the observed altitude of any celestial object, in order to reduce it to the true altitude. In book third is contained a complete system of lunar observations, with an introductory account of this method of finding the longitude at sea. It also contains a new method of finding the longitude and latitude of a ship at sea, together with the apparent time, from the same set of observations; for which the author had the honour of receiving the thanks of the Boards of Longitude of England and of France. Book fourth contains various methods of finding the Longitude of a place; some of which, though scarcely practicable at sea, are yet, perhaps, the very best that can be used for determining the longitude of any place at land. These are, by the Moon's transit over the meridian, by the meridian altitude of the Moon, and by the observed altitude of the Moon when not on the meridian; by lunar eclipses, solar eclipses, occultations of fixed stars by the Moon, eclipses of the satellites of Jupiter, by a chronometer, and by the Variation Chart. Every treatise must be considered as defective, wherein rules are given without demonstrations: and this is particularly the case with most books on the subject of navigation, probably because sea-faring people in general are very indifferent as to the rationale of any. rule given for making an observation, provided it holds good in practice. On the other hand, it not unfrequently happens, that to the persons not very fond of mathematical investigations, the rule and demonstration following each other in regular order, may be disagreeable. For these reasons, and to gratify the scientific seaman in the present highlyimproved state of nautical knowledge, the demonstrations of the rules and formulæ are separately inserted at the end of the work, in the fifth book. And, in order to render this work still more generally useful, there is farther added to what was originally proposed, book sixth, containing various methods of finding a ship's latitude, and the variation of the compass. Volume second contains the tables, together with the necessary explanations of them. This This work, therefore, it is hoped, will be found to comprehend a collection of the best methods of making and reducing all sorts of observations necessary to be made at sea, for ascertaining a ship's place; or for reducing the true course between any two places to the course per compass; or for correcting the course steered. The author, however, flatters himself, that it will not be considered merely as a compilation from the works of others, but that the intelligent reader will discover, in various parts of it, some things that are at least new and of his own invention, however little other merit they may be thought to possess; and in every part of it, a suitable plainness and per spicuity of style, and a warm zeal for the improvement of one of the noblest and most useful of human arts. In a word, he has exerted his best endeavours to render his work worthy the notice of the navigator and astronomer, as well as, in some respects, of the practical geographer and surveyor. The favourable reception of the first edition of this work, has induced the author to revise the whole; in doing which he has made many additions to it, which he flatters himself will be found useful. The size of the page has been also enlarged, in order to prevent that irregularity, or confusion, which would otherwise happen in the arrangement of the calculations, when the page had been too much contracted, And, with a view to render the computations more easy, and to make the work complete, some of the tables in the second volume of the first edition have been altered and extended, and new tables added; which has been a work of considerable labour. To this third edition, still further improvements and additions have been made, as will be evident by comparing it with the former editions, throughout the whole of the work. In the second volume, the Table of Natural Versed Sines is extended to, an arch of 180o, with a view to facilitate the operation of reducing the apparent distance to the true distance, by Method First, page 150: and also for the purpose of employing various other methods of effecting the same thing; see page 157, &c. A table of Logarithmic Tangents, to every ten seconds, with the proportional parts to each second, similar to that of the Logarithmic Sines, is also added. The various other additions, alterations, and corrections, will be obvious; and no expense whatever has been spared, with respect to print, paper, &c. in order to render it worthy of the attention of the public. b2 The The author is extremely sorry to have occasion to observe, that some late authors, or he may rather say, pretended authors, have copied largely from the former editions of this work, without mentioning from whence they obtained their information. Many of the tables they have copied; and, strange to tell! have asserted they were either constructed or altered to the present form by themselves*: many other of the tables they have altered, and to which they have given new titles, with a view to disguise them as much as possible. They have copied several of the various methods, rules, &c. by which they have entailed upon themselves the most opprobrious of epithets, namely, that of plagiarists. This, however, is not the case with foreigners, particularly the French, Dutch, Americans, &c. who have candidly acknowledged from whence they derived their information. The author cannot conclude without mentioning, that many favours have been conferred upon him by his worthy and valuable friend, Mr. BARON MASERES: to whom the public is, in a great measure, indebted for the present edition of this work. His superior knowledge raises him high in the literary and scientific world; and he has, without the least degree of ostentation, an ardent desire to promote true knowledge, and also the happiness of those who are similarly inclined. The Author most respectfully informs the Public, that his signature, as below, is placed to every copy of this work, and which can only be had by application to himself, or by his appointment. George Street, Trinity Square, January 16, 1809. 4. Machey * For the truth of this, the reader is requested to compare a Collection of Matheinatical Tables, printed in London in 1803, and an Epitome of Navigation, of which these Tables make a part, printed in 1805, with the second volume of the author's Treatise on the Longitude, printed in 1801. CONTENTS CONTENTS FIRST VOLUME. Upon the Instruments for measuring angular Distances at Sea, and Of the Manner of taking a complete Set of Lunar Observations Of the Corrections to be applied to the Altitude of an Object observed Of the Method of finding the Longitude of a Ship by Lunar Observations. and the Sun or a fixed Star, from the Effects of Refraction and Parallax 148 V. Of finding the Longitude at Sea or Land by Lunar Observations VI. Of finding the Longitude at Sea or Land by an Observation of the Distance between the Moon and the Sun, or a fixed Star, together VII. A new Method of finding the Longitude and Latitude of a Ship at Sea 194 VIII. Of finding the Longitude at Sea or Land by an Observation of the Distance between the Moon and a Star, not used in the Nautical CHAP. I. Of finding the Longitude by an Observation of the Moon's Transit over the Meridian 213 III. Of finding the Longitude by an Observation of the Moon's Altitude, the apparent Time at the Place of Observation, together with the Latitude, V. The Method of finding the Longitude of a Place by an Eclipse of the Sun 233 VI. The Method of finding the Longitude of a Place by an Occultation of a Containing the Demonstrations of the preceding Rules and Formulæ. 289 Containing various Methods of finding the Latitude of a Place, and CHAP. I. Of finding the Latitude of a Place by single and double Altitudes of the II. Of finding the Variation of the Compass by an Observation of the Ampli- EXPLA |