diff --git "a/Astronomy/a_history_of_nautical_astronomy_1968.md" "b/Astronomy/a_history_of_nautical_astronomy_1968.md" new file mode 100644--- /dev/null +++ "b/Astronomy/a_history_of_nautical_astronomy_1968.md" @@ -0,0 +1,14145 @@ +A HISTORY OF MAGICAL ASTROLOGY + +CHARLES H. COOPER + +HOLLIS & CARTER + +A HISTORY OF NAUTICAL ASTRONOMY +$4=50$ + +BY THE SAME AUTHOR + +The Elements of Navigation +The Principles and Practice of Rope Direction Finding +The Master and His Ship +The Apprentice and His Ship +The Complete Coastal Navigator +The Physical Geography of the Oceans +The Astronomical and Mathematical Foundations of Geography + +A History +of Nautical Astronomy + +CHARLES H. COTTER +EX C BSC F INST NAV +Senior Lecturer in the Department of Maritime Studies +at the University of Wales Institute of Science and Technology + +... but teaching the Theoricks, by observation of +the Heavens, and the stars there, to find out +a passage in the vast Ocean where no paths are, +none is more necessary than this one, since the +first discovery of this Trigonometry. + +From the Epitole Dedicatoria in Ralph Hambone's +*Trigonometry; or the Art of Finding Triangles.* +First written in Latin by Bartholomew Pittonius +of Groningen in Silesia. + +HOLLIS & CARTER +LONDON - SYDNEY +TORONTO + +CLASS 597_COT +VOL. ____________ COPY. ____________ +SUPPLIER ____________ RECD (L) 10-16-99 +ACCESS ____________ + +© Charles H. Gutter, 1986 +ISBN 370 640 + +Printed and bound by The Stationery House for +Hollins & Co., Ltd. +9 Bow Street, Dublin, W.C.2 +by William Chavas & Sons Ltd., Bexley +Set in Imprint +Printed in Ireland + +CITY OF LIVERPOOL +COLLEGE OF TECHNOLOGY LIBRARY + +# Contents + +Preface by Alton B. Moody, ix +Author's Foreword, xi + +## I. The Development of Nautical Astronomy, 1 + +1. Introductory, 1 +2. Babylonians and Phoenicians, 2 +3. The Greeks, 7 +4. Hipparchus, 13 +5. Ptolemy, 16 +6. The Arabs, 19 +7. Early Renaissance Scholars, 20 +8. Copernicus, Tycho Brahe, 22 +9. Kepler, Galileo, Newton, 25 +10. The Dawn of Modern Nautical Astronomy, 28 + +## II. Astronomical Methods of Time-measuring at Sea, 32 + +1. The Units of Time, 32 +2. The Calendar, 35 +3. The Gnomon, 36 +4. The Divisions of the Day, 37 +5. The Nocturnal, 39 +6. Sun Time and the Ring Dial, 41 +7. Mechanical Clocks, 42 +8. Arithmetic of Navigation, 43 +9. The Azimuth Compass, 47 +10. Nautical Tables for Determining Time, 47 +11. The Nautical Almanac, 48 +12. Computing Local Time, 51 +13. The Marine Chronometer, 52 + +## III. The Altitude-measuring Instruments of Navigation, 57 + +1. Introductory, 57 +2. The Seaman's Quadrant, 57 + +vi CONTENTS + +3. The Astrolabe, 60 +4. The Cross-staff, 64 +5. The Kamal, 69 +6. The Back-staff, 70 +7. The Reflecting Quadrant, 74 +8. The Hadley Quadrant, 77 +9. The Reflecting Circle, 83 +10. Perfected Altitude-measuring Instruments, 87 +11. The Artificial Horizon, 91 + +IV. **The Altitude Corrections**, 97 +1. Introductory, 97 +2. Refraction, 97 +3. Depression or Dip of the Sea Horizon, 111 +4. The Sun's Semi-diameter, 117 +5. The Moon's Semi-diameter, 118 +6. Parallax, 119 +7. Irradiation, 121 +8. Personal Error or Equation, 122 + +V. **Methods of Finding Latitude**, 123 +1. Introductory, 123 +2. Latitude by the Pole Star, 130 +3. Latitude by Mean Altitude of the Sun, 137 +4. Latitude by Meridian Altitude of a Star, 139 +5. Latitude by the Southern Cross, 140 +6. Declination Tables, 141 +7. The Double-Altitude Problem, 143 +8. Meridian and Maximum Altitudes, 162 +9. Latitude by Ex-meridian Altitude, 165 + +VI. **Methods of Finding Longitude**, 180 +1. Introductory, 180 +2. Longitude from Eclipse Observations, 182 +3. Longitude from Observations of Jupiter's Satellites, 184 +4. Longitude from Observations of Moon Occultations, 189 +5. Longitude by Lunar Transit Observation, 191 +6. Rewards for Discovering the Longitude, 192 + +CONTENTS + +vii + +7. Longitude by Lunar Distance, 195 + (a) Historical Survey, 195 + (b) Makelyne and the Nautical Almanac, 203 + (c) Practical Astronomical Practice, 205 + (d) Methods for Clearing the Distance, 208 + +8. Finding G.M.T. from a Lunar Observation, 237 + +9. Longitude by Chronometer, 243 + (a) Methods, 243 + (b) Observations for Checking the Chronometer, 254 + (c) Absolute-altitude and Equal-altitudes Observations, 257 + +VII. Position-line Navigation, 268 + 1. Introductory, 268 + 2. Historical Development of Position-line Navigation, 271 + 3. Sumner's Discovery, 275 + 4. Azimuth Tables in Position-line Navigation, 284 + (a) Burdwood and Davis, 284 + (b) Heath and A, B and C Tables, 284 + 5. Marq St Hilaire and the New Navigation, 293 + +VIII. Navigation Tables, 309 + 1. Early Astronomical and Mathematical Tables, 309 + 2. The Nautical Almanac, 313 + 3. Short-method and Inspection Tables, 317 + 4. Graphical Solutions of the PZX Triangle, 334 + 5. Mechanical Aids to Calculation, 340 + +Appendix 1. Spherical Astronomy, 343 +Appendix 2. Spherical Trigonometry, 349 +Bibliography, 357 +Index, 373 + +List of Plates + +Between pages 92 and 93 + +1. Rev. Nevil Maskelyne D.D. +2. Mariner's Astrolabe. Probably Spanish, c. 1585. +3. Nocturnal in Boxwood. English, c. 1646. +4. Azimuth Compass. English, c. 1720. +5. Mariner's Quadrant. c. 1600. +6. Back-staff or Davis Quadrant. By John Gilbert, c. 1740. + +Between pages 236 and 237 + +7. Hadley Octant. By Benjamin Martin, c. 1760. +8. Reflecting Circle. By Edward Troughton, c. 1800. +9. Sextant. By Kelvin Hughes, 1967. +10. Facsimile of Plate 3 of Sumner's Pamphlet. +11. Facing Pages of 1797 Nautical Almanac. +12. Facing Pages 1897 Nautical Almanac. + +Plates 1 to 8 are reproduced by courtesy of the Trustees of the National Maritime Museum, Greenwich. Plate 9 is reproduced by courtesy of Messrs. Kelvin Hughes. + +Preface + +ALTON B. MOODY, +former President of the United States Institute of Navigation + +In this fast-moving age, characterized by a rapid increase in knowledge sometimes referred to as an 'information explosion', it is easy to succumb to the temptation to limit one's reading to new developments in one's own specialty. With the plethora of books and technical periodicals that come to the attention of the professional, it becomes a problem even to keep informed of these. As a result, a certain superficiality pervades much of the thinking of those who exhibit interest in professional matters. + +Navigators is particularly susceptible to this danger because of the turbulent sea which disciplines it encompasses. As a result, a great amount of effort is expended by talented individuals who lack perspective regarding the problems they seek to solve, as attested by the many solutions that look good on paper but do not find a ready acceptance by those who would seem to be beneficiaries of the work. + +The history of nautical astronomy is a fascinating subject involving the hopes, fears, superstitions and thoughtful observations of many individuals over a very long period of time. Early man sensed the value of celestial observations as a means of providing guidance at sea, where no landmarks were available and electronic signals were unknown, but lacked the knowledge and instruments needed fully to utilize this source of guidance. There was something frightening about putting oneself outside these boundaries, and it is understandable how fear might happen if one reached the physical boundary of the earth offered little comfort to those with sufficient curiosity to set forth into the unknown. As a result, only the more intrepid adventurers deliberately attempted long voyages out of sight of land. + +But there were hardy individuals in various periods who sought to widen the horizon of man's knowledge. Little by little nature gradually yielded to these attempts, and the story of this struggle is the story of the progress of man. Certainly there is no + + + + + + +
Alton B. MoodyFormer President of the United States Institute of Navigation
+ +X +PREFACE + +more captivating story than that relating to man's attempt to 'discover' the longitude at sea, and many there were who despised it as a practical science until being found. As late as 1594, nearly two centuries before the problem was solved but many centuries after man had ventured beyond the sight of land, Davis wrote: 'Now there be some that are very inquisitive to have a way to get the longitude, but that is too tedious for seamen, since it requireth the deep knowledge of astronomy, wherefore I would not have any man think that the longitude is to be found at sea by any instrument, so let no seaman trouble themselves with any such thing, which seem keep a perfect account and reckoning of the way of their ship.' + +This book is a history of nautical astronomy. But it is more than that. Captain Charles H. Cotter has done more than trace the sequence of events leading to man's present extensive knowledge of celestial navigation. With discerning care he has delved into representative solutions of various stages of man's developing knowledge and given them their due place, as well as philosophical explanations of methods which are now generally known only by name—or not at all by many members of a rising generation who have been too absorbed in finding solutions to today's problems to take time to learn of those of other ages. + +What then is the value of this book? As a reference, it brings together a wealth of information that would require extensive searching to find and puts it in perspective. As a source of inspiration to those who may be discouraged at the difficulty of conquest over the unknown, it stands as a beacon on a hill. As a repository of cultural information on man's emergence as the master of his environment, it is a worthy addition to any gentleman's library. + +Washington, D.C., 1967 +ALTON B. MOODY + +Author's Foreword + +The several factors, including wind and hidden current, which operate during a voyage to set a vessel from the desired path make it imperative for a navigator frequently to check the progress of his course. Without accurate knowledge of his latitude and longitude a navigator is unable to rectify the course of his ship with confidence. In this present work I have attempted to trace the fascinating story of the development of the astronomical methods used for finding a ship's position at sea when out of sight of land. + +Few would deny that astronomical navigation, or nautical astronomy to give the subject its full name, is an obsolete science. The principal astronomical methods used at present mark the culmination of an evolutionary process which began even before the first of the Phoenician sea-traders navigated their craft in Eastern Mediterranean waters, using the heavenly bodies to guide them, some three thousand years ago. The ancient craft of the nautical astronomer is now rapidly being replaced by sophisticated navigational systems based on the advances made in radio and the distinctive features of the marvellous age in which we live. + +My principal aim in writing this history has been to present in broad outline an historical account of the diverse problems of nautical astronomy and the ways in which they were solved. I have been conscious of a strong desire to associate with these problems the philosophers, scientists and men of letters who compared and explored the paths which led to their solutions. These men occupied prominent positions in a series of actions, events and purposes which, considered collectively, possess a remarkable dramatic unity. + +The history of nautical astronomy spans a long era of many millennia. It had its beginnings when seamen ventured first learnt the rudimentary use of the stars to guide them in their exploratory ocean voyages. That era is ending in our own times when the applications of radio and electronics are superseding the relatively simple techniques of the nautical astronomer. + +xii +AUTHOR'S FOREWORD + +My own intense interest in the history of navigation was sparked off during my formative years as a pupil at Smith Junior Nautical School, Cardiff. From my wife and inspiring teachers I learnt that the history of navigation, and especially navigational astronomy, however important this may be to an officer on the bridge of a ship at sea, does not in itself make a complete and educated navigator. This indeed is a valuable lesson to learn, and I too am firm in the belief that no practitioner can have complete respect for the science or craft he practises without having some acquaintance with the historical development or evolution of his subject. It is also true that many people who do not pursue a career in navigation can have little regard for a science they know all too little about. With these sentiments it is my sincere desire that this work will serve usefully to fill a gap in the literature of the history of the science of navigation. + +I owe a debt of gratitude to several assistants, past and present, at the libraries of the British Museum and the National Maritime Museum, London, and the Public Library of Newcastle upon Tyne, for their kind help during the many years during which I was engaged in making researches. I wish also to record that I have been appreciative of the friendly interest shown by some of my colleagues in the progress of this work. + +It is a very real pleasure for me to place on record my thanks to the staff of the publishers for much advice and many valuable suggestions which have helped considerably in clarifying and otherwise improving my original text. + +Cardiff 1967 +CHARLES H. COTTER + +CHAPTER I + +The development of nautical astronomy + +I. INTRODUCTORY + +John Seller, Hydrographer to the King during the late 17th century, declared, in his popular work on *Practical Navigation*, that the part of navigation + +'. . . which may properly bare the name and principally deserve to be entituled the art of Navigation, is that part which guides the ship in her Course through the Immense Ocean to any part of the known World, which cannot be done until it be determined what is the Latitude and Longitude at that time both in respect of Latitude and Longitude: this being the principal care of a navigator and the Masterpiece of Nautical Science.' + +"To the commendable accomplishment of this knowledge," Seller added, "four things are subordinate Requisites. Viz: + +Geometry. +Trigonometry. +The Doctrine of the Spheres." + +The Doctrine of the Spheres covered the necessary spherical or mathematical astronomy, a knowledge of which was essential for guiding a ship across the pathless ocean for finding her position, and for steering towards her destination. + +The history of Ocean Navigation, or la *Navigation Grande* of the old French navigators, began when astronomy became scientific, that is to say, when men first began to reason about, and speculate upon, the nature of the celestial bodies and their movements. + +Astronomy, which literally means the law of the stars or 'star distribution,' is a branch of knowledge probably as old as mankind itself; and our most ancient ancestors must have, as we + +2 +A HISTORY OF NAUTICAL ASTRONOMY + +have, gazed upon the firmament, rejoiced in its splendour, and pondered about its nature. + +The knowledge, acquired by paintstaking observations, of the relative movements of the Sun, Moon, Planets and stars was, little by little, built into a tradition which provided early man with the means for finding direction, time and season. The earli-est civilizations found it essential to have compass, clock and calendar; and the orderly movements of the heavenly bodies, relative to the Earth, provided for their needs. + +There is reason to suppose that the flourishing civilizations of at least three thousand years ago possessed sound practical knowl-edge of astronomy. The earliest astronomical observations, and the most primitive cosmogonic ideas, belong to an extended period of astronomical prehistory for which no written records exist. During this period the stars were grouped to form the constellations, eclipses were observed, and the apparent paths of the Sun, Moon and planets across the backdrop of fixed stars were delineated. + +The Greek philosophers of the 5th century before the begin-ning of the Christian Era appear to have been the first to enquire into the causes of celestial phenomena. They are, therefore, to be honoured as being the founders of scientific astronomy. + +The astronomical science of the Ancient Greeks of the period between the 5th and 3rd centuries B.C. was based upon observa-tions made by earlier philosophers, notable amongst whom were those of Babylon and Egypt. + +2. BABYLONIANS AND PHOENICIANS + +The Babylonians, who occupied the seaboard of Syria, formed a branch of the Semitic race who cultivated a love for the sea. Some historians have argued that the Phoenicians, these people are said to have dominated on the eastern shore of the Red Sea in the "Land of Edom" (modern Jordan), it is clear we shall see that Phoenician seamen voyaged in the Red Sea as well as in the Mediterranean. + +The notable sea ports of the Phoenicians included Tyre and Sidon. Numerous references are made in the Old Testament to Tyrians and Sidonians, and to Phoenicia and its seamen, and to their nautical—as well as their commercial—skill. + +The earliest biblical account of a long sea voyage appears in the + +THE DEVELOPMENT OF NAUTICAL ASTRONOMY + +First Book of Kings, where we learn that King Solomon made a navy of ships in the land of Edom; and that Hiram, king of Tyre, sent shipmen who had knowledge of the sea to join the servants of Solomon. And the navy, we are told, voyaged to Ophir—believed to be Ceylon—the voyage occupying three years. And from Ophir were brought gold, silver and great plenty of almsug trees, as well as apes, peacocks and precious stones. + +There is every reason to believe that the ancient Phoenician seamen used astronomical methods for navigating their vessels. Reference to Homer's Odyssey, which describes the adventures of the mythical hero Odysseus, reveals that the stars were used for navigational purposes before the days of Homer. The period during which Homer flourished is not known with any degree of certainty, but classical scholars date it between the 12th and 7th centuries bc. + +In Book Five of The Odyssey, which describes how Calypso helps Odysseus to build a craft and gives him sailing directions for his voyage, it is related of Odysseus in Pope's translation that + +'Placed at the helm he sate, and mark'd the skies, +Nor closed in sleep his ever watchful eyes. +There viewed the Pleiads, and the Northern Team, +And great Orion's more refugient beam; +To which he turned his head again, +The Bear revolving, points his golden eye, +Who shines exalted on th' ethereal plain, +Nor bathes his blazing forehead in the main.' + +It was the Bear—the Great Bear—which the fair Calypso bade Odysseus to keep on his port side as he traversed the sea. + +From the time of Solomon, who lived about ten centuries before the birth of Christ there is little recorded in respect of navigation until the year 610 bc when, as we are informed by Herodotus, some Phoenician ships, by order of Necho, king of Egypt, sailed down the Red Sea and, after rounding the African continent, entered the Mediterranean through the Pillars of Hercules after a voyage lasting three years. It was during this voyage that the Phoenicians, it is thought, discovered the Canary Islands. + +From the time of Necho, navigation on the western coast of the + +4 + +**A HISTORY OF NAUTICAL ASTRONOMY** + +African continent was neglected until the rise of the Portuguese in the early 15th century under the sponsorship of Prince Henry the Navigator. + +That the Phoenicians were great seamen there can be no doubt; and for many centuries the ships and mariners of Tyre and Sidon were indispensable to the great powers of the Eastern Mediterranean, including Persia, Greece and Rome. + +Modern historical research has revealed that Phoenician sea power in the Eastern Mediterranean sprang from an earlier Cretan or "Minoan" sea power during the 12th century bc. The rise of the Phoenician colonies in the 8th century bc coincided with considerable disorder in the region principally to invasions of people from the north, and the weakening of Egyptian power in the lands on the seaboard of the Eastern Mediterranean. + +The wide extent of the commercial relations of the Phoenicians during their ascendancy, by both land and sea, may be appreciated from what is written in Chapter 27 of the *Book of the Prophet Ezekiel*. Colonies of Phoenicians were planted in commercially strategic points in the Mediterranean littoral, as well as on islands within the Mediterranean. Colonies of Phoenicians were to be found in North Africa, Spain, Cyprus, Malta and Sicily. Trade with Spain was of the greatest importance, because it was from there that silver was obtained. This silver was then ob- tained. There is no doubt that Phoenician ships traded in the Atlantic, and the shipmen of Tyre and Sidon certainly voyaged to North-west Spain and possibly to Southern Britain in their quest for sea trade. It was the important trade with South-east Spain, however, that doubtless led to the colonization of the Western Mediterranean littoral by Phoenicians as early as the 9th century bc. + +With the fall of Greek culture in the Aegean Sea, the power of Tyre and Sidon declined. The Phoenician colonies, therefore, turned for their protection to Carthage. Carthage, the great Phoenician trading emporium in Tunisia, established during the 8th century bc, emerged as a major power in the 6th century bc. + +Let us now consider briefly the observational astronomy of the Babylonians. The astronomers of Babylon recorded lunar eclipses from as far back as the 5th century bc. Systematic observations of the Moon's apparent movement against the background of + +A diagram showing a lunar eclipse. + +THE DEVELOPMENT OF NAUTICAL ASTRONOMY + +fixed stars reveal the Moon's periodic motions. The same applies to the planets. Moreover, observations of the fixed stars which lie on or near to the Sun's annual apparent revolution on the celestial sphere enable the determination of the length of the year. + +The appearance of a fixed star for the first time in the eastern sky after sunset is known as the *achronical rising* of the star; and its setting at the time of sunset is known as its *achronical setting*. A star rising or setting respectively at the time of sunrise or sunset is said to rise or set *coincidally*. When a star first becomes visible in the morning sky after sunrise, or last disappears before sunset, it is said to rise or set *heliocally*. Observations of the achronical, cosinical and heliacal risings and settings of selected stars, or star groups, were obvious ones to have been made by an ancient astronomer who studied the stars systematically. + +Systematic observations, made by the early astronomers, were related to the practical problems of timeskeeping, so necessary for agricultural purposes. The study of the movements of the stars enabled astronomers to predict astronomical events. Nowadays the astronomical measurement of time is not related to the rising or setting of celestial objects, but to their meridian passages. + +For convenience of civil life a method of fitting days into periods such as months and years to form a calendar is necessary. The details of this calendar system show that certain periods are related to the forming of calendars. Three astronomical periods were of importance in ancient calendar-making. These were: the diurnal rotation of the Earth; the monthly motion of the Moon, in which a complete sequence of phases from New Moon to the next New Moon is exhibited; and the orbital motion of the Earth around the Sun, a reflection of which is the apparent annual motion of the Sun across the sky. The incongruity in nature of these three periods that made the problem of devising a satisfactory calendar one of great complexity to the Ancients. + +The most important astronomical periodic cycle is that of the Earth's revolution around the Sun, this regulating, as it does, the seasons and, therefore, the times for sowing and harvesting. The monthly cycle of the Moon is also an essential element in our lunar calendar. Unfortunately this calendar-making depends the fact that the organization of much of the religious and civil life of the ancient + +5 + +6 + +A HISTORY OF NAUTICAL ASTRONOMY + +peoples, especially that of the Egyptians, was related to the Moon and her phases. + +The astronomical observations made by the Chaldeans—the priestly caste of Babylon—led to the discovery of an important eclipse cycle known as the Saros. The discovery of the Saros, which is a period of 223 lunations, made it possible to predict the times of eclipses. + +The observations which enabled astronomers to predict astronomical events, particularly those observations related to the planets and the Moon, resulted in astronomers being regarded as having supernatural powers. It is not surprising, therefore, that the earliest astronomers, in taking advantage of their knowledge and skill, practised astrology. Events on Earth were regarded as being related to such astronomical events as occultations, eclipses, conjunctions and oppositions. The forecasting of rain and wind phenomena which are related to the seasons (and these are clearly foretold by astronomical means) was also a branch of astronomy. The forecasting of such events as victory or defeat in battle, illness or prosperity (which are unrelated to astronomical events), became the business—and a very lucrative one no doubt—of the astronomer-turned-astrologer. + +The grouping of the stars into asterism or constellations is of great antiquity, and the present names of many of the constellations seen by us today are Babylonian. The Babylonians were responsible for forming and naming them. + +The star groups of the Ancient Greeks were manifestly bor- +rowed from the Phoenicians. The figures or shapes of the con- +stellations were, doubtless, initially simple, and were derived from commonplace things; but the Greek poets metamorphosed these simple figures so that they became hardly recognizable in their original forms. The figures of Orion and Aries, for instance, are taken from the Old Testament books of Job and Amos; that of an armed man; and this immediately directs our attention to Orion the hunter, so prominent a figure in Greek mythology. The figure of the constellation Aish, referred to in the Book of Job, signifies a cluster; and it appears obvious that it is associated with the Pleiades of the Greeks. There are many similar examples of Babylonian constellations having been re- +modelled by the Ancient Greeks to link their fabulous history with the stars. + +THE DEVELOPMENT OF NAUTICAL ASTRONOMY 7 + +The term **Mazzaroth** refers to a broad zone of the celestial sphere on which the tracks of the Sun and Moon are traced. As different star groups within this zone are believed to be different months, the "season" of which is meant by the statement 'bringing forth Mazzaroth in its season," which appears in Chapter 9 of the Book of Job. + +Observations of the fixed stars laid the foundations of astronomy, because, by these alone is it possible to determine the lengths of the year and month and the periods of the planets, as well as other astronomical quantities such as the rate of the precession of the equinoxes; the celestial positions of the stars Sun, Moon and planets; and the irregularities in the apparent motions of these bodies. + +We are expressly told by Herodotus that the Greeks borrowed from the Phoenicians the gnomon and the method of dividing the day into twelve parts. The **gnomon**, which in its simplest form merely a rod planted vertically in the ground, served to mark off the time during the day by means of length and length of the shadow it cast. It also served to mark the succession of the seasons, from the varying length of the shadow cast by successive noontday Suns. The Babylonians are credited with the invention of the so-called *sexagesimal* system of measuring angles, in which the circle is divided into 360 parts. + +In his conquests Alexander crossed over into Egypt under Alexander the Great, the founder of the great Egyptian city and centre of learning, Alexandria. It was a direct result of Alexander's conquests, after which the lands of the entire eastern part of the Mediterranean were welded together into one great political unit, that Babylonian influence upon Greek science became possible. + +3. THE GREEKS + +The ascendancy of Greek science may be regarded as having coincided with the time of Thales, who flourished during the end of the 7th and the early part of the 6th centuries bc. Thales of Mileus travelled to Egypt and learnt much from the Egyptians. He learnt from the priests secret information such as the length of the year, the signs of the Zodiac and the positions of the solstices and equinoxes. Thales is said to have been the inventor of the theorem which is usually known as Pythagoras' Theorem, + +* see p. 125, Chapter V. + +8 A HISTORY OF NAUTICAL ASTRONOMY + +and to have ascertained the heights of the pyramids by measuring the lengths of their shadows when the Sun's altitude was 45°. + +Thales, according to the poet Callimachus, is said to have formed the hypothesis of the Lesser Bear; but this constellation was undoubtedly used by Phoenician navigators before the time of Thales. It is very likely, however, that Thales introduced this constellation to the Greeks. Although Thales left nothing in writing, it is believed that he explained the correct causes of eclipses and the cause of the phases of the Moon. + +The suggestion that the Earth has the form of a sphere, appears to have been made in the 5th century bc by Parmenides. The idea that the Earth is spherical may have suggested that the firmament of heaven is also spherical; and this may have led to the explanation of celestial phenomena by circular motion. + +Some twenty or so years after the death of Thales, during mid-6th century bc, the famous Pythagoras was born. His name is closely linked with the study of geometry; and he is said to have been the inventor of many of the propositions which form the first book of Euclid. Pythagoras and his followers invented a celestial system in which the Earth was regarded as revolving around a central fire called *anlichthos*, which was believed to be located at the centre of the universe. + +The cosmology of Pythagoras was based on fantastic principles. The Pythagoreans were convinced that the total number of moving objects in the heavens must be ten—the perfect number, as they thought. The Sun, Moon, Earth, and the five planets, Mercury, Venus, Mars, Jupiter and Saturn, together with the sphere of the fixed stars made nine. The tenth moving body was a supposed counter-Earth which revolved around the central fire. Because the central fire and the counter-Earth were not visible to them, they concluded that they lived on one part of the Earth on which they lived to be directed away from both. When sea voyages were extended, and observers failed to see either the central fire or the counter-Earth, the hypothesis fell out of favour. + +The deductive methods of the early Greek philosophers, in which ideas of the universe were formulated from general principles rather than from observations and knowledge, are exemplified in the writings of Aristotle (384–322 bc) and Plato (b. 429 + +THE DEVELOPMENT OF NAUTICAL ASTRONOMY 9 + +sc). Plato's ideas on the universe were based on what was thought to be appropriate. The universe he believed to be modelled on a perfect plan and, therefore, because the most perfect shape is a sphere, the circular curve is a circle, the universe must be spherical, and the motions of the heavenly bodies must be circular about the Earth which he believed to be fixed at the centre of the universe. + +Aristotle pictured the universe as comprising a number of concentric spheres with the Earth reposing at the centre. Surrounding the sphere of the Earth were the spheres of water, air and fire. Water, air and fire being wet, dry and warm, were the four elements. Surrounding the sphere of fire were the spheres of the Moon, Sun, and each of the five ancient planets; and beyond these was the sphere of the fixed stars. Aristotle believed that a force was in operation which kept the spheres of the planets, stars, Sun and Moon moving, each at its own allotted speed. The force necessary to do this was thought to be gravitational attraction between all parts of the universe. This was the *Primum Mobile* which was identified with the Creator of the universe. + +Eudoxus of Cnidus (408–355 bc), in reply to Plato's postulate that a set of circular movements would explain the observed planetary motions, devised what is regarded as being first mathematical theory of planetary motion. To fit his theory with observation Eudoxus regarded the universe as consisting of a series of concentric spheres. Surrounding the Earth he postulated a number of revolving spheres enclosing the outermost one corresponding to the sphere of the fixed stars introduced by Aristotle. Secondary spheres were regarded as revolving around points on the inner spheres; and it was these secondary spheres which carried the Sun, Moon and planets. The rotation periods of the many spheres which belonged to the system were made to fit the observed motions of the movements of celestial bodies. This was a preliminary theory of Eudoxus; but a geometrical conception designed to facilitate the compilation of tables of eclipses: there is no suggestion that he thought the universe was constructed in this way. The irregularities of the motions of the heavenly bodies, particularly those of the Moon, which were manifested when observations of astronomical events did not coincide with predictions, resulted in more spheres being added to the system of Eudoxus in an attempt to improve the mathematical laws of prediction. Calippus, a + +I0 A HISTORY OF NAUTICAL ASTRONOMY + +follower of Eudoxus, used thirty-four spheres, compared with twenty-seven in the system of Eudoxus, in order to explain the motions of the Sun, Moon and planets. + +The problems of calendar reckoning were tackled by the Ancient Greeks, and many calendar reforms were suggested by them. The common year of the Greeks consisted of 360 days. It is obvious that had this period been adopted without correction the months and seasons would have fallen out of step with one another. To prevent this from happening it was necessary to intercalate days. The astronomer Meton, who flourished about 430 bc, introduced a reform which regulated the solar year cycle of 235 lunations. The **Metonic Cycle** is a period at the end of which the Sun and Moon occupy the same positions in the celestial sphere, relative to the fixed stars, as they did at the commencement of the cycle. The Metonic Cycle is still used for establishing the date of Easter in the ecclesiastical calendar, the Golden Number of the prayer book being the number in the cycle used for fixing Easter. + +Contemporary with Plato and Eudoxus was Philolaus, who asserted that the Earth revolves around the Sun once in a year. It is not known, however, by what arguments or observations he made this assertion. + +A Syracusan named Nicetas, who lived at about the same time as Philolaus, denied that the Earth rotates once a day. This hypothesis put forward, it is supposed, to overcome the difficulty arising from the common belief which required the celestial sphere to rotate diurnally around the Earth at a fantastic speed. + +Immediately following the vast conquests of Alexander the Great, the principal centre of learning in the Mediterranean region was established in Alexandria; and it was the Alexandrian Greeks who became the foremost scientists, and who were to occupy this fruitful field until its destruction until the year AD 642, when the famous city was sacked by the Arabs, and its splendid library destroyed. + +Before entering upon a discussion on the improvements in astronomical knowledge made by the Alexandrian school, mention must be made of Pytheas of Marsala who, at about the time of Alexander the Great, determined the lengths of the midday shadows cast by the gnomon at the times of the solstices, and found, in effect, that the latitudes of Marsala and Byzantium + +A diagram showing a gnomon casting shadows on a sundial. + +THE DEVELOPMENT OF NAUTICAL ASTRONOMY II + +were roughly the same. Although the latitudes of these places differ by about two degrees, Pythagoras's observations are interesting and original. +Pythagoras is credited with being the first to distinguish climates by the varying lengths of day and night. He is remembered mainly on account of his voyage in the Atlantic, and for his 'discovery' of Iceland. It was during this voyage that Pythagoras found that the Sun was just circumpolar on the day of the summer solstice at Thule, located at the northernmost point of Europe. It seems clear from this that a circle of latitude was recognized by a change in the so-called Arctic Circle—the circle centred at the celestial pole, or axis of the heavens, within which the celestial bodies were circumpolar and did not, therefore, rise or set. +On his death in 325 bc Alexander's great kingdom was divided. The western part, including Egypt, fell to Ptolemy, son of Alexander's generals. Ptolemy chose for his capital the city founded by Alexander—Alexandria. Alexandria at the death of its founder Ptolemy was no less ambitious than Alexander in making Alexandria a great centre of commerce and a great seat of learning and, during the Ptolemaic dynasty—which ended with the death of Cleopatra in 30 bc when the Romans defeated the Egyptians—Alexandria was the cultural centre of the world. +The first amongst the Alexandrian Greeks who applied themselves to the study of astronomy were Timocharus and Aristillus. Instruments were set up by Timocharus and Aristillus, who fixed the positions of the zodiacal stars relative to the ecliptic. This marked a great stride forward in the development of astronomical observations. Hitherto, stars were 'fixed' by determining their heliacal risings and settings, and these determinations of these gave rise to considerable confusion and disagreement. +The division of the celestial sphere into two hemispheres, using the ecliptic, facilitated the determination of planetary motions. The visible planets could now be fixed relative to the ecliptic and to certain fixed stars near their paths. + +To the poet Aratus, who flourished about 270 bc, we are indebted for the description of the constellations in elegant verse. We are reminded by Aratus that Ancient Greeks used the Great Bear to determine direction in their voyaging, whereas the more skilful Phoenicians used the Lesser Bear. Although Helice + +12 +A HISTORY OF NAUTICAL ASTRONOMY + +—the Great Bear—is bright and conspicuous, the Lesser Bear is: ‘better for sailors, for the whole of it turns in a lesser circuit, and by it the men of Sidos steer the straightest course.’ + +It was in the 6th century BC that Thales is believed to have recommended the Phoenician practice of using the Lesser Bear to the ancient Greek navigators. + +Navigation is related closely to geography; and the foundation of geography is based on astronomy. One of the earliest attempts to determine position on the Earth’s surface consisted in observing the movement of the Sun at noon on the days of the solstices and shortest days. We have had occasion to mention the observations made by Pytheas by means of which he determined the ratio between the length of the midday shadow at Marsala, and the length of the gnomon, on the day of the summer solstice. Pytheas is credited with being the first to establish the latitude of Marsala by using this method. + +It is Eratosthenes (276–196 BC) who is credited with being the first to reduce the problem of terrestrial position-fixing to a regular system. He based his system on the gnomon, and imagined a line linking places at which the longest day had the same length. Such a line is a parallel of latitude; and the parallel that was delineated by Eratosthenes passed through the island of Rhodes. This line was always used as a basis for ancient maps. Other parallels were drawn through Alexandria, through Syene in southern Egypt. Eratosthenes also traced a meridian line which he regarded as passing through Rhodes, Alexandria and Syene. + +Eratosthenes, in pursuing his geographical studies, sought to determine the size of the Earth by using a measured length of the arc of a meridian between Alexandria and Syene. He noticed that on the day of the summer solstice, when at noon at Alexandria, in the zenith, and at Syene, on the same day, at noon, the gnomon cast a shadow at Alexandria. He argued that the remote Sun’s rays at the two places were parallel, and that the angle at the centre of the spherical Earth between radii terminating at the two places was equal to the zenith distance of the noon Sun at Alexandria. Knowing this angle and the distance between Alex- andria and Syene, we can see how to calculate the Earth’s circumfer- ence. For him next to this result was the determination of the Earth’s circumference by Eratosthenes is not known, owing to + +THE DEVELOPMENT OF NAUTICAL ASTRONOMY 13 + +uncertainty as to the length of the unit of distance—the stadium—which he used. + +Eratosthenes was also credited with making a very accurate estimation of the obliquity of the ecliptic, that is, the angle of inclination of the Sun's apparent annual path with the plane of the equator. His value for this angle is $11/166$ of a circle, which is $23^{\circ}51'$—a matter of $5'$ too large for the true value at the time. + +4. HIPPARCHUS + +We come now to the prince of ancient astronomers—the great Hipparchus—who flourished about 160 BC. Hipparchus undertook the arduous task of making a star catalogue, having been prompted to do so by the appearance of a new star or nova. This event, according to Pliny, led Hipparchus to wonder if the stars were fixed, and whether or not they had motions peculiar to themselves. + +'Wherefore,' as Pliny says, 'he attempted the task of numbering the stars for posterity and the reduction of the stars to a rule, so that by the help of instruments the particular place of each one may be exactly designed, and whereby men might discern, not only whether they disappear or newly appear, but also whether they change their stations; as likewise whether their motions are uniform or irregular.' He regarded this as an inheritance for the wits of succeeding ages, if any were found acute and industrious enough to comprehend the mysterious order thereof.' + +Pliny remarked further that this was the first time that the fixed stars were catalogued according to their latitudes and longitudes. + +The ends of the axis around which the celestial sphere performs its diurnal rotations are the poles of the equinoctial. The equinoctial is a great circle* on the celestial sphere which is coplanar with the Earth's equator. A semi-great circle extending between the celestial poles is a celestial meridian; and the arc of a celestial meridian between the equinoctial and a star is a measure of the declination of the star. The points of intersection of the + +*A great circle of a sphere is a circle on the sphere's surface, on whose plane the centre of the sphere lies.* + +2 + +14 +A HISTORY OF NAUTICAL ASTRONOMY + +equinoctial and ecliptic are known as the *spring* and *autumnal equinoxes* respectively. When the Sun is at the spring equinox it occupies a point in the sky which, at the time of Hipparchus, was occupied by the zodiacal constellation of Aries. At the time of the spring equinox the Sun entered this constellation, when he is said to be at the *First Point of Aries*. At this time the Sun's declination changed from southerly to northerly. The arc of the equinoctial, or the angle at the celestial pole between the celestial meridian of the First Point of Aries and the meridian of the star, is known as the *first point of Aries*. + +Declination and Right Ascension on the celestial sphere correspond to latitude and longitude on the terrestrial sphere; and it was declination and Right Ascension that Pliny mentioned when he mentioned that Hipparchus was the first to make a star catalogue based on latitude and longitude. + +It was upon this principle of arranging the stars that rested the great improvement in navigation which Hipparchus introduced to the problem of terrestrial position-finding. The rule for defining terrestrial positions was the same as that for defining celestial positions; geography and astronomy were henceforth firmly linked, and this marked an important event in the history of astronomical navigation. + +The determination of latitude is a relatively simple matter, and it has been possible to do so since long before the time of Hipparchus. The problem of finding longitude, however, was one of exceptional difficulty to the Ancients. This problem is related to that of finding time, the difference of longitude between two places being a measure of their difference in local time, reckoning 15° of difference of longitude to one hour difference in local times. + +Hipparchus was the first to suggest a method of determining longitude by eclipses of the Moon. If the times of eclipses are predicted for a particular meridian, the difference between a predicted time and the time of the eclipse at some meridian different from the one for which the predictions apply will give the difference of longitude between the two meridians. + +Eclipses are relatively rare occurrences and it is small wonder that the longitudes of any a few places were determined by the method suggested by Hipparchus. The longitudes of places were determined largely from reports of travellers; and the diffi- + +THE DEVELOPMENT OF NAUTICAL ASTRONOMY 15 + +culties of estimating distances resulted in large discrepancies in all ancient maps. + +Hipparchus determined the obliquity of the ecliptic, and agreed with this figure for this deduced by Eratosthenes. He de- +termined the length of the tropical year, which is the interval between successive instants when the Sun is at the First Point of +Aries, to be 365 days 5 hours 53 minutes. This is about four +minutes short of its true length. The tropical year is slightly +shorter than the time taken for the Sun to make one apparent +revolution around the Earth relative to a fixed star in its path—a +period known as the sidereal year. The difference between +the lengths of the tropical and sidereal years arises from a westward, +or retrograde, movement of the equinoxes known as the +precession of the equinoxes. The discovery of the precession of the equinoxes belongs to Hipparchus, who compared his star positions with +those of Timocharus and Aristylus, which had been determined some 150 years before the date of catalogue of Hipparchus. +The discovery of the precession of the equinoxes was necessary for the proper accurate astronomic observations. + +Another great feat of Hipparchus was the discovery of the +irregular apparent motion of the Sun and the measurement of the +equation of time. To comply with Plato's demand for uniform circular motion, he supposed the circular orbit of the Sun to be centred at a distance from the Earth equal to that from +the Earth at its point of aphelion while imagining the Sun to evolve in a circular path called the apse line and the point, the ex-centric. A circle centred at the ex-centric and radius of length equal to the apse line he called the *equant*; and he supposed that the radius from the ex-centric to the Sun sweeps out equal areas in equal intervals of time. On this basis he computed tables for predicting +the celestial positions of the Sun. + +The plane of the Moon's orbit around the Earth makes an angle +above or below 54° to that plane of the ecliptic. The points of inter- +section of the two planes are known as the nodes. Hipparchus is credited with discovering the retrograde motion of the nodes—a +motion similar to that of the precession of the equinoxes. He also observed that the Moon's motion is irregular, and accounted for this by inventing an ex-centric and apse line and an equant for the +Moon, with the aid of which he was able to compute tables of the +Moon's motion. + +16 +A HISTORY OF NAUTICAL ASTRONOMY + +In addition to the remarkable catalogue of discoveries made by the prince of astronomers, Hipparchus is credited with being the inventor of trigonometry—the mathematics of triangles. He constructed a table of chords, and is believed to have been the first to express the most important theorem of trigonometry: + +\sin(A + B) = \sin A \cos B + \cos A \sin B + +This theorem is usually known as Ptolemy’s theorem. + +Hipparchus is credited with being the inventor of spherical trigonometry as well as plane trigonometry; and the solution of spherical triangles was to play a most important role in the practice of astronomy for many centuries. About 150 years ago, Ptolemy occupied a prominent place amongst those who were responsible for bringing the science of navigation to its state of excellence. + +Hipparchus died about 120 bc. For a space of two centuries or more after this date we find no record of a philosopher of importance emanating from the Alexandrian school. The great wealth of knowledge and discovery which was assembled at Alexandria during the period of the Pharaohs has been lost to mankind in human history for about fifteen hundred years. The decline in Alexandrian learning was not due to any one cause; but the fact that the professors were appointed by the Pharaohs and paid by the State meant that when the Pharaohs lost interest in the progress of science the professors and scholars did likewise, and the spirit of enquiry so necessary for the advancement of learning, came still further into disfavour. The Alexandrian learning was the wide gap between scholar and artisan. Much of the knowledge acquired by the philosophers was never put to practical use. The researches and discoveries of the scholars were, however, recorded; and the vast library of Alexandria became a storehouse of the world’s knowledge. This great assemblage of knowledge was not to bear fruit for many centuries: the world of the practical man, including the seaman, went on for a long while believing that the seeds of science and technology had been sown. + +5. PTOLEMY + +Following the period of relative inactivity which began soon after the death of Hipparchus, the first philosopher of note whom we encounter is the famous Claudius Ptolemy. Ptolemy, who must not be confused with the Pharaohs of the same name, flourished + +THE DEVELOPMENT OF NAUTICAL ASTRONOMY 17 + +during the middle part of the 2nd century AD. We are indebted to Ptolemy, not so much for the part he played in the advancement of science, but for systematizing the astronomical and geographical knowledge of his time. His most well-known and important book is the *Syntaxis or General Composition of Astronomy*, commonly called by its Arabic name the *Almagest*. This work is a veritable cyclopaedia of astronomy. + +In addition to the *Almagest* Ptolemy wrote an important treatise on optics, in which he made a study of atmospheric refraction. He knew that light was bent from one substance to another of different optical density was bent at the common surface of the two sub- +stances. He rightly assumed that light from celestial bodies, on passing downwards through the atmosphere, would be refracted in the same way. Ptolemy is credited for introducing a law of re- +fraction of light in air which, although not true, gives fairly good results for small zenith distances. + +Ptolemy also invented a form of astrolabe and, as we shall see later, this instrument, in a modified form, was adapted for the use of seamen. In the hands of an astronomer the astrolabe was a valuable instrument for determining time, as well as alti- +tudes, azimuths and amplitudes, of heavenly bodies. + +Although the name *astrolabe* (from Greek ἀστρολάβον meaning star-taker) has been used for a variety of astronomical instruments, the word is now applied exclusively to a type known as an armillary astrolabe. Ptolemy's astrolabe consisted of a series of concentric rings, the innermost one carrying a pair of sights. +The outer rings were designed so that when a heavenly body was observed in the sights of the inner ring, the celestial latitude and longitude of the body could be read off, thus saving the consider- +able mathematical labour of converting altitude and azimuth into ecliptic coordinates. The invention was a *planetarium*, employing the stereographic projection for solving astronomical problems, and is believed to have been invented by Hipparchus. + +Ptolemy discussed the principles of map-making; and, in his monumental *Geographia*, which was to have a marked influence on seamen during the Great Age of Discovery, a long list of lati- +tudes and longitudes of places, for the purpose of constructing maps of this kind, was given. Although these maps were con- +ceived, Ptolemy rejected the advice of Hipparchus in respect of the advisability of fixing terrestrial positions by astronomical methods, + +18 +A HISTORY OF NAUTICAL ASTRONOMY + +and pointed out, as did Hipparchus before him, how eclipse observations could be used for the determination of longitude. + +We owe a great debt of gratitude to Ptolemy for it is through him that we learn much about the work of Hipparchus, most of whose writings are lost. Certain it is that the star catalogue found in Ptolemy's *Almagest* is that devised by Hipparchus. + +It is the system of the universe bearing his name for which Ptolemy is widely known. The Ptolemaic system of the universe consists of a fixed Earth at the centre of the system, with the plane of reference being the equator and the ecliptic plane, i.e., the plane of the orbit of the Sun and planets, being the ecliptic. Ptolemy replaced the spheres of Eudexius and Callippus by a system of circles. In his system, the Moon and Sun were regarded as moving in circular orbits around the Earth. The orbits of the planets were regarded as comprising a system of deferents and epicycles. The deferent of a planet is a circular orbit which carries the so-called fictitious planet, or real planet, being supposed to move on a circle called an epicycle as the epicycle of the planet, centred at the fictitious planet. The centres of the epicycles of Mercury and Venus—the inferior planets—were supposed to lie on a straight line joining the Earth and the Sun; and these planets were supposed to revolve in their epicycles in their own periodic times and to revolve in their deferents around the Earth in a year. This was in contrast to the superior planets, Mercury and Venus, which were regarded as revolving in their deferents in their periodic times and in their epicycles once in a year. + +Ptolemy's scheme, although it was wildly erroneous, provided a suitable means for predicting astronomical events. The cumbersome system of deferents and epicycles survived for no less than fifteen hundred years and, until the time of Tycho Brahe and Kepler, there seemed little need for any change. To this day one can observe under modern conditions the knowledge of the periods of the Moon, Sun and planets, and the diameters of their deferents in relation to those of their epicycles. The principal object of Ptolemy's geometrical system was to facilitate the preparation of tables for predicting the places of the Moon and planets. He viewed the problem of astronomical prediction, not as a problem of mechanics but as a problem based on mathematical abstractions. His solution to the problem held the field until the time when the accurate observations of Tycho Brahe, in the hands of + +THE DEVELOPMENT OF NAUTICAL ASTRONOMY 19 + +the illustrious Kepler, demanded a new approach to the problem of planetary motion. + +6. THE ARABS + +During the 6th century AD, a new, and what was to become a powerful, religious movement sprang up in Arabia. This led to the ascendancy of the Arab people, with their vision of world conquest for Islam. With incredible speed a great Arab Empire was formed by Muhammad, the self-proclaimed prophet of the one God, who preached and millions of Arabs became converted to the cause. + +The Arab Empire extended from the boundaries of China in the east to Spain in the west, and it gave the world a new culture. Following the early stage of Arab conquest, when the Koran was considered to contain a complete code of conduct, but all embracing philosophy, a belief was responsible for the destruction of the Roman library of Alexandria when the city was sacked by the Arabs in AD 642—learning was pursued throughout the Arab world and centres of culture were established in Baghdad, Cairo, Cordoba in Spain, and Samarkand in Turkestan. The Jewish communities of the Mediterranean region readily assimilated with their Semitic cousins the Arabs, to the benefit of the learning which was to follow. Moreover, the influence of the Latin language on Arabic was so strong that almost by contrast between Arab and Indian in South-west Asia, was profound. + +Notable amongst the Arabs for the encouragement he gave to the progress of science, and in particular astronomy, was the Caliph Al Mamun. Al Mamun flourished during the 8th century AD, and was the successor to the famous Harun al Rashid who was largely instrumental in having many of the works of Ptolemy translated into Arabic. Proclus and Syntaxis was to form, in its Arabic translation, the foundation of Arabian astronomy. In the mid-9th century Al Battani (Albategnus), a Syrian, produced astronomical tables of the motions of the Moon and the planets which were an improvement in accuracy on those of Ptolemy. + +The greatest interest in astronomy during the brilliant period of the Arabs is reflected in many of our present-day star names: Aldebaran, Algol, Mizar and Alphard, to name but a few. In the study of optics the names of Al Kindi and Al Hazen are + +20 + +**A HISTORY OF NAUTICAL ASTRONOMY** + +noteworthy, Al Hazen (965–1038) found that Ptolemy's law of atmospheric refraction holds good for small angles only, and expressed the view that the refraction for small zenith distances varied directly as the zenith distance. In fact the refraction for small zenith distances varies as the sine of the zenith distance; but it is to be noted that for small angles the angle in radians and the sine of the angle are very nearly equal to one another. + +Not the least of the important contributions to science made by the Arabs was their introduction to the West of a simple numeration system. The Indian or Arabic figure system is now universally used, having done infinite service in mathematics; and mathematics, of course, is the handmaiden of astronomy. The cumbersome Roman numerals were superseded by the Arabic figures; the zero sign was introduced; and a notation in which the value of a digit depended upon its place in a line of digits made the rules of arithmetic accessible even to a child. + +By the 10th century, when the Western Empire began to crumble, in the eastern part many provinces succeeded. The focus of Arab learning was transferred to the Western Mediterranean area, and academies and libraries were set up at Cordoba and Toledo in Spain. It was mainly through, and from, these centres that Arabic learning spread over Western Europe. The Greek works which had been translated into Arabic were now translated from Arabic into Latin. The knowledge of geometry which had been acquired by the Arabs from the East. This, in due course, was to make possible the printed book. Fortunately the Arab impact on Western Europe was made before the great upsurge of Christian religious fervour, manifested by the Crusades, which was to sweep Islam and the infidels from Europe. + +**7. EARLY RENAISSANCE SCHOLARS** + +The legacy of the Arabs paved the way for a great flowering of learning in Western Europe, and many of our own countrymen were to play important roles in this scientific awakening. The Jewish communities in Spain, in particular, were active in the field of mathematics, astronomy and instrument-making. The medieval universities of Paris, Oxford and many others were established in the early 13th century; they attracted scholars who engaged themselves in philosophical discussions. In addition to the European universities, the monastic orders known as Franciscan and + +THE DEVELOPMENT OF NAUTICAL ASTRONOMY 21 + +Dominican respectively, were founded at about the same time; and these religious bodies had a great influence on western science. + +The Franciscan order of Dominicans devoted a great deal of their energies to the acquisition of knowledge in order to refute the numerous heresies of the time. The giant among many monastic scholars was the Franciscan friar Roger Bacon (1214–1294). His principal contribution to science was his insistence upon experimenting in order to further scientific knowledge. Roger has often been regarded as being the founder of the scientific method, a method which he advocated by means of his famous dictum: "Experientia docet." The seeds which Roger had sown were not, however, to bear fruit until about two centuries after his death; but important it is that seeds were sown and fruit was to be borne. + +The year AD 1252 saw the publication of a set of astronomical tables which were sponsored by the Castralian King Alphonso X (the Learned). The *Alphonsine Tables* were compiled by a team of fifty astronomers under the guidance of Roger. The Arabic notation was used in the Alphonsine Tables; and the wide use of the tables was, in no small way, responsible for the Arabic notation becoming widely known and generally adopted. At about the time of the publication of the *Alphonsine Tables* an important textbook on spherical trigonometry and astronomy, the *Sphaera Mundi*, was published. This book, by our compatriot John Holywood—known as John of Holywood—became the standard textbook on the subject for many centuries. + +Purbach (1423–1461) and John Müller, or Regiomontanus as he is familiarly known (1436–1476)—both of Nuremberg—discovered errors in the *Alphonsine Tables*. They set to work to improve observational instruments, so that faithful observations could be made with the aim of improving the tables. This aim was, however, not met, nor through ambition, but through being in the flower of their lives. Both were Waltherus more usually referred to as Waltherus of Nuremberg (1430–1504), devoted much of his great wealth in furthering the study of astronomy. He was instrumental in having an observatory built at Nuremberg for the use of Regiomontanus. Waltherus also established a printing press which gave birth to numerous calendars and ephemerides. These were to be of great value in the hands of readers of the voyages of discovery which were initiated by the Portuguese under the sponsorship of Prince Henry the Navigator. + +A page from a historical text discussing the development of nautical astronomy. + +22 +**A HISTORY OF NAUTICAL ASTRONOMY** + +It is interesting to note that mechanical clocks were used for the first time for astronomical observations at the observatory at Nuremberg. This marked a great stride forward in the science of astronomical observations. + +The German geographer Martin of Bohemia (Martin Behaim) was largely responsible for introducing the ephemerides of Regiomontanus to the Portuguese. It was Martin Behaim who is believed to have first suggested the astrolabe for nautical use. We shall have more to say about this, as well as about the Portuguese navigators, in a later chapter. For the moment, let us return to our discourse on the progress of astronomy as it affected, or was to affect, the art of astronomical navigation. + +**8. COPERNICUS, TYCHO BRAHE** + +The years 1473–1543 mark the birth and death respectively of the Polish scholar Copernicus. After a prolonged education at the universities of Bologna and Padua, his knowledge had great breadth, embracing, as it did, mathematics, astronomy, medicine and theology. His readings of the classical works of the Greeks made him familiar with the views of the universe presented by Pythagoras, Hicetas and Aristarchus, amongst others, who postulated a revolving and/or rotating Earth. + +The apparent diurnal and annual movements of the heavens could be explained by Copernicus, by the real rotation of the Earth about her polar axis, and the real revolution of the Earth around a centrally located Sun. + +Copernicus' great work is entitled *De Revolutionibus Orbium Coelestium*. In the dedication of his book, which was addressed to the Pope, he pointed out that any observed change of position of a heavenly body is due to the motion of the observed body or of the object seen, or both; and that in order to explain it should be noticeable a body outside the Earth, the apparent motion of which would be equal in magnitude but opposite in direction to the real motion of the Earth. + +Copernicus firmly planted the Sun at the centre of a system consisting of circular planetary orbits, the Earth being regarded as a planet. Although the germ of the idea implied the grand simplicity of a system in which all celestial bodies were moved by a complex set of details which the author of the system introduced in an attempt to fit the observed movements of the members of + +A diagram showing Copernicus' model of the solar system. + +THE DEVELOPMENT OF NAUTICAL ASTRONOMY 23 + +the Solar system to his plan. Great difficulty was experienced in endeavouring to achieve compatibility between the plan and the observed motions. The attempt to achieve this was due largely to the deep-rooted belief that circular motion was the natural motion. + +The Copernican system, although simple in essentials, was complicated by the elaborations of epicycles and ex-centricric; and it is for this reason that many writers have regarded Copernicus as the supreme exponent of the epi-cycle theory, and that his system was designed, as were the earlier systems of planetary astronomy, to facilitate the construction of accurate planetary tables. There is every reason to believe that Copernicus' system was not simply a mathematical abstraction; it was evolved along logical lines of argument, and there is no doubt that he believed implicitly in his proposed system. He refuted Ptolemy's argument for a fixed Earth in the clearest and obvious manner; and he proved beyond doubt that man's home in the universe did not, as was generally supposed, occupy an important place which man, in his self-glorification, had believed. + +Copernicus is regarded as having been responsible for the first great change in scientific outlook which came after the Renaissance, the great movement of intellectual development in science as well as in the arts, which swept through Western Europe during the period 1500-1600. + +The method of experiment, advocated by Roger Bacon, was brought to fruition by William Gilbert of Colchester. Gilbert (1540-1603) is the founder of the science of magnetism and electricity. In his famous book *De Magnete*, Gilbert pointed out the value of the results of his experiments with magnets for the purpose of navigation. + +John Werner of Nuremberg is considered to have been the greatest experimenter of his time. He is credited with the introduction of the cross-staff, an instrument adapted specially for seamen for observing altitudes. It is believed that Werner was the first to suggest that longitude could be determined by measuring the angle between the Moon and a fixed star lying in the Moon's monthly path around the celestial sphere. In making this suggestion, in 1514, he argued that the angle between the Moon and a star in its path changes relatively rapidly with time; and that if accurate predictions of the Moon's celestial position could be + +**24** +A HISTORY OF NAUTICAL ASTRONOMY + +furnished for a particular meridian, the time at this meridian could be found and the difference between this and the local time would be a measure of the difference between the longitudes of the local meridian and that for which the predictions were given. This method of finding longitude was to become a standard method as soon as accurate tables of the Moon's motion became available. + +Gemma Frisius (1510–1555), in a tract entitled De Principiis Astronomiae et Cosmographiae, which was printed in Antwerp in 1530, recommended the use of a clock or watch set to the time of a standard meridian, and then to find the time by measuring the longitude between a given meridian and the standard meridian. His suggestion was not to bear fruit until the practical problems related to chronometer-making had been solved; and this was not to be achieved until the time of John Harrison (1693–1776). Gemma is credited with recommending for the use of seamen an improved cross-staff which he had contrived. In his De Principiis he delineated several other axioms, as he called them, and we shall discuss these in due course. + +The year 1545 marked the appearance of an important manual on the subject of navigation. This was the Arte de Navegar, a Spanish treatise which was published in Valladolid by its author Pedro de Medina. Six years later, in 1551, another navigation book, which was composed in Cadiz in the year 1545 by Martin Cortes, was published under the title de la Navegacion y de la Sphera y de la Arte de Navegar con nuevos Instrumentos y Reglas. + +At about the time when these early textbooks of navigation appeared, the most distinguished, diligent and skilful astronomical observer of all time was born. This was the renowned Tycho Brahe (1546–1601), whose birth took place three years after the death of Copernicus. Tycho, after studying mathematics and astronomy at Copenhagen, Leipzig, Basel, received patronage of King Frederick II of Denmark. Tycho was granted a pension and an island in the Danish archipelago on which the famous observatory Uraniborg was built. He was possessed of great mechanical skill and many of the observational instruments with which his observatory was equipped he designed and made. Tycho lived before the days of telescopes and accurate clocks, yet his observations were remarkably accurate with both methods. He recognized that the best instrument is imperfect, and was the first observer to realize the importance of averaging the results of + +THE DEVELOPMENT OF NAUTICAL ASTRONOMY 25 + +several observations to arrive at a value in which observational errors were virtually eliminated. His marvellous, ingeniously contrived instruments, and his method of eliminating errors, resulted in new determinations of the constants of astronomy and stellar positions. + +The appearance of a nova, or new star, in the constellation Cassiopeia in AD 1572 gave occasion (as a similar event did for Hipparchus) for Tycho to compose a catalogue of the stars. During the preparation of this catalogue he used accurate values of atmospheric refraction deduced from his own observations. On comparing his observations of the Moon's position with the values tabulated by Copernicus, and on Copernicus he discovered errors in Copernicus' tables of as much as 2°. + +Tycho maintained that observations should precede theory. He opposed the Copernican theory, being influenced by the Prolemaic objection that the stars did not change their positions, which would be the case if the Earth moved. The great distances of the stars from the Earth are such that stellar parallax is practically undetectable by means of the relatively crude instruments of the early astronomers. Many philosophers, including Prolemaic, believed that if parallax could not be detected it did not exist; therefore the Earth is fixed! False and illogical reasoning to be sure. + +Tycho's great service to astronomy was due to his skill as an observer rather than to his mathematical ability and powers of reasoning. It was largely due to his fruitful work of this marvellous astronomical observer was his record of his observations of the planets - especially those of the planet Mars. This record laid the foundation of the important work of the famous Kepler. + +9. KEPLER, CAILIEO, NEWTON + +Johannes Kepler (1571-1630) of Stuttgart, at the age of twenty-four years published a defence of the doctrines of Copernicus. He was convinced that the plan of the universe was grand but simple; and his work entitled *Prodomus Dissertationum Cosmographarum seu Mysterium Cosmographicum* was drawn to the attention of Tycho Brahe who, recognizing the author's intellectual powers, invited Kepler to become his assistant at Uraniborg. Kepler, no doubt, was quick to see that, with Tycho's accurate observation records, any planetary theory advanced could be put to the test. + +26 +**A HISTORY OF NAUTICAL ASTRONOMY** + +Kepler was entrusted with the compilation of a new set of astronomical tables—the *Rudolphine Tables*—named after the Emperor Rudolf who had authorized their publication. While engaged in this work Kepler's epicyclic theory and his eccentric found that calculations did not agree with the corresponding observed values as determined by Tycho. Having complete confidence in Tycho as an observer, he refused to accept the basis of uniform motion for any theory of planetary motion. + +It is recorded that Kepler, in trying to fit the observations of Tycho to the Copernican doctrine, was left in respect of the planet Mars what he called "a void." Out of this eight minutes, he is reported to have stated, "I will devise a new theory that will explain the motions of all the planets." + +Kepler's discovery that the orbits of the planets are elliptical, and that the Sun is located at one of the focal points of each planetary orbit, was of great moment in the progress of astronomy. + +Although the significance of the laws which Kepler could not be understood until they had been explained by Newtonian mechanics, it is clear that Kepler saw more than the mere geometrical facts of his discovery. He realized that the planets move in their orbits under the action of a force which is directed towards the Sun; and he wondered if this force was similar to the force under which a stone falls to the ground. He postulated universal gravitation which he termed "the law of attraction" or "the law of gravity." He likened this to magnetism and referred to the work of Gilbert. + +The first two laws of planetary motion discovered by Kepler, applied to the planet Mars, are: + +1. Mars moves in an elliptic orbit which has the Sun at one of the foci. +2. The line joining Mars to the Sun sweeps out equal areas in equal time intervals. + +These laws were announced in Kepler's famous work entitled *Astronomia Nova*, published in 1609. In 1618, in another book entitled *Epitome Astronomiae Copernicanae*, he announced the extension of the laws to the other planets, to the Moon, and to the four newly-discovered satellites of Jupiter. In the following year—1619—in his *Harmonices Mundi*, the third law of planetary motion was published. This law is: + +A diagram showing three planets orbiting around a central body (Sun) with varying radii. + +THE DEVELOPMENT OF NAUTICAL ASTRONOMY 27 + +3. The square of the orbital period of a planet is proportional to the cube of its distance from the Sun. + +Kepler's three laws contain the law of universal gravitation which was propounded by Newton, the mathematical genius who was born in 1642, twelve years after the death of Kepler. + +Galileo (1564-1642), who died in the year of Newton's birth, devoted his energies mainly to the subject of mechanics. He discovered the isochronism of the pendulum; and this knowledge, in the hands of Huyghens in the mid-17th century, led to the invention of the pendulum clock. Galileo's work on the motion of bodies under gravity was of great benefit to astronomy. Galileo's close attention to astronomy resulted from his use of the telescope—an invention of a Dutch spectacle-maker named Lippershey in the first decade of the 17th century. Galileo was first to use a telescope for observing the sky. His discoveries were no less amazing than were their consequences far-reaching. Jupiter's satellites proved that the Earth was not alone in having an atmosphere. Jupiter's moons, and especially the phases of Venus proved conclusively that the Ptolemaic hypothesis was wrong. The Moon's surface was observed to be rugged; and a vast multitude of stars, which were invisible to the unaided eyes, were observed. Sun spots were observed, and these were prove to that the Sun rotated about a diameter. + +In reward for his services, he received a prize by King Philip III of Spain to anyone who invented a method of fixing a ship when out of sight of land, Galileo gave much thought to the problem. He pointed out that if the positions of Jupiter's satellites could be predicted for a standard meridian, a seaman provided with these predictions would, in effect, have a means for determining the time at the standard meridian. This, compared with local time, would give a measure of the difference of longitude between the standard meridian and any other. + +The practical application of the process of reasoning in astronomy was greatly facilitated by the invention of logarithms. Logarithms are the uncontestable invention, in the year 1614, of Baron Napier of Merchiston. + +What has been regarded as having been the most important event in the history of astronomy, was the publication, in 1687, of Newton's Principia. Sir Isaac Newton (1642-1727) was born in Lincolnshire. After early school at Grantham, he entered the + +28 + +**A HISTORY OF NAUTICAL ASTRONOMY** + +university at Cambridge in 1661. Newton was the genius of his age, and it was his brilliant investigations that led to the formulation of the laws of motion and the law of universal gravitation. After establishing the law of gravitation Newton proceeded to investigate some of its consequences. He explained: Kepler's laws of planetary motion; the precession of the equinoxes; the ellipsoidal shape of the Earth before the fact was verified; and a theory of tides, all based on the so-called Newtonian principle. + +The dynamical period of astronomy, which was initiated by Galileo, was brought to a close when accurate astronomical tables of the motions of the Moon and the planets were brought to a stage of high accuracy, thus making possible the determination of accurate positions at sea, as well as on land. + +**10. THE DAWN OF MODERN NAUTICAL ASTRONOMY** + +We have already mentioned the need that existed for a suitable method of finding latitude at sea, and this was a reward by Philip III of Spain, in 1598, for the invention of a method. The method of finding longitude by lunar observation had been proposed many times, but imperfections of lunar tables rendered the method unworkable. In 1674 the English King Charles II was pressed by Sir Jonas Moore and Sir Christopher Wren to establish an observatory for the benefit of navigation, and particularly for the making of lunar tables. It was decided that accurate lunar tables could be drawn up for a year in advance. Flamsteed, who was to become the first Astronomer Royal, had pronounced that lunar tables extant were almost useless. Flamsteed also pointed out that the star positions published in the almanacs of the time were erroneous, and that navigators could derive little benefit from them for finding position at sea. The king decided to establish an observatory, mainly for the improvement of nautical charts. On 30th June 1675 Flamsteed was appointed astronomical observer. Signal work was done by Flamsteed and his successors, and we shall deal in some detail with this significant part of our history. + +Lunar tables were improved to a degree sufficient for the needs of ocean navigation, largely through the efforts of Tobias Mayer of Göttingen, whose tables were used by Nevil Maskelyne, who was appointed Astronomer Royal in 1765, for the Nautical Almanac and Astronomical Ephemeris, which was published for the first + +A page from a historical text about nautical astronomy. + +THE DEVELOPMENT OF NAUTICAL ASTRONOMY 29 + +time in 1765 for 1767 by order of the Commissioners of Longitude. +The *Nautical Almanac*, which is published annually, provides the seaman with astronomical data of use for finding position when out of sight of land. The method of finding longitude in the earlier *Nautical Almanacs* was one in which angular distances between the Moon's centre and certain fixed stars and the Sun were given against Greenwich time. By measuring the angle between the Moon and one of the given stars or the Sun, and comparing it—after first calculating the angle between the Moon's centre and the star or Sun at the Earth's centre (a process known as "measuring the distance" between two points on a sphere), the difference in Greenwich time could be found by inspection. This difference in time corres- +ponded to the difference between the longitude of Greenwich and the longitude of the ship at the time of observation. + +The method of finding longitude by timepiece, which had been suggested by Gemma Frisius as far back as 1530, was perfected by John Harrison, the ingenious Yorkshire clockmaker, at about the same time that the lunar method was made by lunar distance reached a state of perfection. We shall discuss these methods of finding longitude in detail in Chapter VI. + +With the introduction of relatively complex mathematical methods of finding longitude at sea grew the need for the better education of seamen ashore. Naval academies sprang up at many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seamen could receive instruction. At many ports at which seemen + +30 +A HISTORY OF NAUTICAL ASTRONOMY + +The principal defect in the method of finding longitude by means of a chronometer was related to the difficulty of finding the local time at the place of the ship. This, as we shall see later, depended upon the fact that a chronometer on one side of which was the complement of an estimated latitude of the observer. The accuracy of the calculated local time was dependent upon how close to the actual latitude of the ship was the estimated latitude used in the calculation of the astronomical—or PZX—triangle. + +Largely as a result of a far-reaching discovery made by an American sea-captain named Thomas Sumner, the longitude-by-chronometer problem was systematized and simplified. It was due to Sumner that the seaman was introduced to the concept of position-line navigation, whereby he is able to fix his position by what are regarded as cross bearings of celestial objects. Sumner's method, which we shall explain in Chapter VII, was discovered in 1837. An improvement in Sumner's method was made by a French navigator, Marcq St Hilaire. + +Notable features in the progress of astronomical navigation, which stemmed from the so-called New Navigation of Sumner and Marcq St Hilaire, were the increasingly popular practice of observing stars for the determination of position at sea, and the standardization of the methods of computation of the astronomical—or PZX—triangle. + +To relieve the navigator of the tedium of calculation, many navigational tables were invented to facilitate the solution of the longitude problem. Many of these tables are based on original and ingenious ideas. In addition to the so-called short-method tables, many mechanical devices have been invented to facilitate astronomical navigation. We shall have occasion to discuss the more important of the short method tables and mechanical navigation machines in Chapters VIII and IX. + +During recent years owing to the application of electronic principles to the requirements of the seaman has resulted in rapid and significant changes in the art of practical navigation. The radio time-signal, first used in 1908, has virtually superseded the mechanical chronometer. Radio direction-finding, which was introduced in 1911, aids the navigator, particularly when he makes his landfall in thick weather. Hyperbolic navigation systems, such as the Decca Navigator, Coned and Loran systems, provide the + +THE DEVELOPMENT OF NAUTICAL ASTRONOMY 31 + +navigator with the means of fixing his ship's position when out of sight of land with an accuracy hitherto thought impossible. Iner- +tial and doppler systems of navigation are being developed; and those when they are perfected will enable a navigator to pinpoint his ship within the nearest yard. + +We live in a technological age when the artist of nautical astro- +nomy is fast becoming a part of history. Few would deny that astronomical navigation is a decaying craft. The story of its de- +velopment through past ages to the present epoch, when the per- +fected methods of astronav are being cast aside for more accurate electronic methods of navigation, is a story that surely can never fail to excite the student of the history of science. + +CHAPTER II + +Astronomical methods of time-measuring at sea + +I. THE UNITS OF TIME + +The passage of time for technical and scientific purposes is perceived by ever recurring astronomical phenomena which occur at regular or nearly regular intervals. The most important recurring astronomical phenomena in respect of time-keeping are events such as: sunrise and sunset; cosmic risings and settings of planets and bright stars; star, Moon and Sun culminations. All of these phenomena are the direct results of the Earth's rotation. + +The period of the Earth's rotation is, for all practical purposes, regarded as being constant. The time taken for the Earth to make one revolution about its axis is called a sidereal day, so named as a sidereal day, because it is manifested by the apparent diurnal revolutions of the fixed stars. The appellation 'fixed' is used because the stars are imagined to lie on the inside surface of a sphere of infinite radius. This imaginary sphere is the celestial sphere and, because it has infinite radius, the distance between the Earth and Sun, for some purposes, is regarded as being of no consequence. Hence, when the Earth is regarded as being located at the centre of the celestial sphere but sometimes the Sun is assumed to occupy the central position. + +The Sun, because of his light and heat, governs, to a large extent, the workaday lives of men; and for ordinary purposes a period of time known as a solar day is the fundamental unit of time. A *solar day* is the rotation period of the Earth relative to the Sun. Because of precession (see below) the direction of the axis of rotation as that of her axial rotation, the interval between successive culminations* of the Sun at any position on the Earth is slightly longer—about four minutes—than the interval between successive meridian passages of a fixed star. That is to say, the day by the Sun, or solar day, is slightly longer than the sidereal day. + +* A celestial body culminates when it is at meridian passage, at which time the body bears due north or south and attains its greatest daily altitude. + +A diagram showing the Earth's rotation around its axis with a line indicating the direction of its spin. + +**ASTRONOMICAL METHODS OF TIME-MEASURING** + +A great circle on the celestial sphere which lies in the plane of the Earth's rotation is the equinoxial. The plane of the Earth's spin and therefore the plane of the equinoxial is inclined to the plane of the Earth's orbit. During the course of a year, during which time the Earth makes one circuit of her orbit, the Sun appears to describe a great circle on the celestial sphere. This great circle is the ecliptic; and the angle between the planes of the equinoctial and the ecliptic—which is about 23$\frac{1}{2}$°—is known as the obliquity of the ecliptic. + +On two occasions each year the Earth occupies positions in her orbit at which she appears to be on the equinoctial. As the name equinoctial suggests, on these occasions the lengths of daylight and darkness all over the Earth are each twelve hours. During the half year when the Sun lies north of the equinoctial he is said to have north declination. For the other half year the name of the Sun's declination is south. Great circles on the celestial sphere which cut the equinoctial at right angles at two points which lie at equal distances from the extremities of its principal axis. These points are the celestial poles, and the semi-great circles which meet at the celestial poles are celestial meridians. The declination of a celestial object is the numerical value of the arc of a celestial meridian intercepted between the object and the equinoctial. The maximum declination of the Sun is numerically the same as that of any point on his path through his orbit. When his declination ceases to increase and commences to decrease he is said to be at a solstitial point. The two solstitial points—one in each of the northern and southern celestial hemispheres—are known as the summer and winter solstices respectively. + +The points of intersection of the ecliptic and the equinoctial are known as the *spring* and *autumn* equinoxes respectively. The spring equinox marks the position of the Sun when his declination changes from northerly to southerly. It is also called a vernal equinox; it marks the position when the Sun's declination changes from northerly to southerly. It is the spring equinox—a fixed point in the celestial sphere more commonly called the First Point of Aries—which serves as a datum point for measuring of sidereal time. A sidereal day is defined as the interval that elapses between successive transits, or culminations, of this First Point of Aries. + +Two factors combine which result in length of the solar day, as determined by successive transits of the Sun over any given + +A diagram showing a celestial sphere with lines representing great circles, including an equinoxial, an ecliptic, and two meridians intersecting at a pole. + +34 +A HISTORY OF NAUTICAL ASTRONOMY + +meridian, being a variable period of time. The two factors are: +first, the apparent annual path of the Sun is along the ecliptic and not along the equinoxes because of the Earth's rotation and revolution inclined to one another at an angle of 23°27'; and second, the Earth's orbit is an ellipse having the Sun at one focus; this resulting in the Earth's rate of motion around the Sun varying, being greatest when she is at perihelion (the point in the orbit nearest to the Sun), and least when she is at aphelion (the point in the orbit most remote from the Sun). + +Mechanical clocks, for case of both manufacture and use, are made to keep Mean time. A fictitious body, known as the Mean Sun, was invented to facilitate this. The Mean Sun is a celestial point which moves, during the course of a year, along the equinoctial at a uniform rate. + +There are four instants each year when time by the Mean Sun coincides with time by the real or True Sun. At all other instants time measured by a clock set correctly to Mean time differs from True Sun time by an amount known as the Equation of Time. True Sun time at any instant is known as the equation of time; and this is essentially the correction to apply to Mean time to give True solar time. Its value lies between +15 and −18 minutes. + +The units of time so far discussed, namely the sidereal and solar days, form the basis of a highly accurate system of time-keeping which demands that the mean Sun and true Sun agree at every instant. From historical evidence that accurate time-measuring, for which star-transit observations were used, began with the Chaldean astronomers of Babylon in the 3rd century bc. + +Just as the solar day is the most important unit of time marked by the Earth's rotation, the most important period of time marked by the Earth's orbital motion is the solar year. The solar year is defined as the period during which the mean Sun returns to its mean position when the Sun occupies a particular point on the ecliptic. To the nearest day, a solar year comprises 365 days. The cycle of the seasons is considered to commence when the Sun is at the spring equinox; so that the solar year is usually defined as being the interval of time which elapses between successive instants when the Sun is at the spring- or vernal-equinox. This period is 365 days 4 hours 49 minutes, that is, about 6 hours or a quarter of a day in excess of 365 days. + +**ASTRONOMICAL METHODS OF TIME-MEASURING** 35 + +In ancient times, the phases of the Moon must have been re- +garded as being spectacles of great interest, and it is little wonder +that the period of the Moon's recurring phases—a period known +as a lunation—was used as an important unit of time. +To the medieval seaman, as well as to seamen of later ages, a +knowledge of the phases of the Moon (which are related to the +rising and falling of sea-level and the related flood and ebb of the +sea) was of great importance. +The solar day, the solar year and the lunation are often regar- +ded as being the natural units of time; and all other divisions, such +as hours, weeks and civil months, are considered to be artificial. + +2. THE CALENDAR + +The problem of calendar-making, in which attempts are made to +fit the days and months into solar years so that with the passage of +time the seasons do not fall out of step with the Sun, was one of +great antiquity on account of the incompatible nature of the +periods involved. + +The earliest method of describing positions of the Sun, Moon +and planets—bodies which have comparatively complex motions +relative to the background of the fixed stars—was to relate posi- +tions in respect of bright zodiacal stars or constellations which lie +on, or near to, the apparent paths of the wandering members of +the Solar System. This method was very laborious and was con- +ducive to numerical computation. An improvement in the method +of describing celestial positions came with the introduction of the +ecliptic as a circle of reference. It is unknown, when, and by +whom, this improvement was made; but it is believed to have been due to Chaldean astronomers of a period five hundred years before +the birth of Christ. + +Associated with the civil calendar of the ancient Egyptians were certain bright stars (or star-groups) which collectively formed a +star-clock system. These stars are located on the celestial sphere +in the vicinity of the ecliptic; and each star or group belonging to +the system rose heliastically ten days before or after the adjacent +member of the clock-system. The ten-day period was known as a +*dekade*, and the star associated with the commencement of each +dekade was known as its *dekadaster*. In this way, a lunar month, +now the Earth rotates about 365\(\frac{1}{4}\) times relative to the Sun during a solar year; but relative to the fixed stars the number of + +36 +A HISTORY OF NAUTICAL ASTRONOMY + +rotations is 3664. In other words 3651 solar days are equivalent to 3664 sidereal days. As the Earth revolves in her orbit the Sun appears to move eastwards across the background of fixed stars at an average rate of $360/365.1^\circ$ per day. It follows that the fixed stars appear to move westwards across the sky relative to the Sun at the rate of about $1^\circ$ per day. And so it is that stars rise, culminate, and set, about four minutes earlier on successive days. + +A particular circumpolar or dekam, which rises heliacally to mark the commencement of a dekam, rises at the time of the dekam, about four minutes earlier, and sets before the beginning of the succeeding dekam, another dekam would, at this time, rise heliacally. + +The consecutive heliacal risings of dekams were used to mark the passage of uniform periods of darkness; and, because the night was marked by the passage across the sky of about twelve dekams between dawn and dawn, each hour was divided into twenty-four units of time, each unit being an hour. The origin of the twenty-four-hour day is of great antiquity and clearly belongs to the Egyptians. The sexagesimal system, which originated in Babylon, was later combined with the Egyptian twenty-four-hour day. Each hour was divided into sixty minutes, each minute being subdivided into sixty seconds. And so the system of time-keeping used at the present time owes its origin to the combination of important aspects of two very ancient cultures. + +3. THE GNOMON + +The simplest method of marking the passage of time during the daytime, when the Sun is not obscured by cloud, is by means of a shadow cast by a rod which is planted vertically in the ground. The length of the shadow cast by this simple gnomon decreases as the Sun's altitude increases (at noon it is shortest), and increases as the Sun's altitude decreases during the afternoon. Not only does the length of the shadow change during the course of the day, but its direction changes as well; and the changing direction of the shadow may be used to mark the passage of the hours of daylight. + +As well as for measuring time during the day the gnomon was used for determining seasons and for navigation. Polewards of the tropics, the midday shadow of a gnomon is shortest and longest on the days of the solstices. + +ASTRONOMICAL METHODS OF TIME-MEASURING 37 + +The simple means afforded by the gnomon for time-measuring led to the invention of Sun-dials of numerous designs. In ancient times there was no great demand for a high degree of accuracy in time-measuring; and the fact that the Sun is a very irregular time-keeper did not detract from its value as a great natural time-piece. Of course, Sun-dials were useless unless the Sun was shining; and when the sky was clouded or during night, water- or sand-clocks were used. + +4. THE DIVISIONS OF THE DAY +For astronomical purposes the ancients made use of horae equinoc-tales or mean time hours. + +We find mention of the term 'watches' as early as the time when, as recorded in the Old Testament Book of Exodus, the Israelites left Egypt. In Roman times the nights were divided into four watches. It was during the fourth watch, it may be remem-bered, that Jesus was walking on the Sea of Galilee, as we are told in St Mark's Gospel. + +On board the Spanish vessels of the 15th century watches were set by 'half-watch' glasses, which ran for two hours. It is clear, therefore, that at this period the seaman's day was divided into six watches of four hours each. Glasses, both half-hour, as well as two-hour glasses, were used for time-measuring in the British Navy, but they were not adopted by the French. The 'half' glasses contained not sand but finely-ground eggshell; and the ex-pressions 'warming the glass' and 'warming the bell' arose from the belief that if the glass was nursed and kept warm, the 'sand' ran more quickly than would otherwise be the case; and if this were done the watch would be shortened. The end of each half-hour after the commencement of the watch was (and still is) marked by ringing the bell—one stroke for each half-hour, so that eight bells mark one hour and it still does) the end of one watch and the beginning of the next. + +Time-keeping at sea in ancient times followed the same general lines as did time-keeping on land. The stars and Sun provided the principal means of checking the running times of sand glasses. + +The star clock of the earliest European navigators involved the use not of any one star but that of three stars—the Pole Star of the Lesser Bear (Ursa Minor) or the Cynosure of the Phoenicians. The most important star of this constellation is Stella Maris—the + +4 + +38 +A HISTORY OF NAUTICAL ASTRONOMY + +sailor's star, also known as Polaris on account of its proximity to the celestial pole, or hub of the celestial sphere. The extent and location of the Lesser Bear are such that it is circumpolar in all but very low latitudes; and in places where it is circumpolar it is always above the horizon, never rising or setting as do constellations or stars which lie near to or on the equinoctial. + +During the course of the year, because of the motion of the stars relative to the Sun, the positions of the constellations, relative to the meridian and horizon of an observer, change by swinging in circles about their respective poles. This rate of change is 360° per year; that is, at the rate of about 1° per day. The change of position of any particular star, because of this, may be interpreted as an angular motion of the celestial meridian on which the star is located, the rate of motion being about 1° per day. If, therefore, the position of a particular star relative to the meridian and horizon (or relative to the celestial pole) is known for a particular time of the year, then this information can be used for determining the time of day on any day of the year. + +The annual retrograde revolution of any fixed star results in the time at which it occupies a given position relative to the horizon and meridian—that is, when it has a given altitude and azimuth—being earlier than the extent of four minutes per day. This is equivalent to two hours per thirty-day month. Thus, for example, if a star crosses the meridian at 12 noon on January 1st, it will cross the same meridian at 2 a.m. a fortnight later; and at 1 a.m. on February 15th. + +The manner in which the star clock was used by early seamen—probably as early as the 13th century AD—was to employ the two stars of the Lesser Bear. These are known as the Guards. The brighter of the two—the foremost guard as it is called—Kuiperah, which has a magnitude of 2-7—is much less brilliant than Polaris. It is necessary to remember the positions of the Guards relative to the celestial pole for midnight on the days which mark the beginnings of the months. To facilitate time-selling by the Lesser Bear without instrumental aid, a human figure was imagined to stand vertically in the heavens looking down on the + +*The magnitude of a star is an expression of its apparent brightness. Magnitude numbers increase as apparent brightness diminishes. A star which is just visible to the naked eye has a magnitude of 6-0 and has one-hundredth of the apparent brightness of a star of magnitude 1-0.* + +A diagram showing two stars labeled "Polaris" and "Lesser Bear," with arrows indicating their positions relative to a meridian and horizon. + +**ASTRONOMICAL METHODS OF TIME-MEASURING** + +observer, having the *Stella Maris* in the region of his midriff. The head of this figure was above, and the feet below, the celestial pole, and his left and right forearms were imagined to extend to the earth, and were respectively at the celestial pole. By relating the mid-night position of the Guards of the Great Bear to this figure, the time of night could be ascertained with reasonable accuracy. For example, the Guards are in the head at midnight during mid-April; and they are in the right arm at midnight in mid-July, so that at 6 a.m. in mid-April the Guards are in the left arm, and so on. + +5. THE NOCTURNAL + +In the 16th century an instrument was invented by the use of which the time by the Lesser Bear could be ascertained with an accuracy greater than that possible by using the eye alone. This instrument is the *nocturnal*, which was first described by Coignet in 1551. The earliest nocturnals consist of two concentric plates of wood or brass. The outermost plate represents on its circular face nine equal divisions corresponding to twelve hours, each part being subdivided into sixths representing five-day periods. The circumference of the inner plate is divided into twenty-four equal parts, each part corresponding to an hour of the day. The outer plate carries a handle, the axis of which corresponds with the axis on which the Guards of the Lesser Bear have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; these Stars have the same Right Ascension as that of the Stars in their date; theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate;theseStarshavethesameRightAscencionasthatofthestarsintheirdate; + +The instrument is then held at arm's length and The long index bar is then read off through a small hole between two sides of The Guards. The time of night is then read off The scale for hours on The inner plate. + +In latitudes where The Great Bear (Ursa Major—the Helice of The Greeks) is circumpolar, The two stars known as The Pointers—Dubhe and Merak, may be used in The same way as The Guards of The Lesser Bear for finding The time of night. Nocturnals of The 17th century had two scales—one for use with The Guards of The + +A HISTORY OF NAUTICAL ASTRONOMY + +Lesser Bear and the other for use with the Pointers of the Great Bear. (See Plate 3.) + +Next to the moon, as well as providing the means for ascertaining the time of night, provided the means for computing the time of High Water at any place for which the establishment was known. The 'establishment of a port', otherwise known as the H.W.F. & C. constant (High Water Full and Change constant), is the interval of time which elapses between the time of meridian passage of the Full Moon or New Moon (midnight and midday respectively) and the time of the following High Water. + +The establishment is obtained by observing European latitudes at midnight, and it crosses the meridian of a stationary observer later each day to the extent of about fifty minutes or four-fifths of an hour. It follows, therefore, that the Moon at the end of the Third Quarter, that is at about a week after the time of Full Moon, bears south at about 6 a.m. At the end of the First Quarter, that is about a week after the time of New or Change of the Moon, the Moon's longitude at about 6 p.m., is equal to one quarter of the number of days since New Moon occurred—by means of circular scales on the nocturnal. By applying the establishment—which was obtained from tide-tables—to the time of the Moon's southing, the approxi- +mate time of High Water, or Full Sea as it was called, could readily be found. + +To find the time of the Moon's meridian passage for any day of the year without instrumental aid demanded knowledge of the epoch—the being the age of the Moon on January 1st. Because twelve lunations amount to 354 days, which is eleven days short of a year, the epoch increases by eleven days each successive year. + +A period known as the cycle of Meton, named after its discoverer, is one of nineteen years consisting of 235 lunations, after which the phases of the Moon recur on the same day of the solar year. The number of years in the Metonic cycle is known as the Golden Number. + +An interesting description of how to find the time of the Moon's meridian passage is contained in Compendium Artis Nauticae, written by John Collier and first published in 1729—a book in which each problem of navigation, according to the author, is rendered intelligible to the meanest capacity.' + +40 + +**ASTRONOMICAL METHODS OF TIME-MEASURING** + +*Rule:* divide the Date of the year by 19, add one to the remainder, and you have the Golden Number; multiply that Golden Number by 11, and divide the product by 30, the Remainder being the epochal number of the month. The epochal numbers of the months are January 0, February 2, March 1, April 0, May 3, June 4, July 5, August 6, September 8, October 8, November 10, December 10, and the Day of the Month, if this sum is less than 30 it is the Moon's age; if greater than 30, take 30 from it, and the Remainder is the Moon's age. Multiply the Moon's Age by 4 and divide the product by 5, the quotient are the hours, and the remainder are the minutes of the Moon's southing.* + +The author adds iconically: + +*Note: While the Moon is in the increase she souths before midnight; while she is decreasing she souths before noon. These things are known by every Cabbin-Boy, Collier's Nag, and Waterman's Servant, therefore needs no farther explanation. + +During the daytime, the Sun, when visible, was used to ascertain the time of day. The Sun crosses the meridian of any observer at midday; and when he is on the meridian he bears north or south. North of the tropic of Cancer the Sun crosses the meridian bearing east; south of it he bears west. A man who has a magnetic compass, could readily find the time of noon by means of a magnetic compass. Near the time of noon, the rate of change of the Sun's bearing, or azimuth, is greatest, because his rate of change of altitude is small, being zero at the instant of noon. + +**6. SUN TIME AND THE RING DIAL** + +On the days of the equinoxes the Sun rises bearing due east and sets bearing due west; that is to say, his bearing amplitude is $0^\circ$. The time of sunrise on the days of the equinoxes is $6 \mathrm{a.m.}$; and the time of sunset is $6 \mathrm{p.m.}$ Sunrise and sunset observations on the days of the equinoxes provided, therefore, the means for finding time. By assuming that the length of daylight changes at a uniform and known rate after the days of the equinoxes, the times of sunrise and sunset could be estimated. It was the practice of some mariners in the 16th century to divide the outer margin of the compass card into twenty-four equal parts which they reckoned as + +42 +**A HISTORY OF NAUTICAL ASTRONOMY** + +hours, at that time could be estimated by compass bearing of the Sun. This was an extremely rough-and-ready method of time reckoning, but it was generally accurate enough at high latitudes. + +The common instrument used to reckon time during daylight, before the advent of mechanical clocks and watches, was the **ring dial**. The invention of the ring dial is sometimes attributed to Gemma Frisius, although Gemma himself, in his dedication of his "astronomical ring" to the secretary of the king of Hungary, admitted that it was not entirely his own invention. + +Gemma's ring dial, which was introduced in 1534, consists of three horizontal rings; one of them is the meridian; and one quadrant of this ring is graduated with a scale of latitude from $0^\circ$ to $90^\circ$. A second ring, fitted at right angles to the plane of the first, represents the equinoctial, and the upper surface of this is divided into twenty-four equal divisions representing hours. On the inner side of this ring are marked the months of the year. A third ring is fitted within the first, this being free to rotate about a polar axis. The third ring is graduated so that when viewed from a favourable sight. + +To find the hour of the day using Gemma's ring, the instrument is suspended from a point corresponding to the latitude of the observer on the meridian ring. The sights on the inner ring are set to the angle of the Sun's declination, and this ring is turned freely to point to the Sun. When this has been done, an index opposite the equal-north scale will indicate the hour of the day. The plane of the meridian ring, at the same time, will indicate the directions of north and south. + +A universal ring dial described in Seller's *Practical Navigation*, which was published towards the end of the 17th century, was thought by the author to have been contrived by Edward Wright a century before Seller described it; and certain it is that Wright described the construction and use of this dial in his *Certaine Errors* (London, 1618), and published it in 1659. + +Ring dials, despite the ingenuity of their inventors, are not reliable indicators of time, especially when used on a lively ship at sea; but no better instrumental means for measuring time was available until mechanical timepieces came into general use. + +**7. MECHANICAL CLOCKS** + +Mechanical clocks were first devised in the 13th century. They were weight-driven and were, therefore, fixed. An invention of + +**ASTRONOMICAL METHODS OF TIME-MEASURING** + +the 15th century, in which the driving mechanism of the clock employed the use of a coiled spring instead of a falling weight, made possible the portable clock. The earliest portable watches were called Nuremberg clocks, a name given to mistranslation of *uhren* (the clock) for *eitel* (little egg). + +To equalize the force transmitted from a clock spring to the gear train of a watch, a mechanism known as a *fuss* was invented in 1477. + +It was not until the early 18th century, when the pendulum was first applied to clock-making, first done by the Dutch scientist Huyghers, that the notion of people generally was brought to the difference between the lengths of the apparent and mean solar days. Tables were, therefore, pasted on the insides of clock-cases by means of which the necessary correction could be lifted in order to find true solar time from the mean solar time registered by the clock were it set correctly and working perfectly. + +Dr John Dee, a figure famous in the history of navigation, produced in 1576 an English translation of *Arte de Navegar* by Martin Osiander. This was the first textbook on navigation printed in the English language. In the preface to his translation Dee mentioned, as part of the art of navigation, the subject of 'Horometrie', which he described as + +'. . . an art mathematical which demonstrate how, at any times appointed, the precise usual denomination of time, may be known, for any place. . .' + +Dee also gave a list of nautical instruments and devices, the use and construction of which should be understood by the pilot. Among these, he included '. . . Clocks with springs, houre, half, and three-hour glasses.' + +**8. ARITHMETICAL NAVIGATION** + +Now the mathematical arts—to use Dee's phrase—incorporating horo-metrie, were arts unknown to seamen by and large. In fact, simple addition and subtraction and the golden rule of three constituted the whole of the mathematical knowledge of the generality of seaman until Elizabethan times when, through necessity, geometry and trigonometry were introduced to—or rather forced upon—the seaman who would navigate his ship across the ocean. + +44 + +**A HISTORY OF NAUTICAL ASTRONOMY** + +The learned doctor played a part of signal importance in introducing to the English seaman of his day the elements of mathematics beyond the stage of the golden rule of three. The application of these elements to practical problems—navigation problems, both terrestrial and celestial—owed much to Dr Dee, who may rightly be regarded as being the founder of arithmetical navigation. His teaching paved the way for the later Elizabethan scholars, including Edward Wright, Richard Hues, Thomas Harriot and Edmund Gunter, who devised methods for calculating navigational problems of importance hitherto impossible of solution. + +Although the invention of trigonometry is attributed to Hipparchus, who used a table of chords for calculating the unknown parts of triangles, modern practical trigonometry dates from the time of the introduction of sine tables in the 15th century. The sine of an angle is equivalent to half the chord of twice the angle in a circle of unit radius; and it is clearly equal to the ratio between two sides of a right-angled triangle. In a right-angled triangle the sides being respectively the side opposite to the angle and the hypotenuse. The ratio between the adjacent to an angle in a right-angled triangle and the hypotenuse of the triangle is equivalent to the sine of the complement of the angle. This is so because the two non-$90^\circ$ angles of a right-angled triangle together make $90^\circ$. This ratio is known as tangent. The cosine of an angle is either ratio of sides of right-angled triangles include tangent, cotangent, secant and cosecant. Tables of these ratios, or trigonometrical functions as they are called, simplify practical computations of the unknown parts of triangles; and the introduction of trigonometrical tables to seamen played a most important part in the advancement of the science of navigation. + +The first systematic trigonometrical tables of sines were introduced by Purbach (1423–1461) and his pupil Müller—better known as Regiomontanus because he came from Königsherg, Müller (1436–1476) and Purbach were mathematicians of the University of Vienna. Their sine tables were constructed on the basis of a radius of 10,000 units, and were designed to facilitate the solution of astronomical problems. Regiomontanus, after the early death of Purbach, published tables of tangents and secants on the same basis as those of the sines. + +The first set of trigonometrical tables published in England + +**ASTRONOMICAL METHODS OF TIME-MEASURING** + +were those of Blundeville, and were printed in his well-known *Exercises . . . necessarie to be read and learned of all young Gentlemen . . .* first published in 1594. Blundeville was responsible, in no small way, for drawing the attention of the public to the relative merits of the celestial problems could be solved arithmetically by using the table of sines. + +Before the introduction of mathematics to seamen, many nautical astronomical problems were solved by means of globes. Several works on the use of globes were published. The first work on the globes was written by Thomas Hood and was published in 1592. Two years later, in 1594, a translation into English of the same work was made under the authorship of Richard Hues. Hues declared, in effect, that although the astronomical problems of navigation could be solved by mathematical methods, the use of the celestial globe provided the essential practical solution without the need to labour with tedious calculations. + +‘The use of the Globe,’ wrote John Davis in his famous work *The Seaman’s Secret*, ‘is of so great ease, certainty and pleasure, as that the commendations thereof cannot sufficiently be expressed: for of all instruments it is the most rare and excellent.’ + +And certainly true it is that the globe enables a navigator to see in his mind’s eye the triangles he has to solve, and this facilitates their solution. However, with the introduction of tables of sines and logarithms, arithmetical functions, the costly, cumbersome and fragile globe, as a practical instrument of navigation—despite its rarity in the eyes of John Davis—naturally became obsolescent. + +One must not be led to believe that by introducing mathematical methods to them, seamen became skilled mathematicians. Nothing of the sort: the mathematical solutions to their problems were reduced to a convenient form by them and applied with little or no understanding of the problems themselves. For a long period of time this applied; and it is only within recent times that seamen have gained some little understanding of the mathematical problems which, by using rules, they have always, since arithmetical navigation was introduced in the 16th century, been able to solve mechanically. + +Mathematical navigation did not gain momentum until the + +45 + +46 + +**A HISTORY OF NAUTICAL ASTRONOMY** + +invention of logarithms by Napier of Merchiston in 1614. Loga- +rithms to base 10—common logs as they are called—are due to +Henry Briggs, a Gresham Professor of Geometry, Edward +Wright, the inventor of the 'logarithmic scale', who was first to +complete tables of meridional parts, was instrumental in translating +Napier's work into English for the benefit of navigation. Napier wrote: + +'... It appears that some of our countrymen well affected to +[mathematical] studies... acquired a most learned mathe- +matician to translate the same [Napier's great work *Mirifici +Logarithmorum Canonis Descriptio*] into our vulgar English +tongue.' Wright's translation was found by Napier to be '... +most exact and precisely conformable to my mind and the +original.' + +Wright's work on logarithms was published poethumously by +his son Samuel in 1616. + +Edmund Gunter was the first to publish, in 1620, a table of +common logs of trigonometrical functions. Gunter, who was a +Gresham Professor of Astronomy, was the inventor of the scale +which bears his name. Gunter's scale and the 'Plain' scale de- +scribed and popularized by John Aspley in his *Speculum Nauticum* +in 1624, were instruments of great value in the hands of seamen +right down to the beginning of the present century for facilitating +the solution of many nautical mathematical problems. + +Before the end of the 17th century, collections of nautical tables included logarithms and natural and logarithmic trigono- +metrical functions, as well as the traditional tables of Sun's de- +clination, amplitudes, Right Ascensions of stars, and tide tables; +and these tables were in general use amongst seamen. + +The 'Descrition of the Sphere', together with his navigational tables, enabled a navigator to solve numerous astrono- +mical problems involving spherical triangles, provided that he +memorized, or had access to, the appropriate rule. Not the least +important of these astronomical problems were those related to +finding the hour of the day or night. Problems such as: + +1. Given latitude, Sun's declination, and Sun's altitude, find +the time of day: + +**ASTRONOMICAL METHODS OF TIME-MEASURING** + +2. Given latitude, Sun's azimuth, and Sun's altitude, find the hour of the day. +3. Given latitude, Sun's Right Ascension, and star's altitude, find the hour of the night; + +are all relatively simple problems of spherical trigonometry, which may be solved readily using the logarithmic trigonometrical functions tables. + +Many nautical astronomical problems involved right-angled spherical triangles, and *Napier's mnemonic rules* for solving right-angled spherical triangles were popular devices amongst seamen. + +**9. THE AZIMUTH COMPASS (See Plate 4)** + +The azimuth compass, in contrast to the steering compass, as its name implies, was used for observing the Sun's azimuth. The instrument, of which there were many designs, consisted of an ordinary magnetic compass, the box of which was fitted on its upper surface with a glass plate, which carried a shadow pin, and which was graduated in degrees. + +The principal use of the azimuth compass was for observing the Sun's magnetic azimuth in order to discover the variation, or the north-easting or north-westing, of the needle, as it was sometimes called. This was necessary in order to rectify the course. But the Sun's azimuth at any time, together with the observer's latitude and the Sun's altitude, enabled the mariner to compute the time. + +**10. NAUTICAL TABLES FOR DETERMINING TIME** + +An interesting table appears in Wakeley's *Mariner's Compass Rectified*, first published at about the middle of the 17th century, by means of which the exact hour of the day could be determined, the Sun being upon any point of the compass,"... fitting all places upon each side and sea that lie between 60 degrees of latitude either north or south." + +Wakeley's book, which ran into many editions, and which was published after his death by his apprentice James Atkinson, was evidently a very popular work with seamen of the day. Each page of Wakeley's table covers a particular degree of latitude, and the table on that page is described as "A sundial for that latitude". Tables often included in collections of navigational tables, by means of which time at night could be ascertained, included: + +A diagram showing a quadrant with a pointer indicating azimuth. + +48 + +**A HISTORY OF NAUTICAL ASTRONOMY** + +1. Table of Star's Right Ascensions. +2. Table of Sun's Right Ascension for noon. + +These two tables, used in conjunction with one another, rendered it possible to find the time of meridian passage of a given fixed star. This is so because the interval between the times of meridian passages of the Sun and the star on any given day is equivalent to the difference between their Right Ascensions expressed in time. The Right Ascension of a heavenly body, it will be remembered, is measured eastwards from the vernal equinox (First Point of Aries), and the celestial meridian of the object, measured eastwards from the spring equinox. Thus, if the Sun's R.A. is less than a given star's R.A., the star will transit later than noon by an amount which is equal to the difference between the R.A.'s of the Sun and the given star. A star whose R.A. is 3 hrs. 20 mins. will cross an observer's meridian on the day of the spring equinox at 6 hrs. 00 mins., whereas a star whose R.A. is 40 mins. at 3 hrs. 20 mins. p.m., whereas a star whose R.A. is 20 hrs. 40 mins. will cross an observer's meridian on the same day at 8 hrs. 40 mins. a.m., that is 3 hrs. 20 mins. before noon. + +A useful table found in many 17th- and 18th-century navigational manuals was the Table of Southing of Selected Stars at Midnight. This table was constructed from tables of star's R.A.s and S.D.S., and was published by John Flamsteed. + +Another table useful for time-measuring at night was one showing the time when a pair of selected stars had the same azimuth. This table, of course, was drawn up for a particular latitude, and was, therefore, not ideally suited for sea use. + +**11. THE NAUTICAL ALMANAC** + +In almost every astronomical computation for nautical purposes, the seaman is dependent upon an ephemeris, or table of astronomical data such as the daily celestial positions of the Sun, Moon and planets, and the R.A.s of the stars. The first official nautical ephemeris, or almanac, was that published by the French—*Connaissance des Temps"—which dates from 1678. The first British Nautical Almanac appeared in 1765 for the year 1767; but ephemerides containing tables of solar eclipses, etc., for the use of sea-men, were published privately long before this time. + +Those elements of nautical astronomical computation which + +**ASTRONOMICAL METHODS OF TIME-MEASURING** + +are in perpetual change, such as the Sun's declination and Right Ascension, are given in a nautical almanac for times corresponding to a particular standard—or prime—meridian; and it is generally necessary to apply corrections to the tabulated elements when the almanac is used in any longitude other than that of the standard meridian. + +The standard time used in the early British Nautical Almanac is that of the Greenwich meridian, and the times given are described as astronomical time. + +The civil day at sea commenced at midnight, and any time (described as the angle at the celestial pole measured westwards from the observer's upper celestial meridian to the meridian of the Sun) was designated a.m. (ante meridiem) if the Sun were east of the observer's meridian. If the Sun were west of the observer's upper celestial meridian the time (in this case described as the angle at the celestial pole between the observer's upper celestial meridian measured westwards to the meridian of the Sun) was designated p.m. (post meridiem). The astronomical day, in contrast to the civil day, commences at noon. The Sun crossed the observer's upper celestial meridian, that is, at noon. The civil day commenced, therefore, at the midnight preceding the noon which marked the beginning of the astronomical day. + +The Mean Solar Day is the interval between successive transits of the Mean Sun across the same celestial meridian. The civil day commenced when the Mean Sun culminated at noon. It is divided into twelve hours, each hour being 60 minutes long, and each minute is divided into twelve divisions, and which is regulated so that the hour hand makes one circuit of the dial in half a mean solar day, and which is set so that the hour hand corresponds to the 12 o'clock position at noon or midnight, is a perfect indicator of civil time. + +Since the civil mode of time reckoning was always twelve hours ahead of astronomical time reckoning, the seaman found it necessary to be able to reduce civil time to astronomical time at the same instant. If the civil time at ship were p.m., the astronomical time would be the same with the p.m. omitted. Thus, March 3rd at 6 p.m. civil time was the same as March 3rd 06.00 hrs. astronomical time. If the civil time were a.m. the astronomical time was found by adding twelve to the hours and subtracting one from the day of the month. Thus, 4 a.m. on March + +49 + +50 +A HISTORY OF NAUTICAL ASTRONOMY + +15th civil time was the same as 16.00 hours March 14th astronomical time. + +The problem of finding the astronomical time and date at Greenwich, in order to extract astronomical elements from the Nautical Almanac, involved expressing the ship's time (civil mode) astronomically, and then applying the longitude expressed in time reckoning 1 hour for each 15° of longitude. + +For the first time in the 1925 almanac, and in every Nautical Almanac published since that year, times styled G.M.T. are reckoned from midnight as in civil usage; and the seaman, since 1925, has no longer been confused by having to use both civil and astronomical times. + +In order to ascertain the ship's longitude when at sea, it is necessary to know both the local and the corresponding Greenwich Mean Time. Longitudes have, since 1834, been reckoned almost universally from the meridian of Greenwich, the Greenwich meridian having been adopted as a prime meridian by the members of an international conference held at New York during that year. The adoption of this meridian was recommended, but in 1911 the French adopted the Greenwich meridian. + +Longitudes are now reckoned eastwards and westwards from the Greenwich meridian to the 180th meridian, which latter is the antipodal meridian to the prime meridian. Thus, if the Greenwich Mean Time (G.M.T.) at any instant is greater than an observer's Local Mean Time (L.M.T.) at the same instant, the observer is located in longitude west of Greenwich. If G.M.T. is less than an observer's L.M.T., at the same instant, the observer is in the eastern hemisphere. This is a direct consequence of the Earth's spin towards the east; a motion which is manifested by the apparent diurnal revolution of the celestial concave towards the west. And so it is that diurnally recurring astronomical events, such as sunrise, sunset and Sun's culmination occur on the Greenwich meridian earlier than in places which have west longitude; and earlier than in places which have west longitude. Hence the seaman's rule: + +Longitude west, Greenwich time best. +Longitude east, Greenwich time least. + +If the L.M.T. at a certain instant is 9 hrs. 20 mins. and the + +**ASTRONOMICAL METHODS OF TIME-MEASURING** 51 + +G.M.T. at the same instant is 10 hrs. 20 mins., the longitude of the local meridian, reckoning 360° per 24 hours, is 15° W. If, on the other hand, the L.M.T. is 10 hrs. 20 mins. and the corresponding G.M.T. is 9 hrs. 20 mins., the longitude of the local meridian is 15° E. + +**12. COMPUTING LOCAL TIME** + +It should be clear, from the foregoing remarks, that the problem of ascertaining longitude by astronomical observations is essential to the finding of Local Time and time at some prime meridian for the same instant. We shall discuss, in a later chapter, the methods available to the mariner since the earliest navigations, by means of which he could find the time corresponding to a particular astronomical event at a particular reference, or prime meridian. The problem of finding Local Time at a particular astronomical event, such as the Local Time at which a star or the Sun has a certain altitude, involves spherical trigonometry involving knowledge of the values of arcs and angles of a spherical triangle known as the **astronomical** or **PZX-triangle**. + +Fig. 1 illustrates a typical astronomical triangle. The pairs of adjacent sides of the triangle meet at: the celestial pole (P), the observer's zenith (Z), and the observed heavenly body (X). It should be evident from the figure that a similar triangle may be projected onto the Earth's surface. If the Earth-triangle is denoted by opx, then: + +$$\begin{align*} +\text{arc PX} &= \text{arc OP} = (90° - \text{latitude of O}, \text{the observer}) \\ +\text{arc PX} &= \text{arc px} = (90° - \text{declination of X}) \\ +\text{arc ZX} &= \text{ox} = (90° - \text{latitude of X}) +\end{align*}$$ + +The three angles of the PZX triangle are: +- P which is known as the Local Hour Angle of X +- Z which is known as the Azimuth of X +- X which is known as the Angle of position or Parallactic Angle + +If three of the six parts of the PZX triangle are known, it is possible to calculate the unknown parts by spherical trigonometry. For calculating Local Time, the angle P is required. If the observed body is east of the observer's meridian, the angle P is a + +52 A HISTORY OF NAUTICAL ASTRONOMY + +measure of the time that must elapse before the body culminates. +If, on the other hand, the body is west of the observer's meridian, +the angle P is a measure of the time that has elapsed since the body +culminated. In the case of Sun observations, the angle P expressed +in hours, minutes and seconds, is the interval of solar time before +or since noon. Thus, if angle P is 30°, and the Sun is east of the +meridian, the Local Time is 10 a.m., that is two hours before + + +A diagram showing the angles P, X, Z, and O. The diagram shows a circle with a central point labeled "O". Around this point, there are three concentric circles labeled "X", "Z", and "P". The outermost circle is labeled "equatorial horizon" and the middle circle is labeled "observer's meridian". The innermost circle is labeled "equator". + + +FIGURE I + +noon. If the angle P is 30°, and the Sun is west of the observer's +meridian, the local time will be 2 p.m., that is to say two hours +will have elapsed since noon. + +13. THE MARINE CHRONOMETER + +In the 18th century, when they were first used on ships for the +purpose of finding longitude at sea, chronometers were instru- +ments of great rarity. The high cost of manufacture and the rela- +tively small number of copies produced were factors that +brought chronometers within reach of only the wealthy. It was not +until the middle of the 19th century that the mechanical construc- + +52 + +**ASTRONOMICAL METHODS OF TIME-MEASURING** + +tion of these timekeepers had attained an unexampled and high standard of efficiency, the improvement in manufacture having, at the same time, accompanied the increase of speed of production. After this time it was not unusual for ocean-going ships to have three, or even more, chronometers on board. The British Admiralty provided all sea-going men-of-war with three chronometers, although a little more than a century ago ships of the Royal Navy were furnished with only one. If, however, a cap- +tain supplied a private chronometer the government provided a third, and if he had two, he had only one, it would be unwise to trust in it implicitly, and therefore, unless it had been necessary, it would be impossible to determine which one (if either) was cor- +rect. If, however, three were carried, then the coincidence of any two would suggest the truth of their results. Moreover, by com- +paring the three, an irregular one could, in some cases, be detected. + +The essential feature of a marine chronometer is the ingenious device known as the *compensated balance* compensation for tem- +perature changes being achieved by means of a bimetallic balance wheel. + +The rate of a chronometer is closely related to temperature. An increase in temperature causes the rate of an uncompensated chronometer to be retarded, whereas a decrease in temperature will cause it to gain. The object of the compensated balance is to correct this defect. + +The effects of a change in temperature are, first, a change in the tension in the balance spring and second, a change in the moment of inertia of the balance wheel due to change in the distribution of the mass of the wheel. The tension in the balance spring varies directly as the temperature, whereas the moment of inertia of the balance wheel varies as the square of the temperature. Accordingly there are two, and only two, ways in which temperature compensation is correct. The aim of a chronometer-maker is to construct instruments which are correctly compensated at two standard temperatures—which are usually 45° F. and 75° F. At temperatures between the standards, a compensated chronometer should gain, and at temperatures above the higher, or below the lower standard temperature, it should lose. A century ago, Cap- +tain Charles Shadwell, R.N., in his classic study on the *Manage- +ment of Marine Chronometers*, observed that the time and trouble + +5 + +54 A HISTORY OF NAUTICAL ASTRONOMY + +expended on compensating a chronometer for temperature was a big factor in the cost of its production. No doubt the same applies at the present time. The author of the 1928 edition of the *Admiralty Manual of Navigation* stated that, + +'The marine chronometer is simply an enlarged watch . . . and its mechanism is by no means complicated, although its construction demands the most accurate workmanship, and its adjustment requires a high degree of skill.' + +Cases have been recorded of elaborate attempts having been made to keep the temperature of the air in the chronometer box as nearly uniform as possible. In a French vessel, for example, during a voyage of survey in the year 1816, the air in the chronometer box was maintained at a uniform temperature of 30° C. by means of an oil lamp, the admission of air into the box being regulated by an aperture, the size of which could be varied by a sliding shutter. This method was rewarded, wrote Shadwell, 'by the watches performing their functions with extreme regularity.' + +It has also been noted that artificially keeping the chronometers at a uniform temperature ensures not only their being kept in dry air, but also the maintenance of their lubricating oils in a state of uniform fluidity. + +The ritualistic routine relating to the management of a chronometer on board a ship, which was formulated during the early history of the instrument, has persisted, at least in part, to the present time. The winding of the chronometers on board is still regarded as something of a ritual, and to forget to wind them at the proper time is a crime which all self-respecting navigating officers live in fear of committing. + +The adoption of a systematic routine for winding the chronometer favouring one end over another in its rate, as well as reducing the chance of it being accidentally allowed to run down. If more than one chronometer were carried they were wound in the same order at the same time each day. The habit of so doing provided a safeguard against the caprice of memory. In winding, the turns or half-turns were counted, and the last turn or part of a turn made gently but deliberately until the key button. All of this procedure is related to the case of the over-cautious warden who, in fear of injury to the chronometer, never wound up to the butt. This re- + +A page from "A History of Nautical Astronomy" by John H. Shadwell. + +**ASTRONOMICAL METHODS OF TIME-MEASURING** + +sulted in a little being lost each day, until after a time the chronometer was found to be stopped at the time it was due to have been wound. There was, of course, an indication on the face of a marine chronometer, by means of which the time of winding might readily be seen, so that it is unlikely for a chronometer to run down if a daily check is made on the indicator. + +To ensure that the daily duty of winding the chronometer was always carried out, it was the practice in men-of-war for a sentry to report to the captain and the officer-of-the-watch when the time had come for the duty to be performed. The sentry subsequently did not return until he had received word from the guard who had ascertained from the officer in charge of the chronometers that the operation had been performed, and had duly reported the same to the officer-of-the-watch and the captain. The corresponding arrangement in merchant ships normally involved a simple inscription afforded by the magic letters CIBON scribbled—usually with a soap tablet on the mirror fitted in the cabin of the officer-in-charge, and such an inscription serving to remind him of an important daily duty. + +When at sea, the regular routine of the ship rendered it relatively difficult to overlook the duty of attending to the chronometers; but when in port, distractions due to a variety of causes, often resulted in the chronometers not being wound, with the consequent possibility of their becoming stopped. + +A chronometer which has an irregular rate is not suitable for the purposes of astronomical navigation, unless its rate can be checked by radio-time-signal soon before or just after an observation has been made. Before the days of radio-time-signals, if the daily rate of a chronometer, even if it were regular, exceeded six or seven seconds a day, it was considered to be unfit for navigational purposes. + +After radio communication had become possible the first electronic aid to navigation was introduced to the seaman in the form of the radio time-signal. At the present time radio-time-signals are available, on request, at any time of the day. + +Since ships have been equipped with radio gear, the need for a chronometer hardly exists. In fact, a reliable wrist watch, having a sweep second hand, may be used to measure the G.M.T. of an astronomical event provided that its error on G.M.T. may be ascertained at a time not much different from the time of + +56 A HISTORY OF NAUTICAL ASTRONOMY + +observation of the event. In other words, the importance of the chronometer as an instrument of navigation has diminished since the advent of the sextant only. + +Now that the sextant is almost non-existent a new type of marine chronometer has been produced. This new chronometer is reputed to have an accuracy of one part in a million. This means that its rate is within a little more than a second per month. This type of chronometer, the functioning of which is no way depends upon the memory of the clock officer, employs electronic techniques and a quartz crystal. It marks a significant advance in the construction and degree of accuracy of the marine chronometer. + +A page from a book with text discussing the history of nautical astronomy. + +CHAPTER III + +The altitude-measuring instruments of navigation + +I. INTRODUCTORY + +In this chapter we shall be concerned with the instruments used by the seaman down the ages for taking sights as a preliminary to calculating his latitude or longitude of his ship when out of sight of land. The earliest of these altitude-measuring instruments were adapted from those used by astronomers and surveyors ashore. + +The most important astronomical observation made at sea—in ancient as well as in modern times—is the altitude observation, in which the arc of a vertical circle contained between an observed celestial body and the sea horizon vertically below it is measured. The process of making such an observation ashore is comparatively simple; but at sea, with an unsteady deck from which to observe, the difficulties of making an accurate altitude observation were not entirely overcome until the advent of Hadley's reflecting quadrant in the 18th century. + +The earliest instruments used by navigators for observing altitudes were the seaman's quadrant and the mariner's astrolabe. + +2. THE SEAMAN'S QUADRANT (See Plate 5) + +It appears that, chronologically, the seaman's quadrant was the first altitude-measuring instrument used by navigators. The instrument is simply a quadrant of wood or metal provided with a plumb-line suspended, when the instrument is in use, from the centre of the arc of the quadrant. One radial edge of the instrument is graduated into degrees. + +The portable seaman's quadrant was, in all likelihood, adapted from the surveyor's quadrant; this, in turn, was adapted from the fixed astronomer's mural quadrant used for measuring altitudes of celestial bodies, and from the astronomer's hand quadrant which, by means of the Sun's altitude and curved lines engraved on the instrument, enabled the observer to find the time of day. + +The mural quadrant of the early astronomers was supported, + +58 + +**A HISTORY OF NAUTICAL ASTRONOMY** + +against a wall of masonry, in the place of the meridian. One radial edge of the quadrant was plumbed vertically. The shadow of a pin at the centre of the quadrant was cast on a plate held close to the graduated arc, so enabling the observer to measure the altitude of the Sun or Moon. + +One disadvantage of the mural quadrant was the difficulty of graduating the arc. The ease with which a straight line could be divided, compared with the difficulty of dividing an arc of a circle, resulted in the *triquet um* being an instrument of greater popularity than the mural quadrant. + +The *triquet um*, also known as Ptolemy's Rule, consists of a vertical post at the top end of which, on a horizontal bearing pin, is pivoted an alidade or sighting rule fitted with upper and lower pinsules. The lower pinule is provided with a tiny hole, and the upper pinule, or *backlight*, is provided with a large hole. A thin lath, which is pivoted at the lower end of the graduated post, provides the means of measuring the length of the chord of the angle equal to the altitude of the celestial body observed in the sights. + +To make an observation with a *triquet um*, the celestial body is sighted through the holes in the pinnules fitted to the sighting rule; and a pin on the alidade, fixed at a distance from the upper pivot equal to the distance on the vertical post between the alidade pivot and its pin, is brought into contact with another pin on a mark on the lath. After the mask has been made the lath is swung up to the graduated post, and the chord, corresponding to the measured altitude, is read off. A table of chords is then used to ascertain the required angle. + +Al Battani, the celebrated Arab astronomer of the 9th century AD, is credited with being the first to suggest graduating the lath to avoid the necessity of transferring the reading of the lath to the scale on which it was marked one source of possible error. + +Fig. 1 illustrates a *triquet um*. + +When a celestial body is observed through the sights of a *triquet um*, the zenith distance of the body is equivalent to the angle between the vertical post and the alidade. Both Copernicus and the famous Tycho Brahe observed with a *triquet um*. + +The arcs of the earliest quadrants used by the Portuguese seamen, during the early part of the Golden Age of Discovery, were not graduated in degrees of altitude (or zenith distance). Angular + +ALTITUDE-MEASURING INSTRUMENTS OF NAVIGATION 59 + +measure was not to play a part in practical navigation until the mathematical ability of seamen had advanced to a stage beyond that possessed by the first ocean navigators. The first practical appearance of the quadrant was that of the Roman's quadrant with the names of important coastal—or island—positions corres- + +A diagram showing a quadrant with a plumb-line attached to it. +FIGURE I + +ponding to the positions of the plumb-line when the Pole Star was observed. In their voyages along the West African coast, during the 14th and 15th centuries, Portuguese navigators knew when they were due west of each of several coastal stations by noting when the plumb-line on the quadrant corresponded with the name of the station engraved on the arc of the quadrant, when observing the Pole Star through the sights. + +The first use of the quadrant appears to have been for measuring altitudes of the Pole Star as a means of finding distance south of Libon or other port of departure. With the introduction of the + +60 +A HISTORY OF NAUTICAL ASTRONOMY + +table of Sun's declination for navigational purposes, the seaman was taught to find his latitude in degrees north or south of the equator by means of altitude observation on the Sun. It is not unlikely that at the time when this method became known to the seaman, angular units were introduced to him, the Sun's declination in the table being given in degrees and minutes of arc north or south of the equinoctial. The seaman's quadrant was, from this time onwards, graduated in angular measure. + +The seaman's quadrant demanded two observers; one to sight the Sun star, and the other to note the position of the plumb-line. It was generally found that this instrument quite unsuitable for observations at sea unless the sea were smooth and the air calm. The degree of accuracy of the measured altitude was coarse—although this could have been improved by employing instruments of larger radius. It was a simple instrument and one that could be made, without difficulty, by the mariner himself. Using the plumb-line to define the vertical, the quadrant could be used for measuring angles up to about 30° when the sea horizon was obscured because of darkness or thick weather. + +3. THE ASTROLABE (See Plate 2) +In about 1480, the astronomer's planispheric astrolabe was adapted for sea use. The word *astrolabe* (ἀστρολάβος) has been used to designate a number of instruments which have been included amongst these are the armillary spheres. An armillary sphere consists of a number of concentric rings, each representing one of the principal great circles of the celestial concave. Armillaries were designed to ascertain celestial positions and celestial angles without having to resort to tedious calculations. Equatorial armillaries were used to determine declinations and Right Ascensions of celestial positions; and zodiacal armillaries were used to determine celestial longitudes and right ascensions of points of the ecliptic system. The armillary sphere is said to have been invented by Eratosthenes in about 250 bc and they appear to have been used by Hipparchus, Ptolemy and Tycho Brahe, amongst others, for mapping the heavens. + +The term *planispheric astrolabe* applies to a 'compendium of instruments' as R. T. Gunther describes it in his *Early Sciences in Oxford*. The planispheric astrolabe consists of an evenly balanced metal disc fitted with a ring or shackle at a point in its circumfer- + +A diagram showing a planispheric astrolabe. + +ALTITUDE-MEASURING INSTRUMENTS OF NAVIGATION 61 + +ence from which the instrument may be suspended. Centrally pivoted to the metal disc is an alidade or diametrical sighting rule fitted with a pair of pinnules. By means of the alidade a celestial body may be sighted and its altitude measured, the metal disc being graduated in degrees from 0° to 90°, one radial edge of the alidade providing the fiducial line or index. + +The metal disc of the astrolabe is recessed to accommodate one of a series of thin metal plates, each engraved with a stereographic projection of the celestial sphere appropriate to a particular latitude. Covering the plate is a metal disc in the form of a star map designed in fret work. Below the zyte, as this openwork star map is called, is a ring for holding the plate in place. + +The planispheric astrolabe is essentially an astronomer's instrument employed for measuring time, using the Sun's altitude during the day and that of one of a small number of bright stars and planets by night. + +The great Hipparchus of Bithynia is usually credited with the invention of the planispheric astrolabe. It is almost certain that all planispheric instruments could not have been used before the time of Hipparchus, since scientific astronomy had not advanced to the stage when such an instrument could have been put to profitable use. If, in fact, Hipparchus did invent the planispheric astrolabe, the instrument of his invention could have been but a primitive form of the complex astrolabes of a later age. If it to the astronomers of India, Persia and Arabia that honour is due for the perfection of these instruments. + +The oldest surviving treatise on the astrolabe was written in the 7th century AD by Severus (Sebokht). It was not until the 13th century, when Ancient Greek learning was revived, that the astrolabe was re-introduced into Europe by Arab scholars. The first treatise on the astrolabe written in Britain was that of Geoffrey Chaucer (c. 1340-1400) which he made to his son that called was Le Meitour (1358). This treatise of 1358 appears to be a re-statement of an Arabic work of the 8th century. + +Unlike the astronomer's astrolabe, which is essentially a time piece, the mariner's instrument (which has no real right to the appellation astrolabe) is a device used simply for measuring the altitude of Sun or star. The varied uses of the astronomer's astro-labe—or mathematical jewel as it was called—were responsible for its popularity. And with increased popularity we find that + +62 +A HISTORY OF NAUTICAL ASTRONOMY + +astrolabes increasingly became objects of rare beauty reflecting the highest degree of art and skill of the instrument-maker. The exquisite and elaborate astrolabes of the period between the 16th and 18th centuries are among the finest examples of the art of metal-working. The mariner's astrolabe was usually more than a dozen or so examples are known to exist, is an instrument having but little beauty, ornamentation or precision. It is simply an astronomer's astrolabe short of its astronomical appendages, leaving only the graduated metal ring and alidade. + +According to Ramond Lull, the famous alchemist and astronomer of Majorca, the astrolabe was in use among the Majorcan pilots as early as 1295; but Puebla, in his Pilgrime, states that Martin Behaim was the first to apply the astrolabe to the art of navigation in the year 1484. + +Martin of Bohemia was commissioned by John II, king of Portugal—who was active in advancing scientific navigation—to teach the pilots of Portugal the rudiments of nautical astronomy. There is no doubt that Martin introduced these rudiments into the Portuguese seamen who participated in the Age of Discovery. + +The earliest record of a description of how an astrolabe is made and used for sea purposes appears in *Arte de Navegar* by Martin Cortes. The earliest seaman's astrolabe was a massive open ring, usually of metal, so that it would hang vertically and steady. It was of relatively small diameter so that, when in use, it offered but little resistance to wind. It was held vertically by means of a metal ring or shackle, which could be adjusted for any amount of movement. An alidade, having two sights, completed the instrument. One of the quadrants was graduated in degrees of altitude (or zenith distance). + +Master Thomas Blundeville, in his *Exercices*, informs us that the astrolabes of the Spanish were: '... not much above 5 inches broad and yet did weigh at least 4 pounds....' + +On the other hand, Blundeville mentions that: + +'English pilots ... that be skillful, do make their sea astrolabes 6 or 7 inches broad and therewith verie massive and heavey, not easie to be moved with winde, in which the spaces be the larger and thereby the truer.' + +The astrolabe was used for measuring altitudes of the noon-day + +**ALTITUDE-MEASURING INSTRUMENTS OF NAVIGATION** 63 + +Sun by day, and the Pole Star by night. Each of the sighting vanes described by Cortes carried two holes: one, a relatively large hole for use when observing the Pole Star; and the other, a very small hole, for use when observing the Sun. When observing the Sun during the altitude of the Sun the ring was held lined up with the plane of the vertical circle through the Sun. The alidade was then turned to a position so + +A diagram showing a circular instrument with a central axis and several concentric circles around it. The innermost circle has markings indicating degrees (0°, 15°, 30°, etc.). The next circle outwards has dashed lines radiating from the center to the edge, creating a series of arcs. The outermost circle has markings indicating minutes (0', 15', 30', etc.). The diagram is labeled "FIGURE 2". + +that a beam of sunlight passed through the hole in the upper sighting vane and fell near the corresponding hole in the lower sighting vane. At times when the Sun was partially obscured by thin clouds, it was necessary to observe him direct through the larger holes in the sighting vanes, as was done when sighting the Pole Star. + +An improvement on the earliest type of astrolabe was the engraving of a second quadrant, thus providing a means of checking and eliminating errors due to faulty graduation and centering of the altitude. The degree of accuracy of altitudes obtained from + +63 + +64 +A HISTORY OF NAUTICAL ASTRONOMY + +astrolabe observations made at sea was coarse, and was unlikely to have been better than to the nearest degree of arc. + +The seaman's astrolabe is often called an *astronomical ring*, although the latter name is often used for a more modified form of mariner's astrolabe. The astronomical ring was used solely for measuring the meridian altitude of the Sun, and was preferred to the simple astrolabe because the divisions on the ring were larger, and, therefore, more accurately cut than those on the astrolabe. + +The astronomical ring consisted of a metal ring of about nine inches in diameter fitted with a ring, shackle, or thread, so that it may be held vertically. At a point on the outer side of the ring, 45° from the point of suspension, is the apex of a conical shaped hole. This lies at the centre of a quadrant which is projected on to the inner surface of the ring as illustrated in Fig. 2. + +When making an observation with an astronomical ring the instrument is set at the point of suspension and lined up with the Sun when he is observed. The observer then falls through the conical shaped hole appears as a bright spot on the graduated scale of altitude (or zenith distance) on the inner side of the ring. + +The seaman's astrolabe was a clumsy instrument and ill-adapted for sea use. One may appreciate how impossible it would be to measure, by its means, the altitude of the Sun or a star from a rolling ship, with any degree of precision. + +4. THE CROSS-STAFF + +An improvement on both seaman's quadrant and astrolabe was the seaman's cross-staff, known variously as baculus Jacobi or Jacob's staff, *vaga victoria, radius astronomus*, and by the Portuguese and Spanish as *ballesta*, meaning cross-bow. + +The invention of the cross-staff is often attributed to Levi ben Gerson (1288-1342), born in Provence. It is true that true that Levi was first to describe the instrument in writing; but the invention appears to belong to Jacob ben Makir, who flourished during the 13th century. + +In its simplest form the *baculus Jacobi* described by Levi ben Gerson consists of a square-sectioned graduated staff having a cross-piece, or transom or transversary, set at right angles to the staff, along which it could be slid. One end of the staff was held at the eye and the two ends of the transom, when correctly set, pro- + +**ALTITUDE-MEASURING INSTRUMENTS OF NAVIGATION** 65 + +vided lines of sight terminating respectively at the two objects between which the angle is required. + +In 1477 Regiomontanus described the cross-staff under the name *radix stella*, and Pedro Nuñez also described the instrument in an essay published in 1537. Our own countryman William Bourne described the use of the cross-staff in his Regiment for the Sea, first published in 1574. Bourne described the cross-staff under the name *balla stella*, and his remarks on its use are interesting: + +'. . . that it is beste to take the height of the Sunne with the crosse staffe, when the Sun is under 50 degrees in heughte above the Horizon, for two causes. The one is this: till the Sunne be 50 degrees in heughte the degrees be largely marked upon the crosse staffe, but when it be over 50 degrees they are lesser marked. The other is, for that the Sunne being under 50 degrees in heughte, you may easie take the height, because you may easie see or viewe the upper end and the nether end of the crosse staffe bothe at one time: but if it doe exceed 50 degrees, then by the meanes of casting your eye upwards and downwards so muche, you may sooone commit error, and then in your next attempt to take the height of the Sunne dothe passe 50 or 60 degrees in heighth, you must leave the cross staffe and use the Mariner's Ring, called by them the Astrolaby which they ought to call the Astrolabe.' + +Bourne goes on to say: + +'The Astrolabe is best to take the height of the Sun, if the Sunne be very high at 60, 70, or 80 degrees, and the cause is this: the Sunne coming so neere unto your zenth, hathie great power of light, for to peacece the two sights of the Alhidada of the Astro- labe, and to make them equal. And therefore if you take a Sunne herturth thoe eyes of a man, and besides that it is to high to occupy the crosse staffe (as before is declared) so that this way you may very well preserve your eyes. If you have not glasses upon your staffe (to save your eyes when taking the heighth of the Sunne) but be unprovided of them, doth: thus take and cover the Sunne with the end of the transtoric of the crosse staffe, unto the very upper edge or brincke of the Sunne (so shall you not need to beholds the brightness of it), and with the other end + +66 +A HISTORY OF NAUTICAL ASTRONOMY + +of the transitoric to take the horizon truely, and that being done, +and that the Sunne is 30 or 31 minutes in diameter or breadth, +therefore you shall rebate 15 minutes from the altitude or +heights of the Sunne. + +The cross-staff does not appear to have been used by seamen +until the early 16th century. It is interesting to note that neither +Columbus nor Vasco da Gama used it, both having navigated by +astrological and quadrants. + +Martinus Coronarius, in *Arta de Navigar*, first published in 1551, +and translated into English by Richard Eden in 1561, appears to +have been the first to describe the cross-staff specifically for the +seaman's use, and to give instructions for making, graduating and +using it. + +The principal defect of the cross-staff rested in the fact that un- +less the eye was placed in the exact position for observation, an +error known as "eye of the instrument" resulted from this deviation. +To avoid or reduce this error many a navigator had his staff +specially shaped so that the observing end fitted snugly over his +cheek-bone when the instrument was held in the correct attitude +for observing. Bourne, in his *Regiment for the Sea*, had given this +advice; but later writers drew attention to the possibility of this +recommended cure making matters worse instead of better. + +Richard Hakluyt, in his *Voyages*, wrote onocular parallax as it applied to the cross-staff, and found that +an error of as much as $1^{\circ}$ may result because of it. + +A difficulty in measuring an altitude with a cross-staff is due to +the need for seeing in two directions simultaneously; and the +greater is the angle to be measured the greater is this difficulty. +When used for measuring the altitude of the Sun, the glare of this +luminary, unless he happened to be partially obscured by a trail of +cloud, would prevent its being observed accurately. This could be overcome by the use of smoked glass, as had been ad- +vised by Bourne; but this remedy often resulted in error due to +the composition of the glass not being uniform, or through the +surfaces not being ground parallel. Hariot, like Bourne, suggested +covering the Sun with the top part of the transom so that, by +measuring the altitude of the Sun's upper limb, the temporary +blindness, which would otherwise result, is prevented. + +One advantage of the cross-staff over both quadrant and astro- + +**ALTITUDE-MEASURING INSTRUMENTS OF NAVIGATION 67** + +labe lies in the relative ease with which the straight staff may be graduated. The distances of the graduations from the zero position on the staff are equivalent to the natural cotangents of the corresponding half altitudes, the half cross being equivalent to the radius or unity. Fig. 3 illustrates the method of graduating the cross-staff. + +The distance $d$ from the eye end of the staff, and the graduated mark corresponding to the altitude of a heavenly body ($a$, $\beta$, $\gamma$, etc.) is given by the formula: + +$$d = r \cot (a/2, \beta/2, y/2, \text{etc.})$$ + +where $r$ = half the length of the transom. + +A diagram showing a cross-staff with graduations along its sides. The diagram includes labels for different parts of the staff. + +**FIGURE 3** + +The cross-staff was most suitable for measuring altitudes of more than about 20° and of less than about 60°. For altitudes greater than about 60°, not only was it difficult to set the transom properly on account of lining up the ends with the horizon and Sun's limb simultaneously, but the distances between the successive graduations become increasingly smaller as the altitude increases. For altitudes of less than about 20°, the length of the transom becomes abnormally great, or that of the staff abnormally great. The cross-staff by itself, therefore, was not sufficient for the navigator's needs. + +Michel Coignet is credited with being the first to describe and illustrate, in 1581, a cross-staff having more than one transverse. From the beginning of the 17th century it became common to provide the cross-staff with three transoms designated the 15°, 30° and 45° transoms respectively. Each transom had one or three scales engraved on each of three sides of the square-sectioned staff. The 15° transom belonged to the side graduated up + +68 +A HISTORY OF NAUTICAL ASTRONOMY + +to 15° and this, the longest transom, was used for measuring small altitudes. The 30° transom was used with the scale that extended from about 10° up to about 35°; and the smallest, or 60° transom, was used with the scale that extended from about 55° upwards to about 80°. + +It appears that John Davis, the famous English navigator, was the first to explain how to deal with ocular parallax as it applied to a cross-staff. He pointed out that the problem of dealing with the problem of avoiding this error is one of finding the exact spot on the cheek-bone at which to place the eye-end of the staff when observing. To find this position, Davis explained that the navigator had merely to take two transoms and to set them on the staff at the correct positions for a common angle that could be measured by either transom; and then to sight along a line to a star or other suitable object at the same time as he took observations. The position of the eye-end of the staff on the cheek bone is then to be noted and remembered for future observations. + +Many an illustration of a cross-staff shows all the transoms belonging to the instrument set on the staff, thus leading to the mistaken idea that more than one transom is used to make an observation. It is to be noted that the cross-staff was used with one transom at a time, the selection being made according to the altitude of the object observed. + +Robert Fluid of Christchurch, Oxford, designed a cross-staff of novel design in 1617. The staff was three feet long; and the transom, which was one foot in length, had a three-inch slot or groove at each end, so that the central part of the transom—of length six inches—could be used for measuring small altitudes. + +In 1638, Edmund Gunter of Oxford published a work entitled *De Sectoris Rectanguli Quadratique Libri Duo*. In this book he described his own invention—the cross-staff. His cross-staff—a yard long so that it could be used as a convenient linear measuring device—provided a convenient instrument on which to engrave his famous scales for facilitating the solutions of navigational problems. + +Although, in many respects, the cross-staff is a better instrument than either a sector or quadrant or astrolabe, the problem of measuring altitudes accurately was recognized as a difficult one; and the attention of many astronomers and navigators was directed towards improving the seaman's altitude-measuring instruments. + +A noteworthy step forward in the technique of measuring the + +**ALTITUDE-MEASURING INSTRUMENTS OF NAVIGATION** 6g + +altitude of the Sun came with a novel cross-staff designed by Thomas Hood, a mathematician who was engaged to lecture on navigation in London in 1588. In 1590 Hood published a small work in which he described his measuring staff. + +Hood's staff consisted of a staff and transom, square sectioned and of equal length. A specially designed socket with two thumb-screws provided the means for setting staff and transom at right angles to one another. The staff was graduated from 90° to 15° and the transom from 0° to 45°. To measure the Sun's altitude when less than 45°, the staff is held horizontally in line with the direction of the Sun, with the transom standing vertically above the staff. The transom is then lowered (one of the thumb-screws in the socket allowing this to be done) until the edge of the shadow cast by a vane fitted at the top of the transom strikes the end of the staff. This indicates that the Sun's altitude is equal to the measured altitude. For measuring the Sun's altitude when greater than 45°, the transom is set to 45° and, with the staff horizontal and pointing in the direction of the Sun, the transom is slid along the staff to a position at which the shadow cast by the vane at the top of the transom strikes the end of the staff. The reading on the staff is then the Sun's altitude. Two observers are needed to make an observation using Hood's staff: one to hold the staff horizontally, and the other to read its scale. + +Hood appears to have been the first to have employed an instrument for measuring the Sun's altitude using a shadow cast by a vane. This idea was used by many other inventors during the decades following the publication of Hood's description of his staff. + +**5. THE KAMAL** + +A navigational instrument of great antiquity, the principle of which is the same as that of the cross-staff, is the kamal (= guide). In its simplest form the kamal consists of a rectangular board to the mid-point of which is fastened a thin cord. The cord is knotted at points corresponding to the positions of trading stations lying on a navigator's route. If, when holding a particular knot at the upper end of the rod, both ends of the rod and its lower edges are coincident with the directions of the Pole Star and the horizon respectively, the navigator knows that his latitude is the same as that of the trading station which lies due east or west of + +6 + +70 +A HISTORY OF NAUTICAL ASTRONOMY + +his ship, and which corresponds to the particular knot. The kamal provided the ancient Arab navigators of the Red Sea and Indian Ocean with the means for navigating by the Pole Star. The instru- +ment became known to European navigators through Vasco da Gama, after he had rounded the African Cape in 1497. + +The ancient kamal, in a modified form, is used by Arab navi- +gators of the present time for navigating the dhowa often to be seen in the Red Sea and off the East African coast. + +6. THE BACK-STAFF + +The most fruitful attempt made during the 16th century to over- +come the difficulties of taking sights at sea was that of John Davis, the inventor of the back-staff. Two variations of a back- +staff were described by Davis in his famous Seaman's Secrets, first published in 1595. + +The more simple of Davis's back-staves consisted of a gradu- +ated staff to which is fitted a sliding half-transom in the form of +an arc of a circle. At the fore end of the staff is fitted a horizon vane +with a slit through which the horizon may be observed. The back- +staff could be used for measuring the altitude of the Sun only when +the horizon was visible. It was an instrument essentially for day- +time use for measuring the meridian altitude of the Sun. To take +a sight on the Sun, the observer stood facing southward, holding +his back to the staff (hence the name back-staff). The half-cross, +which is held vertically above the staff, is then moved along the +staff to a position at which the shadow of the top end of the staff +cast by the Sun struck the horizon vane and made coincidence +with the horizon viewed through the slit in the vane. + +Not only did Davis overcome the difficulty associated with the +temporary nature of daylight by observing when observing the Sun +directly, he also achieved the means for measuring the Sun's alti- +tude without having to look in two directions simultaneously, +as is the case when using the cross-staff. + +The simple back-staff described above was graduated up to 45°, +and was useful for observing the Sun at small altitudes only. For +the northern Atlantic voyages of Sir Francis Drake, English explorers of Eliza- +bethan times required a much larger but less accurate instrument, +in which the Sun's altitude may reach 90°. Davis suggested its use +of a different type of back-staff. The 90° back-staff—as this type +was called to distinguish it from the 45° staff described above— + +**ALTITUDE-MEASURING INSTRUMENTS OF NAVIGATION 71** + +comprised two half-crosses: one straight and the other arcuate. +The straight half-cross is fitted perpendicularly to the staff and is designed to measure along the staff in the same manner as the half- +cross fitted to the 45° back-staff. The curved half-cross is fitted to +the lower side of the staff, and is provided with a sighting vane. The +fore end of the staff is fitted with a horizon vane through which +the horizon is viewed when making an observation. + +The principle of the 90° back-staff is demonstrated in Fig. 4. +In Fig. 4, AH represents the graduated staff fitted with the +horizon vane at H. VB represents the tramom or half-cross, de- +signed to slide along the staff, and which is fitted with the shadow + + +A line segment labeled "To sun" extends from point A to point B on a horizontal line. Below this line, another horizontal line extends from point C to point D. Point E is located between points B and C. + + +FIGURE 4 + +vane V. CD represents the fixed arcuate half-cross fitted with a +sliding eye vane at D. + +To use this instrument with the 90° back-staff the half-cross VB is set +at a graduation on the staff corresponding to a few degrees less +than the Sun's altitude. The observer then holds the instrument +vertically, and, with his back to the Sun and eye at the sighting +vane, he slides the sighting vane to a position on the arc so that he +may view the horizon through the slit in the horizon vane at the +same time as he sees the shadow of the edge of the shadow vane +coincide with the slit. + +From Fig. 4: + +$$\text{Sun's altitude} = \text{VHE}$$ +$$= \alpha$$ +$$= \theta + \phi$$ + +$\theta$ is read off the graduated staff and $\phi$ is read off the graduated arc. + +72 +A HISTORY OF NAUTICAL ASTRONOMY + +As time passed Davia's 90° back-staff was modified, and before the end of the 17th century it had all but replaced the cross-staff (or fores-taff as it was often called) and other primitive measuring instruments. The instrument that replaced the cross-staff was known as the *Davie quadrant*, and by the French and other European seamen as the English quadrant. It was not superseded for nautical astronomical purposes until the reflecting instruments had made their appearance in the middle of the 18th century. + +The Davia quadrant that was in common use during the early part of the 18th century consists of two arches, together making 90° (hence the name), one above and the other below a straight bar in a common plane, one above and the other below a straight bar corres- +ponding to the staff. The length of the straight bar is a little more than the radius of the lower arch and about three times the radius of the upper arch. The upper arch is called the greater arch and it contains 65°. The lower arch is called the smaller—or lesser—arch, and it contains 25°. The greater arch is divided to degrees, and the lesser arch, by means of a diagonal scale, is subdivided to minutes of arc. + +The instrument is fitted with three portable vanes known re- +spectively as the *horizon vane*, the *right vane*, and the *shade vane*. +The horizon vane is fitted at the end of the straight bar close to the centre of the two arches. In the horizon vane is a long slit through which the sea horizon may be observed. The right vane is fitted to slide on the greater arch. The upper part of this vane can be lifted off when making an observation. Some observers used a glass vane instead of the shade vane for measuring altitudes of the Sun. The glass vane is simply a lens which focuses the Sun's rays to a bright spot on the horizon vane when taking a sight. + +The sight vane is fitted to slide along the lesser arch. It has a sharp edge to cut the graduated scale of the lesser arch, to facilitate reading of angles up to 30°. The sight vane is provided with a sighting hole through which the horizon and shadow line (or bright spot when using the glass vane) are observed. + +To take a sight with the Davis quadrant, the shade vane is set to an exact number of degrees on the greater arch, about 10° or 15° less than the Sun's altitude, and the sight vane is placed near the middle of the lesser arch. The observer holds the instrument with its arches in the vertical plane, and with his back to the Sun. + +ALTITUDE-MEASURING INSTRUMENTS OF NAVIGATION 73 + +Then, with his eye at the sight vane, he raises or lowers the quadrant, keeping the eye on the sight vane, until the shadow line is coincident with the one on the horizon, thus adjusting the position of the sight vane if necessary, until the horizon is also sighted through the slit. The altitude of the Sun is obtained by adding the readings on the two arches. The principle of the Davis quadrant is illustrated in Fig. 5. (See also Plate 6.) + + +A diagram illustrating the Davis quadrant. It shows a view looking towards the horizon with a sun's altitude indicator (D.65°) and a horizon vane (H). The observer's eye is at point O'. The sun's altitude is indicated by a line extending from O' to D.65°. The horizon vane is shown as a line extending from H to E. + + +**FIGURE 5** + +In Fig. 5: H represents the horizon vane; E the sight vane; and S the shadow vane. AB represents the lesser arch and CD the greater arch. + +$$\text{Sun's altitude} = \text{SHE}$$ +$$\theta = \alpha$$ +$$\phi = \theta + \alpha$$ + +$\phi$ is measured on the greater arch and $\alpha$ on the lesser arch. + +The Davis quadrant is not capable of adjustment, so that it was necessary for the observer to ascertain the instrumental error of his quadrant. This was usually done by making meridian altitude observations at places of known latitude, or by comparison with angles measured with an instrument the error of which was known. + +Having found the error of his quadrant, the observer would apply it to all altitudes measured with it. According as the error tended to increase or decrease the ship's northern latitude the quadrant was, therefore, said to be "northerly" or "southerly" respectively by its corresponding error. + +74 + +74 A HISTORY OF NAUTICAL ASTRONOMY + +7. THE REFLECTING QUADRANT + +The Davis quadrant and all other forms of altitude measuring instruments were superseded by the reflecting instrument that became known as Hadley's quadrant. + +Although John Hadley is usually credited with the invention of the reflecting quadrant, others before him had designed instruments for measuring altitudes using mirrors. + +A diagram showing a reflecting quadrant. The arms a, b, and c are labeled, and the axis of the telescope T is shown. The mirror M is attached to arm b. + +**FIGURE 6** + +It appears that as early as 1666 Robert Hooke described to the Royal Society of London such an instrument. At the request of the society Hooke constructed the instrument and exhibited it before his fellow members later in the year. Hooke's reflecting instrument is illustrated in Fig. 6. + +Hooke's quadrant consists of three straight arms labelled a, b and c in Fig. 6. Arms a and b are pivoted at joint d. A mirror M is fitted to arm a, one edge of the mirror being coincident with the centre of the pivot d. Arm b is fitted with a telescope T, the axis of which lies in line with the inner edge of the arm. The eyepiece of the telescope is at E in Fig. 6. The third arm c is divided with equidistant graduations so that the angle between arms a and b may be found by measuring the side-angle equivalent to half the altitude of an observed object, as demonstrated in Fig. 7. + +To measure the altitude of a celestial body using Hooke's + +**ALTITUDE-MEASURING INSTRUMENTS OF NAVIGATION** + +75 + +instrument, the arms of the instrument are held in the vertical plane and the telescope is used to sight the reflected image of the celestial body in coincidence with the horizon vertically below it as illustrated in Fig. 7, in which the altitude of the body is found to be 24°, and $\theta$ being equal to the angle between the two arms jointed at M. + +The principal disadvantage of Hooke's instrument rested in the fact that the part of the horizon vertically below the observed object is hidden by the mirror unless the image is at the limit of the reflecting surface. There is no evidence that Hooke's instrument was tried at sea; and it appears that his idea of the use of a mirror fitted to an instrument for measuring altitudes at sea was forgotten. + +Sir Isaac Newton gave some thought to the question of improving on the nautical quadrant; and he seems to have been the first to suggest the use of two mirrors as Hadley was later to employ for his reflecting quadrant. Little attention was given to Newton's suggestion; and it was not until some fifteen years after his death that Newton's design for a quadrant suitable for measuring altitudes at sea received some publicity. It was Edmund Halley who had remembered Newton's suggestion at the time John Hadley had made public his own invention. Newton's design for a sea quadrant is illustrated in Fig. 8. + +Newton's design called for a plate of brass in the form of a sector, the 45° arc of which is graduated from 0° to 90°; each division representing one minute. The pivot or centre of the sector or octant is an arm denoted by $a$ in Fig. 8. The fiducial edge of this arm is used to read off the altitude of an observed heavenly body + + +A diagram showing a quadrant with two arms (M) and a telescope (C). The arms are held in a vertical plane and the telescope is used to sight the reflected image of a celestial body in coincidence with the horizon vertically below it. + + +FIGURE 7 + +to celestial body + +$\theta$ + +to horizon + +$\theta$ + +76 +A HISTORY OF NAUTICAL ASTRONOMY + +in degrees and minutes of arc. Fitted to one radial edge of the octant is a telescope, T, three or four feet long. Two specula A and B, fitted to the plate and B fitted to the arm—are parallel to one another, and the fiducial edge of the arm indicates zero on the graduated scale. But speculum B is inclined at an angle to the plane of the graduated arc. The speculum fitted to the plate is set at an angle of 45° to the axis of the telescope. + +To take an observation with Newton's octant the instrument is held vertically with the arc held towards the observer. The horizon is then viewed below the edge of the fixed speculum through the + +![FIGURE 8](image) + +telescope. The arm is then swung downwards to a position such that a ray of light from the observed object reaches the observer's eye after having been doubly reflected from the index speculum B and the fixed speculum A. + +The geometrical principle of Newton's octant is precisely that of the present-day sextant. Fig. 9 illustrates this principle, viz. the angle denoted by the fiducial edge of the index bar, which is equivalent to the angle between the two reflecting surfaces of the two specula, is equal to half the measured altitude of a celestial body. It follows, therefore, that the 45° arc is divided into 90 divisions each representing 1° of altitude or measured angle. + +ALTITUDE-MEASURING INSTRUMENTS OF NAVIGATION 77 + +In Fig. 9 A and B represent the fixed and index specula respectively. A ray of light from a celestial body X enters the observer's eye at E after having been doubly reflected, the zig-zag ray denoted by XYZ. + +XYN and YZL are normals to the reflecting surfaces at points Y and Z respectively. + +By the first law of optics: + +$$\begin{align*} +XYN &= NYZ \quad (\text{let this be } \theta) \\ +YZL &= LZE \quad (\text{let this be } \phi) +\end{align*}$$ + +A diagram showing the angles between normals to the reflecting surfaces. + +FIGURE 9 + +The angle between the reflecting surfaces is equal to the angle between the normals NY and ZL; and this is equal to $(\phi - \theta)$ (exterior angle $LZY$ of triangle ZNY is equal to the sum of the interior and opposite angles). + +The altitude of the body is denoted by $\alpha$, and this is clearly equal to $2(\phi - \theta)$ (exterior angle $EZY$ in triangle CZY is equal to the sum of the interior and opposite angles). Therefore, the angle between the reflecting surfaces of the specula is equal to half the altitude of the body $\alpha$. + +8. THE HADLEY QUADRANT + +The first account of the reflecting instrument invented by John Hadley (1682-1744) was read before the Royal Society of London on May 13th 1731. The paper was published in Volume 37 of the *Philosophical Transactions of the Royal Society*. Hadley described two reflecting instruments, the design of one of which is very + +78 + +78 +A HISTORY OF NAUTICAL ASTRONOMY +similar to Newton's design described above. Hadley's first instru- +ment is illustrated in Fig. 10. +The instrument illustrated in Fig. 10 consists of a frame in the form of an arc of a circle. The 45° arc is divided into 90 divi- +sions, each representing 1° of arc. An index bar, denoted by I in +the figure, which provides for measuring observed angles, is +pivoted at the centre of the graduated arc. Fixed to the index bar, +perpendicularly to the plane of the arc, is a speculum A. This, the +index speculum, is set so that when the pointer on the index bar +coincides with the zero graduation on the arc scale, it is parallel to + +A diagram showing a telescope with a graduated arc and an index bar. The arc is divided into 90 divisions, each representing 1° of arc. The index bar is pivoted at the center of the arc and is perpendicular to the plane of the arc. There are two specula, A and B, fixed to the index bar. Speculum A is shown in position, while speculum B is shown in position. +FIGURE 10 + +a second speculum B, which is fixed to the frame of the octant. A +telescope T is fitted to one edge of the frame of the octant. This +is set so that, when taking a sight, light from a celestial body is +received at the eye after double reflection from the specula A and +B simultaneously with light from the horizon. + +To take an observation with this type of Hadley's octant, the +instrument is held in such a plane as to lie in the vertical plane, +with the arc towards the observer. The index bar is then set to a +position so that rays of light from the observed object and from +the horizon vertically below the object, are received at the eye +simultaneously. + +The geometrical principle of the octant is the same as that of +Newton's instrument. Fig. 11 illustrates the manner in which the +octant is used. + +9/22 + +**ALTITUDE-MEASURING INSTRUMENTS OF NAVIGATION 79** + +The second type of octant described by Hadley was that adopted by seamen generally. (See Plate 7.) + +The telescope in Hadley's second octant is fitted across, instead of parallel to, the axis of the index bar, as is the case with the first type. This type of octant is illustrated in Fig. 12. + +The speculum fitted to the index bar is called the index speculum; and that fitted to the frame, the horizon speculum. To use the octant the instrument is held with the arc in the vertical plane and directed downwards. The index bar is moved away from the observer along the arc to a position where the rays of light from + +A diagram showing the positions of the observer's body, horizon, and eye relative to the index bar and arc of the Hadley octant. The observer's body is shown on the left, with a line indicating its position relative to the arc. The horizon is shown below the observer's body, with a line indicating its position relative to the arc. The eye is shown on the right, with a line indicating its position relative to the arc. The index bar is shown in the center, with a line indicating its position relative to the arc. The angle between the observer's body and the horizon is labeled as $\alpha$, and the angle between the observer's body and the eye is labeled as $\alpha/2$. The angle between the horizon and the eye is labeled as $\alpha$. A line extends from the center of the arc to the top of the observer's body, and another line extends from the center of the arc to the bottom of the observer's body. A line extends from the center of the arc to the top of the horizon, and another line extends from the center of the arc to the bottom of the horizon. A line extends from the center of the arc to the top of the eye, and another line extends from the center of the arc to the bottom of the eye. + +From obsd body + +O* + +$\alpha/2$ + +From horizon + +$\alpha$ + +To eye + +FIGURE 11 + +the observed celestial body and from the horizon vertically below the body are received at the eye simultaneously. The reading on the arc is then the observed altitude of the body. + +The reflecting surfaces of metallic specula, fitted to the original Hadley octants, suffered the serious disadvantage of tarnishing quickly under the influence of sea-air and salt water. In due course, therefore, they were substituted by polished plates of glass. Good glass specula were scarce at this time. Hadley invented his octant. Caleb Smith designed an instrument on Hadley's pattern using + +80 +A HISTORY OF NAUTICAL ASTRONOMY + +glass prisms instead of specula, but Smith's octants did not become nearly so popular as those of Hadley's design. + +Nevil Maskelyne, associated with Hooke, was the first to suggest a novel use of glass reflecting-surfices instead of specula. Maske-lyne suggested that the under-surface of the glass block should be ground and painted black. In this event light would be reflected only from the polished surface of the block; and the possibility of double reflection (as in the case with a silvered mirror) with the + +FIGURE 12 + +possibility of prismatic error due to the two surfaces of the mirror not being perfectly parallel to one another is, thereby, avoided. + +The great advantage of Hadley's quadrant over those of Hooke and Newton rests in the fact that the direct image of the horizon and the reflected image of the observed celestial object may be brought into coincidence—both the body and the horizon verti- + +from celestial body +index mirror +normal to index mirror +horizon glass +telescope +frame (cansy) +index bar +verrier +scale (cut in ivory) +index + +horizon +horizon glass + +ALTITUDE-MEASURING INSTRUMENTS OF NAVIGATION 81 + +cally below it being in full view during the time the observation is being made. +Hadley's quadrant was tried at sea in 1732 by James Bradley, the Astronomer Royal, in the presence of John Hadley and his brother. The tests were successful and Hadley's sea octant rapidly became popular with navigators. The instrument is especially suited for use at sea. And even when the ship is unsteady, an observer, with but little practice, is able to make an accurate observation. For its purpose it was ideal; and it was not without reason that the instrument has been described as 'the most perfect appliance that has ever been invented.' + +The two mirrors of glass fitted to Hadley octants are known respectively as the index mirror and the horizon glass. The horizon glass is half-silvered, the half farther removed from the plane of the instrument being clear glass. When observing, therefore, the reflected image of the observed celestial body is brought into coincidence with the direct image of the horizon at the line separating the silvered from the unsilvered part of the horizon glass. + +Hadley's quadrant, which could measure angles up to $90^\circ$ and, for this reason, was sometimes described as a quadrant. Hadley did, however, provide the means for measuring angles greater than $90^\circ$, a third mirror being fitted to the instrument for this purpose. + +The great need at the time Hadley's instrument appeared was for an instrument suitable for measuring lunar distances which often exceeded $90^\circ$. This need was met by Godfrey, who, often observed with James Bradley, was prompted to suggest in 1757, enlarging the arc of the octant to $60^\circ$, so that angles up to $120^\circ$ could be measured. It is to Campbell, therefore, that we owe credit for the introduction of the nautical sextant. + +At about the time when John Hadley described his 'new astronomical instrument for making observations of distance (lunar) by reflection', James Bradley had become engrossed in the problem of measuring altitudes and lunar distances at sea. Godfrey is credited with the invention of a reflecting instrument similar in all respects to the first of the two octants described by Hadley to the Royal Society in 1731. + +Hadley's quadrant, unlike Davis's quadrant, was capable of adjustment, so that instrumental errors could be removed. When the instrument is in correct adjustment, the two mirrors are + +82 +A HISTORY OF NAUTICAL ASTRONOMY + +perpendicular to the plane of the instrument and are parallel to one another when the index is set to zero on the arc. + +Following the introduction of the quadrant by the seaman relief early in the 16th century, his altitude-measuring instrument made for him by an instrument-maker ashore. No longer was it neces- +sary for him to be provided with instructions in his manuals for making them himself. The mariner, in his role as a prospective purchaser of a quadrant, was advised to pay particular attention to the construction and accuracy of the instrument before parting with his money. He was advised to examine carefully the instruments in the several shops (the wood being usually mahogany, teakwood, ebony or fruitwood) frame; to see that the graduations on the ivory arc and vernier scale were accurately cut; to ensure that the surfaces of the mirrors were plane; and to see that the coloured shades were free from veins. + +The index bar of the quadrant is provided with a rectangular aperture through which the graduated scale may be seen. One edge of this aperture is used as a dividing scale by means of which angles may be measured to a relatively high degree of accuracy— +usually to the nearest minute of arc. + +The dividing scale on the earlier Hadley instruments was a diagonal scale, but this was soon superseded by a vernier scale. + +The principle of the *vernier* scale was described by Pierre Vernier in his small tract *De la mesure des angles et des distances*, les +*Propriétés du Quadrilatère*, Nouveau de Mathématique, printed at Brussels in 1638. Vernier is usually credited with the invention of the scale which bears his name; but it appears that Clavius, in a treatise on astrolabes, explained the method in 1611. + +The vernier scale on the earlier Hadley quadrants was divided into 20 equal parts, this being equivalent to the space occupied by 21 or 19 equal divisions on the arc. It follows, therefore, that the difference between two adjacent divisions on the arc is $\frac{1}{20}$ or $\frac{1}{19}$ arc minutes. Each of these divisions represents $^{\circ} \frac{1}{20}$ so that angles, therefore, could be measured to the nearest minute of arc. + +The Portuguese mathematician Pedro Nunes (Petrus Nonius) +described a method for measuring angles accurately which ap- +peared in print in his De Arte atque Ratione Navigandi, in 1522. +The name Nonius is sometimes given to Vernier’s scale, although +the principle of the dividing scale described by Nunes is different from that of Vernier’s. + +**ALTITUDE-MEASURING INSTRUMENTS OF NAVIGATION** 83 + +Nufiez' method consists of 45 concentric equidistant arcs described within the same quadrant. The outermost arc is divided into 90 equal parts; the next into 89 equal parts; the next into 88, and so on. When observing, the radial index (or plumb-line) in the case of a mural quadrant provided with a nonius scale) would cross one or other of the graduated arcs at or near a point of division, thus enabling the observer to obtain a very accurate measurement of the observed angle. Nufiez' method of dividing an arc was invented by Thomas Digges, which was described by Thomas Digges in a treatise entitled *Almagestum Sphaerarum Mathematica* published in London in 1573. Digges gives credit for the invention of the diagonal scale to Richard Chancellor. The diagonal scale was used for dividing the arc of the Davis quadrant. + +**9. THE REFLECTING CIRCLE** + +An instrument designed for measuring lunar distances accurately was invented by Tobias Mayer, a figure famous in the history of the method for finding longitude at sea known as the lunar method. Mayer's instrument, called the simple reflecting circle, was improved by the French naval officer Borda and by others. + +The principle of the simple reflecting circle—familiarly called the circle—is the same as that of the reflecting quadrant. The instrument consists of a circular limb with the index bar pivoted at the centre of the circle and with two vernier scales at each extremity. The simple reflecting circle is illustrated in Plate 8. + +The manipulation of the simple reflecting circle is similar to that of the sextant. The mean of the readings of the two diametrically opposite verniers, taken at each observation, will be completely free from errors of eccentricity. This error, which results when the graduation point of the vernier scale is not coincident with the centre of the circle of which the arc forms part, is particularly troublesome in the early octants and sextants. One great advantage of the reflecting circle over the sextant, even for measuring angles of less than $90^\circ$, is therefore the elimination of error of eccentricity. At the same time, effects of errors in reading and accidental errors of graduation were diminished, since every result is derived from the mean of two readings at two different divisions of the arc. + +Some simple reflecting circles, such as those made by the + +84 +A HISTORY OF NAUTICAL ASTRONOMY + +instrument-maker Troughton, have three verniers at distances of 120° apart: but, as the eccentricity is fully eliminated by having two verniers, the third increases the accuracy of a result only by diminishing the effect of errors of reading and graduation. Chauvenet, in his Astronomy, points out that if $e_3$ is the probable error of the mean of two readings, and $e_3$ is that of the mean of three; then: + +$$e_3 = e_3 \sqrt{\frac{2}{3}} = 0.8e_3$$ + +So that if two verniers reduce the error to say 5°, the third will only further reduce the error to 4°, an increase of accuracy which for a single observation is not, according to Chauvenet, worth the additional complication and weight, and the extra trouble of reading. + +Some simple reflecting circles employed glass prisms instead of specula or glass mirrors. The prismatic reflecting circle constructed by the Berlin firm of Pistor and Martini is illustrated in Fig. 13. + +In Fig. 13, ABC represents the arc of the instrument. M is a central mirror on the index arm. m is a glass prism two faces of which are at right angles to one another. The third face of the prism acts as a reflector. The height of the prism above the plane of the arc is half that of the object glass of the telescope; therefore this arrangement causes all objects observed over the prism can be brought to the same focus as that of the reflected ray from the second observed object. When the central mirror is parallel to the longest side of the prism, the two images are in coincidence and the index error is found as with a sextant, except that every reading is here the mean of the readings of the two verniers. + +The repeating reflecting circle is an improvement on the simple reflecting circle. It consists essentially in a telescope which is not attached to the frame of the circle as it is in the simple instrument. It is attached to a separate arm which may be rotated independently about the centre of the instrument. The telescope, which must always be directed through the horizon glass, is also fitted to this arm. In addition to the arm which carries the telescope and horizon glass, a second arm, on which is mounted the centre or index-mirror, may be rotated independently of this arm. + +To use the repeating circle, the instrument is held with the plane of the arc in line with the two objects whose angular distance is required. The index on the index mirror arm is clamped to the + +ALTITUDE-MEASURING INSTRUMENTS OF NAVIGATION 85 + +arc. The right-hand object is then observed direct through the unsilvered part of the horizon glass. The instrument is then rotated, keeping the right-hand object in sight, until the reflected image of the second object is observed through the silvered part of the horizon glass. A fine adjustment is then made to bring the true and reflected images into coincidence. This completes the first part of the observation. The index arm is then unclamped and, + +A diagram showing a telescope with a horizontal axis, a vertical axis, and a horizontal mirror. The telescope is pointing towards the left-hand side of the diagram. The index-mirror arm is shown in two positions, one before and one after the rotation of the telescope. The angle between the index-mirror arm in the two positions is twice the angle between the observed objects. + +FIGURE 13 + +leaving the horizon-glass arm in its clamped position, the telescope is directed to the left-hand object. The index-mirror arm is then rotated to a position in which the reflected image of the right-hand object is brought into coincidence with the direct image of the left-hand object. This completes the second part of the observation. The angle between the index-mirror arm in its two positions is twice the angle between the observed objects. For let $R_1$ and $R_2$ be the readings of the index of the index-mirror arm before the first, and after the second, + +7 + +86 +A HISTORY OF NAUTICAL ASTRONOMY + +contact. At each contact the angle between the index- and horizon- +mirrors is equal to one half the measured angle; and it is evident +that the points R and R₁ are at equal distances on each side of that +point on the arc at which the index of the index-mirror arm would +have stood had its motion been stopped at the instant the mirrors +were parallel. It follows that the angle between R and R₁, in the +direction of the graduations from R, is equal to twice the angle +between the mirrors at either contact. If the measured angle is +denoted by γ, we have: +$$2γ = R_2 - R$$ + +If the observations are now recommenced, starting from the +last position of the index on the index-mirror arm, this index will +be found, after the fourth contact, at a reading $R_0$, which differs +from $R_1$ by twice the angle $γ$, so that we have: +$$2γ = R_3 - R_1$$ + +But, +$$2γ = R_3 - R$$ + +It follows, therefore, that: +$$4γ = R_3 - R$$ + +Continuing the process, we shall have, after any even number $n$ +of contacts, a reading $R_n$. Thus: +$$nγ = R_n - R$$ + +and +$$γ = \frac{R_n - R}{n}$$ + +For any number of contacts, it is necessary to read off only +before the first, and after the last, observed contact. This led to the +great advantage of this instrument, for use on board ship, for +measuring lunar distances. + +When using the repeating reflecting circle for lunar distance +observations, the difference between the first and last readings is +the sum of all the individual measures, and the value of the ob- +served distance is found by dividing this sum by the number of +observations. This distance corresponds to the mean of the times +of the observed positions, provided that the angular distance is changing uniformly. + +The errors of reading and graduation, as well as error of eccen- + +A diagram showing a repeating reflecting circle with indices and mirrors. + +**ALTITUDE-MEASURING INSTRUMENTS OF NAVIGATION** 87 + +tricity, are all nearly eliminated by taking a sufficient number of observations. + +In theory, the repeating circle is very nearly a perfect instru- +ment, capable of eliminating its own errors. This theoretical +perfection is, however, impossible, owing to the mechanical +imperfections arising from the centering of the axes of the two +rotating arms one within the other. + +The most important improvements in the reflecting circle are +due to Chevalier de Borda whose work, *Description et Usage du +Cercle de Réflexion*, was first published in Paris in 1787. +The *Cercle de Réflexion* was used for lunar distance +observations. It provided the highest degree of refinement for this +purpose; and many navigators provided themselves with both +circle and sextant: the latter for altitude observations, and the +former for observing lunar distances. + +**10. PERFECTED ALTITUDE-MEASURING INSTRUMENTS** + +Improvements in sextant design and manufacture, coupled with +the redundancy of the lunar problem, spelt doom for the reflecting +circle during the early part of the 20th century. + +The modern sextant (see Plate 9) is an instrument of precision. +Some of the numerous improvements made to the early sextants +are of great interest and importance. + +The manner of making a fine adjustment of the index bar when +taking sights was effected by first moving the screw to the index +bar. The index bar was clamped to the arc while the reflected +images were observed through the sights or the telescope, +after which the tangent screw was used to make a fine adjustment. +The clamping of the index bar to the limb, in sextants of the last +century, was usually effected by a screw and block piece which +travelled over the smooth surface at the back of the limb. This +clamp block was attached to the tangent screw in such a way that +movement of this screw caused movement of the clamp block. +A disadvantage of the early arrangement is due to the limited +travel of the tangent screw, rendering it necessary, before ob- +serving, to ensure that the tangent screw is not at or near the +end of its travel. An improved form of tangent screw was made +with a spring on each side of the clamp block, so that when the +clamp is released the vernier automatically takes up a central position, and the likelihood of the tangent screw-thread being + +88 +A HISTORY OF NAUTICAL ASTRONOMY + +used up at a critical moment is obviated. This form of tangent screw was ousted during the early part of the present century by the endless tangent screw. The endless tangent screw led to the invention of the tangent sextant, which is still in use today by almost all present-day sextants. Amongst the features of this type of tangent screw, not the least important is that it can be read at arm's length. + +Many early reflecting instruments were fitted with sight vanes instead of telescopes. Improvements in sextant telescopes have been of great significance. It is not uncommon nowadays for a sextant to be fitted with a telescope having a magnification of 30 dio- mumcular. From the middle of the 19th century until the eve of the Second World War most sextants were provided with two telescopes. One, of small magnification and big object glass, was designed for star work. The other, of large magnification and small object glass, was for daylight observations. The star telescope was an erecting telescope, whereas the high-power telescope was an inverting telescope. The reason for using one of these inverting telescopes is doubtless the reason why it fell into disuse. + +Hadley, the inventor of the reflecting quadrant, directed that the line of sight should be parallel to the plane of the instrument; and, for ensuring that this was so, he proposed that two parallel wires should be fixed in the telescope parallel to the plane of the quadrant, and that the centre of the observed objects should be made in the plane of the field view of the telescope between the two parallel wires. These circuits had not previously sufficiently been attended to, although in the inverting telescopes of days gone by, cross-wires were fitted for this purpose and for checking collimation error—error resulting from the line of sight not being parallel to the plane of the sextant arc. + +As soon as it became easy to furnish good glass mirrors, these replaced the metal specula of the older reflecting instruments. As every glass mirror has two surfaces—one from the face and the other from the silvered back surface, double reflections may cause confusion with the reflected rays and error may result in the observation. Moreover, if the front and back surfaces of the mirror are not perfectly parallel to one another, the observation may suffer prismatic error. It was to overcome these difficulties that the Rev. Dr Nevil Maskelyne suggested a reflecting surface of plane glass, the back face of which is rough ground and blackened. By this + +**ALTITUDE-MEASURING INSTRUMENTS OF NAVIGATION** 8g +means only rays falling on the polished surface of the glass are reflected. + +The glass shades, used when observing the bright Sun, owing to their want of uniformity of colour density, often caused error. In some sextants of the last century it was not uncommon to provide a coloured eyepiece for use when observing the Sun. In others, the coloured screens were designed so that they could be instantaneously reversed, so that, by taking half a set of observations with the shades in one position, and the other half with them reversed, errors due to non-parallelism of the surfaces of the shades was eliminated. + +The graduations on the earliest instruments were cut in an arc of ivory. Later arcs were of silver, gold or platinum. The most important operation in sextant manufacture is, undoubtedly, the cutting of the arc. The difficulty of graduating an arc of a sextant or other similar astronomical or surveying instrument was not overcome until about the middle of the 17th century. The famous English engineer Jesse Ramsden (1735–1800) invented a remarkable machine, based on the worm and wheel principle, for dividing circular arcs as well as linear scales with precision. **Ramsden's dividing engine** employed principles that had been published by the French academician Duc de Chaulnes (1714–1769), who was the first to use a tangent screw drive for this type of machine. + +Names associated with early work on the manu- +facture of quadrant and sextant, in addition to Hadley and Ramsden, are George Adams (d. 1773). John Bird (1709–1776), and the opticians Troughton and the Dollond's. + +George Adams specialized in making quadrants for seamen and, soon after Hadley's invention, Adams was producing instruments at a price well below those of other instrument-makers. John Bird, who later travelled through the family's workshops (1753–1751), devoted considerable attention to the determination of the most suitable shapes of the several parts and fittings of the quadrant and the best methods of assembling them. Up to about 1760 the frames of quadrants were made from a combination of iron, brass and wood. It was recognized at about this time that accuracy of measurement was impaired largely on account of differences in expansion coefficients of the materials used. Rectangular instru- +ments having brass frames were manufactured long before the dawn of the 19th century, but metal frames were not common + +90 +A HISTORY OF NAUTICAL ASTRONOMY + +until the middle of that century. To facilitate the dividing of the arc of the early instruments the radius of the limb was kept as large as conveniently practicable, and this was often as much as 20 inches. With the introduction of the metal frame and the dividing engine invented by Ramsden, the length of the radius was reduced to a mere 8 inches or even less; that of most modern sextants being no more than about 6 inches. + +John Dollond (1706-1761) was the optician who is credited with the invention of the *aeromatic lens*. The firm of Dollond's still exists, and has produced many fine sextants, circles and other scientific instruments, especially during the 19th century. + +As far back as 1894, Leckey described an electric lighting system using a small dry cell and incandescent lamp, for use with a sextant for star observations. This appears to have first been fitted to a sextant by the instrument-maker Carl of the Strand in London. Cary was also able to eliminate horizon glare by interrupted thread in order to facilitate the fitting of sextant telescopes. + +The fitting of a *Nicol prism* for eliminating horizon glare appears to have first been suggested by a merchant seaman named Mackenzie who communicated his idea to the Royal Astronomical Society. This device is simply a polarizing prism used like a telescope eyepiece, and so placed that when the telescope is screwed home its axis is parallel to the plane of the horizon. It eliminates the plane of the sextant and, consequently, perpendicular to the horizontal when the sextant is being used for measuring the altitude of a heavenly body. It is used when the glare of the Sun (or Moon) renders it difficult to define the horizon. The prism allows only the "extraordinary" ray to be transmitted to the eye, the intense glare being refracted upwards out of the prism. + +Another device of great value when observing star altitudes is the *Wollaston prism*, which is made up in the form of two wedges of different thicknesses or different refractive indices so that two distinct images of an observed star are formed. The Wollaston prism is fitted between the index mirror and the horizon glass. When using the prism the observer brings the reflected images of the observed star to a position where the true image of the horizon lies centrally between them. + +A similar, but cheaper, device designed to facilitate star observations is the *lenticular or elongating lens*. This, like the Wollaston + +ALTITUDE-MEASURING INSTRUMENTS OF NAVIGATION 91 + +prism, is fitted between the index mirror and the horizon glass and, being a cylindrical lens, the reflected image of an observed star appears as a line, instead of a point, of light. + +To eliminate the uncertainty of the effect of refraction on the dip of the horizon, Commander Bliss of the United States' Navy invented, in the early part of the 20th century, an attachment for a sextant known as the *Bilh prism*. This device has the top and bottom faces bevelled at 45°. It is fitted to the sextant so that the longer of the front and back surfaces faces the observer. This face is provided with two polished surfaces, the lower of which is directly opposite the top of the index mirror, and the higher one is parallel to the horizon. The observer's eye is at a distance of the horizon 180° away from the part the observer is facing. With the index of the sextant set to zero on the arc the observer looks directly at the sea horizon in front of him and sees, at the same time, the back horizon reflected from the prism. When the fore and back horizons are brought into line, the sextant reading is twice the angle of dip, assuming that the sextant is free from index error. + +II. THE ARTIFICIAL HORIZON + +To take a sight with a sextant, the sea horizon vertically below the object whose altitude is required must be clear and distinct. Without the horizon, or a horizontal (or vertical) reference, a sight cannot be taken. "The frequent want of a horizon," wrote Robertson in his famous 18th-century *Elements of Navigation*, "is one great inconvenience that mariners have to struggle with." + +Many attempts have been made to provide means whereby the visible horizon may be dispensed with when taking sights. Hadley himself was the first to provide a remedy for an indistinct or invisible horizon, in the form of a simple spirit level attached to his quadrant. + +In 1732, a description of an improved artificial horizon appeared in *The Philosophical Transactions* of the Royal Society of London under the name of John Elton. Elton's device consisted of two spirit levels at right angles to each other fitted to the frame of a quadrant. The principle of Hadley's and Elton's bubble horizon is simple, but the difficulty of holding the instrument steady and perfectly vertical during observation rendered it, in its earliest form, impracticable. + +The problem of taking a sight with a sextant on dry land without + +92 +**A HISTORY OF NAUTICAL ASTRONOMY** + +a visible horizon is solved by using an artificial horizon in the form of a calm liquid surface such as a puddle of water or a con- +tainer of tar, treacle or oil. The more sophisticated artificial hori- +zon of this type, used extensively during the last century for the +purpose of taking longitudes, consisted of a trough filled with liquid, +chronometers, consisted of a trough of mercury, the surface of +which provided the reflector. The **mercury artificial horizon ap- +pears to have been invented by the London instrument-maker, +George Adams, in about 1738. The equipment consists, in addi- +tion to the trough and mercury, of a glass roof designed to prevent +the troublesome tremulous motion of the mercury due to wind. +The principle on which this horizon depends is based on +the first law of optics, which states that the angle of reflection from a +mirror is equal to the angle of incidence. When using the artificial +horizon the sextant is employed to measure the angle between the +Sun and his image on the mercury surface, this angle being equal +to twice the Sun's apparent altitude. The apparent altitude is the +arc of a vertical circle between the apparent direction of the ob- +served star and the plane of the meridian. The visible horizon or horizontal +plane on which the observer's eye stands. + +The mercury horizon is useless on board ship unless the ship is +perfectly steady. The slightest movement of the ship would cause +the mercury surface to tremor and become useless for observa- +tional purposes. + +Robertson, in his *Elements*, gives the following description of a +mercury horizon for use on board ship: + +*Into a wooden, or iron, circular box, of about 24 or 3 inches +diameter, and about ⅔ inch deep, pour about a pound or more +of quicksilver; and on this lay a metal speculum, or a piece of +plain glass, the diameter of which is about a third of an inch less +than that of the box; this will float in the quicksilver, and shew +the image of any object placed before it. If this be being hung +in jimbals will preserve a tolerable good horizon.* + +*The speculum, or glass, should be homogeneous, and have +parallel sides. There are some workmen who can work the two +planes of a piece of glass, so that they shall be demonstrably parallel.* + +*Or the fine surface of the quicksilver will do of itself, when +the motion is not great.* + +A portrait of Rev. Nevil Maskelyne D.D., the Father of Nautical Astronomy. He is holding an open book with a crest on its cover. +1. Rev. Nevil Maskelyne D.D.: the Father of Nautical Astronomy. From a painting by T. Downman at the National Maritime Museum. + +A circular instrument with a central pivot and two arms extending outward. The upper arm has a heart-shaped handle at its end. +1. Mercator's Astrolabe. Probably Spanish c. 1585. (Found off the Irish coast in 1845.) +2. Neocentral in Boxwood, England, c. 1646. +3. + +4. Azimuth Compass. English, c. 1720. + +5. Mariner's Quadrant, c. 1600. + +6. Buckstaff or Davis Quadrant. By John Gilbert, c. 1740. + +ALTITUDE-MEASURING INSTRUMENTS OF NAVIGATION 93 + +A century or so after Robertson had written the above descrip- +tion, John Merrifield, the headmaster of the navigation school at +Plymouth, described his attempt at using a mercury artificial +horizon on the roof of the school building: + +'We found it quite impossible to take observations with the +artificial horizon . . . owing to the shaking of the walls of the +building by wind or passing vehicles. At the suggestion of the +late Commander Walker, R.N., we had a horizon constructed, +so that a piece of glass, whose surfaces were perfectly plane and +parallel to one another, was placed in a trough, and the glass so +close to the sides of the trough as to prevent any great motion, +yet not so close as to prevent its free action. At first we found +very great discrepancies, owing to the glass not being homo- +geneous, and thus floating slightly deeper at one end than the +other: but the idea occurred to us of taking two sights with the +instrument reversed, using the means of the altitudes and of +the times of the observations. Thus the error due to want of +homogeneity was eliminated, and we have since found this to +be a very efficient instrument.' + +A compact mercury horizon was patented by Captain George, +a merchant service officer, during the seventies of the last century. +This consists of a circular iron trough containing mercury on which +a disc of glass having parallel faces is floated. Before floating the +glass, the surface of the mercury is cleaned with a brush and then rubbed +with a cloth; then the glass was pressed lightly on top of this, the piece of paper +being removed at the same time to ensure a perfectly clean mer- +cury surface. The chief advantage of Captain George's instrument +is that the whole surface of the artificial horizon is available for +observation. + +A form of artificial horizon was introduced into the Royal Navy +during the early part of the 20th century. This consisted of a +shallow trough of metal gilt which was amalgamated, after a first +cleaning of the surface with a drop or two of dilute sulphuric acid, +by the rubbing into it of a small quantity of mercury until the +whole surface was bright. The trough was mounted on three +adjustable screws and was provided with a bubble for levelling. + +In a very interesting and valuable dissertation on the history of +the art of navigation, written by Dr James Wilson, and which + +94 +**A HISTORY OF NAUTICAL ASTRONOMY** + +appeared in the first edition of Robertson's *Elements of Navigation* of 1772, mention is made of a horizontal top, invented by Serres who, Wilson informs us . . . 'unfortunately lost at sea aboard the Victoria.' + +Serres' horizontal top employed the properties of a spinning body. The upper surface of the top was polished metal which formed, when the top was spun, and in obedience to its gyroscopic inertia, a horizontal reflecting surface which could be used as an artificial horizon. + +John Smeaton, the renowned English engineer, improved Serres' top by inscribing his spinning artificial horizon, and how to use it, in the *Philosophical Transactions of the Royal Society* for the year 1752. Smeaton's top had a polished speculum surface of about 34 inches across. The top was fitted with a brass ring placed at right angles to the axis of the top. The sharp spinning point of the top rested on a cup of a hard substance such as flint or agate. Friction was kept to a minimum, and the top could be made to spin for periods of about fifteen minutes. There is no evidence to suggest that Smeaton's gyroscopic horizon met with any success in practical use. + +Raper, in his famous *Practice of Navigation*, refers to the use of a mirror attached to a pendulum which, hanging vertically, provides an 'artificial vertical' which serves the same purpose as an artificial horizon. Raper pointed out the difficulties of using such a device on board ship. + +In about 1838 Lieutenant A. B. Becher R.N. invented an artificial horizon attachment for a sextant, which was subsequently made and sold by Cary of the Strand. Becher's horizon met with some success. The inventor pointed out that ships are not always in violent motion, and that there are many circumstances of weather and sea in which an instrument such as his artificial horizon has value. In particular, he referred to the mouth of the English Channel in which ships often lie for hours obscured by fog and the importance, in these circumstances, of getting a sight for latitude. + +Becher's artificial marine horizon was fitted outside the horizon glass in line with the telescope axis. It consisted of a small pendulum bob of which was suspended in a small cistern of oil, so that the observer could control its movement; which, from the extreme delicacy of its suspension, would otherwise be impossible. + +**ALTITUDE-MEASURING INSTRUMENTS OF NAVIGATION** 95 + +Fitted to the pendulum and at right angles to it (and to the plane of the instrument when a sight is being taken) is a small arm. +Beyond the pendulum, a line for the horizon is formed by the upper edge of a slip of metal at right angles to the plane of the instrument. The pendulum has free motion in any direction; and the observer was required to bring the upper edge of the arm attached to the pendulum, in exact contact with the horizon line formed by the slip of metal. At the same time he was to make his observation by bringing down the image of the reflected object which he saw through the telescope. + +As the observer has thus to form his horizon at the instant of observation, he was advised, when observing on board, to get into that part of the ship where there is the least motion, and especially into a place screened from the wind. + +Becher's marine horizon was fitted with an oil lamp so that observations of stars could be made during hours of darkness. Adam's horizon was invented by Mr. Adam, who made attachments based on similar principles to that of Becher's. Beecher's device consisted of a balance carrying a glass vane which was fitted in the sextant telescope. The lower half of the glass vane was coloured blue, the horizontal line of demarcation between the coloured and uncoloured parts representing the sea horizon. The reflection of the observed object was brought into coincidence with this line, and its position determined by means of divisions below the artificial horizon was indicated by divisions on the glass vane, the values of which were determined by the makers. When taking a sight and using Beecher's horizon the observer brings down the reflection of the Sun's limb to the artificial horizon and leaves it there; and then, as the ship rolls, he catches, with his eye, the upper and lower divisions reached by the Sun's limb, and calls them out to an assistant who notes their readings respectively. After two or more readings have been taken, the altitude is read off and a correction is made according to the mean of the readings of the vane. Beecher's artificial horizon attachment, like that of Becher's, was fitted with a lamp which could be used to illuminate the telescope tube for star observations. + +In the early part of the present century, Paget patented an artificial horizon for attachment to a sextant. The Paget horizon consists of a short, curved spirit level mounted in a tube with a prism + +96 +A HISTORY OF NAUTICAL ASTRONOMY + +above it designed to throw the image of the bubble into the field of view of the telescope. This type of horizon had the advantage of not being affected by vibration or wind. + +Another ingeniously contrived artificial horizon using a small gyroscope was invented by Admiral Fleurais. This was manufactured by the well-known firm of Henry Hughes and Son. + +All the artificial horizons described above, and many others besides, were not entirely satisfactory; and they were, accordingly, short-lived. + +It was not until comparatively recent times that efficient artificial horizon attachments became available for marine sextants. Amongst these, the Booth bubble horizon is noteworthy. The Booth horizon is commonly fitted to air sextants, and good results are obtained by its use. + +CHAPTER IV + +The altitude corrections + +I. INTRODUCTORY + +The fundamental process in position-fixing at sea by astronomical methods is the measuring of the altitude of a celestial body. To find the latitude from a meridian altitude observation, for example, the complement of the altitude of a heavenly body at meridian passage, that is to say, the body's meridian zenith distance, is combined with the declination of the body to give the required latitude. Moreover, in the general nautical astronomical problem in which the astronomical- or PZX-triangle is to be solved, one of the sides of this spherical triangle is the true zenith distance of the observed object; thus it follows that the true distance of a heavenly body is the complement of the true altitude of the body, and this is obtained from the measured—or observed—altitude by applying certain altitude corrections. It is the historical account of these several corrections with which we shall be concerned in this chapter. + +2. REFRACTION + +In astronomical navigation, as in many other scientific activities, the nature and behaviour of light—the phenomenon by which nautical astronomical observations are made possible—are of great importance. + +To the query 'What is light?' there is no simple explanation. The physicist regards light as being electromagnetic radiation—a form of energy which travels at the prodigious speed of 300,000,000 meters per second. + +Many of the Ancient Greeks and other early peoples considered light to be a fundamental property or accident of nature and usually associated it with the Sun, 'the giver of light and life.' Among many speculations relating to the nature of light made by Ancient Greek scholars, that of Empedocles, who flourished during the 5th century BC, is interesting. The theory of light propounded by Empedocles was described by Aristotle, who + +98 +**A HISTORY OF NAUTICAL ASTRONOMY** + +flourished during the following century. 'Light is a streaming sub- +stance,' stated Empedocles, 'of the movement of which, because of its high speed, we are not conscious.' That light generally travels in straight lines was known to the ancients at all times. +The famous Euclid (c. 330-c. 275 bc), in a work on light, laid down the foundations of geometrical optics. Hero of Alexandria (fl. c. 100 bc), amongst many of his scientific activities, produced a work on mirrors. He considered the physical conditions of re- +flecting surfaces and mentioned the desirability of a polished surface to obtain optimum conditions for light reflection. Hero is often credited with being the first to express the law of the +important scientific law known as the 'principle of least action'. +This was expressed in relation to the equality of the angles of inci- +dence and reflection of a ray of light striking a point on a reflecting surface. Hero expressed this observed fact by stating that a ray of light makes the shortest route between object and eye. + +When light travels through a transparent medium of uniform density, it travels in a straight line until it strikes a surface obliquely from one transparent medium to another; its path is bent at the surface of contact. This phenomenon is known as +refraction. The remarkable effects of refraction, such as the appar- +ent bend in a straight stick partly immersed in water, have excited the curiosity of men of all times. There is no doubt that Hero, and scholars before him, understood something of this optical pheno- +menon. Sarton, in his *History of Science*, describes the ancient study of refraction as having been the most remarkable experi- +mental research of antiquity. + +Ptolemy, who flourished during the 2nd century ad, in his +*Optics*, a work known through a 12th-century translation in Latin +which had been translated from Arabic, elucidated certain optical phenomena, amongst which was an approximate law of refraction. +According to Ptolemy, the angles which the incident and refracted rays of light make with each other are inversely proportional to the two transparent media are directly proportional to one another. This relationship holds good only for small angles of incidence and refraction. + +The refraction of light from a celestial body in its passage +through the Earth's atmosphere is known as atmospheric- or +astronomical-refraction. Ptolemy explained atmospheric refrac- +tion as being due to changes in air density. He also concluded that + +THE ALTITUDE CORRECTIONS + +the apparent position of a star did not always correspond to its true position on account of atmospheric refraction. + +After the fall of Alexandria the works on light and optics produced by the Ancient Greek philosophers were developed by the Arabs. Notable amongst the Arab scholars was Al Kindi (c. 800–873), who made a special study of refraction. But the most famous of the Arab physicists who pursued the study of light and optics during the Dark Ages was Ibn al Haitham, better known as Al Hāthām (859–925). In his treatise Optics he showed that the angle of refraction was not proportional to the angle of incidence as had been stated by Ptolemy. Although Al Hazen disagreed with Ptolemy in this respect, he did not give a better law of refraction. + +It is believed that Roger Bacon (1214–1294), a disciple of the famous Franciscan Robert Grosseteste (1175–1253), whose discovery of double refraction of light through a clouded glass proved the way for the invention of spectacles and the telescope, was led to study refraction when he visited Al Hazen. + +The Polish philosopher Vitello (born c. 1230) experimented with the refraction of light passing through air and water, and through air and glass, and determined new values for angles of refraction. Vitello showed that the scintillation of the stars is due to atmospheric effects. + +The Arabian physicist Al Farisi (d. c. 1320) gave an interesting explanation of refraction, attributing the phenomenon to a change of speed of light when passing from one medium to another of different optical density. + +The renowned Tycho Brahe (1546–1601) is often held to have been the first to have employed atmospheric refraction for correcting astronomical observations. He found the value of atmospheric refraction of light from a celestial body on the horizon to be 3°, and that from a body at 45° altitude to be 45°. However, Tycho was not too clear as to the cause of atmospheric refraction, attributing it to 'the gross vapours that float in the atmosphere.' According to Tycho, the refraction of light from the Sun was different from that of starlight. The former he supposed to extend to an altitude of 45°, and the latter to 20°. + +Johannes Kepler (1571–1630), disagreeing with Tycho, stated that atmospheric refraction is the same for all celestial bodies at the same altitude. He also disagreed with the view that refraction is zero for altitude 45°. + +100 +A HISTORY OF NAUTICAL ASTRONOMY + +The discovery, in 1621, of the true law of refraction is due to the Dutch physicist Willebrord Snell (1591–1626). Snell's law asserts that when light passes from one medium to another the planes of the angles of incidence and refraction and the perpendicular or 'normal' to the common surface of the two media are coincident, and that the sines of the angles of incidence and refraction are in a constant ratio for any two media, this being the refractive index for the two media. + +This law was first published in 1637, after Snell's death in 1626, by René Descartes, the French philosopher, who may have made an independent discovery of the same law. The French mathematician Fermat (1601–1665) argued that the law of refraction conformed with the idea that the path of light refracted at the common surface of two media was described in the least time. Snell's law implies, therefore, that the velocity of light in a medium is inversely proportional to the refractive index of the medium. + +The famous Italian philosopher Dominic Cassini (1625–1712), like Kepler, showed the fallacy of Tycho's refraction doctrine. He proved that atmospheric refraction diminishes from a maximum at the horizon to zero at the zenith. The 18th-century French academician Abraham de la Hire produced a table of atmospheric refraction, and he found out that atmospheric refraction of light from stars differs from that of light from planets. + +The law of refraction propounded by Cassini was based on the hypothesis that the atmosphere is spherical and homogeneous. In the simplest investigation of atmospheric refraction, the Earth is regarded as being flat and the atmosphere is considered to be composed of an infinite number of horizontal parallel layers of air; the density of each layer decreases uniformly towards the Earth's centre. On this assumption it is readily proved that the effect of atmospheric refraction is the same as if light entering the atmosphere were refracted directly into the lowest layer of the air without traversing the intervening layers. + +From Snell's law, a ray of light passing through the atmosphere such that $\mu \sin Z$ is constant for every point in its path; $\mu$ being the refractive index at any point, and $Z$ the angle the path makes with the vertical. If $Z_0$ be the value of $Z$ when the ray enters the atmosphere then, since in vacuum the refractive index of light is unity: + +A diagram showing Snell's Law. + +\[ +\frac{\sin i}{\sin r} = \frac{1}{n} +\] + +THE ALTITUDE CORRECTIONS + +101 + +$$\mu \sin Z = \sin Z_0$$ + +If $\mu$ and $Z$ are now taken as referring to the position of an observer's eye; and if $w$ is the atmospheric refraction, then: + +$$Z_0 = Z + r$$ + +Hence: + +$$\mu \sin Z = \sin (Z + r)$$ + +i.e. + +$$\mu \sin Z = \sin Z \cos r + \cos Z \sin r$$ + +Since $r$ is a small angle (never more than about $\frac{\pi}{6}$) + +$$\cos r \approx 1$$ + +and + +$$\sin r = r \text{ radians}$$ + +We may, therefore, write: + +$$\mu \sin Z = \sin Z + r \cos Z$$ + +From which: + +$$r = (\mu - 1) \tan Z$$ + +i.e. + +$$r = U \cdot \tan Z$$ + +where $U = (\mu - 1)$ or coefficient of refraction. + +This result holds good for small zenith distances; but for small altitudes, by treating $\sin r$ and $\cos r$ as radians and 1 respectively, significant error results. Moreover, light from celestial objects at small altitudes has to travel through a considerable length of atmosphere, and we are not justified, therefore, in regarding the layers of air as being bounded by horizontal parallel planes. Cassini recognized this and, accordingly, took into account the Earth's curvature in his formula. + +Cassini's formula for atmospheric refraction is explained with reference to Fig. 1. + +Fig. 1 represents part of a vertical section through the Earth's centre C and an observer O. XO_0O represents a ray of light from a celestial object X entering an observer's eye at O. + +Cassini's hypothesis is that the light undergoes a single refraction on entering the atmosphere at O_1. + +Let the apparent zenith distance of the celestial body be $\theta$; and the true zenith distance $\theta_1$. Let the refraction be $r$ radians. + +8 + +102 +A HISTORY OF NAUTICAL ASTRONOMY + +A diagram showing the angles involved in Cassini's method of determining latitude. The diagram shows the Earth's surface, the atmosphere, and the celestial sphere. The lines represent the angles between the lines of sight to the horizon and the celestial sphere. + +FIGURE 1 + +If $r$ is small, +$$r = (\mu - 1) \tan \theta_1$$ + +Cassini expressed $\tan \theta_1$ in terms of $\tan \theta$. This he did by first drawing CT perpendicular to O$_1$O produced; and O$_1$V perpendicular to COZ. Then: + +$$\frac{O_1T}{T} \tan \theta_1 = OT \tan \theta$$ + +i.e. +$$\tan \theta = \frac{O_1T}{OT}$$ +$$= 1 + \frac{OO_1}{OT}$$ +$$= 1 + \frac{OV \sec \theta}{OC \cos \theta}$$ +$$= 1 + \frac{OV}{OC} \sec^2 \theta$$ + +THE ALTITUDE CORRECTIONS 103 + +Now OV is approximately the vertical height of the atmosphere OW, and is, therefore, x OC, where x is the ratio between the height of the homogeneous atmosphere and the Earth's radius. Therefore: + +$$\tan \theta = 1 + x \sec^2 \theta$$ + +or + +$$\tan \theta_1 = \frac{\tan \theta}{1 + x \sec^2 \theta}$$ + +Expanding the denominator $(1 + x - \sec^2 \theta)^{-1}$ by the Binomial Theorem, we have: + +$$\tan \theta_1 = \tan \theta (1 - x \sec^2 \theta - x^2 \sec^4 \theta - ...)$$ + +Since $x$ is a small quantity, powers of $x$ greater than 1 may be ignored without introducing material error. + +Thus: + +$$\tan \theta_1 = \tan \theta (1 - x \sec^2 \theta)$$ + +and + +$$r = (\mu - 1) \tan \theta (1 - x \sec^2 \theta)$$ + +which is Cassini's formula. + +If the value of $x$ is accurately chosen, Cassini's formula gives good results for altitudes not less than about 10°. + +Newton, Hooke, and Huyghens were the more notable of the 17th-century philosophers who devoted considerable attention to the nature and behaviour of light. To Newton we owe the theory of the spectrum, and to Hooke and Grimaldi the discovery of the phenomenon known as diffraction. The Danish physicist Roemer is credited with being the first to measure the speed of light by comparing the computed and observed times of the eclipses of Jupiter's satellites, and the Dutch physicist Huyghens is credited with being the founder of the wave theory of light. + +The table of refractions given in the early editions of the *Connoissance des Temps*, which was first published under royal patents in 1678, was compiled by Jean Picard. The values given in Picard's table agree closely with the tangent law of refraction $$r = U \tan \sinh^{-1}(\sinh U)$$ + +One of the best 18th-century tables of refraction is that of de la Caille (1713-1762). De la Caille's table is based on observations + +104 +A HISTORY OF NAUTICAL ASTRONOMY + +of circumpolar stars made at Paris and the Cape of Good Hope. He recognized that atmospheric refraction varies with air pressure and temperature, both these properties affecting the air density and, therefore, the refractive index. De la Caille's table gave mean refractions computed for a standard atmosphere having a specified pressure and temperature at sea level. + +Tables of astronomical refraction were compiled by many astronomers during the 18th century, but perhaps the table that + +FIGURE 2 + +was esteemed the best was that of James Bradley (1693-1762). Doctor Bradley's mean refraction table applied to a standard atmosphere having a sea-level pressure and temperature of 29-6 inches of mercury and 50° F, respectively. Bradley furnished an auxiliary table for correcting the mean refraction for use when the atmospheric conditions differed from those for which the mean refractions were tabulated. + +It will be of interest to discuss the methods by which atmospheric refraction may be ascertained. The usual method involved the observations of circumpolar stars, and is explained with reference to Fig. 2. + +THE ALTITUDE CORRECTIONS 105 + +Fig. 2 represents the projection of the celestial sphere on to the plane of the horizon of an observer whose zenith is projected at Z. X and S are the projections of the north and south points of the observed horizon respectively, and P is that of the celestial pole. And X and X₁ are the projections of a circumpolar star at lower and upper meridian passage respectively. + +Let $x$ and $z_1$ be the apparent zenith distances of the star when at lower and upper transit respectively. Let $p$ be the polar distance of the star and the coefficient of refraction $(\mu - 1) = U$. Then: + +i.e. +$$PZ = ZX - PX$$ + +Similarly: +$$PZ = z + U \cdot tan z - p$$ + +i.e. +$$PZ = z_1 + U \cdot tan z_1 + p$$ + +Adding (I) and (II): +$$2 \cdot PZ = z + z_1 + U(tan x + tan x_1)$$ + +In a like manner, if $\bar{x}$ and $\bar{z}_1$ are the apparent zenith distances of another circumpolar star whose declination differs materially from that of the first, we have: + +$$2 \cdot PZ = \bar{x} + \bar{z}_1 + U(tan \bar{x} + tan \bar{z}_1)$$ + +From (III) and (IV) we have: + +$$z + z_1 + U(tan x + tan x_1) = \bar{x} + \bar{z}_1 + U(tan \bar{x} + tan \bar{z}_1)$$ + +From which: + +$$U = (\tan \bar{x} + tan \bar{z}_1) / (\tan x + tan x_1)$$ + +By repeated observations of circumpolar stars, James Bradley found the value of $U$ to be 57-54 seconds of arc. + +Bradley, who was elected a Fellow of the Royal Society in 1718, was appointed Savilian Professor of Astronomy at Oxford in 1721. He made many important contributions to the science of astronomy. He devoted considerable attention to the investigation of stellar parallax. + +The philosopher Robert Hooke, who had invented the zenith sector for the purpose of measuring or detecting stellar parallax, + +106 +A HISTORY OF NAUTICAL ASTRONOMY +had attempted to detect the *annual parallax* of the star λ Draconis. The star transits in London near the zenith, so that its meridian altitude is not affected by atmospheric refraction. The effect of annual parallax on the apparent position of a star depends when at meridian passage should be a annual fluctuation in its meridian altitude about a mean value, the maximum departure from the mean value occurring in December and June. Hooke, in his investigations, discovered a fluctuation; but it was regarded as being due to instrumental error and/or faulty observation, rather than a proof of the star's annual parallax. Half a century after Hooke had published his results, the astronomer Samuel Molyneux, set about tackling the same problem. Their observations showed uncontestably that the meridian altitude of λ Draconis did fluctuate with an annual period, but the maximum departure from the mean value occurred, not in December and June, but in September and March. In 1728 Bradley, who was then Astronomer Royal, demonstrated that the phenomenon that he had observed was caused by the fact that the distance from the star bears a finite ratio to the distance of the Earth from her orbit. When approaching the star its meridian altitude, therefore, is greater than the mean value for the year; and when receding from it, it is less than the mean. This phenomenon, often regarded as being Bradley's greatest discovery, is known as *aberration of light*. Another of Doctor Bradley's great discoveries is the *mutation* of the Earth's axis. This phenomenon is due to the *precession* of the Earth's axis, but due to the effect of the Moon and Sun on the Sun. +Bradley's work on atmospheric refraction is often linked with the later work, in the same field, of Friedrich Bessel. +Bessel (1784–1846) started work at an early age in the old Hansa city of Bremen, where he developed a desire to sail abroad as a supercargo. The fulfilment of his desire led him to a study of navigation and astronomy; and astronomy, branches of science in which he became so remarkable. In 1835 he became Director of the new Königswinter observatory. Bessel was instru- mental in reducing the valuable, but neglected, astronomical ob- servations of Bradley; and it is to Bessel that the discovery of stellar parallax is due. + +*That is*, the angle at a star contained between diametrically opposite points of the Earth's orbit. +? See Chapter v., p. 121, for a brief discussion on precession. + +THE ALTITUDE CORRECTIONS 107 + +A formula for astronomical refraction, more exact than the simple tangent law, was formulated by Bradley. The rule, often stated in navigation and astronomy books of the late 18th century, is: + +'The refraction at any altitude is to 57 seconds, in the direct ratio of the tangent of the apparent zenith distance lessened by three times the estimated refraction, to the radius.' + +Expressed in the common mathematical way, this rule is: + +$$r_e = 57^\circ \cdot \tan(x - 3 \cdot r_a)$$ + +In applying this rule, if the calculated refraction $r_e$ differs materially from the estimated refraction $r_a$, it is necessary to repeat the calculation using $r_e$ in place of $r_a$. + +An alternative method of calculating refraction, due to Bradley, involves observing the Sun's apparent zenith distance at noon on the days of the solstices. If $z_s$ and $z_w$ are the summer and winter solstitial zenith distances of the Sun at noon respectively, then the true zenith distances are respectively: + +$$x_s + (\mu - 1) \tan z_s$$ + +and + +$$x_w + (\mu - 1) \tan z_w$$ + +Now the declination of the Sun, when he is at a solstitial point, is equal to the obliquity of the ecliptic $\epsilon$. Therefore, the observations being made north of northern tropic, in latitude $\lambda$: + +$$x_s + (\mu - 1) \tan z_s = \lambda - \epsilon$$ + +and + +$$x_w + (\mu - 1) \tan z_w = \lambda + \epsilon$$ + +From which: + +$$2\lambda = x_s + x_w + (\mu - 1)(\tan z_s + \tan z_w) \quad (V)$$ + +and + +$$(\mu - 1) = \frac{2\lambda - (z_s + z_w)}{\tan z_s + \tan z_w}$$ + +Alternatively, $\lambda$ may be eliminated from formula (V) by combining it with formula (III) or (IV), so that $(\mu - 1)$ may be found from combined observations of the Sun at each of the solstices and the zenith distances at upper and lower transit of one circumpolar + +108 +A HISTORY OF NAUTICAL ASTRONOMY + +star. Eliminating $\lambda$ in this way, using formulae (III) and (IV), we have: + +$$ (\mu - 1) = \frac{x + x_1 - x_2 - x_3}{\tan x_1 + \tan x_2 - \tan z - \tan x_4} $$ + +Experiments made in the 18th century—notably by the instrument-maker Francis Haukabee who flourished during the first decade of the century—showed that atmospheric refraction is proportional to the density of the air. The density of the air varies directly as its pressure and inversely as its heat. Since the pressure and heat of the air are shown by barometer and thermometer respectively, the difference between them may be reduced to the actual refraction by allowing for the differences between the actual and mean pressures and temperatures of the air. + +The difference of refractions arising from a given difference of temperature may be ascertained by observation. De la Caille made the change of refraction corresponding to a change of $10^\circ$ on Reamur's thermometer to $1/25$ of the whole. Mayer made the change $1/22$ of the whole. According to Bradley: + +'The true refraction is to the mean refraction in a direct ratio of the altitude of the barometer to $29-6$, and in an inverse ratio of the altitude of the thermometer increased by 350 to the number 400.' + +In other words: + +true refraction = mean refraction, $$H = \frac{400}{29.6(T - 350)}$$ + +where $H$ is the height of mercury in inches and $T$ is the temperature on Fahrenheit's scale. Bradley's auxiliary table to the table of mean refraction was based on this rule. + +Andrew Mackay, in his Theory of the Longitude of 1793, explained the physical cause of refraction in a curious, but interesting way: + +'It is demonstrable,' he wrote, 'that every body is endowed with an attractive power, which reaches to some distance beyond its surface, as that of cohesion, magnetism, etc. Now a ray of light from a heavenly body will, at its entrance into the terres- + +THE ALTITUDE CORRECTIONS 109 + +trial atmosphere, be attracted towards the denser parts; and since the density of the atmosphere increases the nearer the Earth's surface, therefore the ray, as it approaches the observer, will be more and more attracted, its velocity accelerated, and of course its rectilinear direction changed. Hence that portion of the ray contained between an observer and the extremity of the atmosphere will be a curve, except in that case when the ray is perpendicular to the refracting medium. + +Let us now turn our attention to atmospheric refraction as a correction to a measured or observed altitude of a celestial body. It must be appreciated that with the earliest instruments em- +ployed by the seaman for measuring altitudes—instruments such as the seaman's quadrant, astrolabe and cross-staff—it was not possible to obtain altitudes to anything but a coarse degree of accuracy. In those circumstances, therefore, the application of altitude corrections to astronomical observations of arc to crude observers had been measured perhaps to the nearest degree of arc would have been unworthy of consideration. + +Edward Wright, in his Certaine Errors of Navigation, first pub- +lished in 1599, brought to the notice of seamen the effect of re- +fraction on altitudes of celestial bodies. He explained the cause of +refraction in the same way as did Tycho Brahe; and, in fact, the +tables he published were based on Tycho's observations. Like Tycho, he believed refraction to be different for the Sun than for the stars; and his table of +refraction for the Sun extended from 0° to 45° altitude, and that +for the stars from 0° to 20°. + +Thomas Harriot (1560–1621), the brilliant Elizabethan mathe- +matician who devoted considerable attention to navigational matters, and who was adviser to Sir Walter Raleigh on mathemati- +cal problems connected with navigation, thought that this effect of re- +fraction was trivial, and need not, therefore, be considered by seamen. + +An early, and interesting, observation made at sea with the +specific object of finding refraction, was made by William Baffin +off the west coast of Spitzbergen in 1614. The account of the voy- +age, during which this observation was made, appears in *Parsch*, +*His Pilgrims*. It was the type of observation in which the famous explorer showed his acute inventive genius. His observations of + +I10 A HISTORY OF NAUTICAL ASTRONOMY + +refraction were based on exact knowledge of the latitude of the place of observation, the declination of the Sun, and the angular diameter of the Sun. Baffin observed the Sun at lower transit at a time at which he estimated that a certain fraction of the Sun's diameter was above the horizon. Knowing his latitude and the declination of the Sun, he calculated the proportion of the Sun's diameter that should be above the horizon. From his results he demonstrated that the angle of atmospheric refraction applicable to his observation was 26°. + +The table of refractions given in Maskelyne's *Requisite Tables*, the first edition of which was published in 1781, was based on Bradley's observations. The table was reproduced in most of the navigation manuals of the late 18th and 19th centuries, particularly, in those of John Hamilton Moore, J. W. Norie, Mrs Janet Taylor, Andrew Mackay, Edward Riddle and John Riddle. + +From about 1825, refraction tables in some navigation manuals were based on a refraction formula due to James Ivory of double altitude fall. In 1830, a more accurate suite of refraction was made by Ivory, and a comprehensive account of this work was printed in the *Philosophical Transactions* for 1823. Ivory's mean refraction table was based on a standard atmosphere having a sea level pressure and temperature of 30 inches of mercury and 50° F., respectively. The celebrated Henry Raper, as well as James Inman, used Ivory's refraction rules in their collections of navigational tables. + +It may be remarked that the actual refraction of light from a heavenly body whose altitude exceeds about 10°, is never more than about half a minute of arc different from the mean refraction. It is for this reason that the table corrections to mean refraction is seldom used by seamen in practice today. In former days however, although the table was of little consequence when finding latitude, it demanded the attention of the seaman who would find longitude by the method of lunar distance. + +Refraction of light from heavenly bodies within a few degrees of the horizon is always very great indeed. Neither the most refined mathematical investigation nor the most careful observations can remove the uncertainty of refraction at small altitudes. Temperature changes--and therefore density changes--of the air along the line followed by a ray of light from an object near the celestial horizon, are almost always taking place. These + +THE ALTITUDE CORRECTIONS III + +changes can never be known with certainty, and no refraction law has yet been formulated which will hold good at all times for altitudes less than about 5°. Testimony to this fact is provided by the frequent investigations made in recent times into the question of refraction at small altitudes. + +3. DEPRESSION OR DIP OF THE SEA HORIZON + +The depression, or dip, of the sea horizon is a measure of the angle contained between the plane of the horizontal surface on which rests an observer's eye, and the direction of the visible or sea horizon, i.e., the line of sight of an observer. The angle of dip is clearly a function of the elevation of the observer's eye: the greater is the height of eye above sea level, the greater is the angle of dip. Moreover, the greater is the height of eye of an observer the greater is the range of his sea horizon. These facts have been recognized since very early times; and they provided simple and compelling evidence of the Earth's rotundity. It is recorded that Eratosthenes, Pythagoras, and others flourished during the 5th century BC, gave as clear proof of the spherical shape of the Earth the changing appearance of a ship as she heaves into the sight of, or sails away from, an observer standing on the shore. + +During the 16th century, when the first of the 'modern' attempts at determining the size of the Earth were made, one method suggested by Sir Isaac Newton was based on mathematical principles. Edward Wright, was related to the height of an observer's eye above sea level and the corresponding range of his sea horizon. The uncertain effect of terrestrial refraction, that is the bending of light in its passage from a point on the visible horizon to an observer's eye, rendered the method unreliable and inaccurate. The principle of the method is described with reference to Fig. 3. + +Fig. 3 illustrates a section of the Earth in the plane of which lies its centre C and the observer's eye at O. A is a point vertically below O at sea level; and AB is the Earth's diameter. The observer sighting a point on his theoretical horizon lying in the plane of the section would be sighting along the straight line OH, which is a tangent to the circle ABD at D. + +The observer's theoretical horizon, the radius of which is AD, is a small circle which limits the observer's view, assuming that terrestrial refraction is non-existent. + +112 +A HISTORY OF NAUTICAL ASTRONOMY + +X +O +Plane of sensible horizon +h + +D +A +B +C +H + +FIGURE 3 + +Because the height of the observer's eye—h units—above sea level is small compared with the distance of the theoretical horizon AD, it may be assumed that tangent OD is equal to arc AD. From a well-known geometrical theorem: + +$$OD^2 = OA \cdot OB$$ + +i.e. + +$$AD^2 = OA \cdot OB$$ + +and + +$$AD = \sqrt{OA \cdot OB}$$ + +or + +distance of theoretical horizon = $$\sqrt{2R \cdot h}$$ (OB = 2R approx.) + +It follows, therefore, that if the Earth's diameter and the height of the observer's eye above sea level are known, AD, the distance of the observer's theoretical horizon, may be computed. Conversely, if the distance of the theoretical horizon and the observer's height above sea level are known, the Earth's radius may be computed. + +THE ALTITUDE CORRECTIONS 113 + +It will be noticed from Fig. 3 that the dip of the theoretical horizon, denoted by $\Delta$, is equal to the angle at the Earth's centre contained between radii terminating at D and A respectively. We have, therefore: + +$$\cos \Delta = \frac{R}{R+h}$$ + +Since the angle of dip is a small angle: + +$$1 - \frac{\Delta^2}{2} = 1 - \frac{h}{R}$$ + +where $\Delta$ is expressed in circular measure, and + +$$\Delta = \sqrt{\frac{2h}{R}}$$ + +The effect of atmospheric refraction is for light coming from the actual horizon, the visible- or sea-horizon as it is called, to follow a path concave to the Earth's surface as illustrated in Fig. 4. + +A diagram showing the relationship between the theoretical horizon, the sensible horizon, and the actual horizon (sea or visible horizon). The diagram includes labels for X, O, A, B, C, D, E, and various lines indicating the different horizons. +X +O +Plane of sensible horizon +dip +A +D +E +theoretical horizon +sea or visible horizon +C +B +FIGURE 4 + +114 +**A HISTORY OF NAUTICAL ASTRONOMY** + +Refraction, therefore, causes the sea horizon to have a greater range than that of the theoretical horizon. It also causes the angle of dip to be smaller than that of the theoretical dip. + +The effect of terrestrial refraction on dip and distance of the sea horizon received the attention of many 18th-century physicists and astronomers, but there was never general agreement as to the exact effect of refraction. Dr Nevil Mankelyne, under whose direction the first British Nautical Almanac was published in 1765, stated that 1/10 of the theoretical dip should be subtracted from the observed dip when given by instruments. The investigators gave fractions generally between 1/9 and 1/15; but according to General Roy (Philosophical Transactions, 1790) it varies from 1/3 to 1/24 of the 'comprehended arc'. + +Andrew Mackay, in his *Theory of the Longitude*, points out that because the Earth is an oblate spheroid the radius of curvature is variable with the latitude. It follows, therefore, that no single table of dip can answer in all places. Mackay wrote: + +'Tables of dip should be calculated for the latitude of the place and the azimuth of the observed object. It however may be observed that the difference of dip arising from the above cause is so inconsiderable as to have been hitherto neglected.' + +In the same book, Mackay gave the following rule for computing dip: + +'To the constant logarithm 0-4236 add the proportional logarithm of the height of the eye above sea level, in feet; half the sum will be the proportional logarithm of the dip of the horizon.' + +When observing the altitude of a heavenly body at sea the measured angle is that contained between the apparent direction of the body and the direction of the point on the visible horizon vertically below the body. This latter point, as we have seen, does not coincide with a point on the true horizon, falling, as it does, below it by an amount which is dependent upon the height of the observer's eye above the sea surface. The angle of depression of the sea horizon must, therefore, be subtracted from the measured altitude to give what is called the apparent altitude. The apparent altitude of a heavenly body is defined as the arc of a vertical circle + +A diagram showing a celestial body at an altitude \(h\) above sea level, with its apparent altitude \(H\), and its true altitude \(H'\). The diagram also shows a line representing the observer's eye level at sea level. + +THE ALTITUDE CORRECTIONS + +contained between the apparent direction of the observed body and the true or sensible horizon. + +The dip varies as the square root of the height of eye of the observer. For a height of eye of 50 feet above sea level, the dip is about 7° of arc; for a 100 feet it is about 10° of arc. For the same reason as the atmospheric refraction correction was not needed by the early mariners who used relatively crude instruments for their observations, the dip also was ignored. Both refraction and dip corrections are subtractive; but even though they are to be applied in the same direction, they do not cancel each other. The correction of dip and refraction for an observation taken from a position on ship often less than about 15 feet from the sea surface was never more than about 10° of arc for altitudes greater than about 10°. + +It appears that the first dip table designed and printed for the use of the seaman was that given by Edward Wright in his remarkable *Certaine Errors in Navigation, Detected and Corrected*, the first edition being published in 1583. Wright's table extended from 5 feet to 90 feet height of eye above sea level with corres- ponding dip corrections of 2° and 11° of arc respectively. These corrections compare favourably with those to be found in a modern dip table in which: + +$$\text{dip for 5 feet} = 2.57°$$ +$$\text{dip for 90 feet} = 10.91°$$ + +Thomas Harriot regarded dip as being sufficiently large to be worth considering for navigational purposes. His table of dip, although ante-dating Wright's table, was never published. Harriot's values for dip are consistently a little too large. + +The dip table in Maskelyne's *Requisite Tables* allowed for the effect of terrestrial refraction based on Maskelyne's rule quoted above. Samples of dip taken from this table are: + + + + + + + + + + + + + + + + + + + + + + +
Height of eyeDip
10 feet3-01'
50 feet6-44'
100 feet9-33'
+ +Maskelyne's dip table, in which the refraction allowance is 1/10 of the theoretical dip, was given in many of the navigation manuals + +116 +A HISTORY OF NAUTICAL ASTRONOMY + +of the late 18th and early 19th centuries. These included those of +Roberson (1788), Norie (1803), John Hamilton Moore (1780), +Andrew Mackay (1796) and James Taylor (1830). +Edward Riddle's dip table which appears in his *Treatise on Navigation* (1828) is based on an allowance of 1/13 of theoretical dip, the factor 1/13 being that suggested by the French physicist Biot. +Raper, in his *Practice of Navigation* (1840), gave a dip table based on an allowance for refraction amounting to 1/14 of the theoretical dip. +Raper drew attention to the facts that the running of the sea in bad weather causes the sea horizon to be in continual vertical motion; and that the rising and falling of an observer due to rolling, pitching, and heaving of the ship, causes the dip to be in perpetual change. To overcome errors due to these causes the seaman was advised to make a series of observations, instead of a single sight, so as to obtain a more reliable result. +Raper also pointed out that the height of eye should be ascertained with some precision, that is to say, within two or three feet; because an error in dip causes a corresponding error of the same amount in altitude. This is of greatest importance when the observer's eye is near the sea surface, because the rate of change of dip with height above sea level is greatest when the height of eye is zero. + +In general, the greater the height of the observer's eye the more distinct will be the sea horizon, provided that the air is clear. In misty weather, however, when celestial observations are possible, it is better to observe from a position as near to the sea surface as practicable so as to 'bring' the sea horizon as near as possible to the observer. + +It is interesting to note that Raper gave a true dip table as well as a table of apparent dips. The true dip table gives the depression of the theoretical horizon, and it is based on the formula: + +$$\text{True dip} = 1 - \frac{h}{k}$$ + +where $h$ is the observer's height of eye in feet. + +A footnote to the explanation of the true dip table is interesting: + +As the lower latitudes are more frequented by shipping than the higher, $40^\circ$ has been assumed as the average latitude. Also, + +THE ALTITUDE CORRECTIONS 117 + +as the curvature of the Earth is different on the prime vertical and on the meridian, the circle of curvature crossing the meridian at 45° of azimuth has been employed. The depression is accordingly, computed to the value 20,909,577 feet which gives the length of the average nautical mile 6082 feet nearly. + +When star observations became popular during the closing decades of the 19th century it became common to provide, in collections of nautical tables such as those of North Iman, etc., a combined table of dip and refraction for each altitude. This table is called the Stars' Total Correction Table, no altitude cor- rections, other than those for dip and refraction, being required for reducing a star's observed altitude to its true altitude. + +4. THE SUN'S SEMI-DIAMETER + +The correction called Sun's semi-diameter is applied to a measured altitude of the Sun, when his limb is observed. It is simply half the angular diameter of the Sun and is to be added to the altitude of the lower limb and subtracted from the altitude of the upper limb, in order to find the altitude of the Sun's centre. + +The Sun's semi-diameter varies during the year, being least when the Earth is at aphelion, when its value is 15·8', and greatest when it is at perihelion, when its value is 17·3'. The traditional value for the Sun's semi-diameter is 16', and this is the angle which the seaman was recommended to apply to the measured altitude of the Sun's limb. In fact, some Davis quadrant were graduated on the back edge of the smaller arch in such a manner as to eliminate the need for applying the semi-diameter correction arithmetically. + +Before the cross-staff was introduced to the navigator, the only instruments used for measuring altitudes were the seaman's quadrant and astrolabe. When either of these were used for measuring the Sun's altitude, the angle given on the graduated arc--by the plumb-line in the case of the quadrant, and the spot of light in the case of the astrolabe--required no correction for semi-diameter, the measured angle being the altitude of the Sun's centre. Similarly, when using a Davis quadrant fitted with a glass vane instead of a shade vane the same applied. + +It was not until the closing decades of the 17th century that + +9 + +118 +A HISTORY OF NAUTICAL ASTRONOMY + +observed altitudes of the Sun were corrected for refraction, dip, and semi-diameter. Although the errors due to not applying alti- +tude corrections were understood, it was realized that the degree of error associated with the measuring instruments used—especi- +ally the seaman's quadrant and astrolabe—was considerably coarser than the small angular values of the corrections. Following +the introduction of Hadley's quadrant, the question of altitude +corrections sprang to the fore, and altitude correction tables were +provided, with corrections often to an unnecessarily high degree +of accuracy, but most useful tables were not available. + +The Sun's semi-diameter and that of the Moon as well, was required when clearing a lunar distance between Sun and Moon; +and for this purpose accurate values of semi-diameter were re- +quired, because a small error in the cleared distance results in a +relatively large error in the calculated longitude. + +From the early part of the 19th century, it became customary to provide in nautical table collections a table giving Sun's total +correction. This included the separate corrections for refraction, +dip and semi-diameter, against observed altitude of Sun's lower +limb and height of observer's eye. The table was constructed +using 16° as the semi-diameter correction, although in later tables +an auxiliary table giving a monthly correction for the variation in +Sun's semi-diameter was provided. In Norie's Nautical Tables, +the first edition of which appeared in 1830, this table was that +of 1828. Robertson, in his Elements of Navigation, gave a com- +bined table of Sun's semi-diameter, dip and refraction, but left +the user to combine the separate corrections. + +5. THE MOON'S SEMI-DIAMETER + +The Moon's semi-diameter is sensibly affected by her altitude. +This follows because the radius of the Earth is a significant pro- +portion of the Moon's distance from the Earth. The Moon's +semi-diameter is least when she is at apogee and has an altitude +of 0°. It is greatest when she is at perigee and is in the zenith of +an observer on Earth. For altitudes other than 0°, both semi-diameter +are given for altitude 0°, so that a small correction, known as the +augmentation, is to be applied to the tabulated value. The aug- +mentation of the Moon's semi-diameter is given by the formula: +Augmentation = Moon's semi-diameter × sine apparent altitude + +THE ALTITUDE CORRECTIONS 119 + +Although the maximum value of the augmentation is no more than about 0-3° of arc, it formed an important correction when using the method of finding longitude by means of the lunar distance method. + +6. PARALLAX + +The point in the heavens which a celestial body occupies when viewed from the Earth's surface, is called the apparent place of the body. The point which it would occupy were it viewed from the + +A diagram showing parallax in altitude. The diagram shows a celestial body at position Xo, with its apparent place Y on the horizon. The true place Z (parallel to Xo) is shown above the horizon. The observer's position is at C. The line OC represents the observer's position and the Earth's centre. The angle between OC and OXo is called the body's parallax-in-altitude. The diagram also shows the plane of sensible horizon. +FIGURE 5 + +Earth's centre is called the true place of the body. The angular difference between the apparent and true places of a celestial body at any given instant is equivalent to the angle at the centre of the body subtended by a line joining its apparent place to the observer's position and the Earth's centre. This angle is called the body's parallax-in-altitude. Fig. 5 illustrates how parallax varies with altitude and distance. + +It should be clear from Fig. 5 that parallax-in-altitude for any given body such as X, is greatest when the body lies on the horizon of the observer. This maximum value is called horizontal parallax. Parallax-in-altitude varies as the cosine of the altitude, being zero when the altitude is 90°. It should also be clear from Fig. 5 that + +120 +A HISTORY OF NAUTICAL ASTRONOMY + +parallax-in-altitude varies inversely as the distance of the body. That for body Y, which has the same apparent place as that of body X, is equal to that of X, but is less. + +For the fixed stars, on account of their great distances compared with the Earth's radius, parallax-in-altitude is infinitesimally small, and is regarded as being zero. For the Sun the value of horizontal parallax varies during the year, being greatest when the distance between the Earth and Sun is least, but it is in never more than about 9° of arc. In practice it is generally ignored when correcting the Sun's altitude, although for clearing a Sun lunar distance the parallax was considered to be a correction of some small importance. + +The Moon's parallax-in-altitude is a correction which is of great importance. The horizontal parallax of the Moon is greatest when the Moon is in perigee and least when she is at apogee, the corresponding values being 62° and 53° respectively. + +In the early British Nautical Almanac, the Moon's horizontal parallax was given as 62° and 53° respectively. The value of horizontal parallax given in the almanac is described as equatorial horizontal parallax, for it is the value applicable to an observer located on the equator. For an observer at any latitude other than the equator, the horizontal parallax of the Moon is less than the equatorial horizontal parallax. This is so on account of the spherical shape of the Earth. The ratio between these two values is tangent in the ratio between the Earth's radius and the distance between the Earth and the centre of the observed object. The Earth's equatorial radius being maximum results in the equatorial horizontal parallax being maximum. The tabulated value of the Moon's horizontal parallax must be corrected by an amount known as the reduction. The reduction of the Moon's horizontal parallax for the figure of the Earth* is found from a table included in most modern collections. The reduction is never more than about 0-3° of arc. + +Parallax is often regarded as being an error due to observing + +\* The term figure of the Earth is a mathematical expression for the ellipsoidal shape of the Earth. This is given in terms of a quantity called compression (c), where $c = m - b/a$ in which $a$ and $b$ are the Earth's equatorial and polar radii respectively. The compression is approximately 1/300, this small fraction indicating that for most practical purposes, the Earth's shape may be taken to be that of a perfect sphere. + +THE ALTITUDE CORRECTIONS 121 + +from the wrong position. The term *ocular parallax* applies to the error due to not placing the eye in the exact position when observing with a cross-staff. Ocular parallax is discussed in Chapter 11. + +As far back as the beginning of the 19th century correction tables for the Moon's semi-diameter and horizontal parallax were combined. Mendoza del Rio included such a table in his nautical tables which were published in 1802. + +Fig. 6 serves to demonstrate the relationship between Moon's horizontal parallax and Moon's semi-diameter. + +A diagram showing the relationship between Moon's H.P., S.D., and M. The Earth's centre is at O, the Moon's centre is at M, and the Sun's centre is at X. The Earth's radius is 4000 miles (approx.), and the Moon's radius is 1000 miles (approx.). The Moon's H.P. is AO/MX = 4/3 (approx.), and its S.D. is AO/MX = 4/3 (approx.). The Sun's distance from the Earth is not shown. +**FIGURE 6** + +In Fig. 6, O and M represent the centres of the Earth and Moon respectively. + +The Earth's radius = 4000 miles (approx.) +The Moon's radius = 1000 miles (approx.) + +Because MOX and AMO are small angles: + +Moon's H.P. = AO/MX = 4/3 (approx.) +Moon's S.D. = AO/MX = 4/3 (approx.) + +Because the Moon's horizontal parallax and her semi-diameter always bear a constant ratio one with the other, it is easy to construct a Moon's altitude correction table in which refraction, semi-diameter and parallax are combined. + +7. IRRADIATION + +When a bright object, such as the Sun, Moon or star, is viewed against a darker background such as the sky, the bright object appears to be larger than it actually is. This phenomenon is known as *irradiation*. Moreover, the sea horizon, because the sky is generally brighter than the sea, appears to be depressed on account of irradiation. The resulting effect is that altitudes of the + +122 +**A HISTORY OF NAUTICAL ASTRONOMY** + +Sun's lower limb are not materially affected by irradiation, because the apparent lowering of the lower limb tends to neutralize the apparent depression of the sea horizon. For upper limb observations, however, the two effects combine to produce an error, the magnitude of which depends upon the relative brightnesses of Sun and sky, and sky and sea. It is only within recent times that an irradiation correction has been made available to seamen. + +**8. PERSONAL ERROR OR EQUATION** + +The timing of an event such as the instant when a star, or the Sun's or Moon's limb makes exact contact with the horizon, is affected by the temperament and nervous and physical condition of the observer. Any error due to this cause is called **personal error** or **equation**. + +Personal error varies not only between observers, but may vary for different observations made by the same observer. + +At the present time little attention is given by practical navigators to the question of personal error. During the last century, however, when great accuracy of observed lunar distances was the aim of all keen navigators, personal equation in nautical astronomy was regarded as a matter of great moment. + +CHAPTER V + +Methods of finding latitude + +I. INTRODUCTORY + +Because the Earth spins she possesses, in common with all spinning bodies, the property of gyroscopic inertia. The gyroscopic inertia of a spinning body is related to the tendency it has to maintain its plane and axis of spin. + +Because of the Earth's gyroscopic inertia, the celestial poles—which are the projections of the Earth's poles from the Earth's centre on to the celestial sphere—tend to be fixed points on the celestial sphere. The celestial poles do not, in fact, remain fixed, because the rotating Earth is itself upon a rotating axis, a couple which is formed between the Sun and Moon and the Earth. This couple results in a phenomenon known as the precession of the equinoxes. The effect of precession is for each of the celestial poles to describe an approximately circular path centred at one of the extremities of the axis of the ecliptic.* The radius of each of these circular paths is equal to the maximum declination of the Sun: that is, about 23°. The movement of the celestial pole around the axis of the ecliptic is called precession, denoted by Right Ascension of every fixed celestial point changing with time. + +The precession of the equinoxes, which was discovered by the renowned Hipparchus, is an extremely slow motion amounting to about 50" per year. It takes about 26,000 years for the equinoxes to swing through 360° of the ecliptic. It is because of the slowness of the precession of the equinoxes that, for most practical purposes, the celestial poles are regarded as being fixed points in space. + +The latitude of a place is its angular distance north or south of the equator. It is equivalent to an arc of the meridian between the equator and the parallel of latitude through the place. All places which have the same latitude lie on a small circle which is parallel + +* The ecliptic is the celestial great circle, co-planar with the Earth's orbit, traced out by the Sun during his apparent annual orbit in the celestial sphere. + +124 +A HISTORY OF NAUTICAL ASTRONOMY + +(a) shows a celestial sphere with a meridian circle drawn on it. The meridian circle is labeled with the letters P, Q, N, S, W, E, Z, Q, and n. The letter P is at the top left of the circle, Q is at the top right, N is at the bottom left, S is at the bottom right, W is at the bottom center, E is at the top center, Z is at the bottom center, Q is at the top center, and n is at the bottom center. + +(a) + +(b) shows a celestial sphere with a meridian circle drawn on it. The meridian circle is labeled with the letters W, E, S, Q, and p. The letter W is at the bottom left of the circle, E is at the top right, S is at the bottom right, Q is at the top center, and p is at the bottom center. + +(b) + +FIGURE I + +to the plane of the equator. These circles are known as parallels of latitude. + +In sketch (a) of Fig. 1, the celestial sphere is projected on to the plane of the celestial meridian of an observer, the larger circle representing the observer's celestial meridian. + +METHODS OF FINDING LATITUDE + +In sketch (b), the celestial sphere is projected on to the plane of the celestial horizon of the observer. The larger circle, in this sketch, represents the celestial horizon. The smaller circle in both sketches represents the horizon. + +q represents the observer and Z his zenith; p represents the Earth's pole and P the celestial pole; N, E, S and W are the projections of the cardinal points of the horizon; wqe represents the equator; and WQE is the projection of the equinoctial. + +Observer's latitude = arc qo = arc QZ +Altitude of celestial pole = arc NP + +Now +arc PQ = arc NZ = 90° + +and +arc QZ = (90 - PZ) + +Also, +arc NP = (90 - PZ) + +Therefore: +arc QZ = arc NP + +or +Latitude of observer = altitude of celestial pole + +During the course of a day the slow spin of the Earth towards the east results in an apparent revolution of the celestial sphere towards the west. As a consequence of this, the celestial objects tend to describe circular paths which are parallel to the equinocial, and which are centred on the axis of the equinoctial. These circular paths are known as diurnal circles. + +The nearer is a star, or other celestial object, to either celestial pole, that is to say, the greater is the declination of a celestial object, the greater is the proportion of its diurnal circle above the celestial horizon of an observer whose zenith lies in the same celestial hemisphere as that in which the object is located. At any place in the northern hemisphere celestial bodies which have south declination will rise at about 6 o'clock each morning and set at about 6 o'clock each evening, having passed through exactly twelve hours each day. Celestial objects which have south declination are, correspondingly, above the horizon for less than twelve hours each day. At any place on the equator, all celestial objects, regardless of the name or magnitude of their declinations, are above and below the horizon for exactly twelve hours each day. + +A celestial object whose diurnal circle lies wholly above the celestial horizon of an observer is said to be *circumpolar* for the + +126 +A HISTORY OF NAUTICAL ASTRONOMY + +observer's latitude. It should be evident that at the equator no celestial object is circumpolar. Likewise, at the North Pole, all celestial bodies which have north declination, and at the South Pole all celestial bodies which have south declination, are circumpolar. + +For a celestial body to be circumpolar at the place of an observer, the polar distance, that is the complement of the body's declination, will have to be less than the observer's latitude. This fact should be clear from Fig. 2. + +A diagram showing a celestial sphere with a line from the center to the top labeled N (North), another line from the center to the bottom labeled S (South), and two lines from the center to points P, Q, Y, Z, E, W on the surface of the sphere. Lines connecting these points form a triangle with sides NP, PQ, and PY. The angle between NP and PQ is labeled as arc NP. The angle between PQ and PY is labeled as arc PY. The angle between NP and PY is labeled as arc QY. + +FIGURE 2 + +Fig. 2 is a projection of the celestial sphere on to the celestial horizon of an observer in the northern hemisphere, whose zenith is projected at Z. The observer's latitude is equal to arc NP. All celestial objects whose north declinations exceed arc QY lie within the parallel of declination (or diurnal circle) centred at P and having a radius equal to arc PY. + +Now, +$$\text{arc PY} = 90 - \text{arc QY}$$ +and +$$\text{arc PY} = \text{arc NP}$$ + +**METHODS OF FINDING LATITUDE** + +Therefore, the limiting declination for an object to be circum-polar in the latitude of the observer is arc QY. In other words, for an object to be circumpolar its polar distance (complement of declination) must equal the latitude of the observer, and the names of the declination and the latitude must be the same. + +An important adjunct to the needs of the ocean navigator is a chart on which his ports of departure and destination are plotted, and on which the position of his ship, at any time during the voyage, may be plotted so that her progress may be checked and her course, if necessary, rectified. + +The early navigator was relied upon knowledge of latitude and estimation of courses and distances made good in order to deduce the ship's longitude. This method of navigation by dead reckoning (D.R.), or by "guessimation" as it has so aptly been described, gave unreliable results because of the difficulty of making accurate estimates of courses and distances made good. + +The early navigators, and the ancient geographers as well, were able to find the latitude of places by astronomical observations; but the practical determination of longitude, apart from that by the method of D.R. navigation, was to remain a mystery until comparatively recent times. + +The chart of the early navigator, on which he plotted his D.R. positions and observed latitudes was, like all maps, a representa-tion of part of the spherical Earth's surface on a plane surface, requiring the use of a projection. The earliest forms of map pro-jections were geometrical and simple in principle, date from the 4th century BC. + +During the 4th century BC the activities of land- and sea-traders led to a great expansion of knowledge of the Earth's sur-face and of the distribution and positions of the major population centres and trading stations. We have, in Chapter 1, referred to Eratosthenes, and noted his attempt to reduce these data into a realistic map. He did this by constructing a network of parallels of latitude and meridians. The principal parallel of latti-tude used by Eratosthenes—a parallel which passes through Gib-raltar and Rhodes—was first suggested by Diocearchus, who flourished during the closing decades of the 4th century BC. Eratosthenes established a meridian line on his world map, this passing through Rhodes and Syene. + +The terms **latitude** and **longitude**, which mean respectively + +128 +**A HISTORY OF NAUTICAL ASTRONOMY** + +breadth and length, sprang from the notion that the habitable part of the Earth is broader in the east–west direction than it is in the north–south direction. This is certainly true of the Eastern Mediterranean region in which the ancient Greek and earlier civilized societies flourished. + +The latitude of a place may be ascertained without reference to the latitude of any other place. In order to find the latitude of a place in this way, recourse must be made to astronomical observation. + +When it became known that the Earth's shape is spherical, and that the Sun's annual path across the celestial concave is inclined at an angle of approximately $23^\circ 27'$ to the plane of the Earth's rotation, it became possible to compare the latitudes of places from the lengths of their midday shadows cast by gnomons set up at the places. + +Essentially, a gnomon provides the means whereby the passage of time during daylight hours may be measured. Because of the changing declination of the Sun, due to the obliquity of the ecliptic, the gnomon may also be used for measuring the march of the seasons. + +Pytheas of Marsala, who flourished during the 4th century BC, bestowed considerable attention upon the measurement of latitude by means of the gnomon. He found the latitude of his native city with great accuracy; and during his famous Atlantic voyage the most northerly latitude reached, according to Pytheas, was located on what would describe as the parallel of latitude of $60^\circ$ N. or the Arctic Circle. + +The term **Arctic Circle** has not always had its present-day meaning. To the ancient astronomers and geographers the term was used to describe a small circle on the celestial sphere which enclosed all the circumpolar stars for any particular latitude. + +To distinguish between **temperate climates** was used to describe zones bounded by parallels of latitude. + +It appears that Hipparchus was the first to suggest that parallels of latitude on world maps should be projected at regular intervals. On the map of Eratosthenes' the parallels of latitude were not plotted systematically and were placed at irregular intervals. Hipparchus introduced the idea of fixing the interval between successive plotted parallels of latitude with reference to the longest—or *solstitial*—day. + +METHOIDS OF FINDING LATITUDE 139 + +Although Hipparchus referred to the latitudinal zones as *climates*, later writers, including Strabo, referred to the boundaries of the zones as *climates*. On this basis *climates* are parallels of latitude. + +The Earth's surface was divided by ancient geographers into twenty-four climatic zones in each of the northern and southern hemispheres. The lengths of the solstitial days on the two boundary parallels of each climatic zone differed by half an hour. The ancient geographers knew no more than nine climatic zones, all of which were named after the principal cities situated within them. The third climate from the equator, for example, was named after Alexandria, the metropolitan city of Egypt; the fourth was named after Rhodes; and the fifth after Rome; and so on. + +We have shown that the latitude of an observer is equivalent to the altitude of the celestial pole; and we have explained how the diurnal circles traced out by the celestial objects, due to the Earth's rotation, are circles centred at the celestial pole. Now the stars which, for purposes of navigation, are regarded as being at an infinite distance from the Earth, maintain their declinations over relatively long periods of time, so that it is an easy matter to find latitude from an observation of a star when it culminates. + +When a star is observed at its greatest altitude, it is at meridian passage. This is so because the greatest daily altitude of a star at any place is attained when it bears due north or south, that is when it is at meridian passage. + +We have noted, in Chapter III, that the Arab navigators of the Red Sea and the Indian Ocean have used the *kamal* for finding latitude from very early times. It is not without significance that the Red Sea, in contrast to the Mediterranean Sea, has great latitudinal variations. We have also noted that some seamen that the meridian altitudes of the visible stars changed appreciably during a voyage extending the full length of the Red Sea. Moreover, it occurred to these seamen that the meridian altitudes of selected stars gave a guide to the positions of certain harbours located on the Red Sea coastlands. Star meridian altitudes were used, and still are used, by the Arab traders of the Red Sea, and provided, as they still do provide, valuable aids to their navigations. + +130 +A HISTORY OF NAUTICAL ASTRONOMY + +There is every reason to believe that the Polynesian seamen of the past navigated their craft for hundreds of miles between the widely-spaced islands of Polynesia by using star observations. These intrepid voyagers used the important fact that the latitude of an observer is equivalent to the declination of a star which culminates at his zenith; or, as D. H. Sadler puts it in a tribute to the late Harold Gatty, "a star in the zenith is a heavenly beacon lighting up the latitude circle which revolves beneath it." This assuredly, 'Mr Sadler goes on to say, "is the simplest principle of all possible nautical rules". The stars in the vicinity of the zenith provided the Polynesian navigators with all necessary astronomical information which enabled them to set their courses and make their desired landfalls. + +During the period of the discovery of the Atlantic coastlands of Africa by the Portuguese seamen under the sponsorship of Prince Henry the Navigator, it was customary to make astronomical observations ashore for the purpose of finding distance south of Labon. Instrumental aid for performing this task on a lively ship at sea was, at the time, not available to the navigator. + +2. LATITUDE BY THE POLE STAR + +An early method for finding latitude was afforded by the Pole Star—the Stella Maris of the early seamen. The Pole Star, on account of its great altitude, appears to be stationary in the course of a day, a tiny circle of angular radius which is equal to the small polar distance of the star and which is centred at the celestial pole. On two occasions each day the altitude of the Pole Star is equal to an observer's latitude; but never does its altitude differ by more than a small angle from the latitude. Henry's navigators were taught to observe the altitude of the Pole Star at the place from which they departed for their voyages of discovery, at a time during the night when they believed that it would be visible. The Pole Star, Guards of the Leeward Bear, was in a particular position relative to the Pole Star. On sailing southwards the altitude of the celestial pole decreases in proportion to the change in latitude. In order to ascertain how far south of the departure point he had sailed, the navigator would observe the altitude of the Pole Star at a time when Kochab occupied the same position relative to the Pole Star as it had when the departure-position observation was made. The difference between the two altitudes in degrees, when multiplied + +METHODS OF FINDING LATITUDE 131 + +by the number of leagues in a degree—164 according to the Portuguese reckoning—gave the distance in leagues between the present and departure positions. The earliest observations were made ashore using the seaman's quadrant and a plumb-line, instruments which have been described in Chapter III. + +After the West African coast and the Atlantic islands of the Azores and Madeiras had been discovered, the Portuguese made a practice of marking the arc of the quadrant at points correspon-ding to the latitude of places known for stations on certain islands and important coastal stations. From this time when this practice began, navigators employed the method of 'running down the latitude.' By this is meant sailing southwards or northwards until the parallel of latitude of the destination is reached, and then sailing along the parallel, that is due east or west, until the required longitudinal is made. + +The charts used by the early Portuguese navigators did not have a scale of latitude and the terms *altura* (altitude or height) and 'running down the altitude' were used in reference to the latitude of a particular place. When seamen became accustomed to using the degree as an angular unit it became convenient to provide a scale of latitude on the maritime chart. When this stage of development had been reached, seamen began to use a *Rule, or Regiment, of the North Star*, by means of which they could traverse any part of the globe with accuracy. + +To use the North Star for finding latitude the seaman was required to memorize, or have access to, the corrections necessary to apply to the altitude of the Pole Star, according to the position of *Kochab* relative to the Pole Star itself. + +When the Pole Star is on the celestial meridian of an observer above the celestial pole, the latitude of the observer is equal to the altitude of the Pole Star. When it is on the meridian below the pole, the latitude is equal to half its value minus the polar distance. When the local hour angle* of*the Pole Star is more than six hours and less than eighteen hours its altitude is less than the latitude; and when the hour angle is more than eighteen hours or less than six hours its altitude is greater than the latitude. The correction to be applied to the altitude of the Pole Star in order to obtain the latitude is clearly dependent upon the position of the Pole Star in its diurnal circle. This, in turn, + +* See Appendix 2. + +132 +A HISTORY OF NAUTICAL ASTRONOMY + +affects the position of Kochab relative to the position of the Pole Star. + +Martin Cortes, in his Breve Compendio de la Sphera, based his Rule of the North Star on the erroneous assumption that the polar distance of the Pole Star was 4° 9'. This value was too great, the + +A diagram showing a circular plate with various markings and indicators. The top of the circle has a pointer labeled "N" pointing north. Below this, there is a smaller circle with a pointer labeled "S" pointing south. Inside these circles, there are additional pointers and markings. The bottom of the circle has a pointer labeled "E" pointing east. There is also a pointer labeled "W" pointing west. The center of the circle has a small compass rose with cardinal directions (N, E, S, W) and additional markings. + +FIGURE 3 + +correct value at the time (1556) being 3° 30'. Cortes devised a simple instrument by means of which the correction to apply to the altitude of the Pole Star to find the latitude could readily be found. The instrument is similar to a nocturnal, and consists of a circular plate having a hole in the middle through which the Pole Star could be sighted. Pivotated at the centre of the plate is a movable indicator in the shape of a hunting horn—"The Horn" being the equivalent of the Portuguese name for the Lesser Bear. The instrument is set up to be level with the horizon, and the Pole Star sighted. The hood indicator was then turned to a position corresponding to the positions of the stars of the Lesser Bear, where- + +METHODS OF FINDING LATITUDE + +upon the mouthpiece end of the horn indicated the correction to apply to the altitude of the Pole Star to find the latitude of the observer. Fig. 3 illustrates the instrument. + +In the commonly-used Rules of the North Star, the position of the foremost guard was described in terms of a compass direction from the Pole Star. Edward Wright devoted Chapter 12 of his *Certaine Errors in Navigation Detected and Corrected*—first published in 1599—to a discussion on the position of the North Star and the Guards. This chapter runs as follows: + +'Among the 48 constellations which the astronomers place in the heavens the nearest unto the Pole of the World is that which they call the lesser Beare, and the Mariner's Bozia or ye Horne, regionally called by some, The Great Bear; which Constellation consisteth of 7 Stars, which are placed after this manner: and of these Stars the three greatest marked A B C doe serve especially for our purpose. And so A is called the North Star; B the foremost Guard, and C the Guard behinde; because by that force of the motion of the first moveable Heaven the star B goeth alwaies before and the starre C behinde. + +'Every of these Starres, as well as al others in the Heavens besides, describe their circles round the Pole with equal motion of the first or highest moveable Heaven; in which motion; sometimes the two Starres A B are just of one height above the Horizon; and when they are said to be E and W one from another. Sometimes they are in a perpendicular line to the Horizon according to our sight; and then they are said to be N and S. And sometimes also the two Guards B C are E and W one from another, and then the foremost Guard beareth from the North Starre NE and SW. And when these two Guards are at right angles together above both, the foremost Guard beareth from the North Starre SE and NW. In so much that from the four Positions do arise eight + +A diagram illustrating a compass direction from a Pole Star to a constellation marked A, B, C, D. +10 + +134 +A HISTORY OF NAUTICAL ASTRONOMY + +Rules for the eight Rhumbes, wherein the foremost Guard may stand, being considered in respect of the North Starre. And so presupposing that the North Starre is distant from the Pole three degrees and a half (according to the opinion of some mariners, who love numbers that have no fractions) sometimes the North Starre shall bee as high as the Pole it selfe, some-times three degrees and one half lower or higher than the Pole, and sometimes three degrees, and sometimes halfe a degree. + +It will be noticed that Wright used 3° 30' as the polar distance of the Pole Star. The same value was used by William Bourne in his Regiment for the Sea, a work which was designed to supplement the work of Cortes, and which was first published in 1577. Other writers, including George Adams, also used this value. + +The declination and Right Ascension of the Pole Star change comparatively rapidly, on account of the precession of the equinoxes. The use of an out-of-date polar distance in the Rule of the North Star, therefore, resulted in an error in the latitude obtained from an observed altitude of the Pole Star. Edward Wright included as one of the errors of navigation, that caused by using a false Rule of the North Star. Wright advised seamen to use a polar distance of 3° 30', but ruled that when they had used 2° 9' Seller, whose Practical Navigations appeared in the late 17th century, used 2° 9' in his 'Rule of the North Starre'. + +Chapter 15 of Wright's book is entitled 'Other Things to be noted in Observing the Height of the Pole'. It reads thus: + +'Next unto the Constellation of the Horne there is a Starre which is called by the Spaniards et Guion signified before [in the accompanying diagram] by the letter D, which standeth east and west from the North Starre giveth you to understand that it and the North Starre, and the very Pole, are east and west. And so taking the height of the North Starre when it is thus situat in regard of the Guion, without making any other account, you have the just height of the Pole and the Distance from the Equinoctial.' + +Accurate Pole Star tables of the type published in seamen's almanacs, were first published in the British Nautical Almanac for + +METHODS OF FINDING LATITUDE + +The year 1834. These tables were calculated using a formula supplied in 1822 by the astronomer Littrow. + +The formula used for finding the latitude from an observation of the altitude of the Pole Star is: + +$$ l = a - p \cos h + \sin 1^\circ (p \sin h)^2 \tan a $$ + +where $l$ is latitude, $a$ is altitude, $h$ is local hour angle and $p$ is polar distance of Polaris. + +The local hour angle $h$, is found by combining the Local Sidereal Time of the observation and the Right Ascension of the star. That is: + +$$ h = \text{L.S.T.} - R . A. $$ + +Both the polar distance $p$ and the Right Ascension of Polaris change rapidly because of precession and nutation, so that average values of $p$ and R.A. are used in compiling Pole Star tables. + +For the year 1834, the average polar distance of Polaris was 61° with a maximum deviation from this value of 0.9°. The corresponding values for the Right Ascension were 01 hrs. 44 mins. and 2 seconds. + +The First Correction, giving values for $-p \cos h$, were tabulated in Table 1, and the Second Correction, giving values for $\pm \sin 1^\circ (p \sin h)^2 \tan a$, were tabulated in Table 2, assuming average values of $p$ and R.A. (see Table 3). Errors in Table 3, allowed for the differences between the true and assumed values of $p$ and R.A. + +The earliest Pole Star tables gave only the First Correction, that is $-p \cos h$. This was sufficient for the early navigators whose instruments were not accurate enough to measure altitudes to a very high degree of accuracy. + +The Portuguese mathematician Núñez pointed out, in the early 19th century, that errors in the Rule of the North Star results from the assumption that the area bounded by the diurnal circle of the Pole Star is a plane circle. The error, however, amounts to no more than a couple of arc; and it is interesting to note that, as late as 1824, Edward Riddle, in his *Treatise on Navigation*, stated that the distance of the Pole Star from the celestial pole is so small that the circle which it describes may, without causing material error, be considered as a plane. His solution to the Pole Star problem involves the First Correction only. + + + + + + + + + + + + + + +
Table 1First Correction
Table 2Second Correction
Table 3Errors in assumed values
+ +A table showing three columns: First Correction, Second Correction, and Errors in assumed values. + +136 +A HISTORY OF NAUTICAL ASTRONOMY + +The famous Elizabethan mathematician Thomas Harriot used a Pole Star table, which was not published for general use, in which he incorporated a correction for the effect of latitude. This correction corresponds to the Second Correction, and it will be noted from the formula given above, that the Second Correction varies as the tangent of the altitude (or latitude). + +We are informed by Staff Commander W. R. Martin in his Navigation and Nautical Astronomy, which was first published in 1888, that good Pole Star tables were published in 1810 by Mr. J. W. Norie of the East India Company's Service. + +The Regiment of the North Star formed part of the stock-in-trade of ocean navigators of the 15th-17th centuries. It is a noteworthy fact that the navigation manuals of the 18th century and the early 19th century give no instructions for finding latitude from an observation of the Pole Star. Robertson, whose Elements of Navigation was probably the most well-used navigation manual of the 18th century, does not mention the Pole Star. Neither does Mackay in his well-known navigation manuals; nor does J. W. Norie in the first edition of his Epitome. It is not without significance that the period when the Pole Star seems to have been out of favour was one during which the double altitude problem became popular. + +It is interesting to note that Carl Zeiss invented an instrument in 1938 for determining longitude at sea by means of an observation of Polaris. The Pol Formahr, as the instrument is called is, in some respects, similar to the ancient nocturnal. It employs a small pane of glass on which are engraved two marks corresponding to a and p Ursa Minoris ('The Guards of the Lesser Bear'). The instrument is held pointing to the north celestial pole and then adjusted so that the two marks correspond with the appropriate stars, upon whom the latitude and Local Sidereal Time may be read off this instrument. + +The Regiment of the North Star and Pole Star tables are applicable to the northern hemisphere only. So that after the Portuguese navigators, in their voyaging southwards along the West African coast, had crossed the equator, an alternative method for finding latitude became a necessity. As early as 1646, Prince Henry's royal father, King John of Portugal, was instrumental in forming a commission to investigate the problems of positioning especially in the southern hemisphere. From the investigations of + +METHODS OF FINDING LATITUDE 137 + +the astronomers and mathematicians who formed this commission, the method for finding latitude by meridian altitude of the Sun was introduced to seamen. + +3. LATITUDE BY MERIDIAN ALTITUDE OF THE SUN +Finding latitude from an observation of the Sun on the meridian demands knowledge of the Sun's declination. Solar Tables were, therefore, prepared for sea use. These tables contained the Sun's declination for every day of the year. Henry's navigators were taught how to apply the Sun's declination to the midday height + +Figure 4 +of the Sun, in order to obtain the latitude, or height of the pole, as it was generally called. + +The latitude of an observer, which is equal to the altitude of the celestial pole at the observer's position, is equal to the angular distance between his zenith and the equinoctial measured in the plane of the observer's celestial meridian. It follows, therefore, that the latitude of an observer is equal to a combination of the declination of a heavenly body and its meridian zenith distance. Fig. 4 illustrates this. + +Fig. 4 is a projection of the celestial sphere on to the plane of the + +138 +A HISTORY OF NAUTICAL ASTRONOMY + +celestial horizon of an observer whose zenith is projected at Z, N, E, S and W are the projections of the cardinal points of the horizon. P is the projection of the elevated celestial pole, and WQE is that of the equinoctial. X₁ represents a celestial body which is culminating north of the observer's zenith; X₂ represents a celestial body which is culminating south of the observer's zenith but north of the equinoctial; X₃ represents a celestial body which is culminating south of the equinoctial. + +Latitude of observer = NZ = ZQ + +Now, + +also, +ZQ = QX₁ - ZX₁ + +also, +ZQ = QX₂ + ZX₂ + +In general: +ZQ = ZX₃ - QX₃ + +or: +Latitude = Meridian Zenith Distance ± Declination + +The Rules, or Regiment, of the Sun were more complex than those of the North Star. The navigator, when using them, had to consider whether the declination of the Sun was north or south; and whether the Sun crossed his meridian altitude north or south of his zenith. In addition to the general cases—three for each hemisphere, making six—there were special cases applying when the Sun's declination was 0°, and when the Sun had a meridian altitude of 90°, to make confusion doubly sure for the non-navigators. + +The Rules of the Sun required knowledge of the meridian altitude, the declination and the 'shadow'. Chapter 3 of Wright's book is entitled 'Of the Shadowes'. He tells us: + +'The shadows being compared with the Sun, may be of three sorts: for at high Noon the shadow falleth either towards that part of ye Worlde to which the Sunne declineth, or towards the contra part, or else we make no shadow at all. The first and second sort are, when the height of the Sunne is less than 90°, and the third is when it is just 90°... The rule of the Shadowes + +METHODS OF FINDING LATITUDE +139 + +is that wee looke well to the lower fane of the Astrolabe when we are taking the height of the Sunne at Noone. . . . + +The earliest Rules of the Sun used meridian altitude instead of meridian zenith distance for finding latitude. Typical of the meridian zenith-distance rules were those of John Seller, which appeared in his *Practical Navigation*, first published in the late 17th century. + +*Rule 1.* If the Sun comes to the meridian in the South, and have South Declination, subtract the Declination from the complement of the Meridian Altitude. The Remainder is the Latitude of the Place of Observation Northerly. But if the Declination is North, then add the Complement of the Zenith Distance from the Declination, the Remainder is the Latitude Southerly. + +*Example 1.* Admit you are at Sea, and the Sun being on the Meridian in the South is 37 degs 30 mins distance from the Zenith, and at the same time hath 12 degs 00 mins South Declination; I demand the Latitude of the Place. +*The Operation.* +\begin{align*} +\text{Complement of the Meridian Altitude} & \quad 37° 30' \\ +\text{The Sun's Declination south subtract} & \quad 12° 00' \\ +\text{The Latitude of the Place} & \quad 25° 30' North +\end{align*} + +*Rule 2.* If the Sun be upon the Meridian in the South, and hath North Declination, then add the Declination to the Zenith Distance, the Sun is the Latitude Northerly. +etc. etc.' + +The quadrant, with its plumb-line, was not a suitable instrument for measuring altitudes at sea from the deck of a heaving ship. The need for a more suitable altitude-measuring instrument resulted in the invention by John Wallis of a telescope called a *seaman's quadrant*. Wright informs us in his *Certaine Errors* . . . that the Portuguese seamen of his time marked their astrolabes 'in reverse,' so that the angle given on the arc was the zenith distance. Hence the need for alternative rules for finding latitude by meridian altitude of the Sun. + +4. **LATITUDE BY MERIDIAN ALTITUDE OF A STAR** + +The method for finding latitude by meridian altitude of the Sun + +140 +A HISTORY OF NAUTICAL ASTRONOMY + +applied equally well to the stars, provided that the navigator was provided with a table of stars' declinations. + +William Bourne, in his *Regimen of the Sea*, explained how star observations could be used to find latitude. Tables of stars' declinations were drawn up for the seaman's attention and these, in due course, made their appearance in navigation manuals. Provided that a mariner could measure the meridian altitude of one of the tabulated stars during morning or evening twilight, he had the means of finding the latitude of his ship. The quadrant and the astrolabe were quite useless for star observations, relying as they did on the sun's position for their measurements, because the normal altitude-measuring instrument for star sights. + +In the northern hemisphere, at least to the north of the tenth parallel of latitude, the Pole Star is always available for observa- +tions made during twilight so long as the sky and horizon are clear. +The North Star, therefore, was regarded by seamen up to the end of the 17th century as being superior to other stars for finding latitude. + +5. LATITUDE BY THE SOUTHERN CROSS + +The constellation of the Southern Cross was used by seamen in southern waters to find latitude. As early as 1505, the Portuguese navigators were provided with a rule for doing so. This rule was copied, its essentials, by many writers of navigation manuals. +Edward Wright's remarks on this use of the Southern Cross, or Crozier as he called it, are interesting. He states that 'the polar distance of the star at the foot of the cross (the Cocke's Foot as he called it) is being $30^{\circ}$. He pointed out that when the stars at the head and the foot of the Cross are perpendicular to the horizon, at which time, Wright informs us, they have their greatest alti- +tude, the latitude may be found by measuring the height or alti- +tude. The rules are: + +*For if the said Heigth be thirtie degrees, then wee are in the very equinoctial, and if it be more than thirtie degrees then we bee so much past the equinoctial towards the South. And if it be less than thirtie degrees, so much as it wanteth are we to the north of the equinoctial.* + +The Right Ascensions of the stars at the head and foot of the + +METHODS OF FINDING LATITUDE 141 + +Southern Cross are approximately equal to one another. This is the reason why they cross the meridian, or culminate, at about the same time. The values of the declinations of these two stars used by John Seller are 28° 45' and 34° 45' respectively. These values are those denoted by Edward Hales. The latter was found by subtracting 28° 45' and 34° 45' from the meridian altitude of the Head of the Cross and the Crow's foot, as Seller called the Foot of the Cross, respectively. The seaman was advised, in order to ascertain the exact time for observing, to hold up a thread and plummet: if the thread cuts both stars simultaneously, they are at meridian passage and suitably placed, therefore, for a latitude observation. + +Observation of the Southern Cross stars for latitude could be used north of the equator as well as in the southern hemisphere. In fact, between the parallels of about 10° and 20° north, the Pole Star and the Southern Cross could be used for this purpose. Within these parallels of latitude the navigator was, therefore, afforded a means of checking his observations. + +6. DECLINATION TABLES + +An advantage of the method for finding latitude by meridian altitude of a star over that for finding latitude by a noon-day Sun sight is that a star's declination is constant (or, at least, practically so) for both time and longitude. The Sun's declination, on the other hand, varies with the time of year and also with longitude or local time of day in different parts of the world. The use of mean declination of the Sun gave declination for noon on each day of the year for a standard—or reference—meridian that of Lisbon or London for example. Because the Sun crosses any given meridian at local noon (which differs from noon at any other meridian by an amount which is proportional to the difference of longitude between the local and standard meridians) it is necessary to apply a correction to the mean declination for each day to obtain true declination for the time of local noon. Edward Wright referred to this correction as the 'aequation of the Sunne's Declination.' It is interesting to note that he remarked that: 'They which sail in the month of June and December, need not much to make an aequation.' + +John Davis, the famous English navigator, in his *Seaman's Secrets*, explained how to correct the tabulated declination for + +142 +A HISTORY OF NAUTICAL ASTRONOMY + +longitude. Davis also repeated the method described by William Bournel for finding latitude from an observation of the Sun when it crossed the observer's celestial meridian below the pole. This method is applicable only in latitudes where the Sun is circum- polar, but it can be used to find the latitude of any point of the Arctic (and Antarctic) circle. It was a method of great importance to the polar navigators of Elizabethan times, during their searches for a northern route to the fabulous spice islands of the East. + +The table of the Sun's declination, so essential for finding lati- +tude from an observation of the Sun on the meridian, reached a stage of perfection only after Kepler had discovered the true form of the orbit of the Earth round the Sun, and that the Earth re- +volves around the Sun. Following the careful observations of the fixed stars by astronomers—notably those of the renowned Tycho Brahe who is generally esteemed as being the most skilful astro- +nomical observer of all time—tables of declinations and Right +Ascensions of the fixed stars were improved and perfected. + +The apparent motions of the Moon and the planets relative to +the background of fixed stars were also studied, and it was not until not till the 18th century, after the illustrious Newton had formul- +lated, and given to the world, his law of universal gravitation, that tables of declinations and Right Ascensions of the Moon and the visible planets became part of the stock-in-trade of the navigator. + +Martin Cortes, in his work on navigation, gave a single table of +the Sun's declination for every day of a leap year. In addition to +this, a table was given for every day of a non-leap year, out of +which the declination of the Sun for any day in a non-leap year could be found. During the 17th and 18th centuries it was the common practice of writers of navigation textbooks to provide four tables of Sun's declination: one for leap years; and a second, +third and fourth for the first, second and third years after leap +years respectively. The seaman, in order to know which of these +four tables he should use, would have to observe whether the year in question was a leap year, or first, second, or third year after leap +year. It is interesting to note the curious and ponderous instruc- +tions given by Edward Wright, pertaining to this simple problem. +'... I will set down a rule whereby you may know whether the +present year be leap year, or whether it bee the first, second or +third years after the leape year.' + +**METHODS OF FINDING LATITUDE** + +143 + +*And the rule is this, that taking from the yeeres of our Lord (which run in our common account) the number of 1600, if the remainder thereof be an even number, and halfe of the remain- +der an even number, then that yeere is leape yeere; and if the remainder be odd, and halfe of the remainder an even number, then is the second yeere after the leape yeare. But if the remainder of the yeeres number be odde we must trie the yeers next going before, to see whether the remainder thereof, and halfe the remainder be even numbers, for then the present yeere is the first yeere after the leape yeare. And if the remainder of the yeeres going before be even, and the halfe thereof odde, then is the present yeere the third yeere after the leape yeare.* + +Pedro Núñez (Nonius), the famous Portuguese mathematician, is credited with being the first to suggest providing tables of the Sun's declination for a four-year period. + +7. THE DOUBLE-ALTITUDE PROBLEM + +Núñez, in his book *De Arte et Ratione Navigandi*, proposed a method for finding the latitude from two observations of the Sun's altitude together with the intermediate azimuth. Núñez's solution to the double-altitude problem (as it later became known) was published in 1537. It involved the use of a globe, as did the solu- +tion to the same problem put forward by our countryman Richard Hues in his *Tractatus de Globis*, published in 1594. An English version of Hues's book on the globes was published in 1618 under that name by John Dee and M. A. M. Christianus van Oosten. + +At the time when Hues published his treatise on the use of +the globes mechanical watches, which kept time with a fair accuracy, +had become available. Hues's solution to the double-altitude prob- +lem—a problem which was to engage the attention of numerous +mathematicians and astronomers in later centuries—required two +altitudes of the Sun together with the elapsed time between +the observations. + +John Davis, in his *Seaman's Secrets*, repeated Hues's method. +The method consisted in drawing two circles on the celestial +globe. The first is centred at the Sun's position on the ecliptic, its +radius being equal to the complement of the first observed alti- +tude. The second circle is centred at the intersection of the parallel +of the Sun's declination and the hour circle which made an angle + +144 +A HISTORY OF NAUTICAL ASTRONOMY + +with the celestial meridian at the time of the first observation equal to the elapsed time between the two observations. The radius of the second circle is to be equal to the complement of the Sun's altitude at the time of the second observation. The declina- +tion of each star is to be measured by its altitude. The distance is equal to the observer's latitude. Which of the two points of intersection is equal to the observer's latitude is obvious from knowledge of the latitude by account. + +Blundeville, in his Exercises . . . which was first published in 1594, described a method for finding latitude from simultaneous altitudes of two stars, one on each side of the meridian. Master Blundeville's method involved adjusting the globe so that the altitude of one star would coincide with the observed alti- +tudes, whereupon the latitude could be measured on the globe: the altitude of the celestial pole, which is equal to the observer's latitude, indicating this. + +The use of globes for navigational purposes was short-lived. +Not only were globes cumbersome and expensive, but the degree of accuracy of the problems solved by their use was crude— even for stars near the horizon. With the development of mathematical methods for the seaman's use, a new breed of navigator developed. Methods of computation for the several astronomical problems related to position-finding at sea were devised and these were adapted for the use of seamen, usually in the form of complex rules. The navigator, having access to the rules from his manual or his memory, was able, therefore, to solve the relatively complex astronomical problems. It was necessary, however, for him to understand the principles involved. Seaman, it appears, at no time in the history of navigation have taken kindly to the mathematical arts—the black Arts, as they were sometimes called. The applica- +tion of mathematical principles of position-finding at sea were forced upon the non-mathematical mariner; and it is only within comparatively recent times that seamen have had any little understanding of the principles of the processes involved in nautical astronomy. By and large, seamen at all times have been content to "proceed according to the Rules." They are concerned more with finding the answer than with knowing the underlying principles of the methods they use for finding it. + +In the preface to his well-known Elements of Navigation, which was the principal English navigation manual of the 18th century, + +**METHODS OF FINDING LATITUDE** + +John Robertson, F.R.S., who wrote his very fine book for the use of 'the children of the Royal Mathematical School,' drew attention to the fact that: + +"The common treatises of navigation, which on account of their small bulk and easy price, are vended among the British mariners, seem not to be written with an intention to excite in their readers a desire to pursue the sciences, farther than they are hinted at by the title. The great majority of seamen in general had so little mathematical knowledge; for the person who could keep a true journal, formed on the most easy occurrences, has been reckoned a good artist; but whenever those occurrences have not happened, the journalist has been at a loss, and unable to find the ship's place with any tolerable degree of precision; and such accidents have probably contributed to the distress which many ships crews have experienced, and which a little more knowledge among them might have prevented, or at least have lessened." + +The problem of finding latitude from two observations of the Sun or star, using mathematical devices other than a globe, received the attention of many scientific men and writers on navigation during the 18th and 19th centuries. The so-called double-altitude principle was first suggested by John Flamsteed (1646-1719), whose various solutions were furnished. The name double altitude was given to a method of finding latitude. In modern navigational practice the term applies to the method of fixing the ship, that is to say, finding her latitude and longitude from astronomical observations. + +The meritorious Raper, author of the 19th-century classic *The Practice of Navigation*, objected to the name double altitude. "...it is defective," he wrote, "inasmuch as the word double means twice the same." He suggested the use of the term combined altitudes. + +The calculations involved in the solution to the double-altitude problem are more complex than those for finding latitude from an observation of the meridian altitude of a heavenly body. Practical seamen never favoured the double-altitude problem until relatively recent times when the method was perfected for finding both latitude and longitude. + +Given two altitudes of a heavenly body and the interval of time + +145 + +146 +**A HISTORY OF NAUTICAL ASTRONOMY** + +between the observations, the latitude may be computed by the so-called direct method of double altitude. In using the direct method, the latitude is computed by the rigorous process of spherical trigonometrical calculation. Many writers, including the famous French astronomer Lalande, whose excellent writings on navigation are well known, preferred the rigorous process and advocated its use in preference to any of the many indirect methods that were devised. + +A diagram showing a circle with points labeled N, P, W, Z, Y, X, C, D, B, E, Q. The lines connecting these points form a triangle with vertices at X, C, and D. +**FIGURE 5** + +The earliest double-altitude problems were related to the Sun. Nicholas Fabio Duillier F.R.S. is credited with being the first to devise a mathematical method for solving the problem. In 1728 Duillier published a pamphlet entitled *Navigation Improved*. He discussed the double-altitude problem in some detail and, in his solution, he took into account the movement of the ship during the interval between the times of observation. Duillier’s method required a considerable amount of tedious computation and, although it was improved by several writers and teachers of navigation, it was not generally considered as providing a practical solution to the problem. The method is explained with reference to Fig. 5. + +**METHODS OF FINDING LATITUDE** + +Fig. 5 is a projection of the celestial sphere on to the plane of the celestial horizon of an observer whose zenith is projected at Z. N, E, S and W, are the projections of the cardinal points of the horizon. WQE is the projection of the equinoctial and P is that of the elevated celestial pole. X and Y are the projections of the object at the times of the first and second observations respectively. The arc XY is a great-circle arc through X and Y. Arcs PC and FD represent the hour circles through the body at the times of the observations. + +The following arcs and angles are known: +PX = polar distance of observed object +PY = polar distance of observed object +ZX = first zenith distance +ZY = second zenith distance +XPY = elapsed time + +The successive steps in the computation in order to find PZ, which is the complement of the observer's latitude, are as follows: + +1. In triangle PXY Using PX, PY and PZ, find XY. +2. In triangle PXY Using PX, PY and XY, find PXY. +3. In triangle ZXY Using ZX, ZY and XY, find ZXY. +4. From PXY and ZXY find PZX. +5. In triangle PZX Using PX, ZX and PZX, find PZ. +6. Latitude = 90° - PZ. + +Duillier's method, which is described in Leadbetter's *Astro-nomy of the Satellites Jupiter*, published in 1729, although involving tedious calculations, provided a direct and unambiguous solution. Moreover, only two common rules of spherical trigonometry were required, these being the rules for: + +1. Finding an angle given the three sides. +2. Finding a side, given the opposite angle and the other two sides. + +John Robertson, in his *Elements of Navigation*, describes Duillier's method—which he calls Facio's method—but states that his solution requires too many trigonometrical operations to make it practicable. + +A solution to the double-altitude problem was published by Richard Graham in 1734. Graham's solution is ingenious, despite + +148 + +A HISTORY OF NAUTICAL ASTRONOMY + +the remark made by the author of the article on navigation in the ninth edition of the Encyclopaedia Britannica to the effect that it was published in the Philosophical Transactions 'with much boasting.' Graham's solution was an instrumental operation requiring the use of a beam compass which could be fitted to the meridian ring of the celestial globe. The beam compass, which was to be fitted so that it could be made to slide along the meridian ring, was used to describe arcs of circles of radii equal to the Sun's zenith distance. In this way, the arc subtended at the centre of the circle by the Sun's position was found and that method was capable of giving the latitude within a few minutes of arc of the truth and, according to John Robertson, '. . . with ease and expedition.' + +In 1740 Cornelius Douwes, an examiner of sea officers and pilots under the College of Admiralty at Amsterdam, devised a method for finding the latitude by means of sextants. Douwes's method became a familiar British navigational tool. + +W. R. Martin, in his Navigation and Nautical Astronomy of 1888, regarded Douwes's method as providing the first practical method for solving the double-altitude problem for the use of seamen. + +According to Robertson, manuscript copies of Douwes's method fell into the hands of English officers who, holding the method in high esteem, caused it to be published in 1759 without any derivation. However, Dr H. Pemberton examined this method and the solar tables devised by Douwes for use with his method. Pemberton communicated his investigations to the Philosophical Society, in whose Transactions it was published. Pemberton demonstrated the method and showed its limitations in 1760. + +Douwes's method requires the use of an estimated latitude. If the latitude yielded by the calculations differs materially from the estimated latitude used in the calculation, it is necessary to resolve the problem using the calculated latitude in place of the estimated latitude. + +As a result of investigations into Douwes's method, improved auxiliary tables were published under the direction of Nevil Maskelyne, Astronomer Royal, in his Tables requisite to be used with the Transit Instrument, second edition, 1781. Maskelyne stated in this work that Douwes transmitted his method to the Lords Commissioners of the English Admiralty, and that he was + +METHODS OF FINDING LATITUDE 149 + +rewarded with £50 by the Commissioners of Longitude. Douwe's method was again investigated in 1797 by Mendosa Rios whose results, together with improved auxiliary tables, were published in the *Philosophical Transactions* of 1797. + +Douwe's method is described with reference to Fig 6. + +Figure 6 + +Fig. 6 represents the projection of the celestial sphere on to the plane of the celestial horizon of an observer whose zenith is projected at Z. N, E, S and W, are the projections of the cardinal points of the horizon, and P that of the elevated celestial pole. X and Y are the projected positions of the Sun at the times of the first and second observations respectively. + +The following arcs and angles are known: + +\begin{align*} +XPY &= \text{clipped distance} \\ +PX &= \text{polar distance of Sun} \\ +PY &= \text{polar distance of Sun (assumed to be equal to PX)} \\ +ZY &= \text{second zenith distance} \\ +ZX &= \text{first zenith distance} +\end{align*} + +In triangle PZX: +$$\cos ZX = \cos PZ \cos PX + \sin PZ \sin PX \cos ZPX.$$ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . + +(1) + +150 +A HISTORY OF NAUTICAL ASTRONOMY + +In triangle PZY: +$$\cos ZY = \cos PZ \cos PY + \sin PZ \sin PY \cos ZPY$$ +(2) + +Subtract (2) from (1): +$$\cos ZX - \cos ZY = \sin PZ \sin PX (\cos ZPX - \cos ZPY)$$ +(3) + +that is: +$$2 \sin \frac{1}{2} (ZX + ZY) \sin \frac{1}{2} (ZX - ZY)$$ +$$\sin PZ \sin PX 2 \sin \frac{1}{2} (ZPX + ZPY) \sin \frac{1}{2} (ZFX - ZPY)$$ +From which: +$$\sin \frac{1}{2} (ZPX + ZPY) = \frac{\sin \frac{1}{2} (ZX + ZY)}{\sin PZ \sin PX}$$ +$$\sin PZ \sin PX$$ +(4) + +The term $(ZFX - ZPY)$ in the denominator of (4), is equal to the elapsed time $XPY$. Thus, having found $(ZFX + ZPY)$ from equation (4), it is an easy matter to find $ZPY$. + +From (2), we have: +$$\cos ZY = \cos PZ \cos PX + \sin PZ \sin PX \cos ZPY$$ +i.e. +$$\cos ZY = \cos PZ \cos PX + \sin PZ \sin PX (1 - 2\sin^2 \frac{1}{2} ZPY)$$ +i.e. +$$\cos ZY = \cos (PZ - PZ) - 2\sin PZ \sin PX \sin^2 \frac{1}{2} ZPY$$ +i.e. +$$\cos (PZ - PZ) = \cos ZY + 2\sin PZ \sin PX \sin^2 \frac{1}{2} ZPY$$ + +Using the latitude by account to give an assumed $PZ$, $(PZ - PZ)$ can be found. From this, $PZ$—and hence the latitude—can be determined. If the calculated latitude differs materially from the latitude by account, the problem must be re-worked using the latitude by account as a new starting point. + +The auxiliary tables provided by Makselyne (and others) gave the logarithms of quantities significant in the computation for 10-second intervals of time. Douwe's original solar tables were given for intervals much coarser than this, thus necessitating troublesome interpolation. + +Should the ship's position at the time of the second observation be different from that at the time of the first observation, it is necessary to adjust the first observed altitude to what it would have been had it been observed at the second position. That is to + +M E T H O D O F F I N D I N G L A T I T U D E 151 + +say, an allowance for run would have had to have been made. +The correction of minutes of arc to apply to the first measured +altitude is equal to the distance in miles either towards or +away from the Sun during the time which elapsed between the +instants of the two observations. Moreover, a change in longitude +between the times of observation necessitated a correction to the +elapsed time. + +A figure famous in the history of astronomical navigation is +Samuel Dunn, one-time teacher of mathematics in London. +Dunn wrote several works on nautical astronomy which were *A New Version of Atwood's New System of Navigation*, which were published in 1776-1777. In both of these works, which were dedicated to the Honourable East India Company, Dunn introduced a new solution to the double-altitude problem. The problem was entitled 'Of a general method whereby the latitude may be found, having any two altitudes of the Sun and the time elapsed.' The method is as follows: + +1. Assume two latitudes differing about a degree or less, and not widely different from the latitude by account. +2. For each co-latitude, together with the co-declination and the co-altitude of the Sun, calculate two hours-angles, making four calculated hour-angles in all. +3. The latitude is then found using the following proportional statement: + 'As the difference of the elapsed times computed from the latitudes is to the difference of those latitudes so is the difference between the true elapsed time and either of the computed elapsed times is to the number of minutes which added to or subtracted from the corresponding assumed latitude (as the case requires) gives the true latitude.' + +Dunn's solution was a remarkable discovery, and it formed the basis of the so-called 'Trial and Error' method for finding longi- +tude. It also paved the way for the development of position-line navigation, which we shall deal with in Chapter VII. + +James Andrew A.M. described a solution to the double-altitude problem in his *Astronomical and Naval Tables*, which were published in 1805. Andrew dedicated his work to the Honourable Court of Directors of the United Company of Merchants of + +A diagram showing two altitudes and their corresponding hour-angles. + +152 +A HISTORY OF NAUTICAL ASTRONOMY + +England trading to the East Indies. The principal feature of his tables was the exclusion of a table of squares of natural semi- +chords, that is to say, a table of $\sin^2 \theta/2$ or Haversine $\theta$, for values +of $\theta$, at intervals of 10 seconds of arc, from $0^\circ$ to $120^\circ$. In his pre- +face, Andrew declared that the table, which occupied the space +of 120 pages, was entirely new. It was designed primarily with +the object of facilitating the computation of lunar distances by +means of a new method of his invention. He was quick to realize +that the table could be employed for solving astronomical prob- +lem besides that for which it was specifically designed. In par- +ticular, precepts were given for solving the double-altitude prob- +lem using the table of squares of natural semi-chords. + +An interesting problem in this connection was furnished by James Ivory, Professor of Mathematics at the Royal +Military College at Sandhurst. Ivory's method which was well +received by seamen, was published in the Philosophical Trans- +actions of the Royal Society in 1821. + +Ivory's rule applies strictly to bodies whose declinations do not +change in the interval between the times of the two observations. +In practice, it is available for Sun sights by using the mean of the + +Figure 7 + +W Z X Y M R P H E S Q H + +METHODS OF FINDING LATITUDE 153 + +two polar distances proper to the times of observation. The method is described with reference to Fig. 7. + +The circle in Fig. 7 represents the projection of the celestial horizon of an observer whose zenith is projected at Z. WQE is the projection of the meridian arc through the observer's zenith and elevated celestial pole. PX and PY are the projections of the hour circles through the Sun at the times of the first and second observations respectively. PM is perpendicular to the great-circle arc through X and Y. ZR is a great-circle arc at right angles to PM. + +In triangle PMX: +$$\sin MX = \sin XPM \sin PX$$ +(1) +$$\cos PM = \cos PX \sec MX$$ +(2) + +In triangle ZMY: +$$\cos ZY = \cos ZM \cos YM + \sin ZM \sin YM \cos ZMY$$ +i.e. +$$\cos ZY = \cos ZM \cos YM + \sin ZM \sin YM \sin ZMR$$ +(3) + +In triangle ZMX: +$$\cos ZX = \cos MX \cos ZM + \sin MX \sin ZM \cos ZMX$$ +i.e. +$$\cos ZX = \cos MX \cos ZM - \sin MX \sin ZM \cos ZMY$$ +i.e. +$$\cos ZX = \cos MX \cos ZM - \sin MX \sin ZM \sin ZMR$$ +(4) +Add (3) and (4): +$$\cos ZY + \cos ZX = 2 \cos ZM \cos YM (YM = MX)$$ +(5) + +Subtract (4) from (3): +$$\cos ZY - \cos ZX = 2 \sin ZM \sin YM \sin ZMR$$ +(6) + +In triangle ZMR: +$$\cos ZM = \cosZR \cos MR$$ +(7) +$$\sinZR = \sin MZ \sin ZMR$$ +(8) + +Substitute cosZR cos MR from (7) for cosZM in (5): +$$\cosZY + cosZX = 2 cosZR cos MR cosYM$$ +(9) + +Substitute sinZR from (8) for sinMZ sinZMR in (6): +$$\cosZY - cosZX = 2 sinYM sinZR$$ +(10) + +154 +A HISTORY OF NAUTICAL ASTRONOMY + +Using complements of ZY and ZX in (9) and (10) we have: +$$\sin HY + \sinHX = 2\cos ZR\cos MR\cos YM$$ +(11) +$$\sin HY - \sinHX = 2\sin YM\sin ZR$$ +(12) + +From (12), by transposition, we have: +$$\sin ZR = \frac{\sin HY - \sinHX}{2\sin YM}$$ +(13) + +From which: +$$\sin ZR = \frac{\cos HY + HX}{2}\sin HY - HX$$ +(14) + +From (11), by transposition, we have: +$$\cos MR = \frac{\sin HY + sinHX}{2\cos ZR\cos YM}$$ +(15) + +Finally, in triangle PZR: +$$\cos PZ = \cos PR\cos ZR$$ +(ZR found from (13)) +or +$$\sin Lat = \cos PR\cos ZR$$ +(16) + +The above method is described in The Theory and Practice of Navigation published in 1900 and written by W. G. Tate, Examiner of Masters and Mates at Tyne and Wear ports. + +The following solution is given in The Extra Master's Guide Book written by Thomas Ainley, and published in 1867: + +Referring to Fig. 7, in triangle PMX: +$$\sin MX = \sin XPM\sin PX$$ +(1) +$$\cos PM = \cos FX$$ +(2) +$$\sin ZR = \frac{\cos ZX + ZY}{2}\sin HY - HX$$ +(3) +$$\cos MR = \frac{\sin HY + sinZX}{2}\cos HY - HX$$ +(4) + +A diagram showing a triangle with sides labeled PM, PX, PY, and angles labeled M, X, Y. + +**METHODS OF FINDING LATITUDE** + +Arc PR is then found by combining PM and MR found from (2) and (4) respectively. Finally, arc PZ which is the co-latitude, is found from FZR. By Napier's rules: + +$$\cos ZR = \cos PR \cos ZR$$ + +or + +$$\sin Lat = \cos PR \cos ZR$$ .......................... (5) + +It will be noticed that Ainsley's formulae (1), (2), (3), (4) and (5) correspond essentially with those given by Tate and numerated (1), (2), (13) (14) and (15) respectively. + +It is interesting to note that Ainsley remarked that the formulae deduced by him are essentially the same as those given for this case by M. Cailliet in his Méthode de la navigation, first published in 1818, that is, three years before Ivory's method made its appearance in the *Philosophical Transactions* of the Royal Society. + +The practice of dividing an oblique spherical triangle into two right-angled triangles, as Ivory (and James Andrew) did in his method for solving the double-altitude problem, is one of great antiquity and is, in fact, almost as old as trigonometry itself. It provides the only practical means of constructing oblique spherical triangles until the time of the invention of the fundamental cosine formula which for any spherical triangle ABC is: + +$$\cos A = \frac{\cos BC - \cos AB \cos AC}{\sin AB \sin AC}$$ + +This formula was known to Battegius, the celebrated astronomer of Batavia, who died in AD 930. + +In the practice of navigation, from the beginning of the second half of the 19th century to the present time, the use of auxiliary right-angled spherical triangles to facilitate the solution of oblique spherical triangles—especially in its application to the construction of the so-called short method tables—has been of prime importance. + +Ivory's method was improved in 1822 by Edward Riddle, the master of the mathematical school at Greenwich (formerly the master at the Trinity House school at Newcastle), the year after it was published. Other teachers of navigation, including Mrs Janet Taylor, Lieutenant Henry Raper, and John Riddle, the son of Edward Riddle, who succeeded his father as master at Greenwich, also published modified versions of Ivory's method. + +156 +A HISTORY OF NAUTICAL ASTRONOMY + +John Riddle is credited with being the first to recognize that an error in the solution to the double-altitude problem may arise through using the polar distance of the Sun for the middle time. He accordingly devised a method for correcting error due to this cause. The first publication of his work was in the "Miscellany" of Edward Riddle's "Tractate on Navigation," which appeared in 1835. + +In 1824 Edward Riddle in his treatise noted that the latitude by double-altitude method could be employed in respect of two altitudes of the same fixed star. Hitherto, Sun altitudes alone had been used for the double-altitude problem. Riddle pointed out that were a star used for this purpose, it was necessary to convert the elapsed interval into sidereal units of time. He supplied a simple table showing the number of minutes per interval of solar time, he wrote, "by one second for every ten minutes." + +Graphical methods of solving the double-altitude problem were devised, although these did not find favour among practical seamen. As early as 1659 John Collins, in his *Mariner's Scale new Plained*, gave a solution using the stereographic projection. P. Kelly, in 1796, published an ingenious solution by construction, again employing stereographic projection. + +The use of simultaneous observations of two stars for finding latitude was suggested by John Brinkley whose method was first published in the *Nautical Almanac* for 1825. + +The great value of the method for finding latitude by simultaneous observations of two stars rested in the fact that the difference between the Right Ascensions of the stars supplied the place of the mean time in the case of the normal double-altitude problem in which the Sun's observed altitude is, therefore, independent of the need for measuring time. + +If the two star altitudes are measured by the same observer it is necessary to reduce the altitude of the first star observed to the time of the observation of the second star. This is a simple matter, especially if a table giving the rate of change of star's altitude is provided. Ideally two observers should make the observations simultaneously. + +A second advantage of the star double altitude applies in cases where the angular distance between the observed stars is known, this arc being required in the solution to the problem. + +We find, in Thomas Lynn's *Astronomical Tables* which were first published in 1825, two tables of angular distances between... + +METHODS OF FINDING LATITUDE 157 + +pairs of selected stars for facilitating the problem of finding latitude from simultaneous star observations. One of these tables was designed for use in the northern hemisphere and the other for use in the southern hemisphere. The same tables were published in the *New Physical Magazine*, in which Brinkley made the following observations: + +*Captain Lynn's tables, lately published, in which the Log Rising is given to seconds of time, will be found extremely convenient for the practice of this method [star's double altitude]. In fact it was his [Lynn's] remarks relative to the opportunities of taking exact altitudes of the brighter stars during the time of twilight, that induced me [Brinkley] to endeavour to investigate an easy process of finding latitude from the altitudes of two stars.* + +Three tables, in addition to the two mentioned above, were given by Lynn to facilitate the solution to the double-altitude problem. These tables are similar to those published by Maske-lyne (and others) to facilitate the solution to the problem using + + +A diagram showing a circle with four points labeled N, P, Z, X, Y, Q, W, E, S. Lines connect these points as follows: N to P, P to Z, Z to X, X to Y, Y to Q, Q to W, W to E, E to S. + + +FIGURE 8 + +158 +A HISTORY OF NAUTICAL ASTRONOMY + +the modified Douwe's method. Values were given to every second of time so that interpolation, when using the tables, was not necessary. + +Commander (later Admiral) Charles Shadwell R.N. favoured the star double-altitude method, and he published tables of angular distances between selected stars in his Star Tables which first appeared in 1839, and which were extended in 1849. + +The principle of the star double-altitude method is explained with reference to Fig. 8. + +The circle in Fig. 5 is a projection of the celestial horizon of an observer whose zenith is projected at Z, N, E, S and W are the projections of the north, south, east and west of the horizon. WQE is the projection of the equinoctial and P that of the elevated celestial pole. X and Y are the projections of two stars whose zenith distances at the time of the observations are ZX and ZY respectively. Arc ZY is the great circle arc joining X and Y. In triangles PZX and PZY: + +Given: +PX, PY (polar distances from Nautical Almanac) +ZX, ZY (co-altitudes of X and Y) +XYP (difference between R.A'S of X and Y) + +To find: +PZ and hence the observer's latitude (90 - PZ) + +In triangle PXY: +Using PX, PY and XYP find XY (arc 1) + +In triangle PXY: +Using PX, PY and XY find PXPY (arc 2) + +In triangle ZXY: +Using ZX, ZY and XY, find ZZY (arc 3) +From arcs (2) and (3) find PZX (arc 4) + +In triangle PZX: +Using PXPX and ZX, find PZ, + +Then: +Latitude of observer = (90 - PZ)* + +**METHODS OF FINDING LATITUDE** + +Should the stars X and Y be a 'selected pair' which appear in the Star Tables (of Lynn's or Shadwell's) arcs (1) and (2) may be found by inspection, and the labour of computation thereby effectively reduced. + +An ingenious method of finding latitude based on the principle of the star double altitude was given by Thomas Lynn in his *Navigation* which was published in 1825. Lynn's ingenious method is explained with reference to Fig. 9. + +A diagram showing a celestial sphere with lines and labels indicating positions of stars and points. The diagram includes labels such as N, P, Z, X₁, Y, Q, E, A, B, S, W, and arrows indicating directions. + +**FIGURE 9** + +Fig. 9 represents the celestial sphere projected on to the plane of the celestial horizon of an observer whose zenith is projected at Z. WQE is the projection of the equinoctial and P that of the elevated celestial pole. X and Y are the projections of two stars whose R.A.'s differ by a small angle YPX, Y being to the west of X. + +The altitude of the westernmost star is observed and, after an interval of time equal to half the difference between R.A.'s of X and Y, the star X is observed. At the time of observation of X, the star will occupy the same hour circle as was occupied by star Y when the first observation was made. The altitude of Y at the time of the first observation is arc BY; and the altitude of X at the time of the second observation is arc AX₂. + +159 + +160 +A HISTORY OF NAUTICAL ASTRONOMY + +In the spherical triangle ZXY, the arc ZXZ and ZYZ are known —these being the zenith distances of the stars observed. The arc XZY is also known—this being the difference between the de- +clinations of the two stars. Given the three sides of the triangle ZXY, the angle ZYX may be calculated. + +In the spherical triangle PYZ, given FY, YZ and PYZ, the arc PZ—which is the complement of the observer's latitude—may be found. + +The accuracy of Lynn's method, as pointed out by Commander W. R. Martin in his *Navigation and Nautical Astronomy* of 1888, is greatest when the mean altitude of bearings of the two stars is 90°, a condition which may be approached by observing stars which have a small difference of Right Ascensions and a large difference of declinations. Martin also pointed out that: + +'In order that errors of altitude may affect the latitude least, it is desirable that, when practicable, one of the stars should have a small altitude near the meridian and the other a large altitude near the prime vertical circle; the altitude of the star first observed requires the usual correction for run in the interval.' + +The reliability of the calculated latitude obtained from the nor- +mal double-altitude problem in which the Sun is employed de- +pends entirely upon the change of bearing of the Sun between the times of the two observations. The Rev. James Inman is credited with being first to point out this very important fact. It is most dependable result from the Sun double-altitude applies when the change in azimuth between the times of the two observations is 90°. This condition was stated by Inman in the first edition of his *Navigation* which appeared in 1826. + +Many writers on navigation give complex rules and advice on limiting conditions for the double-altitude problem. These rules and considerations are sometimes rather quite misleading to time from noon. Lieutenant Rapier, in his review of Captain Sumner's newly-published book, which appeared in the *Nautical Magazine* of 1844, remarked: + +'Projection (of Sumner lines) therefore, affords evidence of the simplest and most convincing kind that the value of a double altitude depends altogether on the difference of azimuth. This' + +A diagram showing a spherical triangle with sides labeled ZXY, PYZ, and FYZ. + +METHODS OF FINDING LATITUDE 161 + +condition, first pointed out by Dr. Inman, has nothing to do with time from noon, which more popular works reiterate as the proper limit of application of the double altitude, to the great detriment of the extensive and successful practice of this important observation.* + +It is interesting to note that in a critique of Bowditch's *Practical Navigator*, which appeared in the *Nautical Magazine* of 1843, the reviewer pointed out that: "Bowditch does not once allude to the difference of azimuth as the criterion of the value of the double-altitude solution." + +He also stated that: + +'Bowditch gives no case of latitude by meridian altitude of a star, and yet he does the double altitude of a star—an observation not taken once in a whole service.' + +Captain J. Trivett, a contributor to the *Nautical Magazine* of 1850, and one of the early Examiners of Masters and Mates, described a method for solving the star double-altitude problem by using a table of angular distances between selected stars. The method, which appears to have been popular at the time, was known as the I O U method. The star nearer the moon was referred to as Inner star, and the angular distance between the two stars was known as U. To facilitate the computation, and at the same time render the method easy to remember, the solution was commenced by writing down: + +Inner altitude + +Outer altitude + +arc U from Star Tables [Lynn's or Shadwell's] + +in this order. Hence the mnemonic I O U. + +A practical seaman, L. T. Fitzmaurice, contributed a brief article to the *Nautical Magazine* of 1854 entitled 'On Finding Position by Double Altitudes with only one Latitude.' The method was described without a proof. In the following number of the magazine, a letter appeared under the signature of John Riddle, the master at the Greenwich Hospital school. Riddle wrote: + +A page from a nautical magazine. + +162 +**A HISTORY OF NAUTICAL ASTRONOMY** + +'I believe the writer [L. T. Fitzmaurice] to have been a pupil at Greenwich Hospital about 25 years ago. I am sorry that he [Fitzmaurice] should have forgotten his old school habit of demonstrating the propositions which he advances. The modi- +fication to which his letter refers was in full practice by our little navigators here in 1852, and since then must have spread far and wide.' + +Riddle then gave a demonstration of the method which is, in effect, an adaptation of the newly-discovered Sumner method, but using only one latitude (instead of two) and calculating two hour angles and two azimuths (instead of four hour angles). (See Chapter VII.) + +Staff Commander J. Burwood, the pioneer of Azimuth Tables, described a method for finding latitude from the altitude of two stars observed at the same instant. The description of his method appeared in the British Magazine in 1864. It is interesting to note that Burwood's Table of Sun's True Bearing (for latitudes $48°$-$56°$ N. and S.) were published the year before, in 1864. In a review of these tables it was written: + +'If J. Burwood had really desired to leave a monument behind him of his labour for the benefit of navigation, he could have done no better than compile these valuable tables.' + +**8. MERIDIAN AND MAXIMUM ALTITUDES** + +In normal circumstances at sea it is usual to regard the meridian altitude of a heavenly body as being identical with the maximum altitude. The usual practice of measuring the meridian altitude of the Sun is to commence observing a little before noon, noting that the Sun rises at an angle equal to its declination, and then to increase and start to decrease his altitude--at which times the Sun is said to dip--the angle read off the sextant is taken as the meridian altitude. + +A celestial body of fixed declination culminates or attains its greatest altitude when it crosses the upper celestial meridian of a stationary observer. Should, however, the observer be changing his position so that the Sun or some other body is not changing its declination the maximum altitude attained is not the meridian altitude. In other words, a celestial body attains its + +METHODS OF FINDING LATITUDE 163 + +maximum altitude when it lies off the meridian, that is before it crosses or after it has crossed the meridian, should the observer be moving northwards or southwards, or should the declination of the body be changing. The hour angle of a celestial body when it is at maximum altitude is dependent upon the rates of change of declination of the body and the latitude of the observer. + +For practical purposes, the rate of change of a star's declination is zero, and that for the Sun, which varies (being zero at the solstices) between 0° and 5° per day, is very small. The rate of change of declination of the Moon, however, may be considerable and, as early as the beginning of the 19th century, the Moon, because of this, was regarded as being an unfit object for meridian altitude observations. + +Chauvenet, in his *Astronomy* of 1896, demonstrates that the interval between the times of meridian and maximum altitudes of a celestial body, when observed by a stationary observer, is given by the formula: + +$$T = 810.000 \sin^2 \cos L \cos D$$ + +where $T$ is the interval in seconds of time, + +$d$ is the rate of change of declination $D$ in seconds of arc per hour and + +$L$ is the observer's latitude. + +It is easy to see that if the declination of a celestial body is changing in the direction of the bearing of the body at meridian passage, the maximum altitude will occur, to a stationary observer, after the meridian altitude, and vice versa. Should the observer be moving northwards or southwards, or should the declination be changing, an effect. It is the algebraic combination of the rates of change of declination and latitude that determines whether the meridian altitude will occur before or after the time of maximum altitude. + +With the advent of fast steam vessels during the last century the problem of the maximum altitude became one of significance. Latitude by maximum altitude became, in effect, a special case of the problem known as latitude by ex-meridian study. + +To overcome this difficulty through treating the maximum altitude as the meridian altitude, seamen have been advised, since the middle of the 19th century, to compute the + +164 +A HISTORY OF NAUTICAL ASTRONOMY + +time of meridian passage using the elements of the Nautical Almanac together with the longitude by account; and to observe the altitude of the body at computed time, instead of waiting until it dips, which was, and still is, the traditional method. + +A formula from which the difference between the meridian and maximum altitudes may be found is: + +$$y = \frac{(157)^{3} sin^{1} cos L cos D}{2 sin (L \pm D)}$$ + +Consider a navigator observing the Moon at a time when this body's declination is zero and its hourly change of declination is 17° to the north. If the observer's ship is in the northern hemisphere and it is steaming due south at the rate of 20 knots, the combined movements of the observer's zenith and the Moon, at the time of meridian passage, would produce the same effect as if the observer's position around and the declination of the Moon were changing at the rate of (20+17), i.e. 37° per hour towards the north. + +When a heavenly body's geographical position and an observer's position are opening, the time of maximum altitude occurs earlier than that of meridian altitude. But when these positions are closing, as they are in the case described above, the time of maximum altitude will be later than that of meridian passage. In the example, the Moon's maximum altitude would be reached at about eleven minutes after noon and it would be about 34° greater than the Moon's meridian altitude. + +After the invention of position-line navigation following Captain Thomas Sumner's discovery in 1837 (see Chapter VII), astronomical navigation was brought to a state of excellence in 1875 by French astronomers. The navigators of France, Marquis de la Grange d'Arquien and others, used a special method of obtaining an astronomical position line by astronomical navigation became merely a special case of the general method of obtaining an astronomical position line. + +The end product of every observation or sight, as the seaman calls it, is a position line which may be drawn on the navigation chart. A **position line** is defined simply as a line on the chart somewhere on which any ship's position is represented. The point of intersection of any two position lines is the projected position of the ship on the chart. + +The astronomical position line obtained with the least effort is + +METHODS OF FINDING LATITUDE 165 + +that which results from an observation of an object on the meridian. The position line, in this case, lies east-west along a parallel of latitude. The meridian altitude observation for latitude became available to European seamen, as we have demonstrated, as soon as tables of the Sun's declination, and a Regiment or Rules for the Sun were devised for the Portuguese navigators of five centuries ago. Since that far-off time the altitude observation of the culminating Sun has been a tradition amongst seamen. There is a distinction between the Sun at every hour these enlightened days, should a cloudy sky or an indistinct horizon prevent the measurement of the Sun's meridian altitude. There has always been something sacrosanct about the so-called noon position by observation; and this position cannot be obtained unless the noon-day height of the Sun is measured. + +9. LATITUDE BY EX-MERIDIAN ALTITUDE + +The troublesome disadvantage of the midday Sun-sight is that the Sun and the horizon vertically below him must be visible at one particular instant of time during the day. In tropical waters it is seldom that these conditions are not satisfied; but in the seas of middle and high latitudes, where rain, fog and cloudy skies are common, it is often impossible for the seaman to find his latitude by meridian altitude of the Sun. It is not surprising, therefore, that in all ages and countries there has been a constant search to provide himself with alternative methods for finding latitude. Latitude, together with the lead, log and look-out, formed one of the four L's of the early mariner's creed. + +The method for finding latitude which we are now about to describe, and which is known as the ex-meridian method, is still practised extensively by seamen. The history of the method, which was first described by John Harrison in 1730, is full of interest, and the ex-meridian problem is almost as celebrated in the history of astronomical navigation as that of the double altitude. + +Considerable attention was devoted to the ex-meridian problem during the 19th century—a period which was, in truth, a golden era of astronomical navigation. Many ingenious solutions were contrived and a diversity of ex-meridian tables were furnished, all aimed to facilitate the process of finding latitude at sea. + +It may be noted that the ex-meridian method became obsolescent as soon as it had been invented. The increasing use and + +12 + +166 +**A HISTORY OF NAUTICAL ASTRONOMY** + +reliability of chronometers—which were scarce instruments for a long time after Harrison had produced his famous timepiece in the 18th century—and the introduction of position-line navigation in the mid-19th century, brought astronomical navigation to a state of near perfection. Had logic prevailed in the chart-room, all the existing methods of astronomical navigation would have been swept away as soon as the ‘New Navigation’ of Marcq St + +A diagram showing a circle with points labeled P, Z, Z₁, X₁, X, and A₁. The line PA₁ forms an angle with the horizontal line through Z₁ and X₁. + +**FIGURE 10** + +Hilaire had made its appearance. This, however, was not the case; many of the old methods of navigation remain with us, even to this day. Old-established practices die hard at sea, and modern navigators tend to maintain their navigation in the same way as did their sea-faring ancestors. + +The essential problem in the ex-meridian method for finding latitude is the comparison of the altitude of a celestial body at a place where the body is culminating (the latitude of the place being the same as that of the observer), with its altitude at the same instant at the observer’s position. + +In Fig. 10 the celestial sphere is projected on to the plane of the equinoctial. Z is the projection of the zenith of an observer, and + +METHODS OF FINDING LATITUDE 167 + +$Z_0$ is that of the zenith of a place whose latitude is the same as that of the observer, and over whose meridian the body X is passing. P is the projection of the celestial pole and the circle is that of the equinox. + +If the arc $Z_0X$ can be found, the latitude of the place whose zenith is at $Z_0$ and, therefore, the observer's latitude, can also be found. + +If $l$, $d$, $x$ and $h$ denote the observer's latitude, the body's declination, the body's zenith distance, and the time of the meridian passage of the body, respectively, we have, from the triangle PZX: + +$$\cos h = \frac{\cos x - \sin l \sin d}{\cos l \cos d}$$ + +when $l$ and $d$ have the same name, and: + +$$\cos h = \frac{\cos x - \sin l \sin d}{\cos l \cos d}$$ + +when $l$ and $d$ have different names. + +When $l$ and $d$ have the same name: + +$$\cos x - \sin l \sin d = \cos l \cos d \cos h$$ + +i.e. + +$$\cos x - \sin l \sin d = \cos l \cos d (1 - \text{vers } h)$$ + +i.e. + +$$\cos x - \sin l \sin d = \cos l \cos d - \cos l \cos d \text{ vers } h$$ + +i.e. + +$$\cos x + \cos l \cos d \text{ vers } h = \cos l \cos d + \sin l \sin d$$ + +i.e. + +$$\cos x + \cos l \cos d \text{ vers } h = \cos (l - d)$$ + +i.e. + +$$\cos x + \cos l \cos d \text{ vers } h = 1 - \text{vers } (l - d)$$ + +i.e. + +$$\text{vers } (l - d) = 1 - \cos x - \cos l \cos d \text{ vers } h$$ + +i.e. + +$$\text{vers } (l - d) = 1 - \text{vers } x - \cos l \cos d \text{ vers } h$$ + +Similarly, when $l$ and $d$ have different names: + +$$\text{vers } (l + d) = 1 - \text{vers } x - \cos l \cos d \text{ vers } h$$ + +168 +A HISTORY OF NAUTICAL ASTRONOMY + +In general: + +$$\text{vers}(\frac{l}{\pm d}) = \text{vers} x - \cos l \cos d \text{ vers } h$$ +(1) + +Now $(l \pm d)$ is the meridian zenith distance of $X$ at the place whose zenith is at $z_1$. Let this be denoted by $x_1$. The latitude of this place and, therefore, the observer's latitude, may thus be found: + +$$\text{Latitude of observer} = (l \pm d) \pm d$$ +that is + +$$l = x_1 \pm d$$ + +The above treatment is a modified form of that first given in 1754 by Cornelius Douwee, a figure famous in the history of the double-altitude problem. Douwee's investigation was modified by the Rev. James Inman D.D., whose famous nautical tables were first published in 1821. Inman's modification, used above, consists in adapting the formula to the tables of natural verses and haversines. + +The ex-meridian method described above requires the use of a latitude by account, which should approximate to the observer's actual but unknown latitude. If the latitude found differs materially from that used it is necessary to repeat the computation, this time using the calculated latitude in place of the one initially used. Moreover, it is necessary for the observer to know his longitude; knowledge of this being required to find $h$, which figures in the computation. + +It may readily be shown that: + +Error in $x_1$ (and therefore error in calculated latitude) is proportional to $\cos$ latitude sine azimuth $\times$ error in $h$. + +It follows, therefore, that the smaller the latitude or the nearer the azimuth to 90°, the greater will be the error in latitude consequent upon an error in $h$. Knowledge of correct time is all important in the ex-meridian problem. + +It was early realized that when using stars for finding latitude in accordance with the ex-meridian method, those with big declinations gave the best results, this because of their relatively slow rates of change of altitude. The Pole Star, therefore, is admirably suited for the purpose; and accurate Pole Star tables have + +METHODS OF FINDING LATITUDE 169 + +been available for the seaman since the early 19th century, although they were not included in the British Nautical Almanac until 1834. + +A method for finding latitude by ex-meridian observation of the Sun, using 'direct spherics,' was given in the later editions of James Robertson's Elements of Navigation. This method is explained with reference to Fig. 11. + +A diagram showing a celestial sphere with a meridian circle. The points P, M, X, Z, Q, E, W, S are labeled on the diagram. The line PX is perpendicular to the plane of the celestial horizon at Z. + +FIGURE II + +Fig. 11 is a projection of the celestial sphere on to the plane of the celestial horizon of an observer whose zenith is projected at Z. P is the projection of the elevated celestial pole; WQE that of the equinoctial; NZS the observer's celestial meridian. Arc XM is a perpendicular from X on to the observer's celestial meridian at M. + +In triangle PMX: +$$\tan \text{PM} = \tan \text{PX} \cos \text{P} \quad \quad (2)$$ +$$\cos \text{PX} = \cos \text{MX} \cos \text{PM} \quad \quad (3)$$ + +In triangle XMZ: +$$\cos \text{ZX} = \cos \text{ZM} \cos \text{MX} \quad \quad (4)$$ + +Divide (3) by (4) to eliminate MX: +$$\frac{\cos \text{PX}}{\cos \text{ZX}} = \frac{\cos \text{PM}}{\cos \text{ZM}}$$ + +170 +A HISTORY OF NAUTICAL ASTRONOMY + +i.e. +$$\cos PX \cos ZM = \cos ZX \cos PM$$ +and +$$\cos ZM = \cos ZX \cos PM \sec PX$$ (5) + +From (2) and (5), arc PM and ZM may be found: and these, when combined, will give arc PZ which is equivalent to the complement of the latitude. + +The direct method is independent of the latitude, and may be used to good effect even when the observed object has a large hour angle and azimuth, provided that the angle P is known with accuracy. A disadvantage of the method applies to cases in which the object's declination is small. In this event the arc PX is nearly 90°. Because the tangent of PX is required in (2), it is necessary, in this case, to use a sextant which was great care. Moreover, any small error in the polar distance would cause, under this circumstance, a relatively large error in secant PX used in (5), and this will lead to error in arc ZM. + +An interesting case of the above method, described by W. R. Martin in his Navigation and Nautical Astronomy, applies when the declination of the observed object is less than about 1°, and the hour angle (P) is less than about half an hour. In this case: + +$$PM = PX$$ +$$XM = P$$ +and +$$\cos PM = 1$$ + +Therefore, for practical purposes: +$$\cos ZM = \cos ZX \sec P$$ + +The latitude is found by combining the computed arc ZM with the object's declination. + +A method alternative to the direct method described above is known as the Reduction to the Meridian Method. This method involves the calculation of a correction to apply to the altitude of the body when out of the meridian to find its altitude when it is on the meridian. The reduction method, like that attributed to Douwe, requires the use of a latitude by account. If this proves to be materially different from the computed latitude it is necessary to repeat the computation using the computed latitude instead of the latitude by account. + +METHODS OF FINDING LATITUDE + +171 + +A diagram showing the reduction to the meridian method. The celestial sphere is projected on to the plane of the celestial horizon of an observer whose zenith is projected at Z. The body's position when it is at meridian passage relative to the observer is represented by arc ZX. The arc ZA is drawn equal to arc ZX. +FIGURE 12 + +The reduction to the meridian method is attributed to the renowned French astronomer Delambre, who published it in 1814. The method is described with reference to Fig. 12. + +Fig. 12 represents the celestial sphere projected on to the plane of the celestial horizon of an observer whose zenith is projected at Z. The body's position when it is at meridian passage relative to the observer is represented by arc ZX. The arc ZA is drawn equal to arc ZX. + +$$\begin{align*} +FX &= PY \\ +ZX &= ZA \\ +YA &= (ZA -ZY) \\ +i.e. & \quad YA = (ZX -ZY) +\end{align*}$$ + +The arc YA is referred to as the reduction. Let this be denoted by $r$. + +In triangle PZX: + +$$\cos ZX = \cos PZ \cos PX + \sin PZ \sin PX \cos P$$ + +or + +$$\cos z = \sin i \sin d + \cos l \cos d \cos h$$ + +Now, + +$$\cos h = 1 - 2 \sin^2 h/2$$ + +172 +A HISTORY OF NAUTICAL ASTRONOMY + +Therefore: +$$\cos x = \sin l \sin d + \cos l \cos d (1 - 2 \sin^2 h/2)$$ +i.e. +$$\cos x = \cos (l \pm d) - 2 \cos l \cos d \sin^2 h/2$$ +(6) +Now, +$$ZX = ZA = (ZY + r)$$ +Therefore: +$$\cos ZX = \cos ZY \cos r - \sin ZY \sin r$$ + +Since $r$ is a small quantity and $ZY$ is $(l \pm d)$ we have: +$$\cos x = (1 - r^2/2) \cos (l \pm d) - r \sin (l \pm d)$$ +(7) + +By equating the values for $\cos x$ from (6) and (7) we have: +$$(1 - r^2/2) \cos (l \pm d) - r \sin (l \pm d) = \cos l \cos d - 2 \cos l \cos d \sin^2 h/2$$ +From which +$$r^2/2 \cos (l \pm d) + r \sin (l \pm d) = 2 \cos l \cos d \sin^3 h/2$$ +(8) + +The first term in (8) is small when the object is near the meridian. It may, therefore, be neglected. Hence: +$$r \sin (l \pm d) = 2 \cos l \cos d \sin^3 h/2$$ +and for practical purposes: +$$r = 2 \frac{\cos l \cos d}{\sin (l \pm d)} \sin^3 h/2$$ +(9) + +Formula (6) may be transposed thus: +$$\cos (l \pm d) = \cos x + 2 \cos l \cos d \sin^3 h/2$$ +(10) +i.e. +$$\cos x_1 = \cos x + 2 \cos l \cos d \sin^3 h/2$$ +(11) +$x_1$ may, therefore, be found using a latitude by account provided that $h$ is known accurately. + +John Hadley, Member in the twelfth edition (1796) of his *Practical Navigator*, described a method for finding 'latitude by one altitude of the Sun when the time is not more distant than one hour from noon.' He gave one rule for finding the time from noon, based on formula (10); and another rule, based on formula (11), for finding the meridian zenith distance, and thence the latitude. + +**METHODS OF FINDING LATITUDE** + +The expression $2\sin^{\frac{1}{2}}$ was known as the *rising*. The rising figured in the double-altitude problem, and existing tables of log risings, therefore, were adapted to the ex-meridian problem. + +Moore pointed out that the rule for finding time from noon should be applied only when the Sun's altitude did not exceed 18°. He also noted that: + +"an error in the supposed latitude can make very small difference in the change of altitude; and the nearer is the altitude taken to noon the better to find the change of altitude." + +He also warned against using the method when the time from noon exceeded one hour, and stated that in cases where the Sun's meridian altitude exceeds 60°, or when the latitude is small, good results cannot be expected unless the time from noon was more than less than one hour. + +Norie, in the 1814 (fourth) edition of his complete Epitome of Navigation, gave the same rule as Moore's for finding latitude from a single observation of the Sun; but he did not specify any limit to time from noon. In the 1828 (ninth) edition of the same work, Norie specified that: 'in this method . . . the time from noon should not exceed 30 minutes.' + +In Lin's first edition, a rule given in Norie's epitome was: "The number of minutes in the time from noon should not exceed the number of degrees in the Sun's meridian zenith distance." This, the common rule quoted by seamen of our time, appears to have stemmed from the celebrated Raper. + +After the ex-meridian method had become established among practical seamen, rules gave way to tables giving limits of time from meridian passage. In Raper's *Practical Astronomer in Navigation*, published in 1885, we find a table similar to those in other text-books of the time, giving limits of time from meridian passage (meridian distances) computed to give the number of minutes of meridian distance, when an error of half a minute in the time will produce an error of one minute of arc in the reduction. This is a reminder of Raper's definition of the term "near the meridian." + +According to Raper: + +"The term implies a meridian distance limited according to latitude and declination, and also the degree of precision with which the time is known." + +174 +A HISTORY OF NAUTICAL ASTRONOMY + +The treatment on limits for ex-meridian by reduction given by J. Merrifield in his *Treatise on Nautical Astronomy*, first published in 1886, is interesting. After showing that: + +Error in lat. = error in mer. dist. (k) sin Az. cos lat. +he concludes that: + +'...as a rule, altitudes for latitude by circummeridional alti- +tude (ex-meridian observations) should be taken within 20 +minutes of the body's transit or when the object's azimuth +is not more than one point.' + +He then quotes the practical rule given by Norie (and others) +quoted above. + +Merrifield, in his discussion on ex-meridian sights, also pointed +out (as others had also done) that, by finding latitude and azimuths +of an object near the meridian both when it is east and when +it is west of the meridian, and then meaning the results, '...the +method is susceptible of very great accuracy.' + +The books of Andrew Mackay, published during the early part +of the 19th century, were among the more comprehensive works +on astronomical navigation. In neither his *Theory of the Longitude*, +in two volumes in 1790 and 1793, nor in his *Nautical Almanac*, second +edition 1810, do we find mention of the ex-meridian problem. +Mackay did, however, describe a method for finding latitude by +equal altitudes of the Sun. In this method half the elapsed inter- +val between the times of the observations is equal to the time of +either from noon (if the observer is stationary). From formula (11), +putting half elapsed time equal to $h$, it is possible to find the +meridian altitude of the Sun. + +The ex-meridian method for finding latitude became in- +creasingly popular with the ever-increasing number of ex-meri- +dian tables which appeared from the middle of the 19th century +onwards. + +If $r$ in formula (9) is expressed in seconds of arc the formula becomes: + +$$r = \frac{\cos^2\cos d}{2}\sin^2h + \sin^2d\sin h$$ + +Values of $(2\sin^2h/2)\sin 1^\circ$ were tabulated for suitable values of $h$; and by means of these tabulated values the computation of + +METHODS OF FINDING LATITUDE + +The reduction $r$ may be performed with great facility. The table 'Reduction to the Meridian' which appears in Riddle's *Treatise on Navigation* sixth edition (1835), gives the value of the expression for values of $h$ from twenty minutes to up to twenty minutes. + +An alternative solution was to use the log sine squared table and to add the constant 5-615455, which is the log of $2\sin 1^\circ$. Another similar solution was to use the log rising table and to add the constant corresponding to the log of $\frac{1}{2}\sin 1^\circ$. + +It will be noticed that $2\sin^2 \frac{1}{2}h$ is equivalent to versine $h$. Inman's rule for the ex-meridian problem, which was given in his *Practical Navigation and Nautical Astronomy* (1835), was based on the formula: + +$$\text{vers } x_1 = \text{vers } z - \text{vers } \theta$$ + +$$\text{hav } \theta = \text{hav } h \cos \frac{1}{2}c \cos d$$ + +One of the earliest of numerous ex-meridian tables was that of Captain J. T. Towson (of great-circle sailing fame), first published by the Hydrographic Office in 1849. The principles of Towson's tables are explained with reference to Fig. 11. In triangle PMX: + +$$\sin MX = \sin P \sin PX \quad (\text{sine rule})$$ + +$$\cos PM = \cos P \cos MX \quad (\text{Napier's rules})$$ + +In triangle ZMX: + +$$\cos ZM = \cos ZX \sec MX \quad (\text{Napier's rules})$$ + +From these three formulae we have: + +$$\sin MX = \sin h \cos d$$ + +$$\cos PM = \sec MX \sin d$$ + +$$\cos ZM = \sec MX \cos z$$ + +Values of $MX$ are tabulated in columns labelled 'Index Number.' Table 1 contains values of $(PX - PM)$. Table 2 contains values of $(ZX - ZM)$. To use the tables, Table 1 is entered using arguments $h$ and $d$ to extract Index Number and Augmentation 1. Augmentation 1 is added to the declination. Table 2 is then entered, using arguments altitude and Index Number, to extract Augmentation 2, which is the required reduction. + +Towson's tables were designed specifically for Sun ex-meridian. + +176 +**A HISTORY OF NAUTICAL ASTRONOMY** + +observations, as were others such as those of James Bairnson (c. 1880). + +The ex-meridian tables of Brent, Walter and Williams, first published in 1886, provided for star sights as well as for Sun sights. When $\theta$ is small sin $\theta \approx \theta$ radians nearly. Formula (12), therefore, may be expressed thus: + +$$r = \frac{\cos l\cos h}{\sin(l + \theta)} \cdot \frac{\sin h\sin^2 15^\circ}{\sin(h + \theta)\sin 1^\circ}$$ + +(13) + +where $r$ is in seconds of arc and $h$ is in minutes of time. The tables of Brent, Walter and Williams were based on this formula. + +It follows, from formula (13), that the change of altitude of a body when it is near meridian passage varies as the square of the meridian distance. That is: + +$$r \propto h^2$$ + +This is the principle of an ingenious method for finding the reduction to the meridian graphically. The curve of $r$ against corresponding meridian distance $h$ is a parabola, which became known as Foscolo's parabola, after Professor Foscolo of Venice who devised this graphical method for solving the ex-meridian problem. Foscolo's parabola was published by the British Hydrographic Office in 1890. + +"Cloudy Weather" Johnson, in his *Brief and Simple Methods for Finding Latitude and Longitude* (third edition 1895), gave a table for finding latitude by ex-meridian observations based on formula (1). He used the term 'reduced versine' for the natural versine corresponding to the log versine of $h$, diminished by the sum of the log secants of $l$ and $d$. Three small tables—all at a single double-page opening—were provided for finding the reduced versine. + +Johnson's rule for finding the meridian zenith distance was simple: "The natural versine of the ex-meridian zenith distance diminished by the reduced versine is the natural versine of the meridian zenith distance." + +In the same work, Johnson introduced his methods for finding longitude by ex-meridian altitudes, and latitude and longitude by double ex-meridian. + +Johnson's explanation of the reduction to the meridian, given + +METHODS OF FINDING LATITUDE + +in his well-known On finding Latitude and Longitude in Cloudy Weather, concludes with the formula: + +$$r = 2C \text{ hav } h \cos I \cos d \sec \text{ altitude}$$ + +where + +$$C = 1 \text{ radian}$$ + +The upper part of Johnson's ex-meridian table in this work gives values of $\cos I$ sec altitude or $N$. The lower part gives values of $2C$ hav $h$ sec $N$. The factor cos $d$ was ignored (presumably because, when used with the Sun, cos $d$ approximates to unity). Johnson justified himself by stating: 'A further correction (for the declination) may be applied when both the reduction and declination are considerable.' + +Formula (15) may be written: + +$$r = A^2$$ + +where + +$$A = \frac{\cos l \cos d \sin^2 15^\circ}{\sin(l \pm d) \sin 2 \sin 1^\circ}$$ + +where $l$ and $d$ may be considered to have their meridian values. + +The ex-meridian tables in present-day nautical tables are based on this relationship. The principle of the method was described by H. B. Goodwin as a kinematic principle applied to navigation. The kinematics in this case is related to the distance travelled by a body in any time $t$ when it is accelerating at the rate of $a$. The same mechanism is employed in the method for finding latitude known as the short double altitude. + +Raper, in his Practice of Navigation, informs us that the first work in which a method occurs of finding the latitude by two altitudes observed near the meridian (but restricted to the same side) with an interval of a few minutes, is *Course d' Observations Nautiques* by Ducum. + +Roberson, in his Elements of Navigation, drew attention to the fact that in the 18th century the problem of finding latitude by three ascending or descending altitudes of the Sun ... exercised the talents of many ingenious persons.' + +Numerous solutions to the problem were given, but only the case of practical value applied when the intervals between the first and second, and the second and third, observations are equal, and the observed object is near the meridian. The Abbé de la Caille is credited with giving a solution to this problem as early as 1760. + +178 +**A HISTORY OF NAUTICAL ASTRONOMY** + +Let $a$ be the meridian altitude and $a_1$, $a_2$ and $a_3$ the ex-meridian altitudes observed at intervals of $t$. If $h$ is the time from meridian passage at which the middle of the three ex-meridian observations are made: + +$$a = a_1 - h(a + t)$$ +$$a = a_2 - h^2(t + 2)$$ +$$a = a_3 - h^3(t + 2)$$ + +where + +$$k = \cos l \cos d \sin^2 15^\circ$$ +$$\sin (l + d) \sin 1^\circ \cos^2 15^\circ$$ + +From these relationships, the meridian altitude, and hence the latitude, may be found algebraically. + +In an interesting paper by John White on the ex-meridian problem, published in the *Nautical Magazine* in 1896, the following formula for the reduction, based on a formula by Godfrey, was given: + +$$r = \frac{A^2}{2(\tan l + \tan d) \sin 1^\circ \cos^2 15^\circ}$$ + +This formula suggested the construction of a table giving values of $2 \tan l \sin 1^\circ \cos^2 15^\circ$, and $2 \tan d \sin 1^\circ \cos^2 15^\circ$, against arguments $t$ in minutes of time from unity to 60. By means of this table the denominator of the formula given above could readily be found and the solution to the ex-meridian problem thereby facilitated. + +The Admiralty published, in 1895, a diagram devised by White for obtaining the reduction in an ex-meridian altitude observation. + +The well-known nautical publishers, Messrs J. D. Potter of the Minoris, published a diagram for solving the ex-meridian problem. This first appeared in 1897, when the inventor, F. Kitchin, a naval instructor on H.M.S. Britannia, described his diagram in *Nautical Magazine* (1897). + +Notable among a profusion of ex-meridian tables published during the present century are those of Captain H. S. Blackburne. In a gentle rebuke, Captain Blackburne stated in the 1918 edition of his *Tables for Asimuth, Great-Circle Sailing, and Reduction to the Meridian*, first published in 1905, that: "the only ex-meridian tables which are at present allowed to candidates for the R.O.T. examinations are those of Towson and Raper." This restriction doubtless limited the use of other ex-meridian tables. + +METHODS OF FINDING LATITUDE 179 + +Blackburne devised several ex-meridian tables, each designed for a special purpose. He was a great advocate of star sights and, in his Excelsior Ex-meridian and Position-Finding Tables first published in 1917, he gave a comprehensive set of tables giving azimuths and reductions for 29 of the brighter stars. + +The ex-meridian method for finding latitude has, since its introduction, been a firm favourite with seamen and ex-meridian tables are still widely used. + +When an object body is on the prime vertical circle its rate of change of altitude is proportional to the cosine of the latitude. It follows, therefore, that: + +$$\cos l = \frac{\text{rate of change of altitude}}{a}$$ + +and + +$$\cos l = k \times \frac{\text{change of altitude } a}{t}$$ + +If $a$ is in seconds of arc and $t$ is in seconds of time, $k = 1/15$. Therefore: + +$$\cos l = \frac{1}{15} \times \frac{a}{t}$$ + +This relationship provides a method for finding latitude. It was described in some textbooks of the late 19th century. Some writers recommended observing the time taken for the Sun, when its true bearing was east or west, to rise or fall an angle equal to its own diameter, this requiring only one setting of the sextant. The change in altitude, in this case, could be checked against the value of the Sun's semi-diameter given in the Nautical Almanac. + +A diagram showing the relationship between the rate of change of altitude and the cosine of latitude. + +CHAPTER VI + +Methods of finding Longitude + +I. INTRODUCTORY + +Anaximander is often venerated by geographers who regard him as being the father of their subject. Credit for the invention of the map is due to this philosopher of Ancient Greece. + +Anaximander, who flourished during the 6th century BC, noted that the stars appear to revolve around the celestial pole—a manifestation of the Earth's axial rotation. He concluded that the Earth must therefore be considered as a sphere, with its surface on which the stars are fixed. Anaximander, therefore, may rightly be claimed to be the father of mathematical astronomy, as well as of geography, by virtue of his invention of the celestial sphere. + +The Greek philosopher Democritus (fl. 450 BC) is credited with being the first to construct a rectangular map. His map was based on his personal travels, which led him to the conclusion that the habitable part of the Earth is divided into two equal parts along the east-west direction as it is broad in the north-south direction. This notion, which became the common view, is perpetuated by the very names we give to the two coordinates used to describe terrestrial positions, the words latitude and longitude being derived from latus and longus signifying, respectively, breadth and length. + +The longitude of a place is a measure of the angle contained between the plane of the meridian of the place and that of a standard meridian from which longitudes are measured east or west. The datum meridian commonly used is that of Greenwich, so that the meridian of Greenwich and its antipodal meridian divide the Earth into the eastern and western hemispheres in much the same way as the equator divides the Earth into the northern and southern hemispheres. + +We have noted, in our discussion on the history of the latitude, that finding the latitude of a place was but a trivial problem to the + +METHODS OF FINDING LONGITUDE 181 + +Greeks of antiquity, Finding longitude, on the other hand, presented considerable difficulty. + +The difference between the longitudes of two places is equivalent to the difference between their local times, reckoning 15° of difference of longitude to one hour difference of local times. The problem of finding longitude, therefore, is one in which the local time at a particular instant is to be compared with the local time at the datum or standard meridian, for the same instant. If, for example, the local time is 6 a.m. at the instant when it is 8 a.m. at the Greenwich meridian, the local longitude is two hours 30'. This is one of the methods of finding longitude by means of the Greenwich meridian, because the spin of the Earth to the east results in clock times at the Greenwich meridian being in advance of those at places in the western hemisphere and behind those in the eastern hemisphere. + +Apart from the rough method of dead reckoning, in which estimations of the ship's courses and distances are made, there are two more exact methods by which the longitude of a terrestrial position, east or west of a given meridian, may be found. One is an astronomical method, and the other is mechanical. + +The astronomical method of finding longitude involves the use of a predicted time of an astronomical event, such as an eclipse or occultation, the predicted time being given for a particular datum moment. If this time is known, then may be found, the longitude of the place east or west of the datum meridian may also be found by comparing the local and predicted times. + +The mechanical method of finding longitude involves the use of a timepiece. If the rate of gaining or losing is known, and if the error of the timepiece on the time at some datum or prime meridian is also known, the correct standard time may be found. This, compared with the local time of any given moment, will give a measure of the local longitude east or west of the standard meridian. + +The ancient philosophers of the Mediterranean world used as a datum line, from which longitude was measured eastwards, the meridian through the 'Fortunate Isles'—believed to be the Canaries. These islands were believed to be the westernmost part of the habitable Earth. Throughout classical times, and during the period of the Renaissance, the meridian through the 'Fortu- nate Isles' continued to be the first or prime meridian. + +A diagram showing a globe with lines indicating latitude and longitude. + +185 + +A HISTORY OF NAUTICAL ASTRONOMY + +Following the Golden Age of discovery, when the peoples of Western Europe emerged as sea-traders, almost every European nation used as a prime meridian one that passed through its State territory. Thus, the French, for example, used Paris as their Prime Meridian; the Dutch, that of Amsterdam; the English, that of London; and so on. + +It is interesting to note that as far back as the late 17th century, when so many prime meridians had already been established, we find a Spanish professor writing to the Royal Society in London proposing a new place for the first meridian. The comment on his proposal was that 'the work was lamentable. . . a thing which could never be accomplished.' + +The difficulties related to the profusion of prime-meridians—difficulties thought by many to be insurmountable—were not brought to an end until the closing decades of the 19th century, when it was decided by international agreement to adopt the meridian through Greenwich as the prime meridian from which longitudes should be measured. + +2. LONGITUDE FROM ECLIPSE OBSERVATIONS + +Ancient maps were constructed on the basis of observed latitudes and estimated longitudes. East–west errors, therefore, were in many cases considerable. We have noted in Chapter I the improvement which the great Hipparchus introduced in relation to mapping the Earth's surface. He is credited with two inventions to suggest how early use of eclipses of the Moon involved finding the longitude of places. Hipparchus explained how the difference of longitude between two places could be found from a comparison of the times at the two places of the occurrence of a lunar eclipse. It appears, however, that during the following three centuries very few eclipse observations had been made for this purpose; and we find the famous Poliomy complaining about this in 1630. + +During his second voyage in 1494, Vasco da Gama, Columbus, and his second voyage in 1494, had recourse to the method of finding longitude by observation of an eclipse of the Moon. It is recorded that the result of his observation, which marked the first attempt to find the position of a place in the New World using astronomical principles, was in error to the extent of about 18°. This error was due to inaccurate predictions of the time of the eclipse through imperfect knowledge of the motion of the Moon, and to the error + +**METHODS OF FINDING LONGITUDE** + +in the determination of the local times of the precise stages of the eclipse. + +Some interesting accounts of finding longitude from eclipse observations are to be found amongst the early *Transactions* of the Royal Society of London. One writer describes how the method was employed for finding the longitude of Moscow in the year 1688. Observations of a lunar eclipse were made at Leipzig and Moscow, the local times being 8.54 p.m. and 10.40 p.m., respectively. This showed that Moscow lies 1 hr. 46 mins. of time or 26° 30' longitude east of Leipzig. From an earlier eclipse observed on the same day at Leipzig, the longitude is 49 mins. of time or 12° 15' of longitude east of London. It was concluded, therefore, that Moscow is 38° 45' east of London. + +In 1719 the astronomer Edmund Halley found the longitude of the Cape of Good Hope from an observation of an eclipse made at sea in latitude 34° 23' S., at a position 180 leagues east of the Cape. From his observation he concluded that the Cape of Good Hope is 1° west of Greenwich. + +With the improvements in accuracy of eclipse predictions concomitant with the advance of astronomical knowledge, it became possible to estimate very accurately the path of the Moon's shadow across the Earth's surface during a solar eclipse. Eclipses of the Sun, therefore, as well as those of the Moon could be used for longitudes throughout Europe. Perhaps the most noteworthy of these observations was that made by the famous navigator Captain James Cook. The account, part which is extracted from the *Philosophical Transactions* of the Royal Society, is of considerable interest. + +'Mr. Cook, a good mathematician and very expert in his business, having been appointed by the Lord Commissioners of the Admiralty to survey the coast of New Found Land, Labrador etc., to obtain such a good apparatus and instru- ments, among them a brass telescope quadrant by Mr. John Bird. + +'Being August 5th 1766 at one of the Burgeo Islands near Cape Ray Latitude 47° 36' 19", the south west extremity of New Found Land, and having carefully rectified his quadrant, he waited for the eclipse of the Sun.... [He] considered the eclipse to have begun at 00 h 04 m 46 apparent time (astro- + +184 +A HISTORY OF NAUTICAL ASTRONOMY + +nomical time] and to have ended at 03 h 45 m 26 s apparent time. +"Mr. George Mitchell had exact observations of the same eclipse taken at Oxford by the Rev. Mr. Hornby, and . . . from the comparison the difference of longitude of the places of observation, making due allowance for parallax and the Earth's spheroidal figure, was computed. . . . +Cook's place of observation became known as Eclipse Island and, according to his eclipse observations, its longitude was found to be 57° 36' 30" W. In 1874 the longitude of the same spot was found by means of electric telegraph to be 57° 36' 52", which speaks highly of the accuracy of the observations made by Cook and Hornby. +We are informed by Andrew Mackay, in his Theory of the Longitude of 1793, that the method of finding longitude from solar eclipse observations is: +". . . the most accurate of any that has hitherto been employed. +The difference of the meridians of two places may be found to the nearest second of time by comparing corresponding observations of the same eclipse." + +3. LONGITUDE FROM OBSERVATIONS OF JUPITER'S SATELLITES +A second astronomical method for finding longitude employed the satellites of Jupiter. Jupiter's four principal satellites were first observed in 1610 by Galileo, the famous mathematical professor of Padua. + +The orbits of the satellites of Jupiter are very nearly co-planar with the equator of Jupiter. The length of Jupiter's shadow cast by the Sun is about 600 times his diameter, whereas the distance of the outermost satellite is a mere 13 diameters from its parent planet. Consequently the satellites, in their orbital movements, are eclipsed. Jupiter's distance from the Sun is about six times the Earth's distance from the Sun, so that when Jupiter's orbit makes an angle of about only 13° to the plane of the ecliptic. Therefore, the times of eclipses of the satellites are almost un-affected by the location of an observer on the Earth. +* Jupiter has at least twelve satellites although only four are relatively bright ones which are plainly visible through a ship's long glass. + +**METHODS OF FINDING LONGITUDE** + +The satellite nearest to Jupiter is called the First Satellite; the next the Second Satellite, and so on. The First Satellite makes one orbital revolution in 42 hours. The periods of the Second, Third and Fourth Satellites are 85, 170, and 400 hours respectively. + +There are four different effects visible from the Earth: eclipses, occultations, transits of satellites, and transits of satellites' shadows. An eclipse occurs when a satellite enters Jupiter's shadow, and an occultation occurs when Jupiter himself hides a satellite. A transit of a satellite occurs when the satellite passes between the Sun and Jupiter, and its shadow appears as a tiny dark spot on the face of the planet. The entrance of the spot on the disc of Jupiter is called its ingress, and its leaving the disc, its egress. The transit of a shadow of a satellite occurs when the satellite lies on the straight line joining the satellite and the Sun. + +There being four principal satellites and four effects for each satellite, the frequency of occasions when observations of the satellites may be made for the purpose of finding longitude is high. + +Galileo, on discovering Jupiter's satellites, was quick to realize that the orbital movements of the satellites conformed to the planetary laws that had been enunciated by his famous contemporary Johannes Kepler. This demonstration provided compelling evidence that the planets revolved around the Sun, and that the other planets revolved around the Sun, for Jupiter and his attendant satellites could be regarded as being a small-scale model of the solar system. + +Galileo seized upon the idea that tables of the eclipses and occultations of Jupiter's satellites could provide a method for finding longitude—especially at sea. In response, therefore, to the handsome reward offered by King Philip III of Spain to any-one who could determine longitude at sea by observing the position of a ship when out of sight of land, Galileo set about the task of preparing suitable tables for the purpose. + +Galileo's tables of predictions were insufficiently accurate for the intended purpose; and it was not until after knowledge of the perturbations of the satellites, due to their mutual interactions, had been acquired that predictions were sufficiently accurate for finding longitude. + +The famous Italian astronomer J. D. Cassini applied himself + +186 +A HISTORY OF NAUTICAL ASTRONOMY + +to the problem of finding longitude by astronomical means and, in 1675, he prepared tables of predicted times of occultations and eclipses of the Moon and Sun. The method was found to give good results for finding the longitude at sea. + +The Danish physicist Roemer (1644-1710), in observing Jupiter's satellites at the Paris Observatory with the object of drawing up eclipse tables, discovered that predicted—and observed—times disagreed. He observed that the eclipses of the satellites were early compared with predicted times when Jupiter was relatively near to the Earth and late when he was relatively remote from it. This led him to the conclusion that light travels at a finite speed; and moreover, he was able to make an approximate estimate of this speed. + +In our own country Robert Hooke devoted some attention to drawing up tables of predictions of eclipses of Jupiter's satellites; but perhaps the most important work done in this connection was that of his celebrated Flamsteed. + +John Flamsteed (1646-1719) acquired an interest in astronomy at an early age. Some papers he had written on the subject attracted attention, and he was appointed a member of a committee set up to report on a proposed method for finding longitude at sea. He became the first Astronomer Royal after the establishment of the Royal Observatory at Greenwich in 1675; and he is often regarded as one of the greatest English astronomical observers, being instrumental in introducing many improvements in observing methods. His *Historia Coelestis Britannica*, in three volumes published in 1725, some years after his death, contains Flamsteed's catalogue of 3,000 stars. + +Flamsteed drew up tables of eclipses and occultations of Jupiter's satellites, and contrived an instrument + +'... whereby with the sole help of the usual catalogue and the table of parallaxes of Jupiter's orbit, their [the satellites'] distance from the axis of Jupiter may be found, to any given time within the compass of the year, and for any future year by the like tables.' + +Numbers 151, 154, 165, 177, 178 of the *Transactions* of the Royal Society of London appear under Flamsteed's name; and all pertain to the problem of finding longitude from observations of + +METHODS OF FINDING LONGITUDE +187 + +eclipses of Jupiter's satellites. Flamsteed confessed it as part of his design to make +'. . . our more knowing seamen ashamed of that refuge of ignorance, their idle and impudent assertion that the longitude is not to be found. . .' +He goes on to say: +'Such of them as pretend to a greater talent of skill than others, will acknowledge that it might be attained by observations of the Moon, if we had tables that would answer her motions exactly; but after 2000 years we find the best tables extant erring sometimes 12 minutes or more in her apparent place which would cause a fault of 1 hour or $\frac{1}{2}$ longitude. I undervalue not this method for I have made it my business to get a large stock of lunar observations for the correction of her theory as well as a ground-work before me, and I think I shall still find it a work of long time; and if we should afterwards attain what we seek, that it will be found much more inconvenient and difficult than that I propose by observing the eclipses of Jupiter's satellites.' +Flamsteed anticipated the seaman's objections to the method he proposed—and very real objections they were. The long telescope required for the observation would be almost unmanageable on board a lively ship at sea; and the difficulty there would be in distinguishing the satellites from one another: these were the principal faults of the method from the seaman's viewpoint. +Flamsteed pointed out the success the French had accomplished using the method, and they managed with telescopes of 14 feet long at most. He also remarked that '. . . the difficulty cannot be known until the trial is made', and that use renders many things easy which our first thoughts conceived impracticable. +Transaction number 214 of the Royal Society is entitled New and Exact Tables for the Eclipses of the First Satellite of Jupiter reduced to the Julian Stile and the Meridian of London. This was the work of Cassini who remarked that the great difficulty in finding longitude from observations of eclipses of Jupiter's satellites had been used for all the principal ports of France. Cassini had + +188 +A HISTORY OF NAUTICAL ASTRONOMY + +employed his skill to make easy and obvious to all capacities the calculations for finding longitude by this method. +Tables of eclipses of Jupiter's satellites were provided in the very first British Nautical Almanac, which made its first appearance in 1765 for the year 1767. Maskelyne, in his explanation to the tables, stated that: + +'The eclipses of Jupiter's satellites are well known to afford the easiest, and for general Practice, the best Method of settling the Longitudes of Places at land; and it is by their means prin- +cipally that Geography has been so much reformed within a Century past.' + +It had been hoped that means might have been found for pro- +viding a Telescope suitable for use on board ship for observing the +eclipses of Jupiter's satellites. Maskelyne described the trial he +made, during the years 1763-4, under the direction of the +Commissioners of Longitude in 1765, of a Telescope designed by a Mr Irwin for the purpose of facilitating the observations. He wrote: + +'... but I could not derive any advantage from the use of it . . . +and considering the great power requisite on a Telescope for +making these observations well, and the Violence as well as the +Irregularities of the Motion of the Ship, I am afraid the com- +plete Management of a Telescope on Shipboard will always +remain among the Desiderata.' + +Maskelyne hastened to add, however, that he would not be +understood to mean this discouragement attempts founded upon good principles to get over this difficulty. + +Many inventors since this time attempted to provide the means of a steady platform suitable for use on board a rolling ship from which observations of Jupiter's satellites could be made. Com- +mander Gould, in his work on the history of the marine chrono- +meter, mentions several of these inventions. He also describes the steam-driven gyroscopic platform which was proposed in 1858 for the Great Eastern that was built by Leviathan of Brunel. + +The inclusion of tables of 'Eclipses of Jupiter's Satellites,' and diagrams of 'Configurations of the Satellites of Jupiter' in the + +**METHODS OF FINDING LONGITUDE** + +189 + +Nautical Almanacs aimed to induce keen nautical observers, in the interests of geography, to ascertain the longitudes of foreign places they visited by observations of Jupiter's satellites at a time when better methods were not available. + +The method for finding longitude by eclipses of Jupiter's satellites involved observing the times of immersions (signifying the instants of disappearance of a satellite on entering the shadow of Jupiter), and emersions (signifying the reappearances of satellites on emerging from Jupiter's shadow). Comparisons of the times of observation with those given in the Nautical Almanac yielded the longitude of the observer. + +For practical use at sea the method suffered, not only from difficulties of observing, but also because the eclipses do not occur instantaneously, this being due to the apparent diameters of the satellites not being appreciable. Atmospheric effects also may affect the observations. Moreover, Jupiter is often near the Sun on the celestial sphere; and for long periods the method is not available at all. The difficulty was recognized by many for observation. Many writers advocated the method for sea use; but practical seamen, and others who appreciated the difficulties of observing from a ship at sea, held little esteem for the method. + +The French astronomer Lagrange (1736-1813) is credited with founding the dynamical theory of Jupiter's satellites; and the French astronomer Delambre (1749-1822) is credited with the discovery of a remarkable numerical relationship between the satellites resulting from their mutual attractions. + +The Swedish astronomer Wargentin, Secretary to the Royal Academy of Sciences at Stockholm, is noted for his tables of eclipses of Jupiter's satellites. His tables were published in the British Nautical Almanac for 1779 and for many succeeding years. From 1824 the predictions of the eclipses of Jupiter's satellites derived from Delambre's tables, which from 1830 onwards were derived from D'Alembert's Table Eclipses des Satellites de Jupiter. + +**4. LONGITUDE FROM OBSERVATIONS OF MOON OCCULTATIONS** + +The next astronomical method for finding longitude at sea, which is to demand our attention, is that in which star occultations by the Moon are employed. + +190 +A HISTORY OF NAUTICAL ASTRONOMY + +The Moon, because of her real motion around the Earth, appears to move across the celestial sphere at a relatively rapid rate towards the east. This retrograde motion of the Moon is, on an average, about 1° per hour. Her angular diameter is also about 1°; so that a star which lies within 1° of the Moon's path will be hidden by the Moon—a phenomenon known as a star occultation. The fact that the Moon has no atmosphere results in an occultation of a star taking place instantaneously. + +The interest which ancient astrologers gave to occultations, particularly those of fixed stars, must surely have led to the suggestion of the use of occultations for finding longitude. + +The accuracy of predictions of occultations of stars or planets by the Moon depends largely upon knowledge of the complex motion of the Moon. It was not until the Moon's motion had been reduced to a reliable rule that accurate predictions became possible. + +The method of finding longitude from an observation of an occultation of a fixed star is reckoned as the best means of finding longitude by astronomical means. The usefulness of the method is increased by the high frequency of star occultations; but the complex computations associated with the method renders it impracticable for finding longitude at sea. + +The parallels of latitude between which particular stars cannot be occulted are tabulated in the early Nautical Almanacs. From this information an observer could ascertain whether or not the phenomenon would occur at his position. + +During the time when the Moon is waxing, that is to say from New Moon to Full Moon, a star lying in the Moon's path will be occulted on the darkened limb of the Moon, because the enlightened edge, which faces the Sun, will be directed to the west, that is in the opposite direction to that in which the Moon is moving. During this period, when the illumination on the Moon is waning, that is during the fortnight from Full Moon to Change of the Moon, a star lying in her path will be occulted at the enlightened limb. + +To find the longitude from an observation of an occultation, the latitude of the observer and the local mean time of the observation must be known. The local mean time was usually ascertained before or after the occultation observation from an observation of the Sun or other object on or near the prime + +**METHODS OF FINDING LONGITUDE** + +191 + +vertical circle. The G.M.T. of the observation was also to be estimated as accurately as possible. + +The problem is one of finding the Right Ascension of the Moon at the time of the observation. It is a problem of great complexity, chiefly on account of the effects of refraction, semi-diameter of the Moon, and parallax of the Moon. After the Moon's R.A. has been computed—the star's R.A. being known and facilitating the operation—it is a simple matter to find the G.M.T. of the observa- tion by interpolation, using the nearest tabulated values of the Moon's R.A. from the Nautical Almanac. Should the computed G.M.T. differ materially from the estimated G.M.T. of the observation, the problem should be re-worked. + +5. **LONGITUDE BY LUNAR TRANSIT OBSERVATION** + +Another lunar method for finding longitude involved finding the local time of the Moon's transit, and comparing it with the time of transit at a prime meridian. This method appears to have been first suggested in 1678 by Herne in a book entitled *Longitude Unveiled*. + +The Moon, because of her retrograde motion across the celestial sphere relative to the fixed stars, crosses the meridian of a station- ary observer later each day by a variable amount of time known as the Moon's *retardation*. Predictions of the times of the Moon's meridian passage for any given latitude and longitude are found by adding to or subtracting from the standard meridian if the local time of meridian passage could be found. The unsuitability of this method for finding longitude is due to the difficulty of finding the exact local time of the Moon's transit. Numerous suggestions have been made for ascertaining local time of Moon's transit, chiefly using equal altitudes on opposite sides of the meridian; but the method has never been brought to a state whereby accurate longitudes could be found. + +An interesting method for finding longitude by means of com- bined altitudes of the Moon and a fixed star was described by John Maurice of Chicago as late as 1900. Maurice's method in- volved measuring the altitudes of the Moon and a fixed star and finding therefrom their local hour angles. The sum or difference of these is equivalent to the difference between their Right Ascen- sions. The R.A. of the star being known, that of the Moon at the time of the observation may, therefore, be found. Having found + +192 +A HISTORY OF NAUTICAL ASTRONOMY + +the Moon's R.A., the G.M.T. may readily be found from the table of the Moon's R.A. given in the Nautical Almanac. + +The method proposed by Maurice is an extension of that pro- +posed by Lemonnier in 1771. Lemonnier proposed finding the +longitude from the hour angle of the Moon. The hour angle, ob- +tained from an altitude and the latitude, enables the observer to +find the sidereal time of the observation from which the Moon's +R.A. may be found. + +6. REWARDS FOR DISCOVERING THE LONGITUDE + +The problem of the longitude, until it was solved in the 18th cen- +tury, engaged the attention of many able astronomers, mathema- +ticians and physicists. The problem of "east-west" navigation, as +it was sometimes called, was one which kept to the fore, largely +as a result of the handsome rewards that were offered from time to +time to anyone who solved the problem. To stimulate competition for +the prize, King Charles II of England issued a Royal Proclamation in +order to give financial support to experimenters and investigators +of this problem of the age. We have mentioned the reward offered +by King Philip III of Spain in 1749. Other rewards were offered +by the governments of France and Venice; and some private indi- +viduals offered prizes for the discovery of the longitude at sea as well. +The most valuable, and the most famous of the prizes, was the +considerable sum of £200 awarded by British Parliament in 1715. This prize appears to have been the only one that was ever paid for the discovery of the longitude. + +Shortly before the passing of the Act of Parliament (12 Anne, +Cap. 15), in which a reward was offered to any person who should +invent or discover a practical method for finding longitude at sea, +a committee was set up to investigate a petition submitted to +Parliament in March 1714 by "Several Captains of Her Majesty's +Ships, Masters and Mariners" complaining about their distresses. +The petition, which was engineered by William Whitson, a dis- +senting clergyman who had held the Lucasian Chair of Mathe- +matics at Cambridge, and Humfrey Ditton, who was a teacher of +mathematics at Christ's Hospital School, set forth: + +"That the discovery of longitude is of such consequence to +Great Britain, for the safety of the Navy, for Merchant Ships, +as well as for improvement of trade, that for want thereof many + +**METHODS OF FINDING LONGITUDE** + +ships have been retarded in their Voyages, and many lost; but if due encouragement were proposed by the public, for such as shall discover the same, some persons would offer themselves to prove the same before the next petition was made, in order to their own satisfaction, for the safety of many lives, her Majesty's Navy, the increase of Trade, and the shipping of these Islands, and the lasting Honour of the British Nation.' + +Whiston and Ditton had proposed an impracticable method for finding longitude at sea which was set out in a pamphlet entitled *A New Method for discovering the Longitude* published in 1714. Their proposal was to use pyrotechnic sound signals fired from vessels permanently moored at precisely defined positions along the oceanic trade routes. Following the publication of their proposal, and probably in order to give it publicity, Whiston and Ditton were instrumental in causing the petition to be submitted to Parliament. + +A committee, which was appointed to examine the petition, consulted the eminent mathematicians and astronomers of the day, amongst whom were Sir Isaac Newton, John Flamsteed and Edmund Halley. + +Newton, in his evidence, reviewed the several methods for finding longitude at sea that had, up to the time, been proposed. He believed that the use of Jupiter's satellites, and held out little hope that the Moon could be employed for finding longitude. He also pointed out that the proposal made by Whiston and Ditton was merely a method of keeping an account of, rather than finding, the longitude at sea. + +The famous bill which was passed following the adoption of the resolution formulated by the committee states + +'... that nothing is so much wanted and desired at sea as the discovery of the longitude.' + +Under the terms of the Bill, commissioners for examining, trying and judging all proposals, experiments and improvements relating to the problem of finding longitude at sea, were to be appointed. The commissioners, who became known as the Board of Longitude, were empowered to grant sums of money for experiments, and to determine the degree of exactness of any proposal. The reward offered for the discovery of the longitude was: + +A historical document page with text about methods of finding longitude. + +194 +A HISTORY OF NAUTICAL ASTRONOMY + +£10,000 for any method capable of determining longitude to an accuracy of 1°. +£5,000 for a method capable of determining longitude to an accuracy of 40'. +£2,000 for a method capable of determining longitude to an accuracy of 30' or 4". + +12 Anne, Cap. 15 stimulated not only eminent men of science but also numerous cranks and crackpots who put forward impracticable, or sometimes ridiculous, proposals. Several pamphlets published in 1714 seem to have had been based on methods of finding longitude at sea, bear testimony to this. There were many who believed that the wit of man would never reach a stage rendering it possible to find the longitude at sea. The phrase "discovery of the longitude" entered common speech at the time, and was used to describe any practical impossibility. Nevertheless the discovery was to be made. Two methods, both of which had been proposed by the end of the 17th century and into the 18th century, were brought within the bounds of practical utility at about the same time during the 18th century. These methods for finding longitude at sea are referred to as the methods of 'longitude by timepiece' and 'longitude by lunar distance,' respectively. + +We shall say little about the development of the mechanical timepiece which made possible the solution to the longitude-by-chronometer problem. This method uses a clock which keeps accurate time for finding longitude has been known since by Gemma Frisius, the famous Flemish astronomer. The proposal had been made, not as a practical suggestion, but as a theoretical possibility, in a work entitled De Principiis Astronomiae et Cosmographiae which was published in Antwerp in 1530. In this same work Gemma gave several nautical axioms, included amongst which was a description of a nautical quadrant. At the time Gemma's ideas were not accepted and his portable watches, using a spring as a prime mover, were of recent invention. His proposal was to be dormant for about two hundred years before the art and science of clock-making had been perfected to render possible the production of a portable watch that could keep time on a moving ship with an accuracy sufficient to find the longitude with a reasonable degree of accuracy. The credit of the invention of the marine chronometer belongs to John + +**METHODS OF FINDING LONGITUDE** + +195 + +Harrison, the Yorkshire carpenter who carried off (although not without a great deal of trouble on his part) the £20,000 prize offered by the British Parliament. + +The history and development of the marine chronometer has been studied in great detail by the late Commander R. T. Gould, the fruits of whose painstaking researches appear in a work published in 1923. More recently, in 1966, Colonel H. Quill in his book *John Harrison: the Man who found Longitude*, has provided his readers with a penetrating study of the life of John Harrison set against the background of the fascinating story of the development of the marine chronometer. + +7. **LONGITUDE BY LUNAR DISTANCE** + +(a) **HISTORICAL SURVEY** + +We now come to a discussion of the method for finding longitude at sea by lunar distances. To open our discussion we shall quote part of the foreword to Gould's *Marine Chronometer*, contributed by Sir Frank W. Dyson, the Astronomer Royal at the time the work was published. + +'The problem of making an almanac of the Moon's position,' wrote Sir Frank, 'is most difficult, as may be seen from the fact that in spite of the attention devoted to the lunar theory by some of the world's greatest mathematicians, it was not until 1767 that the Astronomer Royal was able to give predictions of the Moon's place with sufficient accuracy for practical use for purposes of navigation. From that time to the present day, distinguished mathematicians of England, France, Germany and America have given large portions of their lives to lunar theory. More arithmetic and algebra have been devoted to it than to any other question of astronomy or mathematical physics, but, in the end, the problem has been solved so that observations can be made which will enable one to find longitude within 20 miles.' + +Three important points emerge from Sir Frank's comments. First, the difficulty of predicting the Moon's place; second, the + +A diagram showing a lunar eclipse. + +196 +A HISTORY OF NAUTICAL ASTRONOMY + +difficulty of observation, which limits the degree of accuracy of the method even when perfect predictions are available; and third, the elaborate calculations involved with this method. + +The difficulties of observation were largely overcome by the invention and development of the reflection measuring instruments which have been discussed in Chapter III. Our main purpose in the pages that follow immediately will be related to an historical account of, and the computations involved in, the lunar method for finding longitude at sea. + +The Moon's motion across the background of the fixed stars is more rapid than that of any other visible celestial body. She completes her circuit of the Heavens in a sidereal period of about 274 days. She moves, therefore, at an average rate of 35° of arc in every minute of time. When she is at perigee her rate of travel is greatest and is about 36° per minute: when she is at apogee her rate of motion is slowest and is about 30° per minute. It is the Moon's rapid motion relative to the stars which provides a method for finding absolute time. The Moon's motion relative to the Sun is at the hand of a celestial clock, so that "the mechanism of the Heavens" provides him with a method for measuring absolute time. + +The principle of the method of finding longitude by lunar distance is simple. The angle at the Earth's centre between the Moon's centre and that of the Sun, planet or star, may be predicted if we know their positions at some previous epoch. A measured angle between the Moon and Sun (or other heavenly body), or lunar distance as it is called, may be reduced to what it would have been at the Earth's centre at the time of the observation. The reduced measured distance, when compared with the predicted or geocentric distance, provides a measure of the longitude of the observer east or west of the meridian for which the predictions apply. + +The principle of finding longitude from an observation of the angular distance between the Moon and Sun, planet or zodiacal star, could hardly fail to suggest itself to an astronomer possessing a clear notion of the longitude problem and an understanding of the character of the Moon's motion relative to the fixed stars. The credit for being the first to suggest the method in print is given to Johannes Werner of Nuremberg who mentioned it in the first volume of Ptolemy's Geography which Werner edited in 1514. Werner not only suggested the method but also suggested the use + +METHODS OF FINDING LONGITUDE 197 + +of the cross-staff 'as a very proper instrument for observing lunar distances.' A century later, in 1615, a lunar observation was made by the English explorer William Baffin with the express purpose of finding the longitude of a ship at sea. + +Baffin is credited with being the first Englishman to make a lunar observation at sea. Baffin sailed in 1615 as 'mate and associate' of Robert Bylot, who commanded the *Découverte* on a voyage in quest of the North-west Passage. When the ship was beset in ice off the coast of Greenland, Baffin made lunar observations using the Moon and Sun. The altitudes of the two bodies were measured by means of a cross-staff, and the angle between them was found by means of azimuth observations. This method of finding the lunar distance does not lend itself to great accuracy but, doubtless, was employed in the absence of a suitable instrument capable of measuring large angles. It is not unlikely that the methods of longitude by lunar distances were known to many of the most eminent navigators and astronomers of the period. Peter Apian, in the tenth chapter of his *Cosmographia* published in 1545, and the famous Gemma in his De Principiis, mentioned above, described the method of longitude by lunar distances. + +Werner, Apian, and Gemma Frisius, treated the problem of finding longitude by lunar distance as more of a speculative, rather than a practical, method. At the time when they wrote, neither the cross-staff nor any other single measuring instruments were in existence to make the problem one of practicable possibility. + +Gemma Frisius, in his description of the method, discussed the refraction and parallax corrections that are necessary to apply to the observed lunar distance before it can be compared with a predicted distance in order to find the longitude of the place of observation. He also pointed out that this method is not suited for refraction and parallax is a trigonometrical process far too complex for practical seamen. There is no doubt, judging from his descriptions, that Gemma had a perfectly clear understanding of the lunar problem. Two centuries, however, were to elapse before the lunar method became practicable for finding longitude at sea. + +The principal factor responsible for the delay in the solution to the lunar problem was that longitude was the rudimentary state of lunar theory. This was brought to a state of near-perfection in the middle of the 18th century. + +14 + +108 +A HISTORY OF NAUTICAL ASTRONOMY + +The celebrated Kepler regarded the lunar method for finding longitude in favourable light and, in his Rudolphine Table, he gave directions for observing the distance between the Moon and a star. He also gave directions for making the necessary computations. The method was also recommended by Christian Severin (Longomontanus) (1562–1647), the Danish astronomer and assistant to Tycho Brahe. Master Thomas Blundeville, in his famous *Exercies . . . for Young Gentlemen* . . . first published in 1594, also mentioned the method of the lunar distance, attributing it to Peter Apian. + +We find in 1633 the French professor of mathematics at Paris, Jean B. Morin, proclaiming with much boasting that he had found the solution to the longitude problem. Morin requested Cardinal Richelieu to appoint a commission before whom he [Morin] could have the opportunity of establishing his claim to the honour of finding the solution to the problem and aid of establishing his right to any share of the prize money awarded thereto. The commission was appointed and, in 1634, Morin demonstrated the observations and mathematical computations necessary for finding the longitude by lunar distance. Morin is credited with being the first to provide detailed mathematical rules requisite for the solution to the problem; but, of course, he did not bring the problem any nearer to a practical solution. The remarks made by Captaine de Fréville M. E. Guyot, a member of the French Bureau of Longitude, are interesting and relevant: + +‘The solution,’ wrote Guyou in 1902, ‘required nothing less than the genius of Newton, Clairaut and Euler, and the mass of observations patiently collected for three quarters of a century by Flamsteed, Halley, Lemonnier, and others.’ + +Morin had simply limited himself to an investigation of the spherical trigonometrical computations necessary for the solution of the lunar problem. These calculations, although complex in character, presented no real difficulty and had, hitherto, been passed over on account of their entire want of utility. + +In a work on geography by Carpenter, printed at Oxford in 1635, notice is taken of a method of finding longitude by lunar distances. This is noted by Andrew Mackay in his *Theory of the Longitude*, where he quotes Carpenter as saying: + +A page from a historical text discussing nautical astronomy. + +**METHODS OF FINDING LONGITUDE** + +*This way, though more difficult may seem better than all the rest, for we may as an Eclipse of the Moon seldom happens, and a Westerly Climate, which is rare so will be pre- +served, or at least so well observed in so long a Voyage, whereas +every Night may seeme to give occasion to this Experiment, if +so bee the ayre be freed from Clouds, and the Moone show her +Face above the Horizon.* + +The first effectual step made to promote the solution to the +problem of finding longitude at sea by astronomical methods was +the establishment of the Royal Observatory at Greenwich in 1675, +and the appointment of Flamsteed as the first Astronomer Royal. +Flamsteed's commission enjoined him: + +'. . . to apply himself with the utmost care and diligence to the +rectifying of the tables of the Motions of the Heavens and +the places of the fixed stars, in order to find out the so-much desired +longitude at sea, for perfecting the art of navigation.' + +Up to the time of the setting up of the Royal Observatory at +Greenwich the lunar tables in use were compiled entirely from the +results of observations, and astronomers despairred of ever being +able to predict with certainty the Moon's celestial position, her +movement through the heavens, and her place among the fixed +anxics, rendered possible by the illustrious Newton whose famous +*Principia* was published in 1687, undertaken by several eminent +17th-century astronomers, the theory of the Moon's motion was +gradually brought to a state of perfection. Newton himself under- +took the construction of tables of the Moon's position founded on +the theory of gravitation combined with the observations of Flam- +sted. And Professor Hoyle, in a recent book, *Astronomy 1962*, remarks that '...it is not easy to understand how such a prediction of the Moon ... is', it difficult, and it is the only problem that ever +made my head ache.' + +Edmund Halley, who had had valuable sea-going experience +and who was, therefore, admirably qualified to talk on the matter, +recommended observations of the Moon as providing the most +certain method of finding longitude at sea. He had found, from +experience, that all other methods were impracticable; but he also +pointed out to the Royal Society the defects of the lunar tables + +199 + +200 + +A HISTORY OF NAUTICAL ASTRONOMY + +extent. Halley published an empirical method for reducing the errors in the existing tables. This method he evolved after making careful comparisons of the Moon's motions by observations with those given in the tables. Halley found, however, exceptions from his application of the principles of gravitation, that the errors of the tables recurred with regularity after a period of eighteen years eleven days—a well-known eclipse cycle known to the astronomers of antiquity as the Saros. Halley's paper on the subject appears under No. 420 of the Philosophical Transactions. + +Being encouraged by his discovery of the recurrence of error after a period of eighteen years, Halley determined what error might arise in a period of nine years less nine days, during which period 111 lunations occur. In this manner he deduced a rule for correcting the lunar tables. Unfortunately, Halley was unable to extend his lunar observations over the whole of the eighteen-year period until he succeeded Flamsteed as Astronomer Royal in 1720. The fact that he was well over sixty years of age at the time of his appointment, and that he drew from undertaking the long series of lunar observations with the object of improving lunar theory. In 1731 he announced to the Royal Society that he had taken + +"... with my own eye without any assistant or interruption... 1500 observations of the Moon... more than Tycho, Hevelius and Flamsteed had taken altogether." + +He lived to see the completion of his tremendous project. +Halley discussed in detail the lunar method for finding longitude at sea in his Astronomical Tables in which he gave two complete examples using the distances between the Moon and two different stars. He mentioned, in the same work, that the method is applicable also to determining latitude by the Sun, instead of a fixed star, during the First and Last Quarter of the Moon. + +The French astronomer Lemonnier undertook the task of making an eighteen-year period of lunar observations from which he drew up a table of corrections for the Moon's motion based on Halley's principle. + +The Abbé de la Caille recommended the lunar method as being the only practical method for sea use. In the course of a voyage to the Cape of Good Hope in 1751, de la Caille had used the + +METHODS OF FINDING LONGITUDE +201 + +method for finding longitude. During the same year a French merchant captain, d'Apres de Mannevillette, in the service of the Compagnie des Indes, employed the lunar method using lunar tables based on Halley's rule. +Halley's empirical rule for correcting the Moon's position was not entirely satisfactory; and it was generally believed by astro- +nomers that the inequalities of the Moon's motion were so com- +plex that perfection in the lunar theory was not possible. Some +argued, falsely, that the principle of gravitation was insufficient to +explain the inequalities that affected the Moon's motion. +In 1735, Clairaut published a paper in which he proposed a competition +the object of which was related to lunar theory. The French philo- +sopher Clairaut presented a new theory of the Moon's motion +accompanied by skeleton tables. But, because the work had to be +submitted before a fixed date, the tables were not only incomplete, +but they had a degree of accuracy less than that for which Clair- +aut's theory was susceptible. The comparison of Clairaut's tabu- +les with those of Newton, Halley, and Cassini and others exhibited errors of up to as much as 5° of arc. + +The famous Swiss mathematician Leonhard Euler (1707– +1783), whose work on lunar theory *Theorie Motus Lunae* was +published in 1753, in an appreciation of the work of Clairaut, +published in the *Philosophical Transactions of the Royal Society* +for 1755, wrote: +'. . . It is Clairaut we have to thank for this important discovery +which adds fresh lustre to the theory of the great Newton, and now for the first time, we may hope to have good astronomical +tables for the Moon.' + +Clairaut corrected his lunar tables in 1765, the year of his +death; but ten years before this date Johannes Tobias Mayer had compiled lunar tables which were submitted to the British Parlia- +ment, the author claiming at the same time some reward which +he thought he might merit. + +Tobias Mayer (1723–1762) was a self-taught German mathe- +matician who, in 1751, was appointed Professor of Mathematics +at Göttingen. His fame rests primarily on his skillful develop- +ment of Euler's work in connection with lunar theory. His lunar tables +were communicated in 1752 to the Royal Society of Göttingen. + +203 +A HISTORY OF NAUTICAL ASTRONOMY + +Three years later he submitted an amended table in manuscript to the English Board of Longitude. +Mayer's lunar tables fell into the hands of Dr. James Bradley, who had succeeded John Hadley as Astronomer Royal in 1742. Bradley, after examining the tables and comparing them with a large number of observations of his own, was convinced of their accuracy. In his report to the Admiralty dated February 10th 1765 Bradley wrote: + +'...In obedience to their Lordships' commands I have examined the same, and carefully compared several observations that have been made (during the last five years) at the Royal Observatory at Greenwich, with the places of the Moon computed by the said tables. In more than 230 comparisons which I have already made, I did not find any difference so great as 1° between the observed longitude of the Moon and that which I computed by the tables; although the greatest difference which ever occurred was but a small quantity, yet as it ought to be considered as partly arising from the errors of my observa- +tions, and partly from the errors of the tables, it seems probable that, during this interval of time, the tables generally gave the Moon's place true within one minute of a degree. + +"A more general comparison may perhaps discover larger errors in those which I have hitherto met with being so small, that even the largest could occasion an error of but little more than half a degree; but still it appears evident that the tables of the Moon's motions are exact enough for the purpose of finding at sea the longitude of a ship, provided that the observa- +tions that are necessary to be made on shipboard can be taken with sufficient exactness." + +Long before Bradley had submitted this report he had shown the utility of Mayer's tables from investigation of the trials of the tables carried out by Captain Campbell on board the Royal George cruising within sight of Cape Finisterre in 1757, and again in sight of Ushant in 1758 and 1759. Observations made on board the Royal George were accurate to within 0° 37' of longitude. The lunar distances on these occasions were measured by means of Hadley's quadrant. + +With the completion of the trials of Mayer's tables the problem + +**METHODS OF FINDING LONGITUDE** + +203 + +of finding longitude at sea, to a degree of accuracy within the con- +ditions of precision laid down by the Act of Parliament, had been solved. +It might, therefore, be thought that Mayer was entitled to the reward. It was argued, however, that the method of calculating the latitude by the lunar method was far too complex for practi- +cal use; and since the solution was not given in a form available for the non-mathematical seaman, the practical conditions of the Act were not fulfilled. For this reason the Board of Longitude were led to postpone the granting of the prize. + +The Board of Longitude existed from 1714 until 1823. It was empowered to award the prize on the discovery of the longi- +tude as soon as a majority of the Commissioners agreed that a proposed method was practicable and useful. The remainder of the reward was to be paid as soon as a vessel, on which the method was used, sailed from Britain to a port in the West Indies and back without erring in the longitude more than the amount specified in the terms for prize. It was not the lunar method that was instrumentally responsible for this result; but those who were responsible for perfecting the method did receive rewards. The prize was carried off, as we have already noted, by John Harrison for his ingeniously-conceived chronometer. + +The perfection of lunar tables in itself was not sufficient for finding longitude at sea by the lunar method. Accurate measuring instruments and, perhaps more important than even this, methods of computing which could be reduced to relatively simple rules for all comers were essential before the method was to become practicable. + +(b) **MASKELYNE AND THE NAUTICAL ALMANAC** + +Before entering upon a discussion of the methods of computation of the lunar problem, we shall first describe the development of that essential instrument of nautical astronomy known as the Nautical Almanac. Its author was Ephraim Maskelyne, with special reference to its founder Nevil Maskelyne. + +The earliest of the principal astronomical ephemerides is the +*Comnaissance des Temps ou des Mouvements Celeste*, founded by Jean Picard in 1679 and published at Paris under the auspices of the Bureau de Longitude. From 1761 onwards the *Comnaissance des Temps* gave the celestial positions of the Moon at twelve-hourly intervals, these positions being calculated by Mayer's + +204 +A HISTORY OF NAUTICAL ASTRONOMY + +method. Lunar positions were given using coordinates of the ecliptic system, that is, celestial latitude and celestial longitude respectively. + +A considerable amount of computation is necessary in order to find the longitude from an observation of a lunar distance when the only assistance provided by an ephemeris consists of predicted celestial latitudes and celestial longitudes of the Moon at twelve-hourly intervals. The general practical method, in which much of the computation is dispensed with by providing the seaman with a special 'nautical' almanac, was first proposed by Abbé de la Caillie. This almanac gives the difference between the Moon's centre and that of the Sun and planets and their distances for short equidistant intervals of time. A specimen of this type of lunar table, specially designed for the seaman's use, appeared in the 1761 *Connaissance des Temps*, in which lunar distances were given for four-hourly intervals. Guyot, in his article of 1902, mentions this important fact, and adds: + +*But as has too often happened in our country [France] in connection with the application of science to the art of navigation, this proposal failed to bear fruit. It was the English astronomer Maskelyne who had the honour of carrying out the idea.* + +Nevil Maskelyne, the 'Father of the Lunar Observation', was born in London in 1758. He was educated at Westminster School and Trinity College, Cambridge. He graduated in 1754 and the following year was ordained. In 1761 he was deputed by the Royal Society to observe the transit of Venus at St Helena. He took with him a reflecting quadrant of Hadley's made by the famous instrument-maker Bird with the glass ground by Dollond; and a set of Mayer's tables. During the voyage out and home he determined the longitude of ship using the lunar method. From St Helena he sent back to England a report on his observations (*Philosophical Transactions*, Vol. 52, 1762). He considered that his observations yielded the longitude, in each case, to within $1\frac{1}{4}$° of the truth. On his return to England he set about the task of publishing a work which was to play a prominent part in the development of practical and scientific navigation. This work, entitled *The British Mariner's Guide*, was published in 1763. + +**METHODS OF FINDING LONGITUDE** + +205 + +Maskelyne spoke in high praise of the lunar method for finding longitude at sea. He induced the Commissioners of Longitude to grant the concession that a collection of lunar distances between the Moon and certain bright zodiacal stars, along the lines suggested by Sir Isaac Newton, would suffice for this purpose. + +In 1765 Maskelyne succeeded Nathaniel Bliss as Astronomer Royal and, in his new position of authority, was able to put his proposal into effect. Under Maskelyne's guidance the first British Nautical Almanac and Astronomical Ephemeris was published in 1765 for the year 1767. The method of finding longitude by lunar distance was to become the standard astronomical method for finding longitude at sea and was to remain so for the whole of the 19th century. + +Manuscript tables of the Sun and Moon, drawn up by Tobias Mayer, were received by the Board of Longitude in 1763, the year following Mayer's death. These tables were more accurate and more extensive than the original tables that had been submitted in 1752. Under Maskelyne's superintendence Mayer's lunar tables were revised in 1768. + +The Board of Longitude awarded the sum of £3,000 to Mayer's widow for his lunar tables. The celebrated Euler, who had furnished the theorems used by Mayer, was granted a reward of £300. + +At the same time as the first Nautical Almanac appeared, Maskelyne published a set of tables entitled Tables Requisite to be used with the Nautical Ephemeris for finding the Latitude and Longitude at Sea. In the Requisite Tables this work familiarly became known—two excellent methods, with examples, for finding longitude by the lunar method, were described. + +(c) **PRINCIPLES AND PRACTICE** + +We shall now describe in some detail the principles and practice of the lunar method for finding longitude at sea. + +In order that observations may be made in a regular and accurate manner, three assistants, in addition to the principal observer, are required. The principal observer measures the angle between the Moon's enlightened limb and a star, planet, or the Sun's limb. The first assistant measures the Moon's altitude; the second assistant measures the altitude of the second body; and the third assistant, armed with a watch, records the times of the observations. The three required angles should be measured + +206 +A HISTORY OF NAUTICAL ASTRONOMY + +simultaneously: the principal observer having 'stand-by' a little before each observation to be made; and 'stop' when he obtains perfect coincidence of the enlightened limb of the Moon and the star, planet, or Sun's limb. The observations must be repeated so often that no error is obtained, each set being observed at approximately equal intervals within the space of six or eight minutes. The mean of each particular series of observations is then found: that is to say, the sums of the lunar distances, each of the two sets of altitudes, and the times should be divided by the number of sets. By so doing, small errors of observation are eliminated. + +An expert observer might be capable of making, with a tolerable degree of accuracy, all the necessary observations himself. The normal way in which a single observer would operate would be to take several altitudes of the Moon and the second object in quick succession. Then several lunar distances would be observed; and finally several altitudes of the Moon and second object observed again. The mean of each set of observations would then be found, and the sum of the altitudes reduced to the time of the mean of the distances. + +The altitudes of the Moon and the second object are required in order to ascertain the exact values of refraction and parallax-altitude for the Moon, and refraction for the second body. Moreover, the time of the observation must be known with tolerable accuracy in order to ascertain the declination and Right Ascension of the Moon (or star) at that time; but since these are unknown altitudes where it is not possible to measure the altitudes of the Moon and second object at the time the lunar distance is measured, these angles must be computed by solving the appropriate PZX triangles. + +In Fig. 1, R represents the zenith of an observer, and HO represents that part of his horizon contained between the vertical circles through the Moon and second body (assumed to be a star). + + + + + + + + + + + + + + + + + + + + + + +
arc MOtrue altitude of Moon's centre
arc ZMtrue zenith distance of Moon's centre
arc SHtrue altitude of star
arc ZStrue zenith distance of star
arc MSgreat-circle arc through true positions of Moon's centre and star; that is true lunar distance
+ +METHODS OF FINDING LONGITUDE 207 + +The effect of atmospheric refraction on light from a star is to make it appear to have an altitude greater than its true altitude. Because of the star's immense distance from the Earth its parallax-in-altitude is considered to be nil. The elevating effect of refraction is represented by the arc Zs, which is drawn in the same vertical circle as that through its true position. The apparent position of the star is represented in Fig. 1 at s. + +Refraction has an elevating effect on the Moon but parallax-in-altitude of the Moon is always greater than refraction: so that the + + +A diagram showing the apparent position of a star (M) above the observer's horizon (H). The diagram includes labels for various points and lines: +- S: A point representing the star's apparent position. +- M: The star itself. +- H: The observer's horizon. +- O: The observer's position. +- Z: The zenith. +- m: The moon. + + +FIGURE 1 + +depressing effect of parallax is greater than the elevating effect of refraction. Hence the apparent position of the Moon has a smaller altitude than that of the Moon's true position. The apparent position of the Moon is represented in Fig. 1 at m. + +arc mO apparent altitude of the Moon's centre +arc Zm apparent zenith distance of Moon's centre +arc sH apparent altitude of star +arc Zs apparent zenith distance of star +arc ms great-circle arc through the apparent positions of Moon and star; that is the apparent lunar distance + +The apparent lunar distance of a star or planet is found by applying the Moon's semi-diameter to the measured lunar distance. The apparent lunar distance of the Sun is found by applying a combination of the augmented semi-diameters of the Moon and + +208 +A HISTORY OF NAUTICAL ASTRONOMY + +Sun: the combination being dependent upon which limbs of the Moon and Sun are observed. If the adjacent limbs are observed, the apparent lunar distance is found by applying the sum of the semi-diameters of the Moon and Sun to the observed distance, etc. + +(d) METHODS FOR CLEARING THE DISTANCE + +When finding longitude by the lunar method, the most tedious part of the process is that which is known as clearing the distance. This involves reducing the apparent lunar distance to the true lunar distance, and this is done by assuming the apparent distance from the effects of parallax and refraction, and then calculating the angle at the Earth's centre between the directions of the Moon's centre and the star, planet or Sun's centre. Many eminent astronomers and mathematicians, and many lesser men too, have given commendatory rules, table or diagrams, to facilitate the problem of clearing the lunar distance. Amongst the numerous methods that have been published at different times, are those given by Charles de Borda, Abbe Jean-Baptiste Léonard de la Chambre, Lyman, Witchell, Dunthorne, Kraft and Airy. As long ago as 1797 Meno- doza del Rios, in a paper published in the Philosophical Transactions of the Royal Society, described forty different methods of clearing the lunar distance; and during the following hundred years numerous other methods were proposed, and many of the earlier methods were modified. + +The methods available for clearing the lunar distance reduce themselves into two principal types; and the several rules given by different investigators are obtained by using different trigonometrical transformations of the common fundamental spherical trigonometrical formulae. + +The first type to be described provides a rigorous solution to the problem and no approximations are used in the process. The solution is obtained by means of three fundamental formulae: ZM = mZa + ma; ZS = mSa + ms; and mS, all of which are deduced from observations made on the Earth's surface, the angle at the zenith contained between the vertical circles through m and s is calculated. Using the calculated angle mZa, and the true zenith distances ZM and ZS, the true distance MS is readily found. Typical of the methods based on this type of solution are those of Borda, Delambre, Kraft, Young and Airy, each of which we shall describe. + +METHODS OF FINDING LONGITUDE 209 + +Borda's Method for Clearing the Distance + +Chevalier Jean Borda, the French mathematician and astronomer, was born at Dax in 1733. He entered the French Navy and busied himself with nautical astronomical investigations for which he became famous. His work on lunar distances was pub- +lished in 1787, and to which we have made reference in Chapter III, +that Borda described his method of clearing lunar distances. At the time of its introduction to navigators, and for many decades afterwards, Borda's method was considered by competent authori- +ties to be the best. + +Referring to Fig. 1: + +In triangle ZSM: +$$\cos Z = \frac{\cos SM - \cos ZS \cos ZM}{\sin ZS \sin ZM}$$ +i.e. +$$\cos Z = \frac{\cos SM - \sin HS \sin OM}{\cos HS \cos OM}$$ +(1) + +In triangle Zsm: +$$\cos Z = \frac{\cos sm - \cos Za \cos Zm}{\sin Za \sin Zm}$$ +i.e. +$$\cos Z = \frac{\cos sm - \sin Hs \sin Om}{\cos Hs \cos Om}$$ +(2) + +Let $M$ be the true altitude of the Moon +$m$ be apparent altitude of the Moon +$S$ be true altitude of Sun or star +$s$ be apparent altitude of Sun or star +$D$ be true lunar distance +$d$ be apparent lunar distance + +Equating (1) and (2), we have: +$$\frac{\cos D - \sin S \sin M}{\cos S \cos M} = \frac{\cos d - \sin s \sin m}{\cos s \cos m}$$ + +Add unity to each side we have: +$$1 + \frac{\cos D - \sin S \sin M}{\cos S \cos M} = 1 + \frac{\cos d - \sin s \sin m}{\cos s \cos m}$$ + +210 A HISTORY OF NAUTICAL ASTRONOMY + +i.e. +$$\cos S \cos M + \cos D - \sin S \sin M = \frac{\cos S \cos m + \cos d - \sin S \sin m}{\cos S \cos M}$$ +i.e. +$$\frac{\cos D + \cos (M+S)}{\cos S \cos M} = \frac{\cos d + \cos (m+d)}{\cos S \cos M}$$ +i.e. +$$1-2\sin^2 D/2+2(\cos^2 (M+S)/2-1) = 2\cos (m+x-d)/2\cos (m+x-d)/2$$ +From which: +$$\sin^2 D/2 = \cos^2 (M+S)/2$$ +Let: +$$\frac{\cos S \cos M}{\cos S \cos M} = \frac{\cos (m+x+d)/2\cos (m+x-d)/2}{\cos (m+x+d)/2\cos (m+x-d)/2}$$ +(X) +Then: +$$\sin^2 D/2 = \cos^2 (M+S)/2-\cos^2 \theta$$ +i.e. +$$\sin^2 D/2 = 1(1+\cos (M+S))/2-1(1+\cos 2\theta)$$ +i.e. +$$\sin^2 D/2 = 1(1+\cos (M+S)-\cos 2\theta)$$ +i.e. +$$\sin^2 D/2 = (\sin ((M+S)/2+\theta)\sin ((M+S)/2-\theta))$$ +(Y) + +Equations (X) and (Y) are used in the problem after being adapted to logarithmic computation as follows: + +log cos $\theta$ = $4[\log sec m + \log sec x + \log cos (m+x+d)/2 + \log cos (m+x-d)/2 + \log log M + log cos S]$ + +log sin $D/2$ = $4[\log sin (M+S)/2+\theta] + \log sin (M+S)/2-\theta)$ + +A desirable feature of Borda's method is that it is simple and + +METHODS OF FINDING LONGITUDE 211 + +direct and gives the true lunar distance without embarrassment of algebraic signs. It also has the advantage, when using it, that no special tables are required, apart from those of the common log trig functions. + +The rules for solving the problem of clearing the distance using Borda's method are as follows: + +1. Find $M$, $S$, $m$ and $s$. +2. Correct the observed distance $d$ for index error and semi-diameter. +3. Place under one another the apparent distance $d$ and the apparent altitudes $m$ and $s$; and take half their sum, $L$. From the half sum $L$ subtract the apparent distance $d$. Under this place the difference $M$ and $S$. +4. Take from tables log sine $L$, log secant $L$ and log cosines of $L$, $(L - d)$, $M$ and $S$. Add these six quantities and divide the sum by 2. The result is the log cosine of $\phi$. +5. Take half the sum of the true altitudes $M$ and $S$. Call this $\phi$. Find the sum of and difference between $\theta$ and $\phi$. Add the sines of the sum and difference. Divide by 2. The result is the sine of half the true lunar distance, that is $D/2$. + +A typical problem and its solution by Borda's method is Ex. 519 taken from Merrifield's *Nautical Astronomy* of 1886. Ex. Given the undetermined data to compute the true distance between the Moon and the Sun. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Apparent AltitudesTrue AltitudesApparent Distance
$m = 13° 29' 27''$$M = 14° 18' 32''$$d = 107° 52' 4''$
$s = 31° 11' 34''$$S = 31° 10' 07''$
**
$d$1075204
$m$132927'sec0-012151
$s$311134'sec0-067815
21523305'
L
(L~d)		761632.5'cos-9-375207
313531.5'cos-9-430337
M
d~$\phi$		1418'32'cos-9-868314
S
$\phi$		3110'07'cos-9-332295
+ +2j19-304197 + + + + + + + + + + + + + + + + + + + + + + + + + + +
212A HISTORY OF NAUTICAL ASTRONOMY
θ63 19 08:3 cos 9:652059
θ+φ86 04 17:8 sin 9:998978
θ-φ40 35 38:8 sin 9:813378
D/2219:612356
x2
9:906178
D107 21 26:4 True Distance
+ +With the assistance of an auxiliary table, the operation of clearing the distance using Borda's method may be simplified. +The auxiliary table alluded to gives values called logarithmic differences which are tabulated against arguments 'Apparent Altitude of Moon's Centre' and 'Moon's Horizontal Parallax.' The logarithmic difference is the difference between two logarithms, i.e., the arithmetic difference; and, unlike some auxiliary tables used for abridging the calculations involved in clearing lunar distances, tables of logarithmic differences may be used with confidence. + +Delambre's Method for Clearing the Lunar Distance +Delambre (1749-1822), the famous French astronomer who served on the French Academy of Longitude, succeeded Lalande in 1807 as Professor of Astronomy at the College de France. He devoted considerable attention to nautical astronomical problems. The following method of Delambre's for clearing the distance is not dissimilar to that of Borda'. + +Equating (1) and (2), as in Borda's method, we have: +$$\frac{\cos D - \sin S \sin M}{\cos S \cos M} = \frac{\cos d - \sin s \sin m}{\cos s \cos m}$$ + +From which: +$$\cos D = \frac{(\cos d - \sin s \sin m) \cos S \cos M}{\cos S \cos M} + \sin S \sin M$$ +$$= (\frac{\cos d + \cos (m + r)}{\cos s \cos m}) \cos S \cos M - \sin S \sin M$$ +because $\cos (m + r) = \cos m \cos s - \sin m \sin s,$ +$$\cos D = (\frac{\cos d + \cos (m + r)}{\cos s \cos m}) - 1) \cos S \cos M + \sin S \sin M$$ + +METHODS OF FINDING LONGITUDE + +$$\begin{align*} +&= \left(\frac{\cos d + \cos (M+S)}{\cos s \cos m}\right) \cos S \cos M - (\cos S \cos M - \sin S \sin M) \\ +&= \left(\frac{\cos d + \cos (M+S)}{\cos s \cos m}\right) \cos S \cos M - \cos (M+S) \\ +&= \left[2 \cos \frac{1}{2}((M+S)+d) + 2 \cos \frac{1}{2}((M+S)-d)\right] \\ +&\quad \frac{\cos s \cos m}{\cos S \cos M - \cos (M+S)} +\end{align*}$$ + +The logarithm of the first expression is calculated, and the natural number answering to this logarithm is extracted from log tables. The natural cosine of $(M+S)$ is then subtracted from this to give the natural cosine of the true distance $D$. + +A notable feature of the above method is that tables of natural- and logarithmic-cosines are all that are necessary. + +An example, taken from Professor J. R. Young's book on *Practical Astronomy*, etc. published in 1856, illustrates the operation of calculating the distance using the above method of Delambre's. + +Ex. Find the true distance if: + +$$d = 83° 57' 33'' ; m = 27° 34' 05'' ; s = 48° 27' 32''$$ +$$M = 28° 20' 48'' ; S = 48° 26' 49''$$ + + + + + + + + + + + + + + + + + + + + + + + + +
$d$$s$$m$$M$$S$Ar Comp coslog
83° 57' 33''48° 27' 32''27° 34' 05''215959° 10'33 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .















































































Ar Comp cos log		0-1783835
0-0523390
-0-3010300
9-2399686
-9-9989587
-9-9445275
-9-8217187
0-3434291
-9-5369260
0-228495
-0-1158326
+ +It will be noticed that Ar Comps of cosines are used in the above example. These obviously correspond to secants. 'Ar Comp' is the + +214 +A HISTORY OF NAUTICAL ASTRONOMY + +abbreviation of 'Arithmetical Complement.' The Arithmetical Complement of a logarithm is the number it wants of 10-0000000. +... This artifice was used extensively in the practice of navigation in the 18th and 19th centuries, and was designed to facilitate problems on proportion, and for trigonometrical calculations. In the example above the divisor $\cos x \cos m$ is dealt with by taking the arithmetical complements of the logs of the cosines of $x$ and $m$, and subtracting them from the logarithms of the required log. +via. 9-5369260 observed by addition. The difficulty in finding an arithmetical complement is quite easy of solution. Starting at the left hand every figure is subtracted from 9 except the last which is subtracted from 10. Thus, the Ar Comp of the cosine of $27^\circ 34'$ is 0.05233 to five places of decimals, because the log cosine of the angle is $9\cdot94767$, which, of course, is the log secant of $27^\circ 34'$. + +Young's Method for clearing the distance +Young, whose work we have quoted above, gave the following method for clearing lunar distances. + +From equations (1) and (2), in the development of Borda's method, we have: + +$$\cos Z = \frac{\cos D - \sin S \sin M}{\cos S \cos M} \quad \quad (I)$$ + +and + +$$\cos Z = \frac{\cos d - \sin x \sin m}{\cos s \cos m} \quad \quad (II)$$ + +By adding unity to each side of equations (I) and (II) we have: + +$$1 + \cos Z = \frac{\cos D + \cos M \cos S - \sin M \sin S}{\cos M \cos S}$$ + +$$= \frac{\cos D + \cos (M + S)}{\cos M \cos S} \quad \quad (III)$$ + +and + +$$1 + \cos Z = \frac{\cos d + \cos m \cos s - \sin m \sin s}{\cos s \cos s}$$ + +$$= \frac{\cos d + \cos (m + s)}{\cos m \cos s} \quad \quad (IV)$$ + +**METHODS OF FINDING LONGITUDE** + +215 + +From (III) and (IV) we have: +$$\frac{\cos D + \cos (M+S)}{\cos M \cos S} = \frac{\cos d + \cos (m+r)}{\cos m \cos s}$$ + +From which: +$$\cos D = \left(\frac{\cos d + \cos (m+r)}{\cos m \cos s}\right)\frac{\cos M \cos S}{\cos m \cos s}$$ + +The computation of $\cos D$, using Young's method, is shortened considerably if a table of logarithmic differences is used to evaluate $(\cos M \cos S)/(\cos m \cos s)$. + +Young pointed out that the altitudes of the objects are not required to such a high degree of accuracy as that for the lunar distance; and he remarks: + +*This is a desirable circumstance, because, from the frequent obscurity of the sea horizon, it is more difficult to get the altitudes accurately than the distance.* + +It is evident that, for a given distance $d$, small changes in the values of $m$ and $r$, and the same changes in the values of $M$ and $S$, cannot produce any sensible effect upon the value of $D$. It is obvious that the logarithmic difference is always nearly unity; and this is the principal reason why small errors in the altitudes do not sensibly affect the distance. Young goes on to say that it is important that the proper corrections be carefully applied to the observed altitudes to obtain the true altitudes, even though the formulae should not have been taken with precision—the relative values of the observed and true altitudes must still be preserved. + +**Kraft's Method for Clearing the Distance** + +Kraft's method was published in St Petersburg in 1791. Like Borda's, it is a rigorous method based on the common formulae of spherical trigonometry. As in Borda's and Delambre's methods, the spherical cosine formula is applied to triangles ZMS and Zms for finding $Z$, and the two expressions for $\cos Z$ are equated thus: +$$\frac{\cos D - \sin d \sin M}{\cos S \cos M} = \frac{\cos d - \sin d \sin m}{\cos s \cos m}$$ + +216 + +**A HISTORY OF NAUTICAL ASTRONOMY** + +From which: + +$$\cos D = -\cos (M+S) \frac{\cos M \cos S}{\cos m \cos s} (\cos d + \cos (m+i))$$ + +Let + +$$\frac{\cos M \cos S}{\cos m \cos s} = 2 \cos A$$ + +(X) + +Then: + +$$\cos D = -\cos (M+S) + 2 \cos A (\cos d + \cos (m+i))$$ + +i.e. + +$$\cos D = -\cos (M+S) + 2 \cos A \cos d + 2 \cos A \cos (m+i)$$ + +i.e. + +$$\cos D = -\cos (M+S) + \cos (d+A) + \sin (d-A)$$ + +and $$((m+i)+A) + ((m+i)-A)$$ + +Multiplying both sides by $-1$ and adding $1$ to both sides we get: + +$$1-\cos D = (1+\cos (M+S)) + (1-\cos (d+A)) + (1+\cos (d-A))$$ + +$$= (1-\cos (m+i)+A) + (1-\cos (m+i)-A) - 4$$ + +i.e. + +$$vers D = suvrs (M+S) + vers (d+A) + vers (d-A)$$ + +$$+ vers ((m+i)+A) + vers ((m+i)-A) - 4$$ + +and: + +$$vers D = vem (sum zen. dista.) + vem (d+A) + vem (d-A)$$ + +$$+ vem ((m+i)+A) + vem ((m+i)-A) - 4$$ + +(Y) + +Equations (X) and (Y) give the full solution to the problem. + +The rules for Kraft's method are as follows: + +1. Find $M$, $S$, $m$, $s$ and $d$. +2. Find the sum of the apparent altitudes and of the true zenith distances. +3. From the sum of $\log cos M$, $\log sec m$, $\log cos S$ and $\log sec s$, subtract the log of $2$; the remainder is the log of $A$. +4. Add $(d+A)$, $vers(d-A)$, $vers((m+i)+A)$, $vers((m+i)-A)$, $vers((Z_1+z))$, to the true zenith distances, and subtract from the sum; the remainder is the versine of the true distance. + +Kraft's method for solving the problem on p. 211 is given hereunder. + +METHODS OF FINDING LONGITUDE + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
cos A = 4 cos M sec m cos S sec z
M = 14°18°32°log cos 9-986314
m = 132927log sec 0-012151
S = 311007log cos 9-932295
s = 311134log sec 9-988715
log 2 9-998775
A = 60°06°30°log cos 0-301030
d = 1075204log cos 9-697545
(m + s) = 444101
(Z + z) = 1343112
+ +vers D = vers (Z + z) + vers (d + A) + vers (d - A) + vers ((m + s) + A) + vers ((m + t) - A) + +(Z + z) = 134° +(d + A) = 167 +(d - A) = 47 +((m + t) + A) = 104 +((m + t) - A) = 15 + + + + + + + + + + + + + + ++73+73                                                               +36+36  +36+36  +36+36                                            &nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline ++36+36&nhline +
vers for secsvers for parts of secs
21°27°21°27°
(Z + z)21° veras27° veras21° veras27° veras
(d + A)58 veras34 veras58 veras34 veras
(d - A)45 veras34 veras45 veras34 veras
((m + t) + A)47 veras31 veras47 veras31 veras
((m + t) - A)25 veras29 veras25 veras29 veras
D = 10721° veras27° veras
21° veras27° veras
21° veras27° veras
21° veras27° veras
21° veras27° veras
21° veras
+ +The interesting feature of Kraft's method is that no logarithmic tables are required, and the true distance is deduced from addition of natural verses only. + +The method given by Dr James Inman in his Navigation of 1821, is a modified version of Kraft's method. The angle $A$ in the equation is called, by Inman, the 'auxiliary angle', and being pre-computed, is extracted from Inman's tables. + +Also, the method given by Inman is often called the Inman method for clearing the distance, the method is due, not to Inman, but to Mendosa del Rios. + +Mendoza del Rio was well known to navigators during the early 19 th century through his collection of nautical tables—a massive collection in quarto and containing over 600 pages. Some 300 pages of these tables were designed for facilitating the clearing of lunar distances. Rios undertook the considerable amount of labour of calculating values of $\text{vers}((m+t)+A)$ and $\text{vers}((m+t)-A)$ + +218 +**A HISTORY OF NAUTICAL ASTRONOMY** + +~$A$); and vers $(d+A)$+vers $(d\sim A)$, for every minute of arc. +The Rev. William Hall, a well known naval instructor remarked, +in an essay written in 1902: + +'It was a matter of 300 pages quarto containing 150,000 figures. +I know nothing to equal it except a folio volume by the Board of Longitude containing the correction of the distances for every degree of distance, and of the two apparent altitudes with differences for each and corrections for parallax.' + +We shall have occasion later to refer to the folio volume mentioned by Hall. + +**Airy's Method for Clearing the Distance** + +Sir George Bidell Airy (1801–1892) was born at Alnwick. He was educated at Hereford and Colchester before entering Cambridge where he had a brilliant academic career. He was appointed Astronomer Royal in 1835 in succession to Pond. To scavenge Airy is well known for his work on the theory of corrections for deviations in iron ships. His work on lunar theory was of great importance; and it is interesting to note that Hansen's *Tables of the Moon* are dedicated to Airy. + +Airy's method for clearing a lunar distance dates from 1882, from which time it became customary to place a copy of the method on the chart boxes supplied to H.M. ships. The method is direct, simple, and easy; it requires logarithms to five figures only, and gives results with an accuracy sensibly perfect. It is described by Airy as follows: + +'The characteristic circumstance upon which this treatment depends is the use, in the factors of corrections, not of each apparent element nor of the corresponding correcting element, but of their difference.' + +'The elements which we require are—the apparent altitude and the corrected altitude of the Moon, the apparent altitude and the corrected altitude of the Sun, and the apparent and corrected distance. The first five of these are known accurately. The last (the corrected distance between the Sun and the Moon) must be estimated. There is no difficulty in doing this with accuracy abundantly sufficient for this investigation. With + +A diagram showing lunar distance correction. + +**METHODS OF FINDING LONGITUDE** + +Greenwich time by account, the distance may be rudely com- +puted from the distances in the Nautical Almanac. Or, without +time or calculation, a navigator accustomed to lunar distances +may form a shrewd guess of the probable longitude of departure. +(The moon's distance from the sun is exhibited in diagrams.) We have now all the six elements required for the investigation. + +*Let Moon's corrected altitude + Moon's app. alt. = 2A +Moon's corrected altitude - Moon's app. alt. = 2a +Sun's apparent altitude + Sun's corr. alt. = 2B +Sun's apparent altitude - Sun's corr. alt. = 2b +Corrected distance + apparent distance = 2C +Corrected distance - apparent distance = 2c* + +*Then: +Moon's apparent altitude = A - a +Corrected altitude = A + a +Sun's apparent altitude = B + b +Corrected altitude = B - b +Apparent distance = C - c +Corrected distance = C + c* + +*The essential circumstance which directs the further investi- +gations is the equality of the zenithal angles and consequently +of the cosines of the zenithal angles. The corresponding equation is: + +$$\frac{\cos (C-c) \sin (A-a) \sin (B+b)}{\cos (A-a) \cos (B+b)} = \frac{\cos (C+c) \sin (A+a) \sin (B-b)}{\cos (A+a) \cos (B-b)}$$ + +or, multiplying out the denominators: + +(First side) $\cos (C-c) \cdot \cos (A+a) \cdot \cos (B-b)$ + +$-\sin (A-a) \cdot \sin (B+b) \cdot (-c) \cdot \cos (B-b)$ + += (Second side) $\cos (C+c) \cdot \cos (A-a) \cdot \cos (B+b)$ + +$-\sin (A+a) \cdot \sin (B-b) \cdot \cos (A-a) \cdot \cos (B+b)$ + +*For development of these terms it must be remembered that: + +$$\sin(A+a) = \sin A \cos a + \cos A \sin a$$ +$$\sin(A-a) = \sin A \cos a - \cos A \sin a$$ + +220 + +**A HISTORY OF NAUTICAL ASTRONOMY** + +$$\cos(A+a) = \cos A \cos a - \sin A \sin a$$ +$$\cos(A-a) = \cos A \cos a + \sin A \sin a$$ + +and similarly for $B$ and $C$. + +*We now proceed to develop the first side. Making the substitutions just stated, the first large product gives:* + +Line 1, not containing $\sin a$ or $\sin b$ or $\sin c$: ++ $\cos A \cos B \cos C \cos a \cos b \cos c$ + +Line 2, containing simply $\sin a$, or $\sin b$, or $\sin c$: ++ $\cos A \cos B \sin C \cos a \cos b \sin c$ +- $\sin A \cos B \cos C \sin a \cos b \cos c$ ++ $\cos A \sin B \cos C \sin a \sin b \cos c$ + +Line 3, containing $\sin a$, $\sin b$, or $\sin c$, or $\sin a$ or $\sin c$: +- $\sin A \sin B \cos C \sin a \sin b \cos c$ ++ $\cos A \sin B \sin C \cos a \sin b \sin c$ +- $\sin A \cos B \sin C \sin a \cos b \sin c$ + +Line 4, containing $\sin a$, $\sin b$, or $\sin c$: +- $\sin A \sin B \sin C \sin a \sin b \sin c$. + +*And the second large product gives:* + +Line 7: +- $\sin A \cos A \sin B \cos B$ + +Line 2: +- $\sin A \cos A \sin B \cos B + \sin B \cos B \sin A + \cos A + cos A$ + +Line 3: ++ $\sin A + cos A + sin B + cos B$ + +There is no line 4. + +*We now examine the second side of the equation:* + +The difference between the first side and the second side is this: that in every place where $a$ occurs in the equation, or in $a$ in the development of the first side, $-a$, or $-\sin a$ occurs on the second side; and similarly for $b$, $\sin b$, $c$, $\sin c$. And, moreover, these changes occur simultaneously; so that wherever $\sin a$, $\sin b$ occurs on the first side there will be $-\sin a$, $-\sin b$ on the second side; and where $\sin a$, $\sin b$, $c$, $\sin c$ occurs on the + +METHODS OF FINDING LONGITUDE 221 + +first side there will be $-\sin a$, $-\sin b$, $-\sin c$ on the second side. And thus we see that: + +*For line 1 the first side and the second side are the same. +*For line 2 the first side and the second side are equal but have opposite signs. +*For line 3 the first side and the second side are the same. +*For line 4 the first side and the second side are equal but have opposite signs. + +There is no difference between the second side with sign changed to the first side, the equation becomes the following: +$$\begin{array}{rl}+2.\cos A.\cos B.\sin C.\cos a.b.\sin c.\\ -2.\sin A.\cos B.\cos C.\sin a.b.\cos c.\\ +2.\cos A.\sin B.\cos C.\cos a.b.\cos c.\\ -2.\sin A.\sin B.\sin C.\sin a.b.\sin c.\\ -2.\sin A.\cos A.sin b.c.cos a.\\ +2.\sin B.\cos B.sin a.cos a.\\ \end{array}=0$$ + +This equation is rigorously accurate. + +We will now consider what simplification it will admit, preserving the character of practical accuracy of the highest order. + +$a$, which is half the correction of the Moon's altitude, can never exceed 30'. Cause of this error differs from one by 1/20,000 part, $\sin a.c$ can never differ from one by 1/60,000 part; and therefore for $a$ and $\sin a$ we may put 1 and $a$. The same applies to $b$ and $c$. In the product sin $a$, sin $b$, sin $c$ the factor of sin $c$ can rarely or never amount to 1/50,000, and that term may be neglected. The equation now becomes: +$$\begin{array}{rl}+\cos A.\cos B.sin C.2.c\\ -\sin A.\cos B.cos C.2.a+\cos A.sin B.cos C.2.b.\\ -\sin A.cos A.2.b.+sin B.cos B.2.a.\\ \end{array}=0$$ + +Remarking that $2a$, $2b$, $2c$ are the corrections of Moon's altitude, Sun's altitude, and distance, the result of this equation is: +$$\text{Corr'n of Distance} = (+\tan A.cos C.-\sec A.sin B.cosine C)$$ +$$x \text{ correction of Moon's altitude}$$ +$$(-\tan B.cos C.+ \sin A.sec B.cosine C)$$ +$$x \text{ correction of Sun's altitude}$$ + +222 + +*The only opening in error in this formula is in the estimated value of $C$, as depending on error in the estimated Nautical Almanac distance, or in the estimated correction to the observed distance. Suppose that the time of account was 4 m. in error (implying error of about 1° in angle $A$). Then the correction of distance would be taken out about 2° in error, and $C$ would be about 1° in error. If the value of the distance was about 60°, and error of 1° would produce in cot $C$ an error of about 1/1400 of that term of the computed correction, and in cosec $C$ the error would be 1/1600. There would be hardly sensible. But if, with $C$ corrected by this approximation, the calculation be repeated (repeating only a few minutes), the error of result will be totally insensible. + +'The following is offered as a form proper to be used with this method: + +'Prepare this table, inserting numbers instead of the printed words. + +See Table 1. + +'Then proceed with the following calculations, using 5-figure logarithms: + +See Table 2. + +'This form supposes that $C$ is less than 90°. When $C$ exceeds 90°, the supplement to 180° is to be taken, the cosec and cotan of that supplement are to be used, and the signs of the first and fourth numbers, which are produced by cotan $C$, are to be changed; so that the number will become subtractive, and the fourth number additive. + +'The second approximation will very rarely be required. If, however, the final "correction to apparent distance" differs from that assumed at the beginning by 2° or 3° it may be satisfactory to use the second approximation; it is very easy.' + +Airy's method has been given in full to illustrate the remarkably clever manner in which the lunar problem was analysed and the ingeniously-conceived solution made possible. + +The Approximate Method of Clearing Lunar Distances + +The alternative type of solution to the problem of clearing a lunar distance is often referred to as the approximate method, of + +METHODS OF FINDING LONGITUDE + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Correct to Moon's app. alt.Correct to Sun's app. alt.Fine Approximation (if necessary)Second Approximation
(approximate)(approximate)
Moon's app. alt.Sun's app. alt.Assumed cor. to app. distance (add/ve) [sub/ve]Assumed cor. to app. distance (add/ve) [sub/ve]
Moon's cor. alt.SumApp. distance Cor.App. distance Cor.
$A = \frac{1}{2}$ half sum$B = \frac{1}{2}$ half sumSumSum
$C = \frac{1}{2}$ half sum$C = \frac{1}{2}$ half sum
+ +TABLE 1 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
A HISTORY OF NAUTICAL ASTRONOMY
First ApproximationSecond Approximation
(if necessary)
Additive termsAdditive terms
$\log \tan A$(Repeat) $\log \tan A$
$\log \cot C$$\log \cot C$
$\log$ cor. to Moon's alt.(Repeat) $\log$ cor. to Moon's alt.
Sum and numberSum and number
$\log \sin A$(Repeat) $\log \sin A$
$\log \secant B$(Repeat) $\log \sec B$
$\log \cosec C$$\log \cosec C$
$\log$ cor. to Sun's alt.(Repeat) $\log$ cor. to Sun's alt.
Sum and numberSum and number
Sum additive termsSum additive terms
Subtractive termsSubtractive terms
$\log \sec A$(Repeat) $\log \sec A$
$\log \sin B$(Repeat) $\log \sin B$
$\log \cosec C$$\log \cosec C$
$\log$ cor. to Moon's alt.(Repeat) $\log$ cor to Moon's alt.
Sum and numberSum and number
$\log \tan B$(Repeat) $\log \tan B$
$\log \cot C$$\log \cot C$
$\log$ cor. to Sun's alt.(Repeat) $\log$ cor. to Sun's alt.
Sum and numberSum and number
Sum subtr. termsSum subtr. terms
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Combination of additive and subtractive = correction to apparent distance
+ +METHODS OF FINDING LONGITUDE 225 + +which there are many varieties. The principles of the approximate method are described in relation to Fig. 2. + +Perpendiculars are drawn from M and S to the true places of the Moon and Sun, respectively, on to the arc joining the apparent positions m and s. The difference between the apparent- and true-distances is found from the small triangles Mma and Sab, either by successive approximations (hence the name approximate method), or from a development of an algebraic series of which the smaller terms are neglected. + + +A diagram showing a triangle with vertices labeled Z, H, O, and S. A line segment from Z to S forms one side of the triangle. Another line segment from Z to M forms another side. A third line segment from S to O forms the third side. The angle at Z is labeled b, the angle at S is labeled s, and the angle at O is labeled m. + + +FIGURE 2 + +The invention of the approximate method for clearing a lunar distance is attributed to the English astronomer Israel Lyons. + +Israel Lyons (1739–1775) was born at Cambridge. He showed great aptitude for mathematics; and the Master of Trinity, Dr Robert Smith, offered to provide for his education. He studied botany as well as mathematics, and read a course of lectures on botany at Oxford in 1764. Lyons was engaged by the Board of Longitude in 1768, and received £100 per annum. In 1773, he was appointed by the Board of Longitude as astronomer to Captain Phipps during a polar voyage of discovery. + +Lyons's method for clearing lunar distances was described in the Nautical Almanac of 1767. An alternative method for clearing lunar distances was given in the same work: this being Mr Dunthorne's method. + +225 + +236 +A HISTORY OF NAUTICAL ASTRONOMY + +Dunthorne (1711-1775) of Cambridge, although not fortunate enough to have had an academic education, became an expert in many branches of learning, and particularly in astronomy. Both Lyons and Dunthorne received £50 each as reward from the Commissioners of Longitude for their methods for clearing lunar distances. + +In the Nautical Almanac for 1772 Maskelyne and George Witchell each contributed a method for clearing lunar distances. An improved version of Dunthorne's solution was also given. The same methods were described in the second edition of Maskelyne's *Requisite Tables* which were prepared by William Wales and published in 1781. + +Among the many successful attempts made to shorten the necessary calculations for clearing lunar distances, a notable effort was that made by Witchell, the mathematical master of the Royal Naval Academy at Portsmouth. Witchell conceived the idea of devising a table of corrections, based on the approximate method of clearing a lunar distance. In 1765 the Commissioners of Longitude awarded him £100 to enable him to complete and print 1000 copies of the tables. Later, they advanced a further £200. The work was continued by the Phidian Professor of Astronomy at Cambridge, and they thus became known as the *Cambridge Tables*. These are the tables referred to by the Rev. William Hall, mentioned above. They consist of a large folio volume containing the corrections for every degree of lunar distance, and for the two apparent altitudes, with differences for each degree, and corrections for parallax. It was a costly production—over £3,000 having been spent on it—and was ill-adapted for sea use. + +Those varieties of the approximate method for clearing lunar distances, which are based upon accurate formulae, require a considerable amount of computation. Many usually take time when employed. Such tables are based on a mere approximation, no method being available for correcting the actual refractions which affect the observations. The approximate method, however, is capable of giving correct results; and, in fact, for the same amount of computation, the approximate method gives better results than the rigorous method. + +Referring to Fig. 2: in the small triangles Mma and Ssb, which are right-angled at a and b respectively, the angles at m and s may + +METHODS OF FINDING LONGITUDE 227 + +be calculated using three given sides in triangle mZa. The arcs Mm and Ss are also known, these being the corrections respectively to the Moon's and Sun's (or planet's or star's) altitudes; the triangles are small, the arc Mm never being more than about 1°, so that they may be treated as being plane right-angled triangles. In this case: + +$$\begin{align*} +\text{arc ma} &= \text{correction to Moon's alt.} \times \cos m \\ +\text{arc sb} &= \text{correction to second body's alt.} \times \cos s +\end{align*}$$ + +When the angles at m and s are acute, as they are in the figure, ma is subtractive from, and ab is additive to, the arc sm in order to obtain arc SM. The contrary applies when the angles are obtuse. This principle is sometimes modified by calculating the effects of refraction and parallax-in-altitude of the Moon separately. + +We shall now illustrate a selection of the approximate methods for clearing a lunar distance, commencing with a relatively modern one to best illustrate the method. + +**Merrifield's Method for Clearing a Lunar Distance** + +Dr John Merrifield, L.L.D., was headmaster of the Navigation School at Plymouth for many years during the last century. It is interesting to note that his son W. T. Merrifield, B.A., became headmaster of the same school in 1860, and that of the Navigation School at Liverpool which had been established in the closing weeks of the last century. + +John Merrifield was a well-known author of works on navigation and, at the time of publication of his *Treatise on Nautical Astronomy*, published in 1886, the author had been engaged in teaching navigation for nearly a quarter of a century. In his treatise he described several methods for clearing lunar distances which he had invented, and which had appeared in the *Monthly Notices* of the Royal Astronomical Society for April 1884. The method is direct in its application, requires no special table, and is claimed to be a very close approximation well adapted for sea use. The method is described in relation to Fig. 3. + +In Fig. 3, + +$$M \text{and} S \text{are the true positions of the Moon and Second body respectively; and} m \text{and} s \text{are their apparent positions.}$$ + +Arcs MS and ms are the true- and apparent-distances respectively. + +A diagram showing the positions of the Moon (M), Second Body (S), and their respective apparent positions (ms). The diagram illustrates how to calculate clear lunar distances. + +228 A HISTORY OF NAUTICAL ASTRONOMY + +Q is the point of intersection of arcs MS and ma. +With centre Q describe arcs Mp and Sq. SqQ and QpM are, +therefore, right angles. + +A diagram showing a celestial body (S) at an apparent altitude (m), with a second body (H) at an apparent zenith distance (z). The observer's horizon is shown as a dotted line. The diagram includes labels for various points and lines, such as M, P, S, H, Z, Q, and θ. + +FIGURE 3 + +Let $d$ be the apparent distance +$D$ be the true distance +$m$ the Moon's apparent altitude +$r$ the second body's apparent altitude +y the Moon's measured zenith distance +$x$ the second body's apparent zenith distance + +$$S = \frac{y + z + d}{2} \quad \text{and} \quad S = \frac{m + x + d}{2}$$ + +C = Moon's correction for altitude += parallax in altitude - refraction + +c = second body's correction += parallax in altitude - refraction + +θ = Zms, and φ = Zim + +Then: + +$$D = \text{arc MS}$$ +$$= \text{arc qp}$$ +$$= sm - pm + sq$$ +$$= d - C\cos\theta + c\cos\phi$$ +$$= d - C(1-2\sin^2\theta/2) + c(1-2\sin^2\phi/2)$$ + +i.e. + +$$D = d - (C-c) + 2(C.\sin^2\theta/2 - c.\sin^2\phi/2)$$ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1) + +METHODS OF FINDING LONGITUDE 229 + +Now: + +$$\sin^2 \theta/2 = \frac{\sin (S-d) \cdot \sin (S-y)}{\sin d \cdot \sin y}$$ + +$$= \frac{\sin (y + x - d)/2 \cdot \sin (x + d - y)/2}{\sin d \cdot \sin y}$$ + +$$= \frac{\sin (90 - (m + x + d)/2) \cdot \sin (m + d - x)/2}{\sin d \cdot \cos m}$$ + +i.e. + +$$\sin^2 \theta/2 = \frac{\cos S \cdot \sin (S-t)}{\sin d \cdot \cos m}$$ + +Similarly: + +$$\sin^2 \phi/2 = \frac{\cos S \cdot \sin (S-m)}{\sin d \cdot \cos m}$$ + +Therefore: + +$$C \sin^2 \theta/2 - c \sin^2 \phi/2 =$$ + +$$= \frac{\cos S [C \cdot \sin (S-t) - c \cdot \sin (S-m)]}{\sin d}$$ + +$$= cosec.d.cos S(M-N)$$ + +where $$M = C sec m.sin(S-t),$$ + +and $$N = c sec x.sin(S-m).$$ + +Then, from formula (1): + +$$D = d - (C-c) + 2 cosec.d.cos S(M-N)$$ + +Merrifield's rule is as follows: + +1. Place the Moon's apparent altitude, the second body's apparent altitude and the apparent lunar distance under one another, and take half their sum ($S$); from which subtract the second body's apparent altitude ($S-t$), and the Moon's apparent altitude ($S-m$). Under these place the Moon's correction for altitude ($C$) and the correction for altitude of the second body ($d$). + +2. Add together log secant Moon's apparent altitude, $\sin(S-t)$ and Moon's correction reduced to seconds; the sum is log the number of seconds in $M$. + +$$M = C sec m.sin(S-t)$$ + +230 +A HISTORY OF NAUTICAL ASTRONOMY + +3. Add together log secant second body's apparent altitude, $\sin (S-m)$, and second body's correction reduced to seconds; the sum is log the number of seconds in $N$. +$$N = e \sec s \cdot \sin (S - m)$$ + +4. Find difference between $M$ and $N$, and add together log co-secant apparent distance, cos $S$, and $(M-N)$; the sum is log $C$. $\sin^2 \theta/2 - c \cdot \sin^2 \phi/2$ in seconds. + +5. Double the result in (4), and add to the apparent distance $d$, from which subtract the difference of corrections for altitudes; the remainder is the true distance $D$. + +The solution of the problem given on p. 211, using Merrifield's method, is as follows: + + + + + + + + + + + + + + + + + + + + + +
Apparent AltitudeTrue AltitudeApparent Distance
Moon $m = 13° 29' 27''$$m' = 14° 18' 32''$$d = 107° 52' 04''$
2nd $s = 31° 11' 34''$$s' = 31° 10' 07''$
+ +Here $C =$ correction for Moon's altitude +$$= (m - m')$$ +$$= 49° 05''$$ +$$= 2495''$$ + +and $c =$ correction for second body's altitude +$$= (S - r')$$ +$$= 01° 27''$$ +$$= 87''$$ + +$$D = d - (C - c) + 2(C \cdot \sin^2 \theta/2 - c \cdot \sin^2 \phi/2)$$ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
$m'$$s'$$d'$sec 0-012151sec 0-067815cosec 0-021470
$13° 29' 27''$$31$$11$$34$
$s'$
$107°$$52$$04$
$2152$$33$$05$
(S-r')761632-5
(S-m)450458-5sin 9-850113 cos 9-375207 sin 9-949046 log 3-649085 log 1-939519
C2945
e87
+ +**METHODS OF FINDING LONGITUDE** + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
M2144-6log 3-331349N90+4°log 1-966380
(M-N)2054+2° log 3-312642
C.$\sin^2 \theta/2 - c.\sin^2 \phi/2$512+2° log 2-709319
8° 32' 17" x2
17° 04'2"
(C-c)-47° 38'
Correction for app. dist.-30° 33'8"
d = True Distance107° 52' x2
+ +Dunthorne's Method for Clearing a Lunar Distance + +Dunthorne's method for clearing lunar distances was reproduced in many navigation manuals of the last century, and in particular in Norie's *Epitome*, the firm favourite of seafaring men of the Merchant Service for almost the whole of the 19th century. In the earlier editions of Norie's *Epitome*, of the four solutions for clearing a lunar distance, that of Dunthorne's formed Method I. + +It is interesting to note that tables of 'Logarithmic Differences' appeared in textbooks of nautical astronomy as early as the end of the 18th century. Mr. John Norie published *The Logarithmic Longitude* published in 1793; and so did Norie in the earlier editions of his famous *Nautical Tables*. Norie, in explaining these tables refers to their use in connection with Mr Dunthorne's method for clearing a lunar distance. We shall state Dunthorne's method as given by Norie. No demonstration of the rule was attempted, this being a common feature of works such as Norie's, which were addressed to the inexperienced, and to navigational practitioners who had no desire to understand the methods they used, being content merely to work "according to the rule." Compared with many of the rules for clearing lunar distances, that of Dunthorne's is about the least complex. + +**Rule:** To the correction of the Moon's altitude, add the correction of the Sun's or Star's altitude; their sum, added to the difference of the apparent altitudes when the Moon's altitude is greater, or subtracted from it when the Moon's altitude is less. + +232 +A HISTORY OF NAUTICAL ASTRONOMY + +than the Sun's or star's, will give the difference of their true altitudes. + +2. From the natural cosine of the difference of the apparent altitudes subtract the natural cosine of the apparent distance, when the apparent distance is not less than 90°; but when it is greater add together the natural cosines; and to the logarithm of their sum or difference add the logarithmic difference (Table 31); then the difference between the natural number of this sum and the natural cosine of the difference of the two altitudes will be the natural cosine of the true distance, when the natural number is less than 90°; otherwise it will be equal to the true altitudes; otherwise the remainder will be the natural cosine of the supplement of the true distance or the natural sine of the excess of the true distance above 90°. + +William Hall's Method for Clearing a Lunar Distance + +What is probably one of the last methods proposed for clearing a lunar distance is that described by the Rev. William Hall in a paper printed in the *Nautical Magazine* for the year 1806. 1906. + +Hall was well aware of this problem which clearly had out-lived its limits of its usefulness. It had been announced at about this time that the 1906, and subsequent Nautical Almanacs would not contain tables of lunar distances. A considerable amount of correspondence appeared in the pages of the *Nautical Magazine* over a period of several years during the early part of the 20th century, the writers being divided on their opinions of the lunar method for clearing a lunar distance. Captain John Henry Wrinkle, had been accused—not without justification—of burying the lunar in an unflattering manner, view of the great service it had performed to seamen for well over a hundred years. A handsome 'obituary notice of the lunar', to quote Hall, had prompted him to write on the subject in the August issue of the *Nautical Magazine* for 1906. The writers who had called 'lunarians' were genuinely sorry that the time had come when the lunar problem and finding longitude was beyond resurrection. + +Hall's interesting method is described in relation to Fig. 4. + +Referring to Fig. 4: + +\textit{Mm} = Moon's correction in altitude +\textit{parallax in altitude-refraction} + +METHODS OF FINDING LONGITUDE + +233 + +A diagram showing the positions of the Moon (M), Sun (S), observer's horizon (H), and zenith (Z). The diagram includes labels for various angles and distances. + +**FIGURE 4** + +$$\text{Ss} = \text{second body's correction in altitude}$$ +$$= \text{refraction-parallax in altitude}$$ +$$\text{Mn is plus, and Ss is minus to apparent altitude.}$$ +$$\text{MX and SY are arcs perpendicular to ms produced, so that corrections to apparent distance are -mX and -sY.}$$ + +Let: + +$$m = \text{apparent altitude of Moon}$$ +$$8m+ = \text{correction to apparent altitude of Moon}$$ +$$s = \text{apparent altitude of second body}$$ +$$8s- = \text{correction to apparent altitude of second body}$$ +$$d = \text{apparent distance}$$ +$$tY-mX = bd+$$ +$$= \text{correction to apparent distance}$$ +$$p = \text{Moon's horizontal parallax}$$ +$$q = \text{second body's horizontal parallax}$$ + +Now: + +$$8d = 8s.\cos YtS - 8m.\cos MmX$$ + +Applying the fundamental cosine formula to triangle Zms: + +$$\cos s = \frac{\cos Zm - \cos Zt}{\sin Zt.\sin m}$$ + +i.e. + +$$\cos s = \frac{\sin m - \sin s.\cos d}{\cos s.\sin d}$$ + +234 + +**A HISTORY OF NAUTICAL ASTRONOMY** + +Similarly: + +$$\cos m = \frac{\sin x - \sin y \cdot \cos d}{\cos x \cdot \sin d}$$ + +Considering only that part of the correction due to $\delta z$ we have: + +$$\delta z = \text{Refraction} - \text{parallax in altitude}$$ +$$= \text{Ref.} - p \cdot \cos s$$ +$$= \cos s (\text{Ref. sec. } s - p)$$ + +Hall tabulated the quantity Ref. sec alt. It is obtained by adding to the ordinary tables of refraction the following correction to obtain what may be called 'Horizontal Refraction' on the analogy of 'Horizontal Parallax.' + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 3233343536373839394041424344454647484950515253545556 + +
App. Alt.0°1°2°3°4°5°6°7°8°9°
0°0012334444
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+ +Take the Hor. Ref. and its difference from the Hor. Parallax. + +Call this difference $y$ for second body. + +$$\delta z = y \cdot \cos s$$ + +and + +Correction to distance due to + +$$\delta z = \delta z_0 \cdot \cos s$$ + +$$= \delta z_0 \left( \frac{\sin m - \sin r \cdot \cos d}{\cos x \cdot \sin d} \right)$$ + +METHODS OF FINDING LONGITUDE 235 + +Correction due to $8x = y(\sin m - \sin x \cos d)$ . . . . . (I) +$\sin d$ + +Now consider that part of the correction due to $8m$. + +Again: +Take the Hor. Ref. and its difference from the Hor. Parallax of Moon. Call this difference $x$ for Moon. +$\delta m = x \cos m,$ +and +Correction to distance due to $\delta m = \delta m \cdot \cos m$ +$= \delta m (\frac{\sin x - \sin m \cos d}{\cos m \sin d})$ + +Correction due to $\delta m = \frac{x(\sin t - \sin m \cos d)}{\sin d}$ . . . . (II) + +By combining (I) and (II) we may find the correction to distance $8d$. + +Now, +$8d = 8x \cos x - 8m \cos m$ +$8d = y(\sin m - \sin x \cos d) - y(\sin t - \sin m \cos d)$ +i.e. +$8d \sin d = y(\sin m - \sin x \cos d) - x(\sin t - \sin m \cos d)$ + +From which, because $8d$ is small: +$\sin (8d) = \sin y(\sin m - \cos c - \sin d - \sin x \cos d)$ +$- y(\sin t - \sin m \cos d)$ + +This formula, for a lunar distance, according to Hall, is fairly simple. The footnote to the paper in which he described his method is interesting and relevant to the position of the lunar method at the beginning of the 20th century. + +Footnote: Since this article was penned, an official memorandum has been issued in which notice is given that the lunar is no longer to be included in the course of instruction for junior officers of the Navy. Hence my contribution must be read + +236 +A HISTORY OF NAUTICAL ASTRONOMY + +in the light of pure mathematics, and the lunar can no longer be included in the sphere of practical navigation. Nevertheless I am unable to cancel my expression of opinion that the lunar must remain the supreme test of powers of observation and of computation, and that it will always be found by mathematicians find some other problem to trouble him in place of the lunar. + +'I have a certain satisfaction in the thought that this article is probably the last which will be written on a problem which has engaged the attention of mathematicians and navigators for hundreds of years . . . which in the hands of early pioneers of exploration was the one and only means of fixing longitude, which would be remembered in the future by the coming race, who smile at such an antiquity. I may grudge a modified approval of the industry and ingenuity displayed by disfashioned 'Lanarians.' + +'The Lunar is dead; let us bury it with due respect!' + +Several graphical methods were devised for clearing lunar distances. The Abbé de la Caille devised a graphical method for this purpose as early as 1759. De la Caille's method, by which it was claimed that distances could be cleared graphically with an accuracy of 20' of arc, was published in Connaissance des Temps for 1761. It also appeared in the later editions of Bouguer's Traité de Navigation—a well-known French manual of navigation which the Abbé revised. + +Andrew Mackay, in his Theory of the Longitude, not only refers to a 'Parallèle Ronula' invented by James Ferguson and constructed on the same principles as de la Caille's 'Chassis de Réduction,' but he also describes a graphical method of his own invention for clearing lunar distances. Mackay's method is similar to de la Caille's, and consists of four scales which provide the means for finding the various quantities used in the process of clearing the distance. In addition to these four scales (and similar) graphical method, lunar distances could be cleared accurately and expediently without resort to tables or calculations. + +In 1790, the English mathematician Margretts published tables expressed graphically in curves. Margrett's graphical method for clearing lunar distances was similar in plan to that of the Abbé de la Caille. + +John William Norie devised and published a set of 'linear + +7. Hadley Octant. By Benjamin Martin, c. 1760. + +8. Reflecting Circle. By Edward Troughton, c. 1800. (See page 83.) + + +A sextant, with two lenses and a handle, is shown in the top left corner of the image. + + +9. Sextant. By Kelvin Hughes, 1967. + +PLATE 86 + + + + + + +
+ A map showing the S.E. Coast of Ireland, Wales, Bristol Channel, and the Atlantic Ocean. + The map includes various locations such as Cork Harbour, Limerick Harbour, and the Bristol Channel. + There are also lines indicating latitude and longitude. + The map is labeled "S.E. Coast of Ireland" at the top left corner. + The bottom right corner has a label "X31 North 8 Longitude". + The bottom left corner has a label "WEST from 4. Greenwich". + The bottom center has a label "One Altitude of the Sextant Chronometer Time given to find the true Bearing of the Land". + + A map showing St George's Channel, Wales, Bristol Channel, and the Atlantic Ocean. + The map includes various locations such as Cork Harbour, Limerick Harbour, and the Bristol Channel. + There are also lines indicating latitude and longitude. + The map is labeled "St George's CHANNEL" at the top left corner. + The bottom right corner has a label "X31 North 8 Longitude". + The bottom left corner has a label "WEST from 4. Greenwich". + The bottom center has a label "One Altitude of the Sextant Chronometer Time given to find the true Bearing of the Land". +
+ +10. Facsimile of Plate 3 of Summer's Pamphlet. (See page 277.) + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
DISTANCES of MOON's Center from SUN, and from STARS EAST of line.
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Spois 9
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IX JANUARY 379 - VIII
+ +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + +JANUARY 379 - VIII + + + XIV. + AUGUST, 1897. + AUGUST, 1897. + + + + MEAN DURATION + MEAN DURATION + + + + LOCAL DISTANCE + LOCAL DISTANCE + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 20 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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XIII.
AUGUST, 1897.