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PREFACE.

ALTHOUGH the importance and general utility of the subjects treated of in this work are sufficient to recommend it to public attention, without the aid of prefatory matter, yet, since there is an extensive variety of nautical publications now extant, I think it right to say something relative to what I have done, were it for no other purpose than that of satisfying the reader that the present work is widely different from any former treatise on nautical and mathematical subjects. The following observations will develope my motives for commencing so laborious an undertaking.

In perusing the various nautical publications which have appeared for many years past, I observed that they all fell considerably short of the objects at which they professed to aim ;-some, by being too much contracted, and others by not including all the necessary tables, or by being generally defective : and that, therefore, a great deal remained to be done, particularly in the tabular parts, beyond what had yet been brought before the public.

Of the nautical works that came under my notice, some have proved, on examination, to be so inaccurately executed, as to be entirely unfit for the consultation of any person not sufficiently skilled in the mathematics to detect their numerous errors. Many of the works in question are extremely incomplete, through their want of particular tables, and their logarithms not being extended to a sufficient number of decimal places : such as those by Mendoza Rios, where the decimals are only continued to five places of figures, and where the logarithmic

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tangents are entirely wanting; for, although the addition of a logarithmic sine and a logarithmic secant will always produce a logarithmic tangent, yet there are few mariners so far acquainted with the peculiar properties of the trigonometrical canon, as to be able to find by Rios's tables the arch corresponding to a given logarithmic tangent.* Hence, when the course and the distance between two places are to be deduced from their respective latitudes and longitudes, by logarithmical computation, the mariner is invariably obliged to have recourse to some other work for the necessary table of logarithmic tangents. Besides, since none of the nautical works now in use exhibit the principles upon which the tables contained therein have been constructed, the mariner is left without the means of examining such tables, or of satisfying himself as to their accuracy, though it is to them that he is obliged to make continual reference, and on their correctness that the safety of the ship and stores, with the lives of all on board, so materially depend.

Notwithstanding that Mr. Taylor's Logarithmical Tables are the most extensive, the best arranged, and by far the most useful for astronomical purposes, of any that have ever appeared in print,-yet, since they do not contain the necessary navigation tables, they are but of little use, if of any, to the practical navigator; and, since the same objection is applicable to the very excellent system of tables published by the learned Dr. Hutton, these are, also, ill adapted to nautical purpones, and but rarely consulted by mariners.

Being thus convinced that there was something either deficient or very defective in all the works that had hitherto been published on this subject, I was ultimately led to the conclusion that a general and complete set of Nautical Tables was still a desideratum to mariners : with this sonviction on my mind, I was at length induced to undertake the laborious task of drawing up the following work; in the prosecution of which I found it necessary to exercise the most determined perseverance and industry, in order to surmount the fatigue and anxiety attendant on such a long series of difficult calculations.

These points premised, it remains to present to the reader a familiar and comparative view of the nature of this work, and of the improvements that have been made in the tables immediately connected with the elements of navigation and nautical astronomy: confining the attention to those that possess the greatest claims to originality, or in which the most useful improvements have been made.

Table VI. contains the parallaxes of the planets in altitude; and

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* See Remark, page 98; with diagram and calculations, page 99.

vill be found particularly useful in deducing the apparent time from the altitudes of the planets, and, also, in problems relating to the longitude. The hint respecting this was originally taken from the Copenhagen edition of “The Distances of the Planets from the Moon's Centre, for the Year 1823;" but this design has been considerably enlarged and improved upon.

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 XIII. 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 Almanae, 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:

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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 481, 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 cd-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 above-mentioned 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

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“The Young Navigator's Guide,” &c. : it is arranged in a familiar manner, and, though concise, contains all the riumbers 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 cosecant; 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

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