DETERMINING THE LATITUDE, AND THE LONGITUDE BY LUNAR OBSERVATIONS,

AND

KEEPING A COMPLETE RECKONING AT SEA;

ILLUSTRATED BY

PROPER RULES AND EXAMPLES :

THE WHOLE EXEMPLIFIED IN A JOURNAL,

KEPT FROM BOSTON TO MADEIRA,

IN WHICH

ALL THE RULES OF NAVIGATION ARE INTRODUCED:

ALSO,

THE DEMONSTRATION OF THE USUAL RULES OF TRIGONOMETRY; PROBLEMS IN MENSURATION, SURVEYING, AND GAUGING, DICTIONARY OF SEA TERMS:

AND THE MANNER OF PERFORMING THE MOST USEFUL

EVOLUTIONS AT SEA:

WITH

AN APPENDIX,

CONTAINING

METHODS OF CALCULATING ECLIPSES OF THE SUN AND MOON, AND OCCULTATIONS OF THE
FIXED STARS; RULES FOR FINDING THE longitude oF A PLACE BY OBSERVATIONS
OF ECLIPSES, OCCULTATIONS, AND TRANSITS OF THE MOON'S LIMB OVER
THE MERIDIAN; ALSO A NEW METHOD FOR FINDING THE
LATITUDE BY TWO ALTITUDES.

BY NATHANIEL BOW DITCH, LL. D.

Follow of the Royal Societies of London, Edinburgh, and Dublin; of the Astronomical Society in London; of
the American Philosophical Society, held at Philadelphia; of the American Academy of Arts and
Sciences; of the Connecticut Academy of Arts and Sciences; of the Literary and
Philosophical Society of New York; Corresponding Member of the

Royal Societies of Berlin, Palermo, &c.—and, since

his decease, continued by his son,

J. INGERSOLL BOWDITCH.

SIXTEENTH NEW STEREOTYPE EDITION.

NEW-YORK:

PUBLISHED BY E. & G. W. BLUNT, PROPRIETORS,
No. 179 WATER-STREET, CORNER OF BURLING SLIP.

STEREOTYPED AT THE

BOSTON TYPE AND STEREOTYPE FOUNDRY.

1846.

NOTICE TO THE 16th EDITION.

THE Latitudes and Longitudes of the places on the coast of Ireland, the coast of Louisiana and Texas; some few on the East coast of Mexico, Florida Keys, Salt Key Bank, &c., have been altered to correspond with the latest observations. I have consulted Raper, Blunt, &c.

From Capt. Barnett, R. N., the positions of the following places were ob tained, and of some others on the coast of Yucatan, Florida Keys, and Salt Key, which will be found in the Table.

I have been furnished by Dr. Bache, superintendant of the United States Coast Survey, with the position of the New South Shoal, as surveyed by Lieut. Charles H. Davis:

"The bearing of the New South Shoal from the old is from S. 3° 28′ W. to S. 16° 42′ E, by compass; distance, 6.43 miles.

"The direction of New Shoal is East and West, nearly.

"Greatest length in this direction, 2.3 miles.

"Greatest breadth North and South, nine-tenths of a mile.

"The least depth is 8 feet, which is found in two places, about one-half a mile apart.

"The latitude of the centre of the Shoal is 40° 57′ 50′′, nearly; the longitude, 69° 51′ 40′′, nearly."

Nov. 1846.

J. INGERSOLL BOWDITCH.

Entered according to Act of Congress, in the year of our Lord, 1846, by E. & G. W. Blunt, in the Clerk's Office of the District Court of the Southern District of New York.

PREFACE.

B7 1846

In the Preface to the first edition of this work, it was observed, that the object of the publication was to collect into one volume all the rules, examples, and tables, necessary for forming a complete system of practical navigation. To do this, those authors were consulted whose writings afforded the best materials for the purpose; and such additions and improvements were introduced as were suggested by a close attention to the subject; and the accuracy of the tables accompanying the work was ensured by actually going through all the calculations necessary to a complete examination of them, making the last figure exact to the nearest unit. In performing this, above eight thousand errors were discovered and corrected in Moore's Practical Navigator, and above two thousand in the second edition of Maskelyne's Requisite Tables. Almost all the errors in Maskelyne's collection were in the last decimal place, and in most cases would but little affect the result of any nautical calculation; but when it is considered that most of those tables are useful in other calculations, where great accuracy is required, it will not be deemed an unnecessary improvement to have corrected so great a number of small errors.

Several articles were added in the second edition, particularly the description and use of the circular instrument of reflection, methods of surveying harbors, new tables, &c. In the third, and subsequent editions, several improvements were made, and an Appendix was given, containing methods of projecting and calculating eclipses of the moon and sun, and occultations of the fixed stars or planets by the moon; rules for deducing the longitude of a place from observations of eclipses of the sun or occultations; a new and short method of calculating the altitude and longitude of the nonagesimal degree of the ecliptic; solutions of several useful problems of nautical astronomy, and an improvement of Napier's rules for the solution of spheric triangles. Several new tables. were added. The table of latitudes and longitudes was much increased and corrected.

A new article was given in the sixth and seventh editions, on the method of finding the latitudes by two altitudes of the same or of different objects, being

M537664

M53

an improvement of Mr. Ivory's solution. The method we have given is direct and simple, embracing all the cases of the problem; a point which is not sufficiently attended to in some works of celebrity. This article is an important addition to the work, and it is recommended to the consideration of navigators.

The tables, published separately in the Appendix of the first edition, are introduced into the body of this work, and are extended so as to render the use of them more simple. The first method of working a lunar observation, published in that Appendix, which has one great advantage over all other approximate methods, in the manner of applying the corrections, (all of them being additive,) is here explained and illustrated by several examples. The second is an improvement of Lyons's method, which had been known for many years, but had not been generally used, because the tables were not sufficiently extended. This difficulty is now obviated, by means of Tables XLVII. XLVIII., which have been compared with Thompson's tables, and many of them recomputed by the aid of Shephard's tables. The third method was given by the author of this work, in 1795. The fourth method is an improvement of Witchell's process, in which, without altering materially the calculation, the number of cases is considerably reduced.

Any person who wishes to examine the tables, may do it by the methods used for that purpose, which will here be explained, with some additional remarks:

Tables I. and II. were calculated by the natural sines taken from the fourth edition of Sherwin's logarithms, which were previously examined, by differences; when the proof-sheets of the first edition were examined, the numbers were again calculated by the natural sines in the second edition of Hutton's logarithms; and if any difference was found, the numbers were calculated third time by Taylor's logarithms.

a

Table III. contains the meridional parts for every degree and minute of the quadrant, calculated by the following rule, viz.

M=T× 0.0007915704468,

in which T is the log. tangent less radius of half the latitude, increased by 45°, taken to seven places of figures, reckoned as integers; and M is the meridional parts of that latitude in miles.

Table IV. contains the declination of the sun, which was compared with the Nautical Almanacs for the years 1833, 1834, 1835, and 1836, and marked to the nearest minute.

Table IV. A. The equation of time, for the years 1833, 1834, 1835, and 1836.

Table V. contains the correction of the sun's declination, as published by Dr. Maskelyne. The correction taken from this table will rarely differ more.

Table VI. contains the mean of the sun's right ascension, taken from the Nautical Almanacs for the years 1833, 1834, 1835, and 1836.

Table VI. A. contains the correction for the daily variation of the equation of time.

Table VII. contains the amplitudes of the sun for various latitudes and declinations, calculated by Taylor's logarithms, by this rule:

Log. sec. lat.log. sine declination-10.0000000=log. sine amplitude. Table VIII. contains the right ascensions and declinations of one hundred and eighty stars of the first, second, and third magnitudes, with their annual variations, adapted to the beginning of the year 1830. This table was abridged from that published by the astronomer royal at Greenwich, (Mr. Pond,) in the year 1833.

Table IX. contains the time of the sun's rising and setting, calculated by Taylor's logarithms, by this rule :

Log. cos. hour= log. tang. declin. + log. tang. latitude

10.0000000.

Table X. contains the distances at which any object is visible at sea, calculated by the rule given in § 195 of Vince's Astronomy, in which the terrestrial refraction is noticed. This circumstance was neglected by Robertson. Moore, and others, and of course their tables are erroneous.

by Mr. Vince, expressed in logarithms, is this:

The rule given

0.12155+ half log. of height in feet=log. of dist. in statute miles.

In reducing the rule to logarithms, the radius of the earth was called 20911790 feet, which agrees nearly with the mean value given in De La Lande's Astronomy.

Table XI. is a common table of proportional parts, the construction of which does not need any explanation.

Table XII. contains the refraction of the heavenly bodies, calculated by Dr. Bradley's rule, supposing the refraction to be as the tangent of the apparent zenith distance of the object, decreased by three times the refraction, the horizontal refraction being supposed equal to 33. The rule, expressed in logarithms, is this:

Log. tang. (app. zen. dist.-3. refraction)-8.2438534=log. of ref. in sec. The numbers calculated by this rule agree nearly with those published in Table 1 of Maskelyne's Requisite Tables.

Table XIII. contains the dip of the horizon for various heights, calculated by the rule in 197 of Vince's Astronomy, in which the terrestrial refraction is allowed for. All the numbers of this table differ a little from those published by Dr. Maskelyne, who had made a different allowance for that refraction. The rule given by Mr. Vince, expressed in logarithms, is,

log. dip in seconds.

1.7712711 + half the log. of the height in feet Table XIV. contains the sun's parallax in altitude, calculated by multiplying