longitude, provided the position of the star could be well ascertained. In order to furnish the corrections for each star, ten equations are requisite, depending on the sine et cosine of given ares; so that it would require the computation of upwards of ten thousand