Table VIII. is so arranged that the mean astronomical refraction may be taken out at first sight, without subjecting the mariner to the necessity of making proportion for the odd minutes of altitude. This improvement will have a tendency to facilitate nautical calculations.

Table X.-The arrangement of this table is an improvement of that originally given by the author, in his treatise called "The Young Navigator's Guide to the Sidereal and Planetary Parts of Nautical Astronomy." By this improved table, the correction of the polar star's altitude may be readily taken out, at sight, to the nearest second of a degree, by means of five columns of proportional parts; and, to render the table permanent for at least half a century, the annual variation of that star's correction has been carefully determined to the hundredth part of a second. By means of this table, and that which immediately follows (Table XI.), the latitude may be very correctly inferred at any hour of the night, in the northern hemisphere, to every degree of accuracy desirable for nautical purposes.

Tables XII. and XIV. contain the equations to equal altitudes of the sun: these have been computed on a new principle, so as to adapt them to proportional logarithms, by means of which they are rendered infinitely more simple than those given under the same denomination in other treatises on nautical subjects; they will be found strictly correct, and, from their simplicity, a hope may be entertained that the truly correct and excellent method of finding the error of a watch or chronometer by equal altitudes of the sun, will be brought into more general use.

Tables XV. and XVI., which are entirely new, contain correct equations. for readily reducing the longitudes, right ascensions, declinations, &c. &c., of the sun and moon, as given in the Nautical Almanac, to any given meridian, and to any given time under that meridian.

Table XVII. contains the equation corresponding to the mean second difference of the moon's place in longitude, latitude, right ascension, or declination this table, besides being newly-arranged, will be found more extensive than those under a similar denomination, usually met with in books on navigation.

Table XVIII. is so arranged as to exhibit the true correction of the moon's apparent altitude corresponding to every second of horizontal parallax, and to every minute of altitude from the horizon to the zenith: and will prove very serviceable in all problems where the moon's altitude forms one of the arguments either given or required.

Table XIX. is fully adapted to the reduction of the true altitudes of the heavenly bodies, obtained by calculation, to their apparent central altitudes: the reductions of altitude may be very readily taken out to the decimal part of a second. This table will be found of considerable utility in

deducing the longitude from the lunar observations, when the distance only has been observed.

Table XX. is new; and by its means the operation of reducing the apparent central distance between the moon and sun, a fixed star, or planet, to the true central distance, is very much abridged, as will appear evident by referring to Method I., vol. i., page 433, where the true central distance is found by the simple addition of five natural versed sines.

Table XXI., which is also new, contains the correction of the auxiliary angle when the moon's distance from a planet is observed: this will be of great use in finding the longitude by the moon's central distance from a planet..

Table XXIV. The form of this table is entirely original; and though it is comprised in nine pages, yet it is so arranged that the logarithmic difference may be obtained, strictly correct, to the nearest minute of the moon's apparent altitude, and to every second of her horizontal parallax. This table will be found of almost general use in the problem for finding the longitude by the lunar observations.

Table XXVI., which is original, contains the correction of the logarithmic difference when the moon's distance from a planet is observed: this table will be found of great use in computing the lunar observations whenever the moon's distance from the planets appears in the Nautical Almanac; an improvement which, from the advertisement prefixed to the late Almanacs, may be shortly expected to take place.

Table XXVII., Natural Versed Sines, &c.-The numbers corresponding to the first 90 degrees of this table are expressed by the arithmetical complements of those contained in the Table of Natural Co-sines published by the author in "The Young Navigator's Guide," &c. ; the arithmetical complement of the natural co-sine of an arch being the natural versed sine of the same arch. The numbers contained in the remaining 90 degrees of this table are expressed by the natural sines, from the abovementioned work, augmented by the radius.

This table is so arranged as to render it general for every arch contained in the whole semi-circle, and conversely, whether that arch or its correlative be expressed as a natural versed sine, natural versed sine supplement, natural co-versed sine, natural sine, or natural co-sine.

Table XXVIII. is an extension of that published by the author in "The Young Navigator's Guide," &c.: it is arranged in a familiar manner, and, though concise, contains all the numbers that can be usefully employed in the elements of navigation; for, by means of nine columns of proportional parts, the logarithmic value of any natural number under 1839999 may be obtained nearly at sight, and conversely.

Tables XXX., XXXI., and XXXII., have been carefully drawn up, and proportional parts adapted to them, by means of which the logarithmic

half-elapsed time, middle time, and logarithmic rising may be very readily taken out at the first sight, and conversely.

Table XXXV., Logarithmic Secants.-The arrangement of this table is original, as well as its length: the numbers contained therein are expressed by the arithmetical complements of those contained in the table of logarithmic co-sines published by the author in "The Young Navigator's Guide," &c.

This table is so drawn up as to be properly adapted to every arch expressed in degrees, minutes, and seconds, in the whole semi-circle, whether that arch or its correlative be considered as a secant or a co-secant; and by means of proportional parts, the absolute value of any arch, and conversely, may be readily obtained at sight.

Table XXXVI., Logarithmic Sines.-This table is rendered general for every degree, minute, and second, in the whole semicircle. The Table of Logarithmic Tangents, which immediately follows, is also rendered general to the same extent; and by means of proportional parts, the true value of any arch, and conversely, may be instantly obtained, without the trouble of either multiplying or dividing: this improvement, to the practical navigator, must be an object of great importance, in reducing the labour attendant on computations in Nautical Astronomy.

Table XXXVIII. has been newly computed to the nearest second of time, so that the mariner may be readily enabled to reduce the time of the moon's passage over the meridian of Greenwich to that of her passage over any other meridian. This table will be found very useful in determining the apparent time of the moon's rising or setting, and also in ascertaining the time of high water at any given place by means of Table XXXIX.

Table XLII.-This general Traverse Table, so useful in practical navigation, is arranged in a very different manner from the Traverse Tables given in the generality of nautical books; and although comprised in 38 pages, is more comprehensive than the two combined tables of 61 pages usually found in those books, under the head "Difference of Latitude and Departure." In this table, every page exhibits all the angles that a ship's course can possibly make with the meridian, expressed both in points and de grees; which does away with the necessity of consulting two tables in finding the difference of latitude and departure corresponding to any given course and distance.

Table XLIV. contains the mean right ascensions and declinations of the principal fixed stars. The eighth column of this table, which is original, and is intended to facilitate the method of finding the latitude by the altitudes of two fixed stars observed at any hour of the night, contains the true spherical distance between the stars therein contained and those preceding or abreast of them on the same horizontal line. The ninth or last column of the page contains the annual variation of that

distance, expressed in seconds and decimal parts of a second. Great pains have been taken, in order to find the absolute value of the annual variation of the true spherical distance between the fixed stars; and the author trusts that he has so far succeeded as to render this part of the table permanent for a long period of years subsequent to 1824.

Tables XLV. and XLVI., which are adapted to the reduction of sidereal time into mean solar time, and conversely, have been newly constructed : these will be found considerably more extensive and uniform, than those generally given under the same denomination.

Tables LI. and LII. are entirely new: these will be found exceedingly useful in finding the latitude by the altitude of a celestial object observed at certain intervals from the meridian; and since they are adapted to proportional logarithms, the operation of finding the latitude thereby becomes extremely simple, and yet far more accurate than that resulting from double altitudes, even after repeating a troublesome operation, and then applying correction to correction.

Table LIV. This table will be of service to Masters in the Royal Navy, to officers employed in maritime surveys, and to all others who may be desirous of constructing charts agreeably to Mercator's principles of projection.

Table LVI. will be found essentially useful in reducing the French centesimal division of the circle into the English sexagesimal division, and conversely; and since most of the modern French works on astronomy are now adapted to the centesimal principle, this table will be found of assistance in consulting those works ;-nor will it be of less advantage to the French navigator, in enabling him readily to consult the works of the English astronomers, where the degrees, &c., are expressed agreeably to the original or sexagesimal principle.

Table LVII. is new; and although it may not immediately affect the interest of the mariner, yet it cannot fail to be useful to officers in charge of His Majesty's Victualling Stores, in consequence of the late Act of Parliament for the establishment of a new general standard or imperial gallon measure throughout the United Kingdoms.-See Practical Gauging, page 596 to 606.

Table LVIII. contains the latitudes and longitudes of all the principal sea-ports, islands, capes, shoals, rocks, &c. &c., in the known world; these are so arranged as to exhibit to the navigator the whole line of coast along which he may have occasion to sail, or on which he may chance to be employed, agreeably to the manner in which it unfolds to his view on a Mercator's chart; a mode of arrangement much better adapted to nautical purposes than the alphabetical. But since the table is not intended for general geographical purposes, the positions of places inland, which do not immediately concern the mariner, have, with a few exceptions, been

purposely omitted. The time of high water, at the full and change of the moon, is given at all places where it is known; which will be found considerably more convenient than referring for it to a separate table.

The series of latitudes and longitudes that have been established, astronomically and chronometrically, by Captain William Fitzwilliam Owen, of His Majesty's ship Eden, during his recent and extensive survey along the coasts of Africa, Arabia, Madagascar, Brazil, &c., follow as an Appendix to the last-mentioned table. These series are published by the express permission of Captain Owen; and from his general knowledge as a navigator, hydrographer, and practical astronomer, there is every reason to believe that the geographical positions have been determined with astronomical exactness.

A general Victualling Table forms an addition to the Appendix ; and as this exhibits the full allowance of sea provisions (calculated agreeably to the new Victualling Scale), from one man to any given number of men, it will be found useful to the Pursers of the Royal Navy, to Lieutenants serving as Commanders and Pursers, and to the gentlemen who are officially employed in the auditing of the Naval Victualling Accounts.

The sun's declination is not given in this work; nor is it necessary that it should be, since it is contained, in the most ample manner, in the Nautical Almanac; a work which is so truly valuable to mariners that few now go to sea without it; the judicious never will.

Having thus taken a survey of the principal part of the Tables, I must briefly notice their description and use ;-these will be found at the commencement of the first Volume. The principles and methods of their computation are here fully detailed; and the reader is furnished with the means, in the most simple formula, of examining any part of the Tables; which is far more satisfactory than trusting to the author's mere word for their entire accuracy; though, I flatter myself with the hope that, in this extensive mass of figures, very few errors will be found;-at all events, none of principle.

My original plan had been to close the work with the description and use of the Tables, but being apprehensive that a series of Tables alone, however well arranged, or clearly illustrated, would not be sufficient to ensure general acceptation, I was induced to show their direct application to the different elements connected with the sciences of navigation and nautical astronomy, as well as to other subjects of a highly interesting nature, such as the art of gunnery, &c. &c. In this part of the work, since my design did not extend beyond an ample illustration of the various mathematical purposes to which these tables may be applied, I have restricted myself to the practical parts of the sciences on which I have had occasion to touch; because those are the points which most concern the mariner, and the commercial interests of this maritime nation. Nevertheless, wherever it has