OF NAUTICAL ASTRONOMY AND NAVIGATION. PART II. CONTAINING THE INVESTIGATIONS AND PROOFS OF THE PRINCIPAL With Practical Examples. Designed for Beginners and advanced Students. By H. W. JEANS, F.R.A.S. LATE MATHEMATICAL MASTER AND EXAMINER AT THE ROYAL NAVAL COLLEGE, PORTSMOUTH; AUTHOR OF A WORK ON "PLANE AND SPHERICAL TRIGONOMETRY; HANDBOOK OF THE STARS;" ASTRONOMY, NAVIGATION, ETC. WITH SOLUTIONS:" FORMERLY MATHEMATICAL MASTER IN THE ROYAL MILITARY ACADEMY, WOOLWICH; AND AN EXAMINER OF OFFICERS A NEW EDITION. BIBE LONDON: LONGMANS, GREEN, READER, AND DYER. 1868. 184. h. 53. PREFACE. THE First Part of this Work-consisting of Practical Rules in Navigation and Nautical Astronomy, with a series of examples under each rule was originally drawn up for the use of beginners, and as introductory to some of the larger works on the subject; but recent additions render it complete in itself, and it will now be found to contain ample directions for the guidance of the practical navigator. The rules are adapted to any of the standard Nautical Tables now in se, such as Norie's, Raper's, or Riddle's; but as the collection of Tables published by the late Dr. Inman* is almost universally adopted in the Royal Navy, these tables have in most cases been used in working out the examples. The present volume, Part II., may be considered as the scientific part of the subject, consisting of the analytical investigations and proofs of the principal rules and corrections in Navigation and Nautical Astronomy. The Author trusts that the * This venerable and most useful public servant died on the 8th of February 1859, at his residence at Southsea near Portsmouth, at the advanced age of eighty-three years. The Reverend James Inman, D.D., was upwards of thirty years Professor of Mathematics at the Naval College. He was the oldest of Cambridge senior wranglers, and long possessed a just celebrity in naval circles for his application of science to navigation and ship-building. He laboured very many years unobtrusively but zealously in his country's service. He sailed round the world, having been appointed to the expedition under Flinders as astronomer; was wrecked with him; and returning to Europe in the fleet of East-India ships under the command of Commodore Dance, was present in that celebrated action in which a fleet of merchantmen repulsed the French Admiral Linois. While Professor of Mathematics at the Royal Naval College he reduced to system the previously ill-arranged methods of navigation, and published several valuable works on the subject; but he was best known by his having been the first person in this country who built ships on scientific principles. Dr. Inman's translation of Chapman, with his valuable annotations, is the text-book on which all subsequent writers have proceeded. His pupils, a long list of distinguished naval officers, will remember him as a type of the high-minded scholar,―of the loyal, the truthful, and independent man. plan followed by him in this Part, namely, to exhibit a geometrical figure or diagram of each problem, then to give the analytical investigation from which the rule is derived, followed by a numerical example worked out to show the application of the formula, together with nearly 300 examples for practice dispersed throughout the volume, will render the book useful to beginners (for whom, in fact, it is chiefly intended), as well as deserving the attention of more advanced students, and increasing the confidence of the practical navigator. The Author's wish was to exclude from the present work every problem requiring a mathematical knowledge beyond algebra and trigonometry; he has, however, been induced to depart from this in one or two instances, at the request of several Naval Instructors, who thought that it would render the work more useful to introduce the problems for finding the longitude by an occultation, and for determining the spheroidal figure of the earth from actual measurements on its surface. These two problems belong to the more advanced part of Nautical Astronomy, which the Author contemplated at one time to publish under the title of "Nautical Astronomy, Part III." Part II. however, it is hoped, will enable the naval student to comprehend without difficulty the principal rules in Nautical Astronomy; and this he will more readily do, if he has previously made himself acquainted with the Author's volume of Astronomical Problems, which work may be looked upon as introductory to the more important subject of Nautical Astronomy, explained and illustrated at large in the present volume. Langstone House, Havant, July 10, 1868. CONTENTS. The student is recommended to confine his attention at first to the problems and articles II. Given mean time; to find sidereal time III. Given mean time, or apparent time; to find what heavenly body will pass the meridian the next after that time IV. Given the length of a mean solar day; to express its length in *Construction of table of time equivalents V. Given sidereal time at any instant; to find mean time at the VI. To find at what time a heavenly body will pass the meridian VII. Given the altitude and declination of a heavenly body, and the latitude; to calculate the hour-angle VIII. Given the hour-angle of the sun; to find ship mean time. IX. Given the hour-angle of a star; to find ship mean time X. Given the altitude of a heavenly body, and the time shown by a chronometer; to determine the error of the chronometer *XI. Given a small error in the altitude; to find the corresponding |